Nikiya Anton Bettey 2021: 09-21-2021-1205 - initial condition difference equation linear system mode coupling vector soliton van der Pol oscillator Vlasov equation ball and beam system bounded-input (BIBO) bounded-output stability input output control theory deadbeat controller splitting dynamical system entropy stochastic stochastics geometrical space time ensemble fluids hydrodynamics observable universe connection

Tuesday, September 21, 2021

09-21-2021-1205 - initial condition difference equation linear system mode coupling vector soliton van der Pol oscillator Vlasov equation ball and beam system bounded-input (BIBO) bounded-output stability input output control theory deadbeat controller splitting dynamical system entropy stochastic stochastics geometrical space time ensemble fluids hydrodynamics observable universe connection

In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value,^[1]^{: pp. 160} is a value of an evolving variable at some point in time designated as the initial time (typically denoted t = 0). For a system of order k (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension n(that is, with n different evolving variables, which together can be denoted by an n-dimensional coordinate vector), generally nk initial conditions are needed in order to trace the system's variables forward through time.

In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial value problem. A corresponding problem exists for discrete time situations. While a closed form solution is not always possible to obtain, future values of a discrete time system can be found by iterating forward one time period per iteration, though rounding error may make this impractical over long horizons.

The initial condition of a vibrating string

Evolution from the initial condition

https://en.wikipedia.org/wiki/Initial_condition

In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms of the same function are given; each further term of the sequence or array is defined as a function of the preceding terms of the same function.

The term difference equation sometimes (and for the purposes of this article) refers to a specific type of recurrence relation. However, "difference equation" is frequently used to refer to any recurrence relation.

https://en.wikipedia.org/wiki/Recurrence_relation

In systems theory, a linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be modeled by linear systems.

https://en.wikipedia.org/wiki/Linear_system

In the term mode coupling, as used in physics and electrical engineering, the word "mode" refers to eigenmodes of an idealized, "unperturbed", linear system. The superposition principle says that eigenmodes of linear systems are independent of each other: it is possible to excite or to annihilate a specific mode without influencing any other mode; there is no dissipation. In most real systems, however, there is at least some perturbation that causes energy transfer between different modes. This perturbation, interpreted as an interaction between the modes, is what is called "mode coupling".

Important applications are:

In fiber optics^[1]^[2]
In lasers (compare mode-locking)^[3]
In condensed-matter physics, critical slowing down can be described by a Coupled mode theory.^[4]

https://en.wikipedia.org/wiki/Mode_coupling

In physical optics or wave optics, a vector soliton is a solitary wave with multiple components coupled together that maintains its shape during propagation. Ordinary solitons maintain their shape but have effectively only one (scalar) polarization component, while vector solitons have two distinct polarization components. Among all the types of solitons, optical vector solitons draw the most attention due to their wide range of applications, particularly in generating ultrafast pulses and light control technology. Optical vector solitons can be classified into temporal vector solitons and spatial vector solitons. During the propagation of both temporal solitons and spatial solitons, despite being in a medium with birefringence, the orthogonal polarizations can copropagate as one unit without splitting due to the strong cross-phase modulation and coherent energy exchange between the two polarizations of the vector soliton which may induce intensity differences between these two polarizations. Thus vector solitons are no longer linearly polarized but rather elliptically polarized.

https://en.wikipedia.org/wiki/Vector_soliton

In dynamics, the Van der Pol oscillator is a non-conservative oscillator with non-linear damping. It evolves in time according to the second-order differential equation:

{d^{2}x \over dt^{2}}-\mu (1-x^{2}){dx \over dt}+x=0,

where x is the position coordinate—which is a function of the time t, and μ is a scalar parameter indicating the nonlinearity and the strength of the damping.

https://en.wikipedia.org/wiki/Van_der_Pol_oscillator

The Vlasov equation is a differential equation describing time evolution of the distribution function of plasma consisting of charged particles with long-range interaction, e.g. Coulomb. The equation was first suggested for description of plasma by Anatoly Vlasov in 1938^[1]^[2] and later discussed by him in detail in a monograph.^[3]

https://en.wikipedia.org/wiki/Vlasov_equation

The ball and beam system consists of a long beam which can be tilted by a servo or electric motor together with a ball rolling back and forth on top of the beam. It is a popular textbook example in control theory.

The significance of the ball and beam system is that it is a simple system which is open-loop unstable. Even if the beam is restricted to be very nearly horizontal, without active feedback, it will swing to one side or the other, and the ball will roll off the end of the beam. To stabilize the ball, a control system which measures the position of the ball and adjusts the beam accordingly must be used.

In two dimensions, the ball and beam system becomes the ball and plate system, where a ball rolls on top of a plate whose inclination can be adjusted by tilting it forwards, backwards, leftwards, or rightwards.

Mechanical model of the ball and beam system

https://en.wikipedia.org/wiki/Ball_and_beam

In signal processing, specifically control theory, bounded-input, bounded-output (BIBO) stability is a form of stability for linear signals and systems that take inputs. If a system is BIBO stable, then the output will be bounded for every input to the system that is bounded.

A signal is bounded if there is a finite value $B>0$ such that the signal magnitude never exceeds $B$ , that is

\ |y[n]|\leq B\quad \forall n\in \mathbb {Z}

for discrete-time signals, or

\ |y(t)|\leq B\quad \forall t\in \mathbb {R}

for continuous-time signals.

https://en.wikipedia.org/wiki/BIBO_stability

^[1]^[2]In computing, input/output (I/O, or informally io or IO) is the communication between an information processing system, such as a computer, and the outside world, possibly a human or another information processing system. Inputs are the signals or data received by the system and outputs are the signals or data sent from it. The term can also be used as part of an action; to "perform I/O" is to perform an input or output operation.

I/O devices are the pieces of hardware used by a human (or other system) to communicate with a computer. For instance, a keyboard or computer mouse is an input device for a computer, while monitors and printers are output devices. Devices for communication between computers, such as modems and network cards, typically perform both input and output operations.

The designation of a device as either input or output depends on perspective. Mice and keyboards take physical movements that the human user outputs and convert them into input signals that a computer can understand; the output from these devices is the computer's input. Similarly, printers and monitors take signals that computers output as input, and they convert these signals into a representation that human users can understand. From the human user's perspective, the process of reading or seeing these representations is receiving output; this type of interaction between computers and humans is studied in the field of human–computer interaction. A further complication is that a device traditionally considered an input device, e.g., card reader, keyboard, may accept control commands to, e.g., select stacker, display keyboard lights, while a device traditionally considered as an output device may provide status data, e.g., low toner, out of paper, paper jam.

In computer architecture, the combination of the CPU and main memory, to which the CPU can read or write directly using individual instructions, is considered the brain of a computer. Any transfer of information to or from the CPU/memory combo, for example by reading data from a disk drive, is considered I/O.^[2] The CPU and its supporting circuitry may provide memory-mapped I/O that is used in low-level computer programming, such as in the implementation of device drivers, or may provide access to I/O channels. An I/O algorithm is one designed to exploit locality and perform efficiently when exchanging data with a secondary storage device, such as a disk drive.

https://en.wikipedia.org/wiki/Input/output

Stability[edit]

The stability of a general dynamical system with no input can be described with Lyapunov stability criteria.

A linear system is called bounded-input bounded-output (BIBO) stable if its output will stay bounded for any bounded input.
Stability for nonlinear systems that take an input is input-to-state stability (ISS), which combines Lyapunov stability and a notion similar to BIBO stability.

For simplicity, the following descriptions focus on continuous-time and discrete-time linear systems.

Mathematically, this means that for a causal linear system to be stable all of the poles of its transfer function must have negative-real values, i.e. the real part of each pole must be less than zero. Practically speaking, stability requires that the transfer function complex poles reside

in the open left half of the complex plane for continuous time, when the Laplace transform is used to obtain the transfer function.
inside the unit circle for discrete time, when the Z-transform is used.

The difference between the two cases is simply due to the traditional method of plotting continuous time versus discrete time transfer functions. The continuous Laplace transform is in Cartesian coordinates where the $x$ axis is the real axis and the discrete Z-transform is in circular coordinates where the $\rho$ axis is the real axis.

When the appropriate conditions above are satisfied a system is said to be asymptotically stable; the variables of an asymptotically stable control system always decrease from their initial value and do not show permanent oscillations. Permanent oscillations occur when a pole has a real part exactly equal to zero (in the continuous time case) or a modulus equal to one (in the discrete time case). If a simply stable system response neither decays nor grows over time, and has no oscillations, it is marginally stable; in this case the system transfer function has non-repeated poles at the complex plane origin (i.e. their real and complex component is zero in the continuous time case). Oscillations are present when poles with real part equal to zero have an imaginary part not equal to zero.

If a system in question has an impulse response of

\ x[n]=0.5^{n}u[n]

then the Z-transform (see this example), is given by

\ X(z)={\frac {1}{1-0.5z^{-1}}}

which has a pole in $z=0.5$ (zero imaginary part). This system is BIBO (asymptotically) stable since the pole is inside the unit circle.

However, if the impulse response was

\ x[n]=1.5^{n}u[n]

then the Z-transform is

\ X(z)={\frac {1}{1-1.5z^{-1}}}

which has a pole at $z=1.5$ and is not BIBO stable since the pole has a modulus strictly greater than one.

Numerous tools exist for the analysis of the poles of a system. These include graphical systems like the root locus, Bode plots or the Nyquist plots.

Mechanical changes can make equipment (and control systems) more stable. Sailors add ballast to improve the stability of ships. Cruise ships use antiroll finsthat extend transversely from the side of the ship for perhaps 30 feet (10 m) and are continuously rotated about their axes to develop forces that oppose the roll.

Controllability and observability[edit]

Controllability and observability are main issues in the analysis of a system before deciding the best control strategy to be applied, or whether it is even possible to control or stabilize the system. Controllability is related to the possibility of forcing the system into a particular state by using an appropriate control signal. If a state is not controllable, then no signal will ever be able to control the state. If a state is not controllable, but its dynamics are stable, then the state is termed stabilizable. Observability instead is related to the possibility of observing, through output measurements, the state of a system. If a state is not observable, the controller will never be able to determine the behavior of an unobservable state and hence cannot use it to stabilize the system. However, similar to the stabilizability condition above, if a state cannot be observed it might still be detectable.

From a geometrical point of view, looking at the states of each variable of the system to be controlled, every "bad" state of these variables must be controllable and observable to ensure a good behavior in the closed-loop system. That is, if one of the eigenvalues of the system is not both controllable and observable, this part of the dynamics will remain untouched in the closed-loop system. If such an eigenvalue is not stable, the dynamics of this eigenvalue will be present in the closed-loop system which therefore will be unstable. Unobservable poles are not present in the transfer function realization of a state-space representation, which is why sometimes the latter is preferred in dynamical systems analysis.

Solutions to problems of an uncontrollable or unobservable system include adding actuators and sensors.

Control specification[edit]

Several different control strategies have been devised in the past years. These vary from extremely general ones (PID controller), to others devoted to very particular classes of systems (especially robotics or aircraft cruise control).

A control problem can have several specifications. Stability, of course, is always present. The controller must ensure that the closed-loop system is stable, regardless of the open-loop stability. A poor choice of controller can even worsen the stability of the open-loop system, which must normally be avoided. Sometimes it would be desired to obtain particular dynamics in the closed loop: i.e. that the poles have $Re[\lambda ]<-{\overline {\lambda }}$ , where ${\overline {\lambda }}$ is a fixed value strictly greater than zero, instead of simply asking that $Re[\lambda ]<0$ .

