# Nonequilibrium thermodynamics in a Boltzmann picture

The Boltzman approach to statistical mechanics explains the fact that systems equilibriate by the idea that the equillibrium macrostate is associated with an overwhelming number of microstates, so that, given sufficiently ergotic dynamics, the system is overwhelmingly likely to move into a microstate associated with equilibrium.

To what extent is it possible to extend the story to nonequilibrium dynamics? Can I make concrete predictions about the approach to equilibrium as passing from less likely macrostates to more likely macrostates? (Wouldn't this require us to say something about the geometric positioning of macrostate regions in phase space, rather then just measuring their area? Otherwise you'd think the system would immediately equilibriate rather than passing through intermediate states.) Can the fluctuation-dissipation theorem be explained in this way?

Edit: Upon a bit more poking around, it looks like the fluctuation-dissipation theorem cannot be explained in this way. The reason is that this theorem discusses the time-independent distribution of fluctuations in some macroscopic parameter (e.g. energy of a subsystem) but, as far as I understand, it does not describe the time dependence of such a parameter.

In particular, I'd really like to understand is if it's possible to explain things like Fourier's Law of thermal conduction (that the rate of heat transfer through a material is proportional to the negative temperature gradient and to the cross-sectional area) with a Boltzman story. According to these slides, it's surprisingly hard.

• Good question, I would love to see an answer! You might find the PhD thesis and articles of Gavin Crooks interesting. Apr 4, 2014 at 8:59
• It's my understanding that there are very, very few theorems about non-equilibrium statistical mechanics that can be rigorously derived from first principles - not even results as basic as Fourier's law. May 7, 2017 at 4:43

There are many ways to study approaches to equilibrium, which is obvious as there are many ways to drive a system out of equilibrium. So there is really no unique answer to your question. However, various universal results are known. These include various fluctuation theorems. The most famous of which is usually called just the fluctuation theorem which relates the probability of time-averaged entropy production, $\Sigma_t=A$ over time $t$ to $\Sigma_t=-A$, $$\frac{P(\Sigma_t=A)}{P(\Sigma_t=-A)}=e^{A t},$$ which shows that positive entropy production is exponentially more likely than negative entropy production. Note that the second law follows from this theorem. The fluctuation-dissipation relation may also be derived from it.

There is also, for instance, the Crooks fluctuation theorem which relates the work done on a system, $W$, during a non-equilibrium transformation to the free energy difference, $\Delta F$, between the final and the initial state of the system, $$\frac{P_{A \rightarrow B} (W)}{P_{A \leftarrow B}(- W)} = ~ \exp[\beta (W - \Delta F)],$$ where $\beta$, is the inverse temperature, $A \rightarrow B$ denotes a forward transformation, and vice versa.

A lot of research has been done in this area so for more information I suggest reading some review articles, such as,

Esposito, M., Harbola, U., and Mukamel, S. (2009). Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems. Reviews of Modern Physics, 81(4), 1665. (arxiv)

and

Campisi, M., Hänggi, P., and Talkner, P. (2011). Colloquium: Quantum fluctuation relations: Foundations and applications. Reviews of Modern Physics, 83(3), 771. (arxiv)

There are various master equations (e.g., Fokker-Planck type, Boltzmann, Lindblad, etc.)in physics which will give you more information than theorems like these, but they are derived using various approximations and/or assumptions or are system specif. So, like I said there is no universal answer to your question.

EDIT: Deriving Fourier law is difficult. In fact there is an article from 2000 F. Bonetto, J.L. Lebowitz and L. Rey-Bellet, Fourier's Law: a Challenge for Theorists (arxiv) which states in the abstract: "There is however at present no rigorous mathematical derivation of Fourier's law..."

• Could you explain how your answer relates to my question rather than just the field of non-equilibrium thermodynamics as a whole? In particular, I don't think you addressed the idea of time dependence which I had already mentioned in my edit. Jun 11, 2014 at 17:32
• There is time in the first equation. Your question, as I understood it, seemed very general. I assumed you were interested in results for how macrostates evolve in time. Maybe I misunderstood. Maybe you were looking for something closer to master equation approaches? Jun 11, 2014 at 17:42
• I am indeed looking to understand how macrostates evolve in time, but that fluctuation theorem is really just a (very loose) constraint on that evolution. As I understand it, the theorem says essentially nothing about the chance of evolving to a particular macrostate, only about the chance of evolving to one of the members of a large equivalence class of macrostates characterized by the associated entropy production. Still, I appreciate the info and especially for the link to the Bonetto et al article in your edit. It looks interesting and I am reading it now. Jun 11, 2014 at 17:51
• Yes, fluctuation theorems are just loose constraints, but also hold very generally. For more specif results you need to also specify what types of systems and non-equilibrium transitions you're interested in. For instance, there are various master equations in physics which describe the time evolution of the probability distributions of the microstates (which you can use to compute thermodynamic quantities). Jun 11, 2014 at 17:57

I think people tend to make this sound vastly more mysterious than it is. Boltzmann wrote down an actual equation (which bears his name) that governs the distribution of particles $$\left(\partial_t +\vec{v}\cdot\vec{\nabla}_x +\vec{F}\cdot\vec{\nabla}_p\right)f_p(x,t) = C[f_p]$$ which, among other things, describes the approach to equilibrium. As explained in any text book on kinetic theory (see, e.g. Vol X of Landau), taking moments of this equation and assuming slowly varying distributions gives Fourier's law of heat conduction, Fick's law of diffusion, the Newton-Navier-Stokes law for vicous friction, etc. Not only that, it provides a method for computing the corresponding transport coefficients, and (for reasonably dilute systems) the result agrees with experiment.

The Boltzmann equation does not rely on the classical approximation (it works for quantum fluids, too, as explained by Landau), but it requires the existence of well defined quasi-particles. For systems in which quantum coherence plays a role, quantum analogs of the Boltzmann equation can be derived from non-equilibrium Green functions. These days, Fourier's law (etc) can also be derived for very strongly correlated fluids using the AdS/CFT correspondence, demonstrating that the laws of hydrodynamics are indeed universal low energy, low momentum approximations.

• Boltzmann's equation is phenomenological and isn't derived from first principles. In fact, the inadequacy of this approach was what motivated me to ask the above question in the first place. (At the time I was reading back through Ch 14 of Kittel and Kroemer, and all the other stat mech books I could find, e.g. Landau, had the same argument.) Jun 11, 2014 at 19:34
• The status of the Boltzmann equation (BE) is similar to other effective theories (such as hydro or ChPTh). They can be derived in simple cases, but there is plenty of evidence that the theory is in fact more general. It is certainly not mere phenomenology. I can take the BE with v_p,F,C computed from some underlying QFT and determine transport coefficients systemtically as an expansion in T or the diluteness of the gas (and the result agrees with experiment). Jun 11, 2014 at 21:23
• In simple cases (like phi^4 or the dilute Fermi gas) I can also compare (theoretically) transport coefficients from the BE to calculations based on the Kubo relation and equilibrium Green functions, and the result agrees. Hard to see how that would work if the BE is some kind of phenomenological equation. Jun 11, 2014 at 21:23