# Maximum Principle vs. Minimum Principle in Non-equilibrium Thermodynamics

Prigogine's Min. principle states that in steady-state non-equilibrium systems the entropy generation rate is at a minimum, i.e., a system will seek a steady-state that has min entropy generation. This principle has a well-established proof, but applies to systems so close to equilibrium that there is only one such steady state accessible.

The Maximum Principle has been stated in multiple ways with no well-accepted proof. A common statement is that the system seeks to be in a maximum entropy generation state allowed by constraints always (steady-state or approaching one).

There seems to be a contradiction here with the Min. principle. The way I am trying to resolve this is as follows:

Max. principle says that the system remains at max entropy generation rate all time---steady-state or not. It doesn't say what happens to the time-derivative entropy generation rate, wheres Min. principle says the derivative is negative in the cases that it applies (i.e., in linear non-equilibrium region). Could anyone shed light on this issue? I am unable to find a good explanation/ discussion of this apparent contradiction in papers.

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also have a look at "Extremal principles in non-equilibrium thermodynamics" at Azimuth wiki azimuthproject.org/azimuth/show/… –  Yrogirg Jan 23 '13 at 6:34
This is an interesting link. Thanks! –  Sankaran Jan 23 '13 at 16:38

Part of my PhD thesis was on this stuff, so I hope I can give a satisfactory answer.

Maximum entropy production and minimum entropy production are different types of principle with different domains of application. Before discussing the answer I should make clear that the maximum entropy production principle (which I'll call MaxEP) is really a collection of different hypotheses by different authors, some of which are more plausible than others, and none of which has an accepted theoretical justification. However, there is some empirical evidence in the work of Paltridge from the 70s, e.g. this paper. A very simple one-parameter version of Paltridge's model can be found in this paper by Lorenz et al., and in the discussion below I will keep as close as possible to the version of MaxEP that Lorenz et al. use.

As you say, Prigogine's principle of minimum entropy production (henceforth MinEP) only applies in near-equilibrium situations. It was once hypothesised to be much more widely applicable. This hypothesis has now been disproven, and one must be careful to bear this in mind when reading old material on the subject. (For the moment I've lost track of the paper that disproves this idea, but it's a pretty solid mathematical result. If I find it again I'll update this answer.)

With these caveats out of the way, the basic difference is this:

• For linear, near-equilibrium systems that only admit a single steady state, MinEP says that all of the system's transient states have a higher entropy production than the steady state. A transient state is a temporary state that is not a steady state. MinEP compares steady states with non-steady states.

• For some yet-to-be-determined class of non-linear, far-from-equilibrium systems that admit a continuum of possible steady states, MaxEP says that the system is most likely to be found in the steady state with the greatest entropy production. MaxEP compares steady states to other steady states, but says nothing about transient states.

So aside from the fact that the two principles apply to quite different types of system (linear versus highly non-linear), they also make quite different types of claim. One can imagine a system that admits many possible steady states, but whose transient states all have a higher entropy production than any of its steady states. For such a system, MinEP and MaxEP could apply simultaneously. If so then starting from a non-steady initial state, its entropy production would reduce over time until it reached a steady state and would remain constant thereafter; but nevertheless the steady state that it reaches is most likely to be the one with the highest entropy production.

Unfortunately there is a depressing amount of literature in which these points are not well appreciated. It seems that people often think MaxEP implies that entropy production should increase over time as the system approaches a steady state. But this isn't true for a lot of systems, and I think this mistake in reasoning might be one of the reasons why MaxEP doesn't have a great reputation as a hypothesis.

As for literature that addresses this distinction, I seem to remember there being some fairly readable discussion in this book chapter by Dewar. Another place to look is Edwin Jaynes' criticism of the minimum entropy production principle. It doesn't really mention MaxEP (because Jaynes seems not to have been aware of Paltridge's papers) but it gives some strong hints towards it, and I found it extremely helpful in understanding the nature of MinEP and why a different type of principle is needed. Finally, I suppose I could also humbly point you to my paper on MaxEP, which doesn't discuss MinEP but tries to clarify some points about how MaxEP is applied, and to resolve some serious theoretical problems with the principle. These papers deal with some of the issues I've skipped over above, such as what it means for a system to have "possible" steady states that are different from the actual one.