Another typical specification is the rejection of a step disturbance; including an integrator in the open-loop chain (i.e. directly before the system under control) easily achieves this. Other classes of disturbances need different types of sub-systems to be included.

Other "classical" control theory specifications regard the time-response of the closed-loop system. These include the rise time (the time needed by the control system to reach the desired value after a perturbation), peak overshoot (the highest value reached by the response before reaching the desired value) and others (settling time, quarter-decay). Frequency domain specifications are usually related to robustness (see after).

Modern performance assessments use some variation of integrated tracking error (IAE, ISA, CQI).

Model identification and robustness[edit]

A control system must always have some robustness property. A robust controller is such that its properties do not change much if applied to a system slightly different from the mathematical one used for its synthesis. This requirement is important, as no real physical system truly behaves like the series of differential equations used to represent it mathematically. Typically a simpler mathematical model is chosen in order to simplify calculations, otherwise, the true system dynamics can be so complicated that a complete model is impossible.

System identification

The process of determining the equations that govern the model's dynamics is called system identification. This can be done off-line: for example, executing a series of measures from which to calculate an approximated mathematical model, typically its transfer function or matrix. Such identification from the output, however, cannot take account of unobservable dynamics. Sometimes the model is built directly starting from known physical equations, for example, in the case of a mass-spring-damper system we know that $m{\ddot {x}}(t)=-Kx(t)-\mathrm {B} {\dot {x}}(t)$ . Even assuming that a "complete" model is used in designing the controller, all the parameters included in these equations (called "nominal parameters") are never known with absolute precision; the control system will have to behave correctly even when connected to a physical system with true parameter values away from nominal.

Some advanced control techniques include an "on-line" identification process (see later). The parameters of the model are calculated ("identified") while the controller itself is running. In this way, if a drastic variation of the parameters ensues, for example, if the robot's arm releases a weight, the controller will adjust itself consequently in order to ensure the correct performance.

Analysis

Analysis of the robustness of a SISO (single input single output) control system can be performed in the frequency domain, considering the system's transfer function and using Nyquist and Bode diagrams. Topics include gain and phase margin and amplitude margin. For MIMO (multi-input multi output) and, in general, more complicated control systems, one must consider the theoretical results devised for each control technique (see next section). I.e., if particular robustness qualities are needed, the engineer must shift their attention to a control technique by including these qualities in its properties.

Constraints

A particular robustness issue is the requirement for a control system to perform properly in the presence of input and state constraints. In the physical world every signal is limited. It could happen that a controller will send control signals that cannot be followed by the physical system, for example, trying to rotate a valve at excessive speed. This can produce undesired behavior of the closed-loop system, or even damage or break actuators or other subsystems. Specific control techniques are available to solve the problem: model predictive control (see later), and anti-wind up systems. The latter consists of an additional control block that ensures that the control signal never exceeds a given threshold.

System classifications[edit]

Linear systems control[edit]

For MIMO systems, pole placement can be performed mathematically using a state space representation of the open-loop system and calculating a feedback matrix assigning poles in the desired positions. In complicated systems this can require computer-assisted calculation capabilities, and cannot always ensure robustness. Furthermore, all system states are not in general measured and so observers must be included and incorporated in pole placement design.

Nonlinear systems control[edit]

Processes in industries like robotics and the aerospace industry typically have strong nonlinear dynamics. In control theory it is sometimes possible to linearize such classes of systems and apply linear techniques, but in many cases it can be necessary to devise from scratch theories permitting control of nonlinear systems. These, e.g., feedback linearization, backstepping, sliding mode control, trajectory linearization control normally take advantage of results based on Lyapunov's theory. Differential geometry has been widely used as a tool for generalizing well-known linear control concepts to the nonlinear case, as well as showing the subtleties that make it a more challenging problem. Control theory has also been used to decipher the neural mechanism that directs cognitive states.^[19]

Decentralized systems control[edit]

When the system is controlled by multiple controllers, the problem is one of decentralized control. Decentralization is helpful in many ways, for instance, it helps control systems to operate over a larger geographical area. The agents in decentralized control systems can interact using communication channels and coordinate their actions.

Deterministic and stochastic systems control[edit]

A stochastic control problem is one in which the evolution of the state variables is subjected to random shocks from outside the system. A deterministic control problem is not subject to external random shocks.

Main control strategies[edit]

Every control system must guarantee first the stability of the closed-loop behavior. For linear systems, this can be obtained by directly placing the poles. Nonlinear control systems use specific theories (normally based on Aleksandr Lyapunov's Theory) to ensure stability without regard to the inner dynamics of the system. The possibility to fulfill different specifications varies from the model considered and the control strategy chosen.

List of the main control techniques

Adaptive control uses on-line identification of the process parameters, or modification of controller gains, thereby obtaining strong robustness properties. Adaptive controls were applied for the first time in the aerospace industry in the 1950s, and have found particular success in that field.
A hierarchical control system is a type of control system in which a set of devices and governing software is arranged in a hierarchical tree. When the links in the tree are implemented by a computer network, then that hierarchical control system is also a form of networked control system.
Intelligent control uses various AI computing approaches like artificial neural networks, Bayesian probability, fuzzy logic,^[20] machine learning, evolutionary computation and genetic algorithms or a combination of these methods, such as neuro-fuzzy algorithms, to control a dynamic system.
Optimal control is a particular control technique in which the control signal optimizes a certain "cost index": for example, in the case of a satellite, the jet thrusts needed to bring it to desired trajectory that consume the least amount of fuel. Two optimal control design methods have been widely used in industrial applications, as it has been shown they can guarantee closed-loop stability. These are Model Predictive Control (MPC) and linear-quadratic-Gaussian control(LQG). The first can more explicitly take into account constraints on the signals in the system, which is an important feature in many industrial processes. However, the "optimal control" structure in MPC is only a means to achieve such a result, as it does not optimize a true performance index of the closed-loop control system. Together with PID controllers, MPC systems are the most widely used control technique in process control.
Robust control deals explicitly with uncertainty in its approach to controller design. Controllers designed using robust control methods tend to be able to cope with small differences between the true system and the nominal model used for design.^[21] The early methods of Bode and others were fairly robust; the state-space methods invented in the 1960s and 1970s were sometimes found to lack robustness. Examples of modern robust control techniques include H-infinity loop-shaping developed by Duncan McFarlane and Keith Glover, Sliding mode control (SMC) developed by Vadim Utkin, and safe protocols designed for control of large heterogeneous populations of electric loads in Smart Power Grid applications.^[22] Robust methods aim to achieve robust performance and/or stability in the presence of small modeling errors.
Stochastic control deals with control design with uncertainty in the model. In typical stochastic control problems, it is assumed that there exist random noise and disturbances in the model and the controller, and the control design must take into account these random deviations.
Self-organized criticality control may be defined as attempts to interfere in the processes by which the self-organized system dissipates energy.

People in systems and control[edit]

Many active and historical figures made significant contribution to control theory including

Pierre-Simon Laplace invented the Z-transform in his work on probability theory, now used to solve discrete-time control theory problems. The Z-transform is a discrete-time equivalent of the Laplace transform which is named after him.
Irmgard Flugge-Lotz developed the theory of discontinuous automatic control and applied it to automatic aircraft control systems.
Alexander Lyapunov in the 1890s marks the beginning of stability theory.
Harold S. Black invented the concept of negative feedback amplifiers in 1927. He managed to develop stable negative feedback amplifiers in the 1930s.
Harry Nyquist developed the Nyquist stability criterion for feedback systems in the 1930s.
Richard Bellman developed dynamic programming since the 1940s.^[23]
Warren E. Dixon, control theorist and a professor
Andrey Kolmogorov co-developed the Wiener–Kolmogorov filter in 1941.
Norbert Wiener co-developed the Wiener–Kolmogorov filter and coined the term cybernetics in the 1940s.
John R. Ragazzini introduced digital control and the use of Z-transform in control theory (invented by Laplace) in the 1950s.
Lev Pontryagin introduced the maximum principle and the bang-bang principle.
Pierre-Louis Lions developed viscosity solutions into stochastic control and optimal control methods.
Rudolf Kálmán pioneered the state-space approach to systems and control. Introduced the notions of controllability and observability. Developed the Kalman filter for linear estimation.
Ali H. Nayfeh who was one of the main contributors to nonlinear control theory and published many books on perturbation methods
Jan C. Willems Introduced the concept of dissipativity, as a generalization of Lyapunov function to input/state/output systems. The construction of the storage function, as the analogue of a Lyapunov function is called, led to the study of the linear matrix inequality (LMI) in control theory. He pioneered the behavioral approach to mathematical systems theory.

https://en.wikipedia.org/wiki/Control_theory#Stability

In discrete-time control theory, the dead-beat control problem consists of finding what input signal must be applied to a system in order to bring the output to the steady state in the smallest number of time steps.

For an Nth-order linear system it can be shown that this minimum number of steps will be at most N (depending on the initial condition), provided that the system is null controllable (that it can be brought to state zero by some input). The solution is to apply feedback such that all poles of the closed-loop transfer function are at the origin of the z-plane. (For more information about transfer functions and the z-plane see z-transform). Therefore the linear case is easy to solve. By extension, a closed loop transfer function which has all poles of the transfer function at the origin is sometimes called a dead beat transfer function.

For nonlinear systems, dead beat control is an open research problem. (See Nesic reference below).

Dead beat controllers are often used in process control due to their good dynamic properties. They are a classical feedback controller where the control gains are set using a table based on the plant system order and normalized natural frequency.

The deadbeat response has the following characteristics:

Zero steady-state error
Minimum rise time
Minimum settling time
Less than 2% overshoot/undershoot
Very high control signal output

Transfer functions[edit]

Consider the transfer function of a plant

\mathbf {G} (s)={\frac {B(z)}{A(z)}}z^{-d}

with polynomials

A(z)=a_{0}+a_{1}z^{-1}+a_{2}z^{-2}+\cdots a_{m}z^{-m},

B(z)=b_{0}+b_{1}z^{-1}+b_{2}z^{-2}+\cdots b_{n}z^{-n}

and discrete time delay $e^{-d}$ .

The corresponding dead-beat controller is noted as

\mathbf {C} (s)={\frac {A(z)}{B(1)-z^{-d}B(z)}}

and the closed-loop transfer function is calculated as

\mathbf {L} (s)={\frac {B(z)}{B(1)}}.

https://en.wikipedia.org/wiki/Dead-beat_control

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

At any given time, a dynamical system has a state given by a tuple of real numbers (a vector) that can be represented by a point in an appropriate state space (a geometrical manifold). The evolution rule of the dynamical system is a function that describes what future states follow from the current state. Often the function is deterministic, that is, for a given time interval only one future state follows from the current state.^[1]^[2] However, some systems are stochastic, in that random events also affect the evolution of the state variables.

In physics, a dynamical system is described as a "particle or ensemble of particles whose state varies over time and thus obeys differential equations involving time derivatives".^[3] In order to make a prediction about the system's future behavior, an analytical solution of such equations or their integration over time through computer simulation is realized.

