Maximum entropy principle: a note

Some years ago I wrote a paper debunking several claims related to the so-called “maximum entropy production principle”:

Polettini M., Fact-checking Ziegler’s maximum entropy production principle beyond the linear regime and towards steady states, Entropy 15, 2570-2584 (2013) [pdf], arXiv:1307.2052

This is one of a few critical contributions in a field that is otherwise flourishing with papers very optimistic about the fact that the principle has anything to say about anything at all, each citing each other in a self-inconsistent way. The central reference in this business is the following review:

Martyushev, L. M., and V. D. Seleznev. Maximum entropy production principle in physics, chemistry and biology, Physics reports 426, 1-45 (2006).

I have been very critical of many claims made in this review (which is a patchwork, so it is not consistently wrong all over the place, only here and there…). In response to a criticism by Andresen et al.

Andresen, B.; Zimmermann, E. C.; Ross, J, Objections to a proposal on the rate of entropy production in systems far from equilibrium, J. Chem. Phys. 81, 4676−4677 (1984).

the authors of this review wrote a reply trying to show that such criticisms fall outside of the range of application of the principle:

Martyushev, L.M., Seleznev, V.D.: The restrictions of the maximum entropy production principle, Phys A: Stat. Mech. Appl. 410, 17–21 (2014).

In a footnote, they also address my paper by claiming that

“[The entropy production rate] is not negative for any values of [the] variables. The critics of MEPP often forget about this obvious corollary of the second law”.

This outrageous little sentence has been used by many authors to dismiss my work altogether, without actually reading it. Some examples (I have seen more…):

M. Amiri and M. Modarres, An Entropy-Based Damage Characterization, Entropy 16, 6434-6463 (2014)

Ž. Bonačić Lošić, T. Donđivić, and D. Juretić, Is the catalytic activity of triosephosphate isomerase fully optimized? An investigation based on maximization of entropy production, J. Biol. Phys. (2017).

So, apparently, I don’t understand the very foundations of thermodynamics! A similar observation had been made by one of the referees of my paper:

“Author’s forces (Eq. 15) contradict the foundation of non-equilibrium thermodynamics and common sense”

to whom I replied:

“Eq.(15) is just a generic Taylor expansion of the forces in terms of the fluxes, something one should take into account if one wants to go beyond the linear regime.”

So, do I know that the entropy production should be positive? Of course, I do! It’s a positive functions of several variables, some of which are called the fluxes flowing through the system. The linear regime is that situation in which it makes sense to approximate the entropy production as a positive quadratic form of the fluxes. Among several other observations, in my paper I show that this supposed over-arching principle only makes sense in the linear regime (I also argue that it does not say what people attribute to it, that it is actually much weaker, but let’s leave this aside…). In fact, I show that by expanding in Taylor series to (say) third order the entropy production, the predictions of the theory already fail.

However, the fact that I expand to third order does not mean that I assume that the entropy production is a cubic function! This would be like saying that the exponential function is not positive because it’s cubic:

exp x = x + x^2/2 + x^3/6 +

So, while I don’t know if and how Andresen et al.’s objections fall within the realm of application of MaxEP (if there is any), I know exactly that my objections fall perfectly within the assumptions of the review and I know very well that the entropy production rate is positive, thanks so much. Which makes the observation of Mr. Martyushev and Seleznev an impolite and stupid mockery.

I know that all of this is not really worth my time, but I’m sick and tired of seeing how this work is being mistreated, just for trying to make a little clarity in a field that is otherwise as foggy as 19th century London.

And I have not even started explaining in how many ways is the MaxEP a complete bluff…

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