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Extremal principles in non-equilibrium thermodynamics



This page discusses ‘maximum entropy production’, ‘minimimum entropy production’, and related extremal principles that have been proposed in non-equilibrium statistical mechanics. It is not about E. T. Jaynes’ MaxEnt Principle in statistical inference, except insofar as it may be related to the topic at hand. (As we shall see, Jaynes has tried to develop a relationship.)

This is a controversial and confusing subject, as a brief perusal of this page makes clear:

It begins:

According to Kondepudi (2008), and to Grandy (2008), 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), 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) state that “In non-equilibrium … it is generally not possible to construct thermodynamic potentials depending on the whole set of variables”. Šilhavý (1997) 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.

Minimum entropy production

This book offers a concise and fairly rigorous discussion of Ilya Prigogine’s principle of minimum entropy production, which applies only to a limited class of systems:

  • Georgy Lebon and David Jou, Understanding Non-equilibrium Thermodynamics, Springer, Berlin, 2008.

It starts on page 51 in section 2.5.1, “Minimum Entropy Production Principle”. The authors say that the main assumptions behind Prigogine’s theorem are:

  1. Time-independent boundary conditions
  2. Linear phenomenological laws
  3. Constant phenomenological coefficients
  4. Symmetry of the phenomenological coefficients

As we shall see, the argument really shows not that rate of entropy production is minimized in the steady state, but that the rate of entropy production decreases as time passes. It takes some extra assumption to conclude that when the system has reached a steady state, its entropy production has reached the minimum possible.

Let us try to distill the argument to its mathematical essence. Warning: the rest of this section will be extremely abstract, not very well motivated, and deeply flawed - but it can be salvaged. So, keeping an example in mind will be very helpful. Suppose we have a possibly inhomogeneous ball of metal whose temperature is some function X(t,x)X(t,\vec{x}) of time tt and space x\vec{x}. Suppose the temperature is independent of time at the boundary of the ball: that’s condition 1 above, ‘time-independent boundary conditions’. Let dXd X stand for the exterior derivative of XX, regarded as a function of x\vec{x} at a fixed time tt. (In physics exterior derivative dXd X is usually treated as a vector field and denoted X\vec{\nabla} X, but we are about to engage in a massive generalization). The rate of entropy production is

P=dX,dX P = \langle d X, d X \rangle

where the angle brackets denote a certain inner product on 1-forms. Moreover, the heat equation can be written

X˙=d *dX \dot X = d^* d X

where d *d^* is the adjoint of dd with respect to this inner product. Combining these facts, the argument we’re about to exhibit will show that

P˙0 \dot{P} \le 0

meaning the rate of entropy production always decreases. The above authors conclude:

This result proves that that the total entropy production PP decreases in the course of time and that it reaches its minimum value in the stationary state.

However, this is a leap of logic: just because a function is decreasing (P˙0\dot P \le 0), we can’t conclude that when the function is constant (P˙=0\dot P = 0) it has reached its minimum value. However, in practice they are right! So, there is some true assumption that they are failing to make explicit… but their argument can be rescued somehow. We shall see how in a while.

Now let us exhibit the argument leading to P˙0\dot{P} \le 0 in a very general context. We shall assume the state of a system is described by an nn-chain XX in some chain complex equipped with an inner product. We also assume the rate of entropy production is

P=dX,dXP = \langle d X, d X \rangle

where the brackets are the inner product. Then taking the time derivative of both sides:

P˙=2dX,dX˙ \dot{P} = 2 \langle d X , d \dot X \rangle

or using the adjoint of the operator dd:

P˙=2d *dX,X˙ \dot{P} = 2 \langle d^* d X , \dot X \rangle

The analogue of the heat equation in this situation says that:

d *dX=LX˙ d^* d X = L \dot{X}

for some linear operator LL. We thus obtain

P˙=2LX˙,X˙ \dot{P} = 2 \langle L \dot{X} , \dot{X} \rangle

In many cases the operator LL is negative, so that

P˙0 \dot{P} \le 0

In other words, the rate of entropy production decreases with the passage of time.