The study of dynamical systems is the focus of dynamical systems theory, which has applications to a wide variety of fields such as mathematics, physics,^[4]^[5] biology,^[6] chemistry, engineering,^[7] economics,^[8] history, and medicine. Dynamical systems are a fundamental part of chaos theory, logistic map dynamics, bifurcation theory, the self-assembly and self-organization processes, and the edge of chaos concept.

https://en.wikipedia.org/wiki/Dynamical_system

Dynamical systems theory is an area of mathematics used to describe the behavior of complex dynamical systems, usually by employing differential equationsor difference equations. When differential equations are employed, the theory is called continuous dynamical systems. From a physical point of view, continuous dynamical systems is a generalization of classical mechanics, a generalization where the equations of motion are postulated directly and are not constrained to be Euler–Lagrange equations of a least action principle. When difference equations are employed, the theory is called discrete dynamical systems. When the time variable runs over a set that is discrete over some intervals and continuous over other intervals or is any arbitrary time-set such as a Cantor set, one gets dynamic equations on time scales. Some situations may also be modeled by mixed operators, such as differential-difference equations.

This theory deals with the long-term qualitative behavior of dynamical systems, and studies the nature of, and when possible the solutions of, the equations of motion of systems that are often primarily mechanical or otherwise physical in nature, such as planetary orbits and the behaviour of electronic circuits, as well as systems that arise in biology, economics, and elsewhere. Much of modern research is focused on the study of chaotic systems.

This field of study is also called just dynamical systems, mathematical dynamical systems theory or the mathematical theory of dynamical systems.

https://en.wikipedia.org/wiki/Dynamical_systems_theory

Chaos theory is an interdisciplinary theory and branch of mathematics focusing on the study of chaos: dynamical systems whose apparently random states of disorder and irregularities are actually governed by underlying patterns and deterministic laws that are highly sensitive to initial conditions.^[1]^[2] Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization.^[3] The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions).^[4] A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas.^[5]

Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general.^[6] This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution^[7] and is fully determined by their initial conditions, with no randomelements involved.^[8] In other words, the deterministic nature of these systems does not make them predictable.^[9]^[10]This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:^[11]

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather and climate.^[12]^[13]^[7] It also occurs spontaneously in some systems with artificial components, such as the stock market and road traffic.^[14]^[3] This behavior can be studied through the analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plotsand Poincaré maps. Chaos theory has applications in a variety of disciplines, including meteorology,^[7] anthropology,^[15]sociology, environmental science, computer science, engineering, economics, ecology, pandemic crisis management,^[16]^[17]philosophy and jazz music. The theory formed the basis for such fields of study as complex dynamical systems, edge of chaostheory, and self-assembly processes.

https://en.wikipedia.org/wiki/Chaos_theory

Self-assembly is a process in which a disordered system of pre-existing components forms an organized structure or pattern as a consequence of specific, local interactions among the components themselves, without external direction. When the constitutive components are molecules, the process is termed molecular self-assembly.

AFM imaging of self-assembly of 2-aminoterephthalic acid molecules on (104)-oriented calcite.^[3]

Self-assembly can be classified as either static or dynamic. In static self-assembly, the ordered state forms as a system approaches equilibrium, reducing its free energy. However, in dynamic self-assembly, patterns of pre-existing components organized by specific local interactions are not commonly described as "self-assembled" by scientists in the associated disciplines. These structures are better described as "self-organized", although these terms are often used interchangeably.

https://en.wikipedia.org/wiki/Self-assembly

In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for each moment i in time, the times at which a phase space trajectory visits roughly the same area in the phase space as at time j. In other words, it is a graph of

{\vec {x}}(i)\approx {\vec {x}}(j),

showing $i$ on a horizontal axis and $j$ on a vertical axis, where ${\vec {x}}$ is a phase space trajectory.

https://en.wikipedia.org/wiki/Recurrence_plot

Determinism is the philosophical view that all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have sprung from diverse and sometimes overlapping motives and considerations. The opposite of determinism is some kind of indeterminism (otherwise called nondeterminism) or randomness. Determinism is often contrasted with free will, although some philosophers claim that the two are compatible.^[1]^[2]

https://en.wikipedia.org/wiki/Determinism

Predictability is the degree to which a correct prediction or forecast of a system's state can be made, either qualitatively or quantitatively.

https://en.wikipedia.org/wiki/Predictability

Determinism often is taken to mean causal determinism, which in physics is known as cause-and-effect. It is the concept that events within a given paradigm are bound by causality in such a way that any state (of an object or event) is completely determined by prior states. This meaning can be distinguished from other varieties of determinism mentioned below.

https://en.wikipedia.org/wiki/Determinism#Causal_determinism

Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state or object (a cause) contributes to the production of another event, process, state or object (an effect) where the cause is partly responsible for the effect, and the effect is partly dependent on the cause. In general, a process has many causes,^[1] which are also said to be causal factors for it, and all lie in its past. An effect can in turn be a cause of, or causal factor for, many other effects, which all lie in its future. Some writers have held that causality is metaphysically prior to notions of time and space.^[2]^[3]^[4]

Causality is an abstraction that indicates how the world progresses,^[5] so basic a concept that it is more apt as an explanation of other concepts of progression than as something to be explained by others more basic. The concept is like those of agency and efficacy. For this reason, a leap of intuition may be needed to grasp it.^[6]^[7] Accordingly, causality is implicit in the logic and structure of ordinary language.^[8]

In English studies of Aristotelian philosophy, the word "cause" is used as a specialized technical term, the translation of Aristotle's term αἰτία, by which Aristotle meant "explanation" or "answer to a 'why' question". Aristotle categorized the four types of answers as material, formal, efficient, and final "causes". In this case, the "cause" is the explanans for the explanandum, and failure to recognize that different kinds of "cause" are being considered can lead to futile debate. Of Aristotle's four explanatory modes, the one nearest to the concerns of the present article is the "efficient" one.

David Hume, as part of his opposition to rationalism, argued that pure reason alone cannot prove the reality of efficient causality; instead, he appealed to custom and mental habit, observing that all human knowledge derives solely from experience.

The topic of causality remains a staple in contemporary philosophy.

https://en.wikipedia.org/wiki/Causality

A dissipative system is a thermodynamically open system which is operating out of, and often far from, thermodynamic equilibrium in an environment with which it exchanges energy and matter. A tornado may be thought of as a dissipative system. Dissipative systems stand in contrast to conservative systems.

A dissipative structure is a dissipative system that has a dynamical regime that is in some sense in a reproducible steady state. This reproducible steady state may be reached by natural evolution of the system, by artifice, or by a combination of these two.

A dissipative structure is characterized by the spontaneous appearance of symmetry breaking (anisotropy) and the formation of complex, sometimes chaotic, structures where interacting particles exhibit long range correlations. Examples in everyday life include convection, turbulent flow, cyclones, hurricanes and living organisms. Less common examples include lasers, Bénard cells, droplet cluster, and the Belousov–Zhabotinsky reaction.^[1]

One way of mathematically modeling a dissipative system is given in the article on wandering sets: it involves the action of a group on a measurable set.

Dissipative systems can also be used as a tool to study economic systems and complex systems.^[2] For example, a dissipative system involving self-assembly of nanowires has been used as a model to understand the relationship between entropy generation and the robustness of biological systems.^[3]

The Hopf decomposition states that dynamical systems can be decomposed into a conservative and a dissipative part; more precisely, it states that every measure space with a non-singular transformation can be decomposed into an invariant conservative set and an invariant dissipative set.

https://en.wikipedia.org/wiki/Dissipative_system

Energy dissipation and entropy production extremal principles are ideas developed within non-equilibrium thermodynamics that attempt to predict the likely steady states and dynamical structures that a physical system might show. The search for extremum principles for non-equilibrium thermodynamics follows their successful use in other branches of physics.^[1]^[2]^[3]^[4]^[5]^[6] According to Kondepudi (2008),^[7] and to Grandy (2008),^[8] there is no general rule that provides an extremum principle that governs the evolution of a far-from-equilibrium system to a steady state. According to Glansdorff and Prigogine (1971, page 16),^[9]irreversible processes usually are not governed by global extremal principles because description of their evolution requires differential equations which are not self-adjoint, but local extremal principles can be used for local solutions. Lebon Jou and Casas-Vásquez (2008)^[10] state that "In non-equilibrium ... it is generally not possible to construct thermodynamic potentials depending on the whole set of variables". Šilhavý (1997)^[11] offers the opinion that "... the extremum principles of thermodynamics ... do not have any counterpart for [non-equilibrium] steady states (despite many claims in the literature)." It follows that any general extremal principle for a non-equilibrium problem will need to refer in some detail to the constraints that are specific for the structure of the system considered in the problem.

Fluctuations, entropy, 'thermodynamics forces', and reproducible dynamical structure[edit]

Apparent 'fluctuations', which appear to arise when initial conditions are inexactly specified, are the drivers of the formation of non-equilibrium dynamical structures. There is no special force of nature involved in the generation of such fluctuations. Exact specification of initial conditions would require statements of the positions and velocities of all particles in the system, obviously not a remotely practical possibility for a macroscopic system. This is the nature of thermodynamic fluctuations. They cannot be predicted in particular by the scientist, but they are determined by the laws of nature and they are the singular causes of the natural development of dynamical structure.^[9]

It is pointed out^[12]^[13]^[14]^[15] by W.T. Grandy Jr that entropy, though it may be defined for a non-equilibrium system, is when strictly considered, only a macroscopic quantity that refers to the whole system, and is not a dynamical variable and in general does not act as a local potential that describes local physical forces. Under special circumstances, however, one can metaphorically think as if the thermal variables behaved like local physical forces. The approximation that constitutes classical irreversible thermodynamics is built on this metaphoric thinking.

As indicated by the " " marks of Onsager (1931),^[1] such a metaphorical but not categorically mechanical force, the thermal "force", $X_{{th}}$ , 'drives' the conduction of heat. For this so-called "thermodynamic force", we can write

X_{{th}}=-{\frac {1}{T}}\nabla T

Actually this thermal "thermodynamic force" is a manifestation of the degree of inexact specification of the microscopic initial conditions for the system, expressed in the thermodynamic variable known as temperature, $T$ . Temperature is only one example, and all the thermodynamic macroscopic variables constitute inexact specifications of the initial conditions, and have their respective "thermodynamic forces". These inexactitudes of specification are the source of the apparent fluctuations that drive the generation of dynamical structure, of the very precise but still less than perfect reproducibility of non-equilibrium experiments, and of the place of entropy in thermodynamics. If one did not know of such inexactitude of specification, one might find the origin of the fluctuations mysterious. What is meant here by "inexactitude of specification" is not that the mean values of the macroscopic variables are inexactly specified, but that the use of macroscopic variables to describe processes that actually occur by the motions and interactions of microscopic objects such as molecules is necessarily lacking in the molecular detail of the processes, and is thus inexact. There are many microscopic states compatible with a single macroscopic state, but only the latter is specified, and that is specified exactly for the purposes of the theory.

It is reproducibility in repeated observations that identifies dynamical structure in a system. E.T. Jaynes^[16]^[17]^[18]^[19] explains how this reproducibility is why entropy is so important in this topic: entropy is a measure of experimental reproducibility. The entropy tells how many times one would have to repeat the experiment in order to expect to see a departure from the usual reproducible result. When the process goes on in a system with less than a 'practically infinite' number (much much less than Avogadro's or Loschmidt's numbers) of molecules, the thermodynamic reproducibility fades, and fluctuations become easier to see.^[20]^[21]

According to this view of Jaynes, it is a common and mystificatory abuse of language, that one often sees reproducibility of dynamical structure called "order".^[8]^[22] Dewar^[22] writes "Jaynes considered reproducibility - rather than disorder - to be the key idea behind the second law of thermodynamics (Jaynes 1963,^[23] 1965,^[19] 1988,^[24] 1989^[25])." Grandy (2008)^[8] in section 4.3 on page 55 clarifies the distinction between the idea that entropy is related to order (which he considers to be an "unfortunate" "mischaracterization" that needs "debunking"), and the aforementioned idea of Jaynes that entropy is a measure of experimental reproducibility of process (which Grandy regards as correct). According to this view, even the admirable book of Glansdorff and Prigogine (1971)^[9]is guilty of this unfortunate abuse of language.

