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Is Maximum Entropy Production (MEP) a physical principle? In this paper I tentatively suggest it is not, on the basis that MEP is equivalent to Jaynes’ Maximum Entropy (MaxEnt) inference algorithm that passively translates physical assumptions into macroscopic predictions, as applied to non-equilibrium systems. MaxEnt itself has no physical content; disagreement between MaxEnt predictions and experiment falsifies the physical assumptions, not MaxEnt. While it remains to be shown rigorously that MEP is indeed equivalent to MaxEnt for systems arbitrarily far from equilibrium, work in progress tentatively supports this conclusion. In terms of its role within non-equilibrium statistical mechanics, MEP might then be better understood as Messenger of Essential Physics.

What is the current status of Maximum Entropy Production (MEP)? Is it a working hypothesis about the macroscopic behaviour of non-equilibrium systems whose domain of application has yet to be firmly established? Or is it an accepted physical principle with a well-defined range of validity? Or has MEP already been shown to be fundamentally wrong? If it is a working hypothesis, what further critical tests would help to define its range of validity? If it is an established principle with a well-defined range of validity, what is its physical basis? And if it is just plain wrong, what if anything can be salvaged from the wreckage? Much of the discussion at the 2009 and previous MEP workshops, and in the literature [

Behind all of these questions, however, lies the tacit assumption that MEP

To an extent, the view that MEP is a physical principle has been encouraged by the way in which thermodynamics is often presented in text books. The terms

However, the statistical nature of thermodynamic laws (e.g., the relationship between molecular diffusion and concentration gradients) means that the analogy between thermodynamic and mechanical forces,

This last statement is a consequence of Jaynes’ information-based formulation of statistical mechanics [

Two contrasting interpretations of MEP. (a) MEP as an inference algorithm (MaxEnt) that converts physical assumptions into macroscopic predictions, with no physical content itself. Only the physical assumptions are falsifiable by observations (dotted arrows). (b) MEP as a physical principle,

If it can be shown that MEP is equivalent to MaxEnt when applied to non-equilibrium systems (and there lies a challenge), it follows that MEP is not a physical principle after all but, rather, a passive algorithm for translating given or assumed physical information about non-equilibrium systems into macroscopic predictions (

The paper is structured as follows.

The practical problem we are trying to address is how to predict the macroscopic behaviour of non-equilibrium physical systems (e.g., the large-scale structure of turbulent heat flow within the Earth’s atmosphere and oceans, or the mean fluid velocity and temperature fields in a Rayleigh-Bénard cell). One approach would be to try to calculate the microscopic trajectory of the system directly, by (numerically) integrating the microscopic equations of motion forward in time from some initial microstate, typically requiring some approximate computational scheme involving discretization in space and time. Practical challenges here include the treatment of sub-grid scale dynamics, symmetry-breaking of the equations of motion by the computational scheme [

Nature herself suggests that there is: at least for large

We may view Jaynes’ information-based formulation of statistical mechanics as a response to these two challenges, within which the MaxEnt algorithm plays a central role. Broadly speaking, statistical mechanics takes an informed guess as to the essential physics and applies MaxEnt to make macroscopic predictions from that guess. Disagreement between prediction and experiment informs a new guess and so on until, by trial and error, acceptable agreement is reached.

Operationally, the MaxEnt algorithm is very simple, although its interpretation remains a subject of controversy. As a concrete and familiar example, consider the application of MaxEnt to a closed system in thermal equilibrium with a heat bath at temperature _{i}_{i}_{i}_{i}_{i}_{i}

The solution to this constrained optimisation problem is:

The value of the maximised Shannon entropy is then:
_{i}_{i}

What then is MaxEnt? What are we doing when we apply it? And what does it mean when MaxEnt predictions agree or disagree with experiment? In Jaynes’ interpretation, MaxEnt is a passive inference algorithm that converts given information into predictions; it has no physical content itself (_{i}_{i}

Experimental falsification of MaxEnt is thus a meaningless concept because, in Jaynes’ interpretation, MaxEnt is not a physical principle. When MaxEnt predictions disagree with experiment, it is the assumed essential physics (the message) that is falsified, not MaxEnt (the messenger). Jaynes’ interpretation also makes it clear that MaxEnt can be applied to any system, equilibrium or non-equilibrium, physical, biological or otherwise. In all cases, MaxEnt makes no claim as to the reality of its predictions; its role within the general programme of statistical mechanics is to ensure that the assumed essential physics (and no other physical assumptions) is faithfully represented in the predictions that are compared with experiment. MaxEnt thus plays a central role in the identification of the actual essential physics, which would otherwise be obscured by the implicit introduction of extra physical assumptions.