However, if P˙=0\dot P = 0, the equation P˙=2LX˙,X˙\dot{P} = 2 \langle L \dot{X} , \dot{X} \rangle implies that LX˙=0L \dot{X} = 0. In this case the equation d *dX=LX˙ d^* d X = L \dot{X} implies that d *dX=0d^* d X = 0, which further implies that

P=dX,dX=d *dX,X=0P = \langle d X, d X \rangle = \langle d^* d X, X \rangle = 0

So now we are getting that P˙=0\dot{P} = 0 implies P=0P = 0! This in turn implies that PP is minimized, but somehow we’ve gone too far: we want P˙=0\dot{P} = 0 to imply that PP is minimized, not that it’s zero. So, there is something about our assumptions here that are too strong, and need to be corrected.

This idea, that entropy production decreases with the passage of time, is also what the following authors focus on:

So it could be that this is the really interesting content of Prigogine’s theorem.

To wrap up (for now):

  1. “Time-independent boundary conditions” are what let us do the “integration by parts” here: dX,dX˙=d *dX,X˙\langle d X , d \dot X \rangle = \langle d^* d X , \dot X \rangle .

  2. “Linear phenomenological laws”, “constant phenomenological coefficients” and “symmetric phenomenological coefficients” are what let us write the rate of entropy production as a quadratic form P=dX,dXP = \langle d X, d X \rangle . We need to break this down into steps, but this seems roughly right.

  3. Another crucial step is the equation d *dX=LX˙ d^* d X = L \dot{X} . This again can be derived from smaller assumptions, but it’s worth noting that this is formally just a generalization of the heat equation.

  4. Another crucial assumption is that LL is negative.

  5. However, the assumptions used so far are too strong; we need to fix them to get PP minimized but not necessarily zero when P˙=0\dot{P} = 0. The trick is probably to use ideas from here:

After all, the ‘principle of least power’ discussed there is closely related to the ‘principal of minimum entropy production’ we’re discussing here.

References on minimum entropy production

Christian Maes has written some papers on minimum entropy production:

We explain the (non-)validity of close-to-equilibrium entropy production principles in the context of linear electrical circuits. Both the minimum and the maximum entropy production principles are understood within dynamical fluctuation theory. The starting point are Langevin equations obtained by combining Kirchoff’s laws with a Johnson-Nyquist noise at each dissipative element in the circuit. The main observation is that the fluctuation functional for time averages, that can be read off from the path-space action, is in first order around equilibrium given by an entropy production rate. That allows to understand beyond the schemes of irreversible thermodynamics (1) the validity of the least dissipation, the minimum entropy production, and the maximum entropy production principles close to equilibrium; (2) the role of the observables’ parity under time-reversal and, in particular, the origin of Landauer’s counterexample (1975) from the fact that the fluctuating observable there is odd under time-reversal; (3) the critical remark of Jaynes (1980) concerning the apparent inappropriateness of entropy production principles in temperature-inhomogeneous circuits.

The minimum entropy production principle provides an approximative variational characterization of close-to-equilibrium stationary states, both for macroscopic systems and for stochastic models. Analyzing the fluctuations of the empirical distribution of occupation times for a class of Markov processes, we identify the entropy production as the large deviation rate function, up to leading order when expanding around a detailed balance dynamics. In that way, the minimum entropy production principle is recognized as a consequence of the structure of dynamical fluctuations, and its approximate character gets an explanation. We also discuss the subtlety emerging when applying the principle to systems whose degrees of freedom change sign under kinematical time-reversal.

I. W. Richardson has an interesting letter on minimum entropy production versus steady state, which could probably be used to formulate these ideas using differential forms and Laplacians, or more general elliptic operators. He mentions a no-go theorem due to Gage, saying that steady state cannot always be described using an extremal principle:

Maximum entropy production

While Ilya Prigogine has a successful principle of least entropy production that applies to a special class of linear steady-state systems, other people talk about a principle of ‘maximum entropy production’! Is there a contradiction here? This paper begins to address the issue:

Martyusheva and Seleznev write:

1.2.6. The relation of Ziegler’s maximum entropy production principle and Prigogine’s minimum entropy production principle

If one casts a glance at the heading, he may think that the two principles are absolutely contradictory. This is not the case. It follows from the above discussion that both linear and nonlinear thermodynamics can be constructed deductively using Ziegler’s principle. This principle yields, as a particular case (Section 1.2.3), Onsager’s variational principle, which holds only for linear nonequilibrium thermodynamics. Prigogine’s minimum entropy production principle (see Section 1.1) follows already from Onsager–Gyarmati’s principle as a particular statement, which is valid for stationary processes in the presence of free forces. Thus, applicability of Prigogine’s principle is much narrower than applicability of Ziegler’s principle.