Local thermodynamic equilibrium[edit]

Various principles have been proposed by diverse authors for over a century. According to Glansdorff and Prigogine (1971, page 15),^[9] in general, these principles apply only to systems that can be described by thermodynamical variables, in which dissipative processes dominate by excluding large deviations from statistical equilibrium. The thermodynamical variables are defined subject to the kinematical requirement of local thermodynamic equilibrium. This means that collisions between molecules are so frequent that chemical and radiative processes do not disrupt the local Maxwell-Boltzmann distribution of molecular velocities.

Linear and non-linear processes[edit]

Dissipative structures can depend on the presence of non-linearity in their dynamical régimes. Autocatalytic reactions provide examples of non-linear dynamics, and may lead to the natural evolution of self-organized dissipative structures.

Continuous and discontinuous motions of fluids[edit]

Much of the theory of classical non-equilibrium thermodynamics is concerned with the spatially continuous motion of fluids, but fluids can also move with spatial discontinuities. Helmholtz (1868)^[26] wrote about how in a flowing fluid, there can arise a zero fluid pressure, which sees the fluid broken asunder. This arises from the momentum of the fluid flow, showing a different kind of dynamical structure from that of the conduction of heat or electricity. Thus for example: water from a nozzle can form a shower of droplets (Rayleigh 1878,^[27] and in section 357 et seq. of Rayleigh (1896/1926)^[28]); waves on the surface of the sea break discontinuously when they reach the shore (Thom 1975^[29]). Helmholtz pointed out that the sounds of organ pipes must arise from such discontinuity of flow, occasioned by the passage of air past a sharp-edged obstacle; otherwise the oscillatory character of the sound wave would be damped away to nothing. The definition of the rate of entropy production of such a flow is not covered by the usual theory of classical non-equilibrium thermodynamics. There are many other commonly observed discontinuities of fluid flow that also lie beyond the scope of the classical theory of non-equilibrium thermodynamics, such as: bubbles in boiling liquids and in effervescent drinks; also protected towers of deep tropical convection (Riehl, Malkus 1958^[30]), also called penetrative convection (Lindzen 1977^[31]).

Historical development[edit]

W. Thomson, Baron Kelvin[edit]

William Thomson, later Baron Kelvin, (1852 a,^[32] 1852 b^[33]) wrote

"II. When heat is created by any unreversible process (such as friction), there is a dissipation of mechanical energy, and a full restoration of it to its primitive condition is impossible.

III. When heat is diffused by conduction, there is a dissipation of mechanical energy, and perfect restoration is impossible.

IV. When radiant heat or light is absorbed, otherwise than in vegetation, or in a chemical reaction, there is a dissipation of mechanical energy, and perfect restoration is impossible."

In 1854, Thomson wrote about the relation between two previously known non-equilibrium effects. In the Peltier effect, an electric current driven by an external electric field across a bimetallic junction will cause heat to be carried across the junction when the temperature gradient is constrained to zero. In the Seebeck effect, a flow of heat driven by a temperature gradient across such a junction will cause an electromotive force across the junction when the electric current is constrained to zero. Thus thermal and electric effects are said to be coupled. Thomson (1854)^[34] proposed a theoretical argument, partly based on the work of Carnot and Clausius, and in those days partly simply speculative, that the coupling constants of these two effects would be found experimentally to be equal. Experiment later confirmed this proposal. It was later one of the ideas that led Onsager to his results as noted below.

Helmholtz[edit]

In 1869, Hermann von Helmholtz stated his Helmholtz minimum dissipation theorem,^[35] subject to a certain kind of boundary condition, a principle of least viscous dissipation of kinetic energy: "For a steady flow in a viscous liquid, with the speeds of flow on the boundaries of the fluid being given steady, in the limit of small speeds, the currents in the liquid so distribute themselves that the dissipation of kinetic energy by friction is minimum."^[36]

In 1878, Helmholtz,^[37] like Thomson also citing Carnot and Clausius, wrote about electric current in an electrolyte solution with a concentration gradient. This shows a non-equilibrium coupling, between electric effects and concentration-driven diffusion. Like Thomson (Kelvin) as noted above, Helmholtz also found a reciprocal relation, and this was another of the ideas noted by Onsager.

J. W. Strutt, Baron Rayleigh[edit]

Rayleigh (1873)^[38] (and in Sections 81 and 345 of Rayleigh (1896/1926)^[28]) introduced the dissipation function for the description of dissipative processes involving viscosity. More general versions of this function have been used by many subsequent investigators of the nature of dissipative processes and dynamical structures. Rayleigh's dissipation function was conceived of from a mechanical viewpoint, and it did not refer in its definition to temperature, and it needed to be 'generalized' to make a dissipation function suitable for use in non-equilibrium thermodynamics.

Studying jets of water from a nozzle, Rayleigh (1878,^[27] 1896/1926^[28]) noted that when a jet is in a state of conditionally stable dynamical structure, the mode of fluctuation most likely to grow to its full extent and lead to another state of conditionally stable dynamical structure is the one with the fastest growth rate. In other words, a jet can settle into a conditionally stable state, but it is likely to suffer fluctuation so as to pass to another, less unstable, conditionally stable state. He used like reasoning in a study of Bénard convection.^[39] These physically lucid considerations of Rayleigh seem to contain the heart of the distinction between the principles of minimum and maximum rates of dissipation of energy and entropy production, which have been developed in the course of physical investigations by later authors.

Korteweg[edit]

Korteweg (1883)^[40] gave a proof "that in any simply connected region, when the velocities along the boundaries are given, there exists, as far as the squares and products of the velocities may be neglected, only one solution of the equations for the steady motion of an incompressible viscous fluid, and that this solution is always stable." He attributed the first part of this theorem to Helmholtz, who had shown that it is a simple consequence of a theorem that "if the motion be steady, the currents in a viscous [incompressible] fluid are so distributed that the loss of [kinetic] energy due to viscosity is a minimum, on the supposition that the velocities along boundaries of the fluid are given." Because of the restriction to cases in which the squares and products of the velocities can be neglected, these motions are below the threshold for turbulence.

Onsager[edit]

Great theoretical progress was made by Onsager in 1931^[1]^[41] and in 1953.^[42]^[43]

Prigogine[edit]

Further progress was made by Prigogine in 1945^[44] and later.^[9]^[45] Prigogine (1947)^[44] cites Onsager (1931).^[1]^[41]

Casimir[edit]

Casimir (1945)^[46] extended the theory of Onsager.

Ziman[edit]

Ziman (1956)^[47] gave very readable account. He proposed the following as a general principle of the thermodynamics of irreversible processes: "Consider all distributions of currents such that the intrinsic entropy production equals the extrinsic entropy production for the given set of forces. Then, of all current distributions satisfying this condition, the steady state distribution makes the entropy production a maximum." He commented that this was a known general principle, discovered by Onsager, but was "not quoted in any of the books on the subject". He notes the difference between this principle and "Prigogine's theorem, which states, crudely speaking, that if not all the forces acting on a system are fixed the free forces will take such values as to make the entropy production a minimum." Prigogine was present when this paper was read and he is reported by the journal editor to have given "notice that he doubted the validity of part of Ziman's thermodynamic interpretation".

Ziegler[edit]

Hans Ziegler extended the Melan-Prager non-equilibrium theory of materials to the non-isothermal case.^[48]

Gyarmati[edit]

Gyarmati (1967/1970)^[2] gives a systematic presentation, and extends Onsager's principle of least dissipation of energy, to give a more symmetric form known as Gyarmati's principle. Gyarmati (1967/1970)^[2] cites 11 papers or books authored or co-authored by Prigogine.

Gyarmati (1967/1970)^[2] also gives in Section III 5 a very helpful precis of the subtleties of Casimir (1945)).^[46] He explains that the Onsager reciprocal relations concern variables which are even functions of the velocities of the molecules, and notes that Casimir went on to derive anti-symmetric relations concerning variables which are odd functions of the velocities of the molecules.

Paltridge[edit]

The physics of the earth's atmosphere includes dramatic events like lightning and the effects of volcanic eruptions, with discontinuities of motion such as noted by Helmholtz (1868).^[26] Turbulence is prominent in atmospheric convection. Other discontinuities include the formation of raindrops, hailstones, and snowflakes. The usual theory of classical non-equilibrium thermodynamics will need some extension to cover atmospheric physics. According to Tuck (2008),^[49] "On the macroscopic level, the way has been pioneered by a meteorologist (Paltridge 1975,^[50] 2001^[51]). Initially Paltridge (1975)^[50] used the terminology "minimum entropy exchange", but after that, for example in Paltridge (1978),^[52] and in Paltridge (1979)^[53]), he used the now current terminology "maximum entropy production" to describe the same thing. This point is clarified in the review by Ozawa, Ohmura, Lorenz, Pujol (2003).^[54] Paltridge (1978)^[52] cited Busse's (1967)^[55] fluid mechanical work concerning an extremum principle. Nicolis and Nicolis (1980) ^[56] discuss Paltridge's work, and they comment that the behaviour of the entropy production is far from simple and universal. This seems natural in the context of the requirement of some classical theory of non-equilibrium thermodynamics that the threshold of turbulence not be crossed. Paltridge himself nowadays tends to prefer to think in terms of the dissipation function rather than in terms of rate of entropy production.

Speculated thermodynamic extremum principles for energy dissipation and entropy production[edit]

Jou, Casas-Vazquez, Lebon (1993)^[57] note that classical non-equilibrium thermodynamics "has seen an extraordinary expansion since the second world war", and they refer to the Nobel prizes for work in the field awarded to Lars Onsager and Ilya Prigogine. Martyushev and Seleznev (2006)^[4] note the importance of entropy in the evolution of natural dynamical structures: "Great contribution has been done in this respect by two scientists, namely Clausius, ... , and Prigogine." Prigogine in his 1977 Nobel Lecture^[58] said: "... non-equilibrium may be a source of order. Irreversible processes may lead to a new type of dynamic states of matter which I have called “dissipative structures”." Glansdorff and Prigogine (1971)^[9] wrote on page xx: "Such 'symmetry breaking instabilities' are of special interest as they lead to a spontaneous 'self-organization' of the system both from the point of view of its space order and its function."

Analyzing the Rayleigh–Bénard convection cell phenomenon, Chandrasekhar (1961)^[59] wrote "Instability occurs at the minimum temperature gradient at which a balance can be maintained between the kinetic energy dissipated by viscosity and the internal energy released by the buoyancy force." With a temperature gradient greater than the minimum, viscosity can dissipate kinetic energy as fast as it is released by convection due to buoyancy, and a steady state with convection is stable. The steady state with convection is often a pattern of macroscopically visible hexagonal cells with convection up or down in the middle or at the 'walls' of each cell, depending on the temperature dependence of the quantities; in the atmosphere under various conditions it seems that either is possible. (Some details are discussed by Lebon, Jou, and Casas-Vásquez (2008)^[10] on pages 143–158.) With a temperature gradient less than the minimum, viscosity and heat conduction are so effective that convection cannot keep going.

Glansdorff and Prigogine (1971)^[9] on page xv wrote "Dissipative structures have a quite different [from equilibrium structures] status: they are formed and maintained through the effect of exchange of energy and matter in non-equilibrium conditions." They were referring to the dissipation function of Rayleigh (1873)^[38] that was used also by Onsager (1931, I,^[1] 1931, II^[41]). On pages 78–80 of their book^[9] Glansdorff and Prigogine (1971) consider the stability of laminar flow that was pioneered by Helmholtz; they concluded that at a stable steady state of sufficiently slow laminar flow, the dissipation function was minimum.

