# Consistent application of Microscopic evolution with the definition of Macrostates?

How does one use the concept of microstate evolution and definition of macrostate in a consistent manner. As my understanding goes the thermodynamic concepts are defined in equilibrium. How does one apply these concepts in situations out of equilibrium.

I am looking for a simple example involving thermodynamics out of equilibrium. Such as 2 system previously in equilibrium with Temperatures T1 and T2, are coupled. Given the details microscopic coupling, initial statistical distribution of micro-states, what can one say about the evolution and definition Temperature in