For the relation between maximum entropy production and Jaynes’ MaxEnt principle, see:

David Corfield has noted that Dewar’s paper relies on a paper by E. T. Jaynes in which he proposes something called the ‘Maximum Caliber Principle’:

  • E. T. Jaynes, Macroscopic prediction, in H. Haken (ed.) Complex systems – operational approaches in neurobiology, Springer, Berlin, 1985, pp. 254–269.

This paper delves further into the relation:

Abstract: Jaynes’ maximum entropy (MaxEnt) principle was recently used to give a conditional, local derivation of the “maximum entropy production” (MEP) principle, which states that a flow system with fixed flow(s) or gradient(s) will converge to a steady state of maximum production of thermodynamic entropy (R.K. Niven, Phys. Rev. E, in press). The analysis provides a steady state analog of the MaxEnt formulation of equilibrium thermodynamics, applicable to many complex flow systems at steady state. The present study examines the classification of physical systems, with emphasis on the choice of constraints in MaxEnt. The discussion clarifies the distinction between equilibrium, fluid flow, source/sink, flow/reactive and other systems, leading into an appraisal of the application of MaxEnt to steady state flow and reactive systems.

On the n-Category Café, David Lyon wrote:

Most of the posters here study beautiful subjects such as the quantum theory of closed systems, which has time reversal symmetry. I have a little bit of experience with open dissipative systems, which are not so pretty but may interest some of you for a moment. My advisor has experimentally explored entropy production in driven systems. Although I haven’t been personally involved in most of the experiments, we’ve had many interesting discussions on the topic.

There is a very simple maximum entropy production principle in systems with a linear response. In this case the system evolves towards its maximum entropy state along the gradient, which is the direction of maximum change in entropy. This principle applies in practice to systems which are perturbed a small amount away from equilibrium and then allowed to relax back to equilibrium. As Tomate said, if you take a closed system, open it briefly to do something gentle to it, and then wait for it to relax before closing it again, you’ll see this kind of response.

However, the story is very different in open systems. When a flux through a system becomes large, (e.g. close to the Eddington Limit for radiating celestial bodies, when heat flow follows Cattaneo’s Law, etc), the response no longer follows simple gradient dynamics and there is no maximum entropy production principle. There have been many claims of maximum or minimum entropy production principles by various authors and many attempts to derive theories based on these principles, but these principles are not universal and any theories based on them will have limited applicability.

In high voltage experiments involving conducting spheres able to roll in a highly resistive viscous fluid, there is a force on the spheres which always acts to reduce the resistance RR of the system. This is true whether the boundary condition is constant current II or constant voltage VV. Since power dissipation is I 2RI^2 R in the first case and V 2/RV^2 / R in the second case, one can readily see that entropy production is minimized for constant current and maximized for constant voltage.

In experiments involving heat flow through a fluid, convection cells (a.k.a. Benard Cells) form at high rates of flow. For a constant temperature difference, these cells act to maximize the heat flow and thus the entropy production in the system. For a constant heat flow, these cells minimize the temperature difference and thus minimize the entropy production in the system.

If one were to carefully read “This Week’s Finds in Mathematical Physics (Week 296)” one would be able to find several more analogous examples where the response of open systems to high flows will either maximize or minimize the entropy production for pure boundary conditions or do neither for mixed boundary conditions.

Tomate added:

As David points out, many variational principles for nonequilibrium systems have been proposed. They only hold in the so-called “linear regime”, where the system is slightly perturbed from its equilibrium steady state. We are very far from understanding general non-equilibrium systems, one major result being the “fluctuation theorem”, from which all kinds of peculiar results descend; in particular, the Onsager-Machlup variational principle for trajectories. For the mathematically-minded, I think the works by Christian Maes et al. might appeal to your tastes.