These advances have led to proposals for various extremal principles for the "self-organized" régimes that are possible for systems governed by classical linear and non-linear non-equilibrium thermodynamical laws, with stable stationary régimes being particularly investigated. Convection introduces effects of momentum which appear as non-linearity in the dynamical equations. In the more restricted case of no convective motion, Prigogine wrote of "dissipative structures". Šilhavý (1997)^[11] offers the opinion that "... the extremum principles of [equilibrium] thermodynamics ... do not have any counterpart for [non-equilibrium] steady states (despite many claims in the literature)."

Prigogine's proposed theorem of minimum entropy production for very slow purely diffusive transfer[edit]

In 1945 Prigogine^[44] (see also Prigogine (1947)^[60]) proposed a “Theorem of Minimum Entropy Production” which applies only to the purely diffusive linear regime, with negligible inertial terms, near a stationary thermodynamically non-equilibrium state. Prigogine's proposal is that the rate of entropy production is locally minimum at every point. The proof offered by Prigogine is open to serious criticism.^[61] A critical and unsupportive discussion of Prigogine's proposal is offered by Grandy (2008).^[8] It has been shown by Barbera that the total whole body entropy production cannot be minimum, but this paper did not consider the pointwise minimum proposal of Prigogine.^[62] A proposal closely related to Prigogine's is that the pointwise rate of entropy production should have its maximum value minimized at the steady state. This is compatible, but not identical, with the Prigogine proposal.^[63] Moreover, N. W. Tschoegl proposes a proof, perhaps more physically motivated than Prigogine's, that would if valid support the conclusion of Helmholtz and of Prigogine, that under these restricted conditions, the entropy production is at a pointwise minimum.^[64]

Faster transfer with convective circulation: second entropy[edit]

In contrast to the case of sufficiently slow transfer with linearity between flux and generalized force with negligible inertial terms, there can be heat transfer that is not very slow. Then there is consequent non-linearity, and heat flow can develop into phases of convective circulation. In these cases, the time rate of entropy production has been shown to be a non-monotonic function of time during the approach to steady state heat convection. This makes these cases different from the near-thermodynamic-equilibrium regime of very-slow-transfer with linearity. Accordingly, the local time rate of entropy production, defined according to the local thermodynamic equilibrium hypothesis, is not an adequate variable for prediction of the time course of far-from-thermodynamic equilibrium processes. The principle of minimum entropy production is not applicable to these cases.

To cover these cases, there is needed at least one further state variable, a non-equilibrium quantity, the so-called second entropy. This appears to be a step towards generalization beyond the classical second law of thermodynamics, to cover non-equilibrium states or processes. The classical law refers only to states of thermodynamic equilibrium, and local thermodynamic equilibrium theory is an approximation that relies upon it. Still it is invoked to deal with phenomena near but not at thermodynamic equilibrium, and has some uses then. But the classical law is inadequate for description of the time course of processes far from thermodynamic equilibrium. For such processes, a more powerful theory is needed, and the second entropy is part of such a theory.^[65]^[66]

Speculated principles of maximum entropy production and minimum energy dissipation[edit]

Onsager (1931, I)^[1] wrote: "Thus the vector field J of the heat flow is described by the condition that the rate of increase of entropy, less the dissipation function, be a maximum." Careful note needs to be taken of the opposite signs of the rate of entropy production and of the dissipation function, appearing in the left-hand side of Onsager's equation (5.13) on Onsager's page 423.^[1]

Although largely unnoticed at the time, Ziegler proposed an idea early with his work in the mechanics of plastics in 1961,^[67] and later in his book on thermomechanics revised in 1983,^[3] and in various papers (e.g., Ziegler (1987),^[68]). Ziegler never stated his principle as a universal law but he may have intuited this. He demonstrated his principle using vector space geometry based on an “orthogonality condition” which only worked in systems where the velocities were defined as a single vector or tensor, and thus, as he wrote^[3] at p. 347, was “impossible to test by means of macroscopic mechanical models”, and was, as he pointed out, invalid in “compound systems where several elementary processes take place simultaneously”.

In relation to the earth's atmospheric energy transport process, according to Tuck (2008),^[49] "On the macroscopic level, the way has been pioneered by a meteorologist (Paltridge 1975,^[50] 2001^[69])." Initially Paltridge (1975)^[50] used the terminology "minimum entropy exchange", but after that, for example in Paltridge (1978),^[52] and in Paltridge (1979),^[70] he used the now current terminology "maximum entropy production" to describe the same thing. The logic of Paltridge's earlier work is open to serious criticism.^[8] Nicolis and Nicolis (1980) ^[56] discuss Paltridge's work, and they comment that the behaviour of the entropy production is far from simple and universal. Later work by Paltridge focuses more on the idea of a dissipation function than on the idea of rate of production of entropy.^[69]

Sawada (1981),^[71] also in relation to the Earth's atmospheric energy transport process, postulating a principle of largest amount of entropy increment per unit time, cites work in fluid mechanics by Malkus and Veronis (1958)^[72] as having "proven a principle of maximum heat current, which in turn is a maximum entropy production for a given boundary condition", but this inference is not logically valid. Again investigating planetary atmospheric dynamics, Shutts (1981)^[73] used an approach to the definition of entropy production, different from Paltridge's, to investigate a more abstract way to check the principle of maximum entropy production, and reported a good fit.

Prospects[edit]

Until recently, prospects for useful extremal principles in this area have seemed clouded. C. Nicolis (1999)^[74] concludes that one model of atmospheric dynamics has an attractor which is not a regime of maximum or minimum dissipation; she says this seems to rule out the existence of a global organizing principle, and comments that this is to some extent disappointing; she also points to the difficulty of finding a thermodynamically consistent form of entropy production. Another top expert offers an extensive discussion of the possibilities for principles of extrema of entropy production and of dissipation of energy: Chapter 12 of Grandy (2008)^[8] is very cautious, and finds difficulty in defining the 'rate of internal entropy production' in many cases, and finds that sometimes for the prediction of the course of a process, an extremum of the quantity called the rate of dissipation of energy may be more useful than that of the rate of entropy production; this quantity appeared in Onsager's 1931^[1] origination of this subject. Other writers have also felt that prospects for general global extremal principles are clouded. Such writers include Glansdorff and Prigogine (1971), Lebon, Jou and Casas-Vásquez (2008), and Šilhavý (1997). It has been shown that heat convection does not obey extremal principles for entropy production^[65] and chemical reactions do not obey extremal principles for the secondary differential of entropy production,^[75] hence the development of a general extremal principle seems infeasible.

Creation of order[edit]

Background[edit]

The second law of thermodynamics states that the disorder (entropy) of a physical or chemical system and its surroundings (a closed system) must increase with time. Systems left to themselves become increasingly random, and orderly energy of a system like uniform motion degrades eventually to the random motion of particles in a heat bath.

There are, however, many instances in which physical systems spontaneously become emergent or orderly. For example, despite the destruction they cause, hurricanes have a very orderly vortex motion when compared to the random motion of the air molecules in a closed room. Even more spectacular is the order created by chemical systems; the most dramatic being the order associated with life.

This is consistent with the Second Law, which requires that the total disorder of a system and its surroundings must increase with time. Order can be created in a system by an even greater decrease in order of the system's surroundings.^[4] In the hurricane example, hurricanes are formed from unequal heating within the atmosphere. The Earth's atmosphere is then far from thermal equilibrium. The order of the Earth's atmosphere increases, but at the expense of the order of the sun. The sun is becoming more disorderly as it ages and throws off light and material to the rest of the universe. The total disorder of the sun and the earth increases despite the fact that orderly hurricanes are generated on earth.

A similar example exists for living chemical systems. The sun provides energy to green plants. The green plants are food for other living chemical systems. The energy absorbed by plants and converted into chemical energy generates a system on earth that is orderly and far from chemical equilibrium. Here, the difference from chemical equilibrium is determined by an excess of reactants over the equilibrium amount. Once again, order on earth is generated at the expense of entropy increase of the sun. The total entropy of the earth and the rest of the universe increases, consistent with the Second Law.

Some autocatalytic reactions also generate order in a system at the expense of its surroundings. For example, (clock reactions) have intermediates whose concentrations oscillate in time, corresponding to temporal order. Other reactions generate spatial separation of chemical species corresponding to spatial order. More complex reactions are involved in metabolic pathways and metabolic networks in biological systems.

The transition to order as the distance from equilibrium increases is not usually continuous. Order typically appears abruptly. The threshold between the disorder of chemical equilibrium and order is known as a phase transition. The conditions for a phase transition can be determined with the mathematical machinery of non-equilibrium thermodynamics.

Temporal order[edit]

A chemical reaction cannot oscillate about a position of final equilibrium because the second law of thermodynamics requires that a thermodynamic systemapproach equilibrium and not recede from it. For a closed system at constant temperature and pressure, the Gibbs free energy must decrease continuously and not oscillate. However it is possible that the concentrations of some reaction intermediates oscillate, and also that the rate of formation of products oscillates.^[5]

Idealized example: Lotka–Volterra equation[edit]

The Lotka–Volterra equation is isomorphic with the predator–prey model and the two-reaction autocatalytic model. In this example baboons and cheetahs are equivalent to two different chemical species X and Y in autocatalytic reactions.

Consider a coupled set of two autocatalytic reactions in which the concentration of one of the reactants A is much larger than its equilibrium value. In this case, the forward reaction rate is so much larger than the reverse rates that we can neglect the reverse rates.

A+X\rightarrow 2X

X+Y\rightarrow 2Y

Y\rightarrow E

with the rate equations

{d \over dt}[X]=k_{1}[A][X]-k_{2}[X][Y]\,

{d \over dt}[Y]=k_{2}[X][Y]-k_{3}[Y]\,

Here, we have neglected the depletion of the reactant A, since its concentration is so large. The rate constants for the three reactions are $k_{1}$ , $k_{2}$ , and $k_{3}$ , respectively.

This system of rate equations is known as the Lotka–Volterra equation and is most closely associated with population dynamics in predator–prey relationships. This system of equations can yield oscillating concentrations of the reaction intermediates X and Y. The amplitude of the oscillations depends on the concentration of A (which decreases without oscillation). Such oscillations are a form of emergent temporal order that is not present in equilibrium.

Another idealized example: Brusselator[edit]

Another example of a system that demonstrates temporal order is the Brusselator (see Prigogine reference). It is characterized by the reactions

A\rightarrow X

2X+Y\rightarrow 3X

B+X\rightarrow Y+D

X\rightarrow E

with the rate equations

{d \over dt}[X]=[A]+[X]^{2}[Y]-[B][X]-[X]\,

{d \over dt}[Y]=[B][X]-[X]^{2}[Y]\,

where, for convenience, the rate constants have been set to 1.

The Brusselator in the unstable regime. A=1. B=2.5. X(0)=1. Y(0)=0. The system approaches a limit cycle. For B<1+A the system is stable and approaches a fixed point.

