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starburst-pokieEntropy production selects nonequilibrium states in multistable systems | Scientific Reports

Securing Wireless Communications at the Physical Layer - Google Books

Although entropy is often used as a characterization of the information content of a data source, this information content is not absolute: it depends crucially on the probabilistic model. A source that always generates the same symbol has an entropy rate of 0, but the definition of what a symbol is depends on the alphabet.
Boltzmann realised the connection between entropy and probability. Watch Eann Patterson explain more using the behaviour of penguins as an analogy.

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Best candidate theories based on the maximum entropy production principle could not be unequivocally proven, in part due to complicated physics, unintuitive stochastic thermodynamics, and the existence of alternative theories such as the minimum entropy production principle.
Here, we use a simple, analytically solvable, one-dimensional bistable chemical system to demonstrate the validity of the maximum entropy production principle.
To generalize to multistable stochastic system, we use the stochastic least-action principle to derive the entropy production and its role in the stability of nonequilibrium steady states.
This shows that in a multistable system, all else being equal, the steady state with the highest entropy production is favored, with a number of implications for the evolution of biological, physical, and geological systems.
In an isolated system, e.
However, in open systems, characterized by fluxes of energy and matter, order can arise as long as the entropy of the surrounding system increases enough so that the total entropy from the two parts of the system together increases.
Note also that the second law does not make any predictions about how fast a system approaches equilibrium, except at the stationary nonequilibrium steady state.
Under this circumstance, the Carnot efficiency limits the rate at which entropy is produced by the heat flux into the system .
This qualitative nature of the second law makes the prediction of dynamical systems based on thermodynamics notoriously difficult.
For the last 150 years, there has been speculation that universal extremal principles determine what happens in nature ,, most prominent being the maximum entropy production principle MaxEPP by Paltridge, Ziegler and otherssee alsofor reviews.
Its most important conclusion is that there is life on Earth, or the biosphere as a whole, because ordered living structures help dissipate the energy from the sun on our planet more quickly as heat than just absorption of light by rocks and water .
However, such principles have never rigorously been proven, and conflicting results exist.
MaxEPP apparently explains Rayleigh-Bénard convection, flow regimes in plasma physics, the laminar-turbulent flow transition in pipes, crystallization of ice, certain planetary climates, and ecosystems ,.
For instance, in plasma physics large-scale dissipative structures can increase the impedance and thus sustain high temperature gradients while producing large amounts of entropy at a smaller scale.
Confusingly, MaxEPP seems to contradict the minimum entropy production principle MinEPP as promoted by Onsager, Prigogine and others .
To make things worse, both MinEPP and MaxEPP can apply simultaneouslyor one of the two can be selected depending on the boundary conditions, i.
These wide-ranging results, along with the broad range of applications from biochemistry, fluid mechanics, ecosystems, and whole planets, leave the question of extremal principles wide open.
What about more general theoretical approaches?
A promising direction for proving the MaxEPP is based on information-theoretic approaches related to the maximum-entropy inference method , but previous attempts relied on overly strong assumptions reviewed in .
Recently, the stochastic least-action principle was also established for dissipative systems .
Information theory and the stochastic least-action principle are important corner stones of modern stochastic thermodynamics.
Here, with an interest in the emergence of protocells and life, we focus on stochastic biochemical systems and ask whether MaxEPP provides a mechanism for selecting states in a multistable system.
We demonstrate that previous attempts to disprove MaxEPP suffered from misinterpretations of unintuitive aspects of stochastic systems, and that with modern approaches in stochastic thermodynamics, MaxEPP can be proven.
Initially, we focus on the single chemical species, one-dimensional 1D bistable Schlögl model, but then generalize to stochastic multistable systems.
We find that if multiple steady states exist, then the first MaxEPP predicts that the steady state with the highest entropy production is the most likely to occur.
Furthermore, we demonstrate that MinEPP simply corresponds to the second MaxEPP near equilibrium.
These findings should clarify the role of thermodynamics in selecting nonequilibrium states in biological, chemical, and physical systems.
A short primer on entropy production Imagine a system undergoing state changes due to external driving as shown in Fig.
However, at a nonequilibrium steady state, the time-averaged rate of entropy change is zero, i.
Put differently, the time-averaged entropy production rate is the negative of the entropy flow rate.
Since entropy flow is often easier to calculate, it can be used in place of the entropy production at steady state.
At thermodynamic equilibrium, both quantities are zero.
Now, we need a microscopic model to investigate this further.
Illustration of a driven system with sources of entropy production and flow.
A system is called closed if only energy is exchanged with the surroundings, and open if also matter is exchanged.