Funnily enough, there exists a “minimum entropy production principle” and a “maximum entropy production principle”. The apparent clash is due to the fact that while minimum entropy production is an ensemble property, that is, it holds on a macroscopic scale, the maximum entropy production principle is believed to hold for single trajectories, single “histories”. I think the first is well-established, indeed a classical result due to Prigogine, while the second is still speculative and sloppy; it is believed to have important ecological applications. [Similarly, a similar confusion arises when one defines entropy as an ensemble property (Gibb’s entropy) or else as a microstate property (Boltzmann entropy)]

Unfortunately, that I know, there is not one simple and comprehensive review on the topic of variational principle in Noneq Stat Mech.

Tomate also wrote:

I went through Dewar’s paper some time ago. While I think most of his arguments are correct, still I don’t regard them as a full proof of the principle he has in mind. Unfortunately, he doesn’t explain analogies, differences and misunderstandings around minimum entropy production and maximum entropy production. In fact, nowhere in his articles does a clear-cut definition of MEP appear.

I don’t think, like Martyusheva and Seleznev, that it is just a problem of boundary conditions, and the excerpt you take does not explain why these two principles are not in conflict in the regime where they both are supposed to hold.

Let me explain my own take on the minEP vs. maxEP problem and on similar problems (such as Boltzmann vs. Gibbs entropy increase). It might help sorting out ideas.

By “state” we mean very different things in NESM, among which: 1) the (micro)state which a single history of a system occupies at given times 2) the trajectory itself 3) the density of microstates which an ensemble of a large number of trajectories occupies at a given time (a macrostate). One can define entropy production at all levels of discussion (for the mathematically-inclined, markovian master equation systems offer the best set up where all is nice and defined). So, for example, the famous “fluctuation theorem” is a statement about microscopic entropy production along a trajectory, while the Onsager’s reciprocity relations are a statement about macroscopic entropy production. By “steady state”, we mean a stationary macrostate.

The minEP principle asserts that the distribution of macroscopic currents at a nonequilibrium steady state minimizes entropy production consistently with the constraints which prevent the system from reaching equilibrium.

As I understand it, maxEP is instead a property of single trajectories: most probable trajectories are those which have a maximum entropy production rate, consistently with constraints.

As a climate scientist, you should be interested in the second as we have not an ensemble of planets among which to maximize entropy or minimize entropy production. We have one single realization of the process, and we’d better make good use of it.

Maximum entropy production in climate science

The above paper by Niven cites some papers applying these ideas to climate change. Here’s a review article on entropy maximization in climate physics:

As mentioned by Ozawa et al., Lorenz suspected that the Earth’s atmosphere operates in such a manner as to generate available potential energy at a possible maximum rate. The available potential energy is defined as the amount of potential energy that can be converted into kinetic energy. Independently, Paltridge suggested that the mean state of the present climate is reproducible as a state with a maximum rate of entropy production due to horizontal heat transport in the atmosphere and oceans. Figure 2 shows such an example. Without considering the detailed dynamics of the system, the predicted distributions (air temperature, cloud amount, and meridional heat transport) show remarkable agreement with observations. Later on, several researchers investigated Paltridge’s work and obtained essentially the same result.

On the other hand, some climate scientists are deeply skeptical of work based on the principle of entropy maximization. For example, Garth Paltridge has done work based on this principle, but others say his theories would imply that the Earth’s climate is independent of its rate of rotation, in blatant contradiction to what more detailed models show. So, it seems that climate models based on some principle of entropy maximization are highly controversial at best, at this time.

The fact that Garth Paltridge has also written a book entitled The Climate Caper, arguing that the “case for action against climate change is not nearly so certain as is presented to politicians and the public”, adds an extra political aspect to this controversy.

The Wikipedia article on Garth Paltridge says:

Paltridge was involved in studies on stratospheric electricity, the effect of the atmosphere on plant growth and the radiation properties of clouds. Paltridge researched topics such as the optimum design of plants and the economics of climate forecasting, and worked on atmospheric radiation and the theoretical basis of climate. In terms of scientific impact, his most significant contribution has been to show that the earth/atmosphere climate system may have adopted a format that maximises its rate of thermodynamic dissipation, i.e. entropy production. This suggests a governing constraint by a principle of maximum rate of entropy production. According to this principle, prediction of the broad-scale steady-state distribution of cloud, temperature and energy flows in the ocean and atmosphere may be possible when one has sufficient data about the system for that purpose, but does not have fully detailed data about every variable of the system.