The Brusselator has a fixed point at

[X]=A\,

[Y]={B \over A}\,

The fixed point becomes unstable when

B>1+A^{2}\,

leading to an oscillation of the system. Unlike the Lotka-Volterra equation, the oscillations of the Brusselator do not depend on the amount of reactant present initially. Instead, after sufficient time, the oscillations approach a limit cycle.^[6]

Spatial order[edit]

An idealized example of spatial spontaneous symmetry breaking is the case in which we have two boxes of material separated by a permeable membrane so that material can diffuse between the two boxes. It is assumed that identical Brusselators are in each box with nearly identical initial conditions. (see Prigogine reference)

{d \over dt}[X_{1}]=[A]+[X_{1}]^{2}[Y_{1}]-[B][X_{1}]-[X_{1}]+D_{x}\left(X_{2}-X_{1}\right)\,

{d \over dt}[Y_{1}]=[B][X_{1}]-[X_{1}]^{2}[Y_{1}]+D_{y}\left(Y_{2}-Y_{1}\right)\,

{d \over dt}[X_{2}]=[A]+[X_{2}]^{2}[Y_{2}]-[B][X_{2}]-[X_{2}]+D_{x}\left(X_{1}-X_{2}\right)\,

{d \over dt}[Y_{2}]=[B][X_{2}]-[X_{2}]^{2}[Y_{2}]+D_{y}\left(Y_{1}-Y_{2}\right)\,

Here, the numerical subscripts indicate which box the material is in. There are additional terms proportional to the diffusion coefficient D that account for the exchange of material between boxes.

If the system is initiated with the same conditions in each box, then a small fluctuation will lead to separation of materials between the two boxes. One box will have a predominance of X, and the other will have a predominance of Y.

Real examples[edit]

Real examples of clock reactions are the Belousov–Zhabotinsky reaction (BZ reaction), the Briggs–Rauscher reaction, the Bray–Liebhafsky reaction and the iodine clock reaction. These are oscillatory reactions, and the concentration of products and reactants can be approximated in terms of damped oscillations.

The best-known reaction, the BZ reaction, can be created with a mixture of potassium bromate ${\ce {(KBrO3)}}$ , malonic acid ${\ce {(CH2(COOH)2)}}$ , and manganese sulfate ${\ce {(MnSO4)}}$ prepared in a heated solution with sulfuric acid ${\ce {(H2SO4)}}$ as solvent.^[7]

Optics example[edit]

Another autocatalytic system is one driven by light coupled to photo-polymerization reactions. In a process termed optical autocatalysis, positive feedback is created between light intensity and photo-polymerization rate, via polymerization-induced increases in the refractive index. Light's preference to occupy regions of higher refractive index results in leakage of light into regions of higher molecular weight, thereby amplifying the photo-chemical reaction. The positive feedback may be expressed as:^[8]

{\text{polymerization rate}}\to {\text{molecular weight}}/{\text{refractive index}}\to {\text{intensity}}

Noting that photo-polymerization rate is proportional to intensity^[9] and that refractive index is proportional to molecular weight,^[10] the positive feedback between intensity and photo-polymerization establishes the auto-catalytic behavior. Optical auto-catalysis has been shown to result on spontaneous pattern formation in photopolymers.^[11]^[12]^[13] Hosein and co-workers discovered that optical autocatalysis can also occur in photoreactive polymer blends, and that the process can induce binary phase morphologies with the same pattern as the light profile.^[8] The light undergoes optical modulation instability, spontaneous dividing into a multitude of optical filaments, and the polymer system thereby forms filaments within the blend structure.^[8] The result is a new system that couples optical autocatalytic behavior to spinodal decomposition.

Biological example[edit]

It is known that an important metabolic cycle, glycolysis, displays temporal order.^[14] Glycolysis consists of the degradation of one molecule of glucose and the overall production of two molecules of ATP. The process is therefore of great importance to the energetics of living cells. The global glycolysis reaction involves glucose, ADP, NAD, pyruvate, ATP, and NADH.

{\ce {glucose{}+2ADP{}+2P_{\mathit {i}}{}+2NAD->2(pyruvate){}+2ATP{}+2NADH}}

The details of the process are quite involved, however, a section of the process is autocatalyzed by phosphofructokinase (PFK). This portion of the process is responsible for oscillations in the pathway that lead to the process oscillating between an active and an inactive form. Thus, the autocatalytic reaction can modulate the process.

Shape tailoring of thin layers[edit]

It is possible to use the results from an autocatalytic reaction coupled with reaction–diffusion system theory to tailor the design of a thin layer. The autocatalytic process allows controlling the nonlinear behavior of the oxidation front, which is used to establish the initial geometry needed to generate the arbitrary final geometry.^[15] It has been successfully done in the wet oxidation of $Al_{x}Ga_{1-x}As$ to obtain arbitrary shaped layers of $AlO_{x}$ .

Phase transitions[edit]

The initial amounts of reactants determine the distance from a chemical equilibrium of the system. The greater the initial concentrations the further the system is from equilibrium. As the initial concentration increases, an abrupt change in order occurs. This abrupt change is known as phase transition. At the phase transition, fluctuations in macroscopic quantities, such as chemical concentrations, increase as the system oscillates between the more ordered state (lower entropy, such as ice) and the more disordered state (higher entropy, such as liquid water). Also, at the phase transition, macroscopic equations, such as the rate equations, fail. Rate equations can be derived from microscopic considerations. The derivations typically rely on a mean field theory approximation to microscopic dynamical equations. Mean field theory breaks down in the presence of large fluctuations (see Mean field theory article for a discussion). Therefore, since large fluctuations occur in the neighborhood of a phase transition, macroscopic equations, such as rate equations, fail. As the initial concentration increases further, the system settles into an ordered state in which fluctuations are again small. (see Prigogine reference)

Asymmetric autocatalysis[edit]

Asymmetric autocatalysis occurs when the reaction product is chiral and thus acts as a chiral catalyst for its own production. Reactions of this type, such as the Soai reaction, have the property that they can amplify a very small enantiomeric excess into a large one. This has been proposed as an important step in the origin of biological homochirality.^[16]

Role in origin of life[edit]

In 1995 Stuart Kauffman proposed that life initially arose as autocatalytic chemical networks.^[17]

British ethologist Richard Dawkins wrote about autocatalysis as a potential explanation for abiogenesis in his 2004 book The Ancestor's Tale. He cites experiments performed by Julius Rebek and his colleagues at the Scripps Research Institute in California in which they combined amino adenosine and pentafluorophenyl ester with the autocatalyst amino adenosine triacid ester (AATE). One system from the experiment contained variants of AATE which catalyzed the synthesis of themselves. This experiment demonstrated the possibility that autocatalysts could exhibit competition within a population of entities with heredity, which could be interpreted as a rudimentary form of natural selection, and that certain environmental changes (such as irradiation) could alter the chemical structure of some of these self-replicating molecules (an analog for mutation) in such ways that could either boost or interfere with its ability to react, thus boosting or interfering with its ability to replicate and spread in the population.^[18]

Autocatalysis plays a major role in the processes of life. Two researchers who have emphasized its role in the origins of life are Robert Ulanowicz ^[19] and Stuart Kauffman.^[20]

Autocatalysis occurs in the initial transcripts of rRNA. The introns are capable of excising themselves by the process of two nucleophilic transesterification reactions. The RNA able to do this is sometimes referred to as a ribozyme. Additionally, the citric acid cycle is an autocatalytic cycle run in reverse.

Ultimately, biological metabolism itself can be seen as a vast autocatalytic set, in that all of the molecular constituents of a biological cell are produced by reactions involving this same set of molecules.

Examples of autocatalytic reactions[edit]

Photographic processing of silver halide film/paper
DNA replication
Haloform reaction
Formose reaction (also known as Butlerov reaction)
Tin pest
Reaction of permanganate with oxalic acid^[21]
Vinegar syndrome
Binding of oxygen by haemoglobin
The spontaneous degradation of aspirin into salicylic acid and acetic acid, causing very old aspirin in sealed containers to smell mildly of vinegar.
The α-bromination of acetophenone with bromine.
Liesegang rings
Autocatalytic surface growth of metal nanoparticles in solution phase^[22]

Physical processes[edit]

The features of Bénard convection can be obtained by a simple experiment first conducted by Henri Bénard, a French physicist, in 1900.

Development of convection[edit]

Convection cells in a gravity field

The experimental set-up uses a layer of liquid, e.g. water, between two parallel planes. The height of the layer is small compared to the horizontal dimension. At first, the temperature of the bottom plane is the same as the top plane. The liquid will then tend towards an equilibrium, where its temperature is the same as its surroundings. (Once there, the liquid is perfectly uniform: to an observer it would appear the same from any position. This equilibrium is also asymptotically stable: after a local, temporary perturbation of the outside temperature, it will go back to its uniform state, in line with the second law of thermodynamics).

Then, the temperature of the bottom plane is increased slightly yielding a flow of thermal energy conducted through the liquid. The system will begin to have a structure of thermal conductivity: the temperature, and the density and pressure with it, will vary linearly between the bottom and top plane. A uniform linear gradient of temperature will be established. (This system may be modelled by statistical mechanics).

Once conduction is established, the microscopic random movement spontaneously becomes ordered on a macroscopic level, forming Benard convection cells, with a characteristic correlation length.

Convection features[edit]

Simulation of Rayleigh–Bénard convection in 3D.

The rotation of the cells is stable and will alternate from clock-wise to counter-clockwise horizontally; this is an example of spontaneous symmetry breaking. Bénard cells are metastable. This means that a small perturbation will not be able to change the rotation of the cells, but a larger one could affect the rotation; they exhibit a form of hysteresis.

Moreover, the deterministic law at the microscopic level produces a non-deterministic arrangement of the cells: if the experiment is repeated, a particular position in the experiment will be in a clockwise cell in some cases, and a counter-clockwise cell in others. Microscopic perturbations of the initial conditions are enough to produce a non-deterministic macroscopic effect. That is, in principle, there is no way to calculate the macroscopic effect of a microscopic perturbation. This inability to predict long-range conditions and sensitivity to initial-conditions are characteristics of chaotic or complex systems (i.e., the butterfly effect).

turbulent Rayleigh–Bénard convection

If the temperature of the bottom plane was to be further increased, the structure would become more complex in space and time; the turbulent flow would become chaotic.

Convective Bénard cells tend to approximate regular right hexagonal prisms, particularly in the absence of turbulence,^[4]^[5]^[6] although certain experimental conditions can result in the formation of regular right square prisms^[7] or spirals.^[8]

The convective Bénard cells are not unique and will usually appear only in the surface tension driven convection. In general the solutions to the Rayleigh and Pearson^[9] analysis (linear theory) assuming an infinite horizontal layer gives rise to degeneracy meaning that many patterns may be obtained by the system. Assuming uniform temperature at the top and bottom plates, when a realistic system is used (a layer with horizontal boundaries) the shape of the boundaries will mandate the pattern. More often than not the convection will appear as rolls or a superposition of them.

Rayleigh–Bénard instability[edit]

Since there is a density gradient between the top and the bottom plate, gravity acts trying to pull the cooler, denser liquid from the top to the bottom. This gravitational force is opposed by the viscous damping force in the fluid. The balance of these two forces is expressed by a non-dimensional parameter called the Rayleigh number. The Rayleigh number is defined as:

\mathrm{Ra}_{L} = \frac{g \beta} {\nu \alpha} (T_b - T_u) L^3

where

T_u is the temperature of the top plate

T_b is the temperature of the bottom plate

L is the height of the container

g is the acceleration due to gravity

ν is the kinematic viscosity

α is the Thermal diffusivity

β is the Thermal expansion coefficient.

As the Rayleigh number increases, the gravitational forces become more dominant. At a critical Rayleigh number of 1708,^[2] instability sets in and convection cells appear.