For completeness, in an isolated system, there is no exchange at all with the surroundings.
In a special case of the latter, everything is included, i.
At steady state, both time-averaged contributions are equal in magnitude but of opposite sign see text for details.
The high state produces high amounts of entropy as indicated by heat radiation, warm glow, directedness and oscillatory fluxes projected onto the concentration axiswhile the low state is cold and close to equilibrium with little entropy production.
The hypothesis of MaxEPP is that the state with high entropy production also is the more likely to occur right.
How do we obtain the probability distribution P Xt based on these transition rates?
The master equation describing the exact time evolution of P Xt is given by , 5 where the first second term on the right-hand side corresponds to an increase decrease in probability P X, t by jumps towards away from the state with X molecules.
The first term on the right-hand side of Eq.
The passive contribution in Eq.
Note, that the active part of Eq.
This final expression is sometimes called the medium entropy or action functionalbut in our terminology corresponds to the entropy flow.
However, this also shows that small stochastic systems can violate the second law of thermodynamics!
In the following, we use both the molecule number and trajectory-based pictures.
Minimal nonequilibrium bistable model To understand the validity of the MaxEPP, why not investigate it with a simple exactly solvable model?
This was indeed attempted using article source well-known chemical Schlögl model of the second kind .
This paradox highlights the fact that microscopic master equation and macroscopic mean-field descriptions can yield very different results .
The Schlögl model only depends on one chemical species X with interesting features such as bistability two different stable steady statesfirst-order phase transition energy-assisted jumps between statesand front propagation in spatially extended systems .
Biochemically, the model converts species A to B and vice versa via intermediate species X 11b with rate constants as shown note we use same capital letter symbols for species names and molecule numbers.
The model recently attracted renewed interest due to its mapping onto biologically relevant models with bistability, e.
Nevertheless, the Schlögl model with spatial dependence and diffusion is believed to describe front propagation in CO oxidation on Pt single crystals surfaces, and the nonlinear generation and recombination processes in semiconductors.
In terms of the master equation, the transition rates are , 12d where the molecule numbers of species A and B are fixed.
Such a chemical system can be simulated by the Gillespie algorithm, which is a dynamic Monte Carlo method and reproduces the exact probability distribution from the master equation upon long enough simulations sampling Fig.
Vertical grey dashed lines for guiding the eye.
For large but finite volumes Ω, an analytical formula for the probability distribution can be derived, i.
However, for easier analytical calculations, click to see more are a number of possible simplifying assumptions to the master equations.
This demonstrates bistability in a regime of intermediate concentrations of b two stable steady states and one intermediate unstable state.
However, the deterministic model does not predict the weights of the steady states, i.
This entropy production is illustrated in Fig.
Hence, near equilibrium the entropy production rate is minimized with respect to changes in the rates or their parameters.
It is important to note that Eq.
To remedy this problem, we can rewrite Eq.
Imposed flux F ensures that the molecule concentrations a and b are maintained and that the dynamics of species x are driven out of equilibrium.
The entropy production rate is plotted in Fig.
The argument goes as follows: In the ODE model, the high state always has the higher entropy production see Fig.
This can easily be understood since overall in the Schlögl model species A is converted to species B and vice versa.
As a result, if MinEPP is the rule, then the low state should be selected, while MaxEPP would dictate that the high state is more stable.
This argument can be made sharper when we consider the results from the master equation learn more here the large volume limit.
This indicates a first-order phase transition and loss of bistability Fig.
Hence, for b b c the high state is selected.
Since these correspond respectively to the low and high entropy-production rates Fig.
What is the issue with this conclusion?
Ultimately, this paradox is caused by the switching of the order of the limits.
In the macroscopic description, the infinite volume limit is taken first to derive the ODEand then the infinite time limit is taken for obtaining the steady stateswhile in the microscopic description the opposite order is applied.
Since the above argument combines the entropy production from the bistable ODE model with the weights from the master equation, which is mono-stable in the infinite volume limit, 60 symbols entropy mixing of models may have led to the wrong conclusion regarding MaxEPP.
MaxEPP in a simple bistable model Having identified an inconsistency in the argument to disprove MaxEPP in the Schlögl model, we now proceed to rescue MaxEPP as a valid principle for determining the weights of steady states.
Specifically, we would first like to demonstrate that the order of the weights of the two stable steady states Fig.
This would confirm the MaxEPP in this particular case.
Now, using the expression for the average stochastic entropy production rate at steady state Eq.
MaxEPP in Schlögl model.
Remaining parameters as in Fig.
Even without the Gaussian approximation, we can obtain the weights of the states and their entropy production rates using the master equation.