This article argues against maximum entropy production in the Earth’s weather, but says some data are compatible with ‘maximum kinetic energy dissipation’:

They write:

Lorenz (1960) proposed that the atmospheric general circulation is organized to maximise kinetic energy dissipation (MKED), or, equivalently, the generation of APE (available potential energy). Similarly Paltridge (1975, 1978) suggested that Earth’s climate structure might be explained from a hypothesis of maximum entropy production (MEP). Closely related principles have been popular also in biology and engineering. For example the ‘‘maximum power principle’’, advocated by Odum (1988) for biological systems, is consistent with the maximum dissipation conjecture; the ‘‘constructal law’’ of Bejan and Lorente (2004) is very closely related to MEP as discussed by Kleidon (2009). A broad discussion on the maximizing power generation and transfer for Earth system processes can be found in Kleidon (2010).


We conclude that the maximum entropy production conjecture does not hold within the climate system when the effects of the hydrological cycle and radiative feedbacks are taken into account, but our experiments provide some evidence in support of the conjecture of maximum APE production (or equivalently maximum dissipation of kinetic energy).

Roderick Dewar on maximum entropy production

On his website, Roderick Dewar writes:

Theory and application of Maximum Entropy Production

Cells, plants and ecosystems – like all open, non-equilibrium systems – import available energy from their environment and export it in more degraded (higher entropy) forms. But at what rate is energy degraded? According to the hypothesis of Maximum Entropy Production (MEP) – as fast as possible. MEP provides a new guiding principle for modelling the flows of energy and matter between plants, ecosystems and their environment, and offers a novel thermodynamic perspective on the origin and evolution of life.

MEP has reproduced key features observed in a diverse range of non-equilibrium systems across physics and biology, from the large-scale distributions of temperature and cloud cover in Earth’s climate system to the functional design of the ubiquitous biomolecular motor ATP synthase. But in the absence of a fundamental explanation for MEP, it has remained something of a scientific curiosity.

Our aim is to elucidate the theoretical basis of MEP in order to underpin and guide its wider practical application. We are exploring the idea that MEP can be derived from the fundamental rules of statistical mechanics developed in physics by Boltzmann, Gibbs and Jaynes – implying that MEP is a statistical principle that describes the most likely properties of non-equilibrium systems. Ultimately our goal is to extend the application of MEP from climate modelling – where previous MEP work has mostly focused – to plant and ecosystem modelling.

We collaborate with an international network of MEP researchers including: Axel Kleidon (Max-Planck Institute for Biogeochemistry, Jena), Peter Cox & Tim Jupp (Exeter University), Amos Maritan (Padua University), Robert Niven (UNSW@ADFA), Hisashi Ozawa (Hiroshima University), Davor Juretić & Pasko Zupanović (Split University).

His website also includes this reading list:

  • R. C. Dewar, Information theoretic explanation of maximum entropy production, the fluctuation theorem and self-organized criticality in non-equilibrium stationary states, Journal of Physics A (Mathematical and General), 36 (2003), 631-641. Summary here.

  • R. C. Dewar, Maximum entropy production and non-equilibrium statistical mechanics, in Non-Equilibrium Thermodynamics and Entropy Production: Life, Earth and Beyond, eds. A. Kleidon and R. Lorenz, Springer, New York, 2004, 41-55.

  • R. C. Dewar, D. Juretić, P. Zupanović, The functional design of the rotary enzyme ATP synthase is consistent with maximum entropy production, Chem. Phys. Lett. 430 (2006), 177-182.

  • R. C. Dewar, Maximum entropy production and the fluctuation theorem, J. Phys. A.: Math. Gen. 38 (2005), L371-L381.

  • R. C. Dewar, Maximum entropy production as an inference algorithm that translates physical assumptions into macroscopic predictions: don’t shoot the messenger, Entropy 11 (2009), 931-944. Contribution to Special Issue (eds. Dyke J, Kleidon A): What is Maximum Entropy Production and how should we apply it?