The critical Rayleigh number can be obtained analytically for a number of different boundary conditions by doing a perturbation analysis on the linearized equations in the stable state.^[10] The simplest case is that of two free boundaries, which Lord Rayleigh solved in 1916, obtaining Ra = 27⁄4 π⁴ ≈ 657.51.^[11] In the case of a rigid boundary at the bottom and a free boundary at the top (as in the case of a kettle without a lid), the critical Rayleigh number comes out as Ra = 1,100.65.^[12]

Effects of surface tension[edit]

In case of a free liquid surface in contact with air, buoyancy and surface tension effects will also play a role in how the convection patterns develop. Liquids flow from places of lower surface tension to places of higher surface tension. This is called the Marangoni effect. When applying heat from below, the temperature at the top layer will show temperature fluctuations. With increasing temperature, surface tension decreases. Thus a lateral flow of liquid at the surface will take place,^[13] from warmer areas to cooler areas. In order to preserve a horizontal (or nearly horizontal) liquid surface, cooler surface liquid will descend. This down-welling of cooler liquid contributes to the driving force of the convection cells. The specific case of temperature gradient-driven surface tension variations is known as thermo-capillary convection, or Bénard–Marangoni convection.

History and nomenclature[edit]

In 1870, the Irish-Scottish physicist and engineer James Thomson (1822–1892), elder brother of Lord Kelvin, observed water cooling in a tub; he noted that the soapy film on the water's surface was divided as if the surface had been tiled (tesselated). In 1882, he showed that the tesselation was due to the presence of convection cells.^[14] In 1900, the French physicist Henri Bénard (1874–1939) independently arrived at the same conclusion.^[15] This pattern of convection, whose effects are due solely to a temperature gradient, was first successfully analyzed in 1916 by Lord Rayleigh (1842–1919).^[16] Rayleigh assumed boundary conditions in which the vertical velocity component and temperature disturbance vanish at the top and bottom boundaries (perfect thermal conduction). Those assumptions resulted in the analysis losing any connection with Henri Bénard's experiment. This resulted in discrepancies between theoretical and experimental results until 1958, when John Pearson (1930– ) reworked the problem based on surface tension.^[9] This is what was originally observed by Bénard. Nonetheless in modern usage "Rayleigh–Bénard convection" refers to the effects due to temperature, whereas "Bénard–Marangoni convection" refers specifically to the effects of surface tension.^[1] Davis and Koschmieder have suggested that the convection should be rightfully called the "Pearson–Bénard convection".^[2]

Rayleigh–Bénard convection is also sometimes known as "Bénard–Rayleigh convection", "Bénard convection", or "Rayleigh convection".

Design[edit]

The S5G was a pressurized water reactor^[1] plant with two coolant loops and two steam generators. It had to be designed with the reactor vessel situated low in the boat and the steam generators high in order for natural circulation of the primary coolant to be developed and maintained.^[2]

Reactor primary coolant pumps are one of the primary sources of noise from submarines, and the elimination of coolant pumps and associated equipment would also reduce mechanical complexity and the space required by propulsion equipment.

The S5G had primary coolant pumps, but they were only needed for very high speeds. And since the reactor core was designed with very smooth paths for the coolant, the coolant pumps were smaller and quieter than the ones used by the competing S5W core. They were also fewer in number. In most cases the submarine could be operated without using coolant pumps at all. The quiet design resulted in a larger hull diameter but required larger primary coolant piping than the competing S5W reactor.^[3] Due to the larger size, the S5G was not used in subsequent attack submarines, but was a precursor to the S8G reactordesign used in the larger Ohio-class submarines.

To further reduce engine plant noise, the normal propulsion setup of two steam turbines driving the screw through a reduction gear unit was changed instead to one large propulsion turbine with no reduction gears. This eliminated the noise from the main reduction gears, but the cost was to have a huge main propulsion turbine. The turbine was cylindrical, about 12 feet in diameter, and about 30 feet long. This massive size was necessary to allow it to turn slowly enough to directly drive the screw and be fairly efficient in doing so. The same propulsion setup was used on both the USS Narwhal and the land-based prototype.

The concept of a natural circulation plant was relatively new when the Navy requested this design. The prototype plant in Idaho was therefore given quite a rigorous performance shakedown to determine if such a design would work for the US Navy. It was largely a success, although the design never became the basis for any more fast-attack submarines besides the Narwhal. The prototype testing included the simulation of essentially the entire engine room of an attack submarine. Floating the plant in a large pool of water allowed the prototype to be rotated along its long axis to simulate a hard turn, accomplished by torquing large gyroscopes mounted forward of the reactor compartment. This was necessary to determine whether natural circulation would continue even during hard turns, since natural circulation is dependent on gravity whereas submarines are known to maneuver at various angles.

History[edit]

This nuclear reactor was installed both as a land-based prototype at the Nuclear Power Training Unit, Idaho National Laboratory near Arco, Idaho, and on board the USS Narwhal (SSN-671); both have been decommissioned. It was intended to test the potential contribution of natural circulation technology to submarinequieting.^[4]^[5]^[6]

The S5G prototype was permanently shut down in May 1995.^[7]

https://en.wikipedia.org/wiki/S5G_reactor

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportionalto the change of the input.^[1]^[2] Nonlinear problems are of interest to engineers, biologists,^[3]^[4]^[5] physicists,^[6]^[7]mathematicians, and many other scientists because most systems are inherently nonlinear in nature.^[8] Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems.

https://en.wikipedia.org/wiki/Nonlinear_system

In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors which emerge only when the parts interact in a wider whole.

Emergence plays a central role in theories of integrative levels and of complex systems. For instance, the phenomenon of life as studied in biology is an emergent property of chemistry, and many psychologicalphenomena are known to emerge from underlying neurobiological processes.

In philosophy, theories that emphasize emergent properties have been called emergentism.^[1]

https://en.wikipedia.org/wiki/Emergence

In artificial intelligence, genetic programming (GP) is a technique of evolving programs, starting from a population of unfit (usually random) programs, fit for a particular task by applying operations analogous to natural genetic processes to the population of programs. It is essentially a heuristic search technique often described as 'hill climbing'^{[according to whom?]}^{[citation needed]}, i.e. searching for an optimal or at least suitable program among the space of all programs.

The operations are: selection of the fittest programs for reproduction (crossover) and mutation according to a predefined fitness measure, usually proficiency at the desired task. The crossover operation involves swapping random parts of selected pairs (parents) to produce new and different offspring that become part of the new generation of programs. Mutation involves substitution of some random part of a program with some other random part of a program. Some programs not selected for reproduction are copied from the current generation to the new generation. Then the selection and other operations are recursively applied to the new generation of programs.

Typically, members of each new generation are on average more fit than the members of the previous generation, and the best-of-generation program is often better than the best-of-generation programs from previous generations. Termination of the recursion is when some individual program reaches a predefined proficiency or fitness level.

It may and often does happen that a particular run of the algorithm results in premature convergence to some local maximum which is not a globally optimal or even good solution. Multiple runs (dozens to hundreds) are usually necessary to produce a very good result. It may also be necessary to increase the starting population size and variability of the individuals to avoid pathologies.

https://en.wikipedia.org/wiki/Genetic_programming

In computational intelligence (CI), an evolutionary algorithm (EA) is a subset of evolutionary computation,^[1] a generic population-based metaheuristic optimization algorithm. An EA uses mechanisms inspired by biological evolution, such as reproduction, mutation, recombination, and selection. Candidate solutions to the optimization problem play the role of individuals in a population, and the fitness function determines the quality of the solutions (see also loss function). Evolution of the population then takes place after the repeated application of the above operators.

Evolutionary algorithms often perform well approximating solutions to all types of problems because they ideally do not make any assumption about the underlying fitness landscape. Techniques from evolutionary algorithms applied to the modeling of biological evolution are generally limited to explorations of microevolutionary processes and planning models based upon cellular processes. In most real applications of EAs, computational complexity is a prohibiting factor.^[2] In fact, this computational complexity is due to fitness function evaluation. Fitness approximation is one of the solutions to overcome this difficulty. However, seemingly simple EA can solve often complex problems;^{[citation needed]} therefore, there may be no direct link between algorithm complexity and problem complexity.

https://en.wikipedia.org/wiki/Evolutionary_algorithm

Artificial intelligence (AI) – intelligence exhibited by machines or software. It is also the name of the scientific fieldwhich studies how to create computers and computer software that are capable of intelligent behaviour.

https://en.wikipedia.org/wiki/Outline_of_artificial_intelligence

In mathematics, the spiral optimization (SPO) algorithm is a metaheuristic inspired by spiral phenomena in nature.

The first SPO algorithm was proposed for two-dimensional unconstrained optimization^[1] based on two-dimensional spiral models. This was extended to n-dimensional problems by generalizing the two-dimensional spiral model to an n-dimensional spiral model.^[2] There are effective settings for the SPO algorithm: the periodic descent direction setting^[3]and the convergence setting.^[4]

https://en.wikipedia.org/wiki/Spiral_optimization_algorithm

Parity problems are widely used as benchmark problems in genetic programming but inherited from the artificial neural network community. Parity is calculated by summing all the binary inputs and reporting if the sum is odd or even. This is considered difficult because:

a very simple artificial neural network cannot solve it, and
all inputs need to be considered and a change to any one of them changes the answer.

https://en.wikipedia.org/wiki/Parity_benchmark

A schema is a template in computer science used in the field of genetic algorithms that identifies a subset of strings with similarities at certain string positions. Schemata are a special case of cylinder sets; and so form a topological space.^[1]

https://en.wikipedia.org/wiki/Schema_(genetic_algorithms)

In the developmental biology of the early twentieth century, a morphogenetic field is a group of cells able to respond to discrete, localized biochemical signals leading to the development of specific morphological structures or organs.^[1]^[2]The spatial and temporal extents of the embryonic field are dynamic, and within the field is a collection of interacting cells out of which a particular organ is formed.^[3] As a group, the cells within a given morphogenetic field are constrained: thus, cells in a limb field will become a limb tissue, those in a cardiac field will become heart tissue.^[4] However, specific cellular programming of individual cells in a field is flexible: an individual cell in a cardiac field can be redirected via cell-to-cell signaling to replace specific damaged or missing cells.^[4] Imaginal discs in insect larvae are examples of morphogenetic fields.^[5]

https://en.wikipedia.org/wiki/Morphogenetic_field

A Bayesian network (also known as a Bayes network, Bayes net, belief network, or decision network) is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor. For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.

Efficient algorithms can perform inference and learning in Bayesian networks. Bayesian networks that model sequences of variables (e.g. speech signals or protein sequences) are called dynamic Bayesian networks. Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called influence diagrams.

https://en.wikipedia.org/wiki/Bayesian_network

In statistics, the likelihood function (often simply called the likelihood) measures the goodness of fit of a statistical model to a sample of data for given values of the unknown parameters. It is formed from the joint probability distribution of the sample, but viewed and used as a function of the parameters only, thus treating the random variables as fixed at the observed values.^[a]

The likelihood function describes a hypersurface whose peak, if it exists, represents the combination of model parameter values that maximize the probability of drawing the sample obtained.^[1] The procedure for obtaining these arguments of the maximum of the likelihood function is known as maximum likelihood estimation, which for computational convenience is usually done using the natural logarithm of the likelihood, known as the log-likelihood function. Additionally, the shape and curvature of the likelihood surface represent information about the stability of the estimates, which is why the likelihood function is often plotted as part of a statistical analysis.^[2]

The case for using likelihood was first made by R. A. Fisher,^[3] who believed it to be a self-contained framework for statistical modelling and inference. Later, Barnard and Birnbaum led a school of thought that advocated the likelihood principle, postulating that all relevant information for inference is contained in the likelihood function.^[4]^[5] But in both frequentist and Bayesian statistics, the likelihood function plays a fundamental role.^[6]

https://en.wikipedia.org/wiki/Likelihood_function

The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy, in the context of precisely stated prior data (such as a proposition that expresses testable information).