This produces the same qualitative result that the entropy production rates and weights of the two states are correlated cf.
Nevertheless, article source are quantitative differences between the two approaches as the curves of the weights and the entropy production rates do not cross exactly at the same b value.
This is because the order of the weights shown in Fig.
Consider the ratio of the transition rates between low 1 and high 2 states, given by20 see Supplementary Information for details.
The prefactor alone suggests that the lower the curvature the higher the weight of a state but this weak curvature dependence outside exponential only reflects the attempt frequency to escape the attractor.
In particular, the more stable state i.
A higher curvature may imply a larger depth of the potential Φ O D E x k and hence increased stability.
However, a proper treatment requires the inclusion of noise, which is done next.
General MaxEPP for nonequilibrium steady states Can we establish a formal link between the weight of a state and its entropy production and curvature in general?
Basically, we wish to find the probability of a certain configuration for a system in a way that neither assumes something we do not know, nor contradicts something we do know.
For this purpose, we define the caliber for the probability p Γ of observing a trajectory of duration t 21 where the first term on the right-hand side is the Shannon information entropy and the second term is a constraint.
Now, using the Evans-Searles fluctuation theorem23 where the ratio of the probabilities of forward and backward time-reversed trajectories corresponds to the exponential of the entropy produced along trajectory Γ at steady state.
Hence, the entropy production 24 is the difference between the backward and forward actions at steady state see also.
To gain insight into the problem we derive in the following the entropy production for steady states explicitly.
To combine the best of ODEs and master equations, we extend Eq.
There are a link of different ways of how to model the noise.
The usage of a constant effective temperature is a suitable approximation when the noise or ε is small and the system is settled into its steady state rare switching only; however, these effective temperatures can be different for the different states in a multistable system.
To reconnect with Eq.
According tothe action for the system of Langevin Eq.
Note that the integral or time average over F q ̇ is zero and hence this term does not appear in Eq.
Two comments are in order.
First, for the backward action, all time derivatives need reversing in sign, such as for a ̇, b ̇ and x ̇.
The change in sign of d t ˜ in the integral is canceled by the change in order of integration.
In summary, the action in Eq.
Hence, trajectories do not only minimize the classical and stochastic actions equivalent to solving the dynamical equations but also maximize the entropy production due to negative sign in front of the entropy production term.
Simplified MaxEPP for nonequilibrium steady states The entropy production appearing in the action can be brought to a more familiar form, at least heuristically.
We need to be careful with the last term of Eq.
However, this ensemble average is technically restricted to sampling from a particular steady state, as we 60 symbols entropy check this out switching between states here.
Now, introducing ensemble averages throughout, we can, at least heuristically, introduce the potential from the master equation 30 using the detour of the Fokker-Planck potential as another approximation to the master equation see Supplementary Information for details.
Keeping only the highest order terms in Ω or lowest order terms in ε and using Eq.
Hence, MaxEPP is a principle for multistable systems in which the entropy production biases the evolution of the system towards the highest-entropy producing state.
We showed that MaxEPP is applicable when comparing the two states in the simple bistable Schlögl model using the master equation; Fig.
MaxEPP applies in the former because the weights of the low and high states shift in the exact stochastic approach due to a first-order phase transition.
MaxEPP applies in the latter because trajectories minimize the action two-fold: First, the classical action is minimized, meaning that the dynamic system takes on its appropriate solution, i.
Second, the entropy production from the fluxes between the reservoir and the reaction volume is maximized.
Our analytical derivations show that MaxEPP is a consequence of the least-action principle applied to dissipative systems stochastic least-action principle.
Note however the discrepancy in how the MaxEPP is achieved in the two approaches: using the master equation we observe a first-order phase transition and state switching at the critical point, while using the Langevin approximation, the high state is selected.
This statement is simply a result of the average of Eq.
Equation is minimally zero at equilibrium due to detailed balance and is the larger the more the forward and backward trajectories differ.
Hence, the earlier discussed MinEPP is not really a separate principle, but simply a different perspective of the global Possible thewizardofoz com blackjack well />Figure summarizes the two MaxEPPs and the MinEPP for a bistable system at steady state.
MaxEPP 1 simply states that the more a system is driven away from equilibrium the more entropy is produced.
MaxEPP 2 is more subtle, describing how states are selected in a multistable system.
Our results can be connected to recent results in fluid systems.
Similar to the Schlögl model with a nonequilibrium first-order phase transition, flow systems undergo a laminar-turbulent flow transition as the Reynolds number Re increases.
In both systems, MaxEPP applies and can be used to predict the critical transition point Fig.