  • R. C. Dewar, Maximum entropy production and plant optimization theories, Phil. Trans. Roy. Soc. B 365 (2010) 1429-1435. Contribution to Theme Issue (eds. Kleidon A, Cox PM, Mahli Y): Maximum entropy production in ecological and environmental systems: applications and implications.

The principle of least action

While the principle of least action most commonly appears in classical mechanics and classical field theory, it also shows up in nonequilibrium statistical mechanics. See for example:

Kohler’s variational principle

“Kohler’s variational principle” appears in the kinetic theory of gases. It is mentioned in this book:

  • J. H. Ferziger and H. G. Kaper, Mathematical Theory of Transport Processes in Gases, 1972.

Here is a quote:

Thus we can formulate the following maximum principle: In non-equilibrium systems the distribution of the molecular velocities is such that, for given temperature and velocity gradients, the rate of change of the entropy density due to collisions is as large as possible. This maximum principle, together with a similar minimum principle, was first given by Kohler (1948). A discussion of these and other variational principles can be found in a paper by Ziman (1956) or in a paper by Snider (1964a).

It may have been proved within the Chapman–Enskog theory of gases.

In the appendix to Chapter 5 of this book:

  • D. N. Zubarev, V. G. Morozov, G. Röpke, Statistical Mechanics of Nonequilibrium Processes, 1996.

another (more general?) principle is discussed, that is claimed to be similar to Kohler’s variational principle.

The minimal free energy principle

Karl Friston introduced his Free Energy Principle back in 2010: “The free-energy principle says that any self-organizing system that is at equilibrium with its environment must minimize its free energy”.

Abstract: “A free-energy principle has been proposed recently that accounts for action, perception and learning. This Review looks at some key brain theories in the biological (for example, neural Darwinism) and physical (for example, information theory and optimal control theory) sciences from the free-energy perspective. Crucially, one key theme runs through each of these theories — optimization. Furthermore, if we look closely at what is optimized, the same quantity keeps emerging, namely value (expected reward, expected utility) or its complement, surprise (prediction error, expected cost). This is the quantity that is optimized under the free-energy principle, which suggests that several global brain theories might be unified within a free-energy framework.”

We can find in the article the connection with entropy extremal principles:

Motivation: resisting a tendency to disorder.

The defining characteristic of biological systems is that they maintain their states and form in the face of a constantly changing environment. From the point of view of the brain, the environment includes both the external and the internal milieu. This maintenance of order is seen at many levels and distinguishes biological from other self-organizing systems; indeed, the physiology of biological systems can be reduced almost entirely to their homeostasis.

More precisely, the repertoire of physiological and sensory states in which an organism can be is limited, and these states define the organism’s phenotype. Mathematically, this means that the probability of these (interoceptive and exteroceptive) sensory states must have low entropy; in other words, there is a high probability that a system will be in any of a small number of states, and a low probability that it will be in the remaining states.

Entropy is also the average self information or ‘surprise’ (more formally, it is the negative log-probability of an outcome). Here, ‘a fish out of water’ would be in a surprising state (both emotionally and mathematically). A fish that frequently forsook water would have high entropy. Note that both surprise and entropy depend on the agent: what is surprising for one agent (for example, being out of water) may not be surprising for another. Biological agents must therefore minimize the long-term average of surprise to ensure that their sensory entropy remains low. In other words, biological systems somehow manage to violate the fluctuation theorem, which generalizes the second law of thermodynamics.“

The constructal law

The ‘constructal law’ is yet another maximum principle that has been proposed in biology:

This article outlines the place of the constructal law as a self-standing law in physics, which covers all the ad hoc (and contradictory) statements of optimality such as minimum entropy generation, maximum entropy generation, minimum flow resistance, maximum flow resistance, minimum time, minimum weight, uniform maximum stresses and characteristic organ sizes.

Other references

From the abstract:

We show how common principles of entropy maximization, applied to different ensembles of states or of histories, lead to different entropy functions and different sets of thermodynamic state variables. Yet the relations of among all these levels of description may be constructed explicitly and understood in terms of information conditions. The example systems considered introduce methods that may be used to systematically construct descriptions with all the features familiar from equilibrium thermodynamics, for a much wider range of systems describable by stochastic processes.

Markov processes in non-equilibrium thermodynamics