Nevertheless, one may still question whether MaxEnt is the appropriate algorithm to use in the first place. For example, maximum

This question arises at both a theoretical and a practical level. As an alternative to Jaynes’ information-based formulation of statistical mechanics, the Maximum Probability (MaxProb) formulation (e.g., [_{i}

The macrostate {_{i}_{i}_{i}_{i}_{i}_{i}

Therefore, if one adopts MaxProb as the fundamental basis of statistical mechanics, there is a theoretical issue regarding the validity of MaxEnt for small

Nevertheless there remains the practical issue of whether MaxEnt still requires the assumption that

If

Different scenarios for MaxEnt-predicted (

Thus, regardless of whether

In summary, MaxEnt remains practically useful whether

Here I give a brief status report. In two papers by this author [

More recently, Niven [

In work in progress [_{R}—with the central result that for non-equilibrium systems

In summary, work in progress tentatively suggests that MEP is the expression of MaxEnt applied to physical systems arbitrarily far from equilibrium (for which the standard conservation laws remain valid). If this is indeed the case, then MEP is free of additional physical assumptions beyond those already used as input to MaxEnt, and is therefore an inference algorithm rather than a physical principle (

This question arose during discussions at the 2009 MEP workshop in relation to the fact that MEP predictions of the large-scale structure of heat transport in Earth’s climate obtained from Paltridge’s 10-zone climate model do not depend on the physical nature of the transport medium [

But let us suppose that MEP indeed passively translates physical assumptions into macroscopic predictions (

The apparent success of MEP in revealing the essential physics governing planetary-scale heat flows [

Several features of the empirical velocity fields in turbulent shear flow and thermal convection have been predicted—with varying degrees of success—by maximising various dissipation-like functionals of the flow under a restricted number of dynamical constraints derived from the Navier-Stokes equation (e.g., global energy balance, mean momentum balance) plus a small number of additional assumptions and constraints (e.g., incompressibility, boundary conditions). The search for the relevant constraints using UBT is exactly in the spirit of the search for the essential physics using MEP. Moreover, in UBT it remains an open question which dissipation-like functional is the most appropriate to maximise, analogous to the question of which expression for EP to use in applications of MEP. Maximum dissipation rate of the mean flow was derived as a statistical stability criterion for turbulent flow [

“

The close relationship between UBT and MEP suggests the possibility that such a universal functional might be Shannon information entropy, and that UBT, like MEP, may ultimately be an expression of MaxEnt. If this is indeed the case, then one might anticipate that previous [

If MEP is indeed equivalent to MaxEnt and has no physical content itself (

In this paper I have suggested that MEP is not a physical principle but, rather, an inference algorithm (MaxEnt) that passively translates physical assumptions into macroscopic predictions (

If this suggestion is correct, then previous statements by various authors (including this one) regarding “the evidence or otherwise for MEP” are misleading, and ought to be refocused on “the evidence or otherwise that the physical assumptions capture the essential physics governing the macroscopic phenomena in question.” It might then be more informative to reinterpret the acronym MEP as Messenger of Essential Physics or Messenger of Erroneous Physics, according to whether its predictions agree or disagree with observations.

However, it remains to be seen whether indeed MEP is a faithful expression of MaxEnt, particularly for systems that are far from equilibrium, although work in progress tentatively supports that view [

I thank the organisers of the MEP workshop held at the Max-Planck-Institute for Biogeochemistry, Jena, Germany, from 18-20 May, 2009 for their kind hospitality and financial support towards attendance, and the participants for stimulating discussions. Additional financial support was provided under the Australian National University’s Travel Grant Scheme.

Caticha [