What are the weights of the states in the fluid system?
As a macroscopic system, the system is largely monostable - below the critical Re, laminar flow is dominant, while above it, turbulent flow is the result.
This is analogous to the macroscopic Schlögl model, where the bistable region disappears for increased system size and a first-order phase transition results Fig.
However, even in the fluid system, the laminar state can be metastable even for relatively large Re if unperturbed.
This is a sign of hysteresis and hence bistability, and so both laminar and turbulent flows may coexist with the turbulent flow the more stable state turbulent flow never switches back to laminar flow when Re is above the critical value.
Another previously investigated physical system is the fusion plasma, where a thin layer of fluid is heated from one side.
Models of heat transport in the boundary layer predict that the MaxEPP MinEPP applies when the heat flux temperature gradient is fixed .
Similarly in the Schlögl model, when the concentrations of species A and B are fixed like the temperature in the fusion plasmathe system becomes indistinguishable from an equilibrium system see comments above Eq.
However, once fluxes are fixed Eq.
We believe that flux constraints correspond to the more physically correct scenario as now the entropy production of the macroscopic ODE model matches the entropy production of the microscopic model as described by the exact master equation cf.
In both above described flow systems, dissipative structures form when strongly driven.
In the former fluid system, turbulent swirl structures appear while in the latter plasma system a shear flow is induced.
What do such dissipative structures correspond to in the Schlögl model?
There are large fluctuations and inhomogeneities in the spatial Schlögl model with diffusion for increasing system size, although these may represent the approach of the critical point and less actual dissipative structures.
Paradoxically, work in the fluid system raised the possibility that both MinEPP and MaxEPP apply simultaneously.
MinEPP appears to predict the flow rates in parallel pipes while MaxEPP seems to predict the flow regime laminar versus turbulent.
This can potentially be explained 60 symbols entropy our Eq.
In this case, the entropy production second term is less crucial to fulfill.
In contrast, if the noise is large, the classical action becomes a less important constraint, and the entropy production becomes important, leading necessarily to MaxEPP.
The MaxEPPs was previously also applied to ecosystem functioning, which aims to predict the evolution of large-scale living systems in terms of thermodynamics also called ecological thermodynamics.
Considering simple food-web models of predators, preys, and other resources, the state-selection and gradient-response principles were found to break down in more complicated models with multiple trophic hierarchical levels.
However, the stability of the steady states was assessed with linear stability analysis, i.
However, as we showed, the macroscopic Schlögl model predicts the wrong stability and only in the thermodynamic limit of the this web page master-equation model the MaxEPP is predicted correctly.
Our interpretation of MaxEPP is in line with the recent finding that the entropy production, by itself, is not a unique descriptor of the steady-state probability distribution.
In fact, far-from-equilibrium physics has many pitfalls.
Whether a trajectory is actually selected depends on the underlying chemical rules or physical laws see classic action in Eq.
Future work may investigate applications of MaxEPP in models of nonequilibrium self-assembly, climate, and the emergence of molecular complexity or life.
Imagine there are two stable steady states, one with high complexity and high entropy production, and another one with low complexity and low entropy 60 symbols entropy />We speculate that the high complexity state is more likely as long as the extra cost from the entropy reduction due to complexity is offset by a significantly larger entropy production.
Another issue to keep in mind is that evolution of our biosphere may not be at steady state, and so transient 60 symbols entropy may need to be investigated.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Entropy (information theory) - Wikipedia

In thermodynamics the entropy of mixing is the increase in the total entropy when several initially separate systems of different composition, each in a thermodynamic state of internal equilibrium, are mixed without chemical reaction by the thermodynamic operation of removal of impermeable partition(s) between them, ...
The average number of bits per symbol would be 141/60 = 2.35 bits/symbol. For the optimal case, this would be 100/60 = 1.67 bits/symbol. So the KL-divergence would 2.35 – 1.67 = 0.68 bits/symbol. Ideally, we want this to be as low as possible when we estimate our model to build the entropy coder.
Entropy, Signs, Symbols and Natural. Languages. Christian Bentz. “Entropy as possibility is my favorite short description of entropy because. 0. 5. 10. Colour. Frequency. Games n=80. 0. 30. 60. 90. Colour. Frequency. Games n=800. 0. 2500. 5000. 7500. 10000. Colour. Frequency. Games n=80000.
The concept of Fundamental Scale is tested for English and musical instrument digital interface (MIDI) music texts using an algorithm developed to split a text in a collection of sets of symbols that minimizes the observed entropy of the system. This Fundamental Scale reflects more details of the complexity of the language ...

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