# What are some good articles on trend to equilibrium?

I am interested in studying systems out of equilibrium that are trending to equilibrium. Trend to equilibrium, entropy production, etc. seem to be very tricky topics. Any suggestions will be appreciated.

I am mainly interested in stuff like Cercignani conjecture, etc. Extra points for references in which estimates for entropy production are in terms of information theoretic arguments.

I have done the google searches and hence asking for suggestions from an expert.

• Are you asking about near-equilibrium thermodynamics? Commented May 10, 2018 at 8:46
• Not just near equilibrium. Any general state trending towards equilibrium.
– user29978
Commented May 10, 2018 at 12:50
• Cercignani's conjecture and modern trends in mathematical kinetic theory: cmouhot.wordpress.com/2011/02/01/cercignanis-conjecture. Some references for out-of-equilibrium thermodynamics physics.stackexchange.com/a/780147/226902 Commented Nov 7, 2023 at 20:25

Handbooks on statistical mechanics/physics seem to be some starting "space" for discussion - there are many among them, which are formulated in maximally mathematical way, which really TRY to overview ALL interesting problems for gases (whole modern physics begun from equation for ideal gas PV=RT), which discuss many entropies, such as Von Neumann, Kolmogorov-Sinaj; V.I.Arnold - there are at least 10-20 of them, especially if we want to come to information problems - below.

Assumptions (wiki): a. The rapidly moving particles constantly collide among themselves. All these collisions are perfectly elastic, which means the molecules are perfect hard spheres. b. Except during collisions, the interactions among molecules are negligible. They exert no other forces on one another. Thus, the dynamics of particle motion can be treated classically,

and the equations of motion are time-reversible.

These assumptions are too simplified, and rather physical, not mathematical, as e.g. chaos theory (unpredictability for 3 (!!) such particles, so time become irreversible) demonstrated after 1960s.

This is also where (time reversible and predictable, but then entropy is (normally) constant - good Reviews on this were in Modern Physics: see RMP, by e.g. Wehrl, in 1970s):

the main problems jump into all theories, because time is irreversible even in most simple kinetics: gas kinetics, as e.g. all the well-known fights between Boltzmann (great man!) and his mathematical opponents (Poincare, Zermelo, etc.) demonstrated.

We still do not know "correct" system of axioms, as e.g. Prigogine analyzed in his books (his popular overview: From Being to Becoming). In fact, this is here quantum problems, formulations, quasi-"axioms" suddenly arise, as was predicted, usually on intuitive levels, also by Planck, Bohr, Russell + Whitehead Principia of Mathematics (yes!). BUT e.g. Einstein still believed in time reversible as basic time, so he tried to reject quantum physics as long as he could. He was last really great determinist, rejecting probability in really basic laws, as well as entropy with information (which are +/- ln P).

More about physics and other sciences:

May be you can look at Ilya Prigogine first (chemical physics, Nobel Prize, 1977; he wrote many books, very popular around 1970-1995, he influenced e.g. extremely important 2018 (medical!) Nobel Prize on immunology). This is of course NOT about gases, on which Cercignani worked (surely most of his life - below I copied intro to the article, on which you refer). Boltzmann was main inspiration for Prigogine, or e.g. Erwin Schroedinger!

Moreover, Alan Turing himself become extremely interested in these problems at the end of his short life! He was well aware of Shannon- von Neumann - Wiener etc. works on information and entropy, and wanted to answer e.g. Schroedinger's questions: what LIFE means for physicists. Turing even asked Prigogine for discussion, who did not understand the importance (it was 1949, or 1950): because Prigogine at that time was working on the similar problems to Cercignani, looking for such interesting, but relatively too much simple cases, which allow already some mathematical solutions: Prigogine hoped for all his life to prove e.g. minimum entropy production theorem, even though closer to 2003 (when he died) maximum entropy production was becoming popular as well!

Min. Entr. Prod (MinEP) is important for living systems. MaxEP should be good for gases and it tell, that system drops to some STEADY state (very) fast; but then different scenarios can happen (if it is NOT GAS, but e.g. liquid, not mentioning more complicate systems, like chemistry, biology, computing and information sciences; information = infoEnergy).

MinEP also put forward question: are there interesting situations, when system tend to equlibrium in extremely slow or long way, such as spin glasses, which Hopfield applied to neuron networks, modelling brain (see. e.g. David Mackay's complicated Cambridge handbook for this - free in Internet). Hopfiled, Mackay, Friston, Bishop, etc. formulated mathematical problems for neurosciences, networks, ML, RL, AI, similar as Turing, who was studying origins of organisms, morphogenesis - this his work was well-understood in 1970s, and Prigogine was returning to their discussion many times. After 1990-2000 this Turing's work is becoming extremely popular!

So we are coming to what is most interesting to me as well (if I understood your intentions): "Extra points for references in which estimates for entropy production are in terms of information theoretic arguments". I think, we should forget here about Cercignani (at least for several hours!). Information still is NOT defined completely, so e.g. life is not understood as well!!

Can we define this for gases, as e.g. quantum information? Why not! But: a. mathematics still rather unknown (see below). b. all best scientists tried to understand these "tricky problems" (agree!!), surely after Boltzmann's, Schroedinger's, Turing's works. But now these are considered as different, even opposite systems/sciences etc.: c. gases. d. applications of information: computers, internet, life, economical and other "highest life" sciences/theories/models.

To understand this in most simple way, let us consider that information = (equals) minus entropy (with Boltzmann, Planck, cosmological, etc. constants everywhere, where possible), like Schroedinger's negentropy (many other attempts were done as well!).

This is here all existing axiomatics suddenly refuse to work! For this reason Feynman, and even modern "computational" physicsts tried to limit themselves to time reversible: very nice solutions, excellent 19th Century technology. Feynman did not like to write all this equations with entropy (see e.g. his Statistical Mechanics (compare with Prigogine), or Path Integrals (Schulman's or Kleinert's books are much better in this sense!), preferring e.g. free energy, which still means running away from the problems (Karl Friston even proposed Free Energy principles for neurosciences!).

p.s. Cercignani’s conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann’s nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, they survey the positive and negative results which were obtained since the conjecture was proposed in the 1980s. This paper is dedicated to the memory of the late Carlo Cercignani, one of the founders of the modern theory of the Boltzmann equation.

Have to finish now (late) and hope to discuss more.

Keywords: Cercignani’s conjecture; spectral gap; Boltzmann equation; relative entropy; entropy production; relaxation to equilibrium; Landau equation; logarithmic Sobolev inequality; Poincare inequality.