Another way of stating this: Take precisely stated prior data or testable information about a probability distribution function. Consider the set of all trial probability distributions that would encode the prior data. According to this principle, the distribution with maximal information entropy is the best choice.

Since the distribution with the maximum entropy is the one that makes the fewest assumptions about the true distribution of data, the principle of maximum entropy can be seen as an application of Occam's razor.

https://en.wikipedia.org/wiki/Principle_of_maximum_entropy

Radical probabilism is a doctrine in philosophy, in particular epistemology, and probability theory that holds that no facts are known for certain. That view holds profound implications for statistical inference. The philosophy is particularly associated with Richard Jeffrey who wittily characterised it with the dictum "It's probabilities all the way down."

https://en.wikipedia.org/wiki/Radical_probabilism

Certain and uncertain knowledge[edit]

That works when the new data is certain. C. I. Lewis had argued that "If anything is to be probable then something must be certain".^[9] There must, on Lewis' account, be some certain facts on which probabilities were conditioned. However, the principle known as Cromwell's rule declares that nothing, apart from a logical law, if that, can ever be known for certain. Jeffrey famously rejected Lewis' dictum.^[10] He later quipped, "It's probabilities all the way down," a reference to the "turtles all the way down" metaphor for the infinite regress problem. He called this position radical probabilism.^[11]

https://en.wikipedia.org/wiki/Radical_probabilism

Cromwell's rule, named by statistician Dennis Lindley,^[1] states that the use of prior probabilities of 1 ("the event will definitely occur") or 0 ("the event will definitely not occur") should be avoided, except when applied to statements that are logically true or false, such as 2+2 equaling 4 or 5.

The reference is to Oliver Cromwell, who wrote to the General Assembly of the Church of Scotland on 3 August 1650, shortly before the Battle of Dunbar, including a phrase that has become well known and frequently quoted:^[2]

I beseech you, in the bowels of Christ, think it possible that you may be mistaken.

As Lindley puts it, assigning a probability should "leave a little probability for the moon being made of green cheese; it can be as small as 1 in a million, but have it there since otherwise an army of astronauts returning with samples of the said cheese will leave you unmoved."^[3] Similarly, in assessing the likelihood that tossing a coin will result in either a head or a tail facing upwards, there is a possibility, albeit remote, that the coin will land on its edge and remain in that position.

If the prior probability assigned to a hypothesis is 0 or 1, then, by Bayes' theorem, the posterior probability(probability of the hypothesis, given the evidence) is forced to be 0 or 1 as well; no evidence, no matter how strong, could have any influence.

A strengthened version of Cromwell's rule, applying also to statements of arithmetic and logic, alters the first rule of probability, or the convexity rule, 0 ≤ Pr(A) ≤ 1, to 0 < Pr(A) < 1.

https://en.wikipedia.org/wiki/Cromwell%27s_rule

"Turtles all the way down" is an expression of the problem of infinite regress. The saying alludes to the mythological idea of a World Turtle that supports a flat Earth on its back. It suggests that this turtle rests on the back of an even larger turtle, which itself is part of a column of increasingly large turtles that continues indefinitely.

The exact origin of the phrase is uncertain. In the form "rocks all the way down", the saying appears as early as 1838.^[1]References to the saying's mythological antecedents, the World Turtle and its counterpart the World Elephant, were made by a number of authors in the 17th and 18th centuries.^[2]^[3]

The expression has been used to illustrate problems such as the regress argument in epistemology.

The saying holds that the world is supported by a chain of increasingly large turtles. Beneath each turtle is yet another: it is "turtles all the way down".

Four World Elephants resting on a World Turtle

https://en.wikipedia.org/wiki/Turtles_all_the_way_down

The "world-elephants" are mythical animals which appear in Hindu cosmology. The Amarakosha (5th century) lists the names of eight male elephants bearing the world (along with eight unnamed female elephants). The names listed are Airavata, Pundarika, Vamana, Kumunda, Anjana, Pushpa-danta, Sarva-bhauma, and Supratika. The names of four elephants supporting the earth from the four directions are given in the Ramayana : Viroopaaksha (east), Mahaapadma (south), Saumanasa (west), Bhadra (north).^[1]^[2]

Brewer's Dictionary of Phrase and Fable lists Maha-pudma and Chukwa are names from a "popular rendition of a Hindu myth in which the tortoise Chukwa supports the elephant Maha-pudma, which in turn supports the world".^[3] The spelling Mahapudma originates as a misprint of Mahapadma in Sri Aurobindo's 1921 retelling of a story of the Mahabharata,

The popular rendition of the World Turtle supporting one or several World Elephants is recorded in 1599 in a letter by Emanual de Veiga.^[4] Wilhelm von Humboldt claimed, without any proof, that the idea of a world-elephant maybe due to a confusion, caused by the Sanskrit noun Nāga having the dual meaning of "serpent" and "elephant" (named for its serpent-like trunk), thus representing a corrupted account of the world-serpent.^[5]^[6]^[7]

Love and Death
On the wondrous dais rose a throne,
And he its pedestal whose lotus hood
With ominous beauty crowns his horrible
Sleek folds, great Mahapudma; high displayed
He bears the throne of Death. There sat supreme
With those compassionate and lethal eyes,
Who many names, who many natures holds;
Yama, the strong pure Hades sad and subtle,
Dharma, who keeps the laws of old untouched.^[8]

https://en.wikipedia.org/wiki/World_Elephant

Conditioning on an uncertainty – probability kinematics[edit]

In this case Bayes' rule isn't able to capture a mere subjective change in the probability of some critical fact. The new evidence may not have been anticipated or even be capable of being articulated after the event. It seems reasonable, as a starting position, to adopt the law of total probability and extend it to updating in much the same way as was Bayes' theorem.^[12]

P_new(A) = P_old(A | B)P_new(B) + P_old(A | not-B)P_new(not-B)

Adopting such a rule is sufficient to avoid a Dutch book but not necessary.^[13] Jeffrey advocated this as a rule of updating under radical probabilism and called it probability kinematics. Others have named it Jeffrey conditioning.

Alternatives to probability kinematics[edit]

Probability kinematics is not the only sufficient updating rule for radical probabilism. Others have been advocated including E. T. Jaynes' maximum entropy principle, and Skyrms' principle of reflection. It turns out that probability kinematics is a special case of maximum entropy inference. However, maximum entropy is not a generalisation of all such sufficient updating rules.^[14]

Selected bibliography[edit]

Jeffrey, R (1990) The Logic of Decision. 2nd ed. University of Chicago Press. ISBN 0-226-39582-0
— (1992) Probability and the Art of Judgment. Cambridge University Press. ISBN 0-521-39770-7
— (2004) Subjective Probability: The Real Thing. Cambridge University Press. ISBN 0-521-53668-5
Skyrms, B (2012) From Zeno to Arbitrage: Essays on Quantity, Coherence & Induction. Oxford University Press (Features most of the papers cited below.)

Categories:

https://en.wikipedia.org/wiki/Radical_probabilism

In Bayesian inference, the Bernstein-von Mises theorem provides the basis for using Bayesian credible sets for confidence statements in parametric models. It states that under some conditions, a posterior distribution converges in the limit of infinite data to a multivariate normal distribution centered at the maximum likelihood estimator with covariance matrix given by $n^{-1}I(\theta _{0})^{-1}$ , where $\theta _{0}$ is the true population parameter and $I(\theta _{0})$ is the Fisher information matrix at the true population parameter value.^[1]

https://en.wikipedia.org/wiki/Bernstein–von_Mises_theorem

The principle of indifference (also called principle of insufficient reason) is a rule for assigning epistemic probabilities. The principle of indifference states that in the absence of any relevant evidence, agents should distribute their credence (or 'degrees of belief') equally among all the possible outcomes under consideration.^[1]

In Bayesian probability, this is the simplest non-informative prior. The principle of indifference is meaningless under the frequency interpretation of probability,^{[citation needed]} in which probabilities are relative frequencies rather than degrees of belief in uncertain propositions, conditional upon state information.

https://en.wikipedia.org/wiki/Principle_of_indifference

In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain a sample of the desired distribution by recording states from the chain. The more steps are included, the more closely the distribution of the sample matches the actual desired distribution. Various algorithms exist for constructing chains, including the Metropolis–Hastings algorithm.

https://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo

Genetic architecture is the underlying genetic basis of a phenotypic trait and its variational properties.^[1] Phenotypic variation for quantitative traits is, at the most basic level, the result of the segregation of alleles at quantitative trait loci (QTL).^[2] Environmental factors and other external influences can also play a role in phenotypic variation. Genetic architecture is a broad term that can be described for any given individual based on information regarding gene and allele number, the distribution of allelic and mutational effects, and patterns of pleiotropy, dominance, and epistasis.^[1]

There are several different experimental views of genetic architecture. Some researchers recognize that the interplay of various genetic mechanisms is incredibly complex, but believe that these mechanisms can be averaged and treated, more or less, like statistical noise.^[3] Other researchers claim that each and every gene interaction is significant and that it is necessary to measure and model these individual systemic influences on evolutionary genetics.^[1]

https://en.wikipedia.org/wiki/Genetic_architecture

Blog Archive

Tuesday, September 21, 2021

Stability[edit]

Controllability and observability[edit]

Control specification[edit]

Model identification and robustness[edit]

System classifications[edit]

Linear systems control[edit]

Nonlinear systems control[edit]

Decentralized systems control[edit]

Deterministic and stochastic systems control[edit]

Main control strategies[edit]

People in systems and control[edit]

Transfer functions[edit]

Fluctuations, entropy, 'thermodynamics forces', and reproducible dynamical structure[edit]

Local thermodynamic equilibrium[edit]

Linear and non-linear processes[edit]

Continuous and discontinuous motions of fluids[edit]

Historical development[edit]

W. Thomson, Baron Kelvin[edit]

Helmholtz[edit]

J. W. Strutt, Baron Rayleigh[edit]

Korteweg[edit]

Onsager[edit]

Prigogine[edit]

Casimir[edit]

Ziman[edit]

Ziegler[edit]

Gyarmati[edit]

Paltridge[edit]

Speculated thermodynamic extremum principles for energy dissipation and entropy production[edit]

Prigogine's proposed theorem of minimum entropy production for very slow purely diffusive transfer[edit]

Faster transfer with convective circulation: second entropy[edit]

Speculated principles of maximum entropy production and minimum energy dissipation[edit]

Prospects[edit]

See also[edit]

Creation of order[edit]

Background[edit]

Temporal order[edit]

Idealized example: Lotka–Volterra equation[edit]

Another idealized example: Brusselator[edit]

Spatial order[edit]

Real examples[edit]

Optics example[edit]

Biological example[edit]

Shape tailoring of thin layers[edit]

Phase transitions[edit]

Asymmetric autocatalysis[edit]

Role in origin of life[edit]

Examples of autocatalytic reactions[edit]

See also[edit]

Physical processes[edit]

Development of convection[edit]

Convection features[edit]

Rayleigh–Bénard instability[edit]

Effects of surface tension[edit]

History and nomenclature[edit]

See also[edit]

Design[edit]

History[edit]

Certain and uncertain knowledge[edit]

Conditioning on an uncertainty – probability kinematics[edit]

Alternatives to probability kinematics[edit]

Selected bibliography[edit]

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