Cedric Villani recently wrote an article on Landau damping in collisionless plasmas, where at least one topic discussed confused me. Besides discussing the issue of how a process can be microscopically reversible and macroscopically irreversible, he mentions something regarding a "entropy-preserving relaxation mechanism" that is irreversible.

Am I missing something? I think the article is suggesting that something can preserve entropy but still be irreversible.

This begs the following questions:

  • Are entropy and irreversibility synonymous?
  • Or are they distinct entities each with their own fundamental properties?
  • $\begingroup$ I would think that the entropy of the system may be conserved but not the entropy of the universe. Said in a different manner, the entropy is not the 'right' thermodynamic potential for this transformation. $\endgroup$
    – gatsu
    Sep 24, 2014 at 14:22

2 Answers 2


Landau damping seems mysterious because it is derived from completely reversible equations, yet it gives a "damping" behaviour. But the funny thing is that when you go through the equations you find even an "antidamping" exponentially growing solution$^1$!

The reason for this is that the Landau damping isn't really a damping caused by particle collisions. It is caused by the electrons in the plasma "getting out of sync" and causing the wave to disappear because every electron creates a slightly different wave at a different phase. This can be shown through an ensemble of harmonic oscillators representing the electrons.

The exponentially growing solution corresponds to the oscillating electrons on the contrary "getting into sync" and superposing again into a macroscopic wave. However, the original Landau analysis is done in linear order, so the exponential growth of the wave will be altered as soon as the wave grows large enough.

Landau damping is an "entropy preserving process" in the sense that the single electrons do not get randomized but the direction of the wave is still reflected in their movement - there is a slight hidden anisotropy in the plasma in the direction of the damped wave even after it's disappearance. This anisotropy is indeed macroscopic in the sense of a deviation from the isotropy of the average velocities of particles.

The question is whether to count these hidden oscillations into heat gained by the plasma or not. If not, then entropy was indeed preserved, but if yes, entropy grows just by the decoherence of oscillations. Truth is entropy tends to be badly defined for non-equilibrium ensembles, so this is why Cedric Villani uses the quotation marks in talking about an "entropy-preserving process".

However, in practice real damping and collisions are happening along Landau damping and this takes quickly care of the hidden anisotropy.

$^1$ This is done by taking the lower counterpart of the "Landau contour", see the original paper by Landau On the vibration of the electronic plasma..

  • $\begingroup$ I have read Landau's original paper and the paper by J.D. Jackson in 1960 [in J. Nucl. Energy, Part C] that showed the correct direction from which to close the contours in Landau's paper (this is how Jackson made his name). I understand linear and nonlinear Landau damping/growth, that isn't the issue. I am confused about the distinction between entropy and irreversibility. $\endgroup$ Sep 24, 2014 at 19:11
  • $\begingroup$ I am wondering who was the one to downvote and what was their issue (Anyone?). It is hard to guess the level of expertize just from the question and it is good to give a broader introduction anyways so that the question is useful for anyone stumbling upon it. The distinction between the process being isentropic or not is exactly the one between counting the decoherent oscillations into heat or not. The most well agreed upon definitions of entropy do not work here so an isentropic process would be loosely defined as one where there were no transfers of heat. $\endgroup$
    – Void
    Sep 24, 2014 at 19:31
  • $\begingroup$ yeah sorry, that was my idiotic mistake. It wasn't an issue with your answer, I was screwing around moving between windows and must have hit it by accident. I did not realize I had done this until you made a comment about it. If you make a minor edit to your answer, I can replace the 1up vote (it won't let me change this unless you edit your answer). $\endgroup$ Sep 25, 2014 at 0:10
  • $\begingroup$ @Void Dear Void, any insight from you on this post would be most valuable. physics.stackexchange.com/questions/222014/… $\endgroup$
    – user929304
    Dec 3, 2015 at 14:39

Entropy and irreversibility are related to each other. Irreversibility is the one because of which process takes place and due to this energy degradation takes place that is exergy (availability) decreases. And basically irreversibility(I) is defined as I = To * ( del s )

That is when you multiply change in entropy of universe with the reference temperature you'll get irreversibility. Energy degradation increases because of irreversibility and exergy decreases. I is directly proportional to entropy change of universe and inversely proportional to exergy

  • $\begingroup$ The article I referenced seemed to suggest that one could have an entropy preserving process that was irreversible, which is inconsistent with your definition. Thoughts? $\endgroup$ Sep 30, 2014 at 15:56
  • $\begingroup$ Oh sorry. You are right my definition is not for entropy preserving process. It is a general relationship between entropy and irriversibility. My mistake. – $\endgroup$ Sep 30, 2014 at 16:27

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