The OP has commented that maybe the above implies that systems always choose the most entropy-producing state they "could" be in, regardless of whether this is a transient or a steady state, but for the transient states the maximum possible entropy production can reduce over time as the system converges to a steady state.

There are several ways I can address this. The first possibility is to say that above I was talking only about the version applied by Paltridge and by Lorenz et al., because this is the only version with even the tiniest little sliver of empirical evidence. It's very, very important to note that this version of MaxEP doesn't say anything at all about transient states. As Paltridge has said (as the OP points out), his version of MaxEP is just an empirical observation and not a theoretical claim, and it's an observation of the atmosphere's steady state, not its transient ones.

It's also important to note that there are few if any systems other than atmospheres that have been observed to obey a principle similar to Paltridge's. (There are claims for other systems, mostly in the Earth sciences, but I don't find these very convincing. There are no laboratory-based observations of Paltridge's principle as far as I know, although this is partly because the experimental crowd have their own completely different "principle of maximum entropy production" that they like to play with, in which systems choose between a finite number of steady states instead of a continuum.) So we already know that MaxEP as an empirical principle is not broadly applicable to all non-linear systems, and it shouldn't be surprising that we get contradictions if we try to imagine it applying too broadly. It might well be that MaxEP, if it is a valid principle at all, will turn out to apply only to thermally-driven turbulent fluids in steady state with very large Reynolds numbers, and not to any other type of system.

(As a side note, I should say that although I think Dewar's work is the closest thing we have to a theoretical explanation of Paltridge's observations, I don't think it's quite correct. My paper, linked above, attempts to resolve what I see as a serious logical contradiction in his approach. This is a different contradiction from the one we've been discussing so far, and has to do with the fact that Dewar's version of MaxEP makes different predictions depending on where you draw the system's boundary.)

I could just leave it there. However, in my paper I do make the claim that Dewar's version of MaxEP (or something like it) can be extended to transient states, in something quite similar to the way you suggest. Like Dewar, I try to extend Jaynes' MaxEnt thermodynamics to deal with non-equilibrium states. Briefly, the idea is that if we maximise the information entropy of the system's microscopic state at time $t_1$, subject to the knowledge we have about the system from measurements made at time $t_0$ then, trivially, we've maximised the rate of increase of information entropy between times $t_0$ and $t_1$. Identifying this information entropy with the thermodynamic entropy is trickier than it might seem at first, but if we can do that then we've reached a version of MaxEP that does indeed apply to all states, transient or otherwise.

However, I don't think it leads to a contradiction if you look at it in this way. The reason is that, given the knowledge constraints formed by the measurements at $t_0$, there is exactly one macrostate at every time $t>t_0$ that maximises the (information) entropy subject to those constraints; it cannot be any other way. This means, I think, that within this framework it is not possible for the situation you suggest to arise, and transient states with high entropy productions must always lead to steady states with high entropy productions. (But, having thought about it a bit more just now, this is all subject to an additional constraint of reproducibility that I don't think I spelt out very clearly in the paper. This needs more thought on my part.)

Important Note

For the sake of it not getting lost, there is an in-depth and (currently) on-going discussion of this answer and related issues in this chat room.

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Thanks Nathaniel. Could we have a have a chat about this. I have looked at many papers that you have mentioned. Including a later paper by Paltridge who claims that his previous work was just an observation and shouldn't be construed as support to MaxEnP in any way. In terms of thinking it appears that you are confirming my understanding in that at any point in time my system will choose to be in the most entropy producing state it could be in (transient or steady) and with time its ability to produce entropy will reduce (if transient) so MinEnp has a sense of the derivative and MaxEnP not. –  Sankaran Jan 22 '13 at 16:36
What happens if the system is in non-linear non-eq state, has options of sets of transient states all leading to different steady state, but the most entropy-producing transient state leads to a less entropy producing steady state? Now what might it choose? It seems that MaxEnP might face self-contradiction. –  Sankaran Jan 22 '13 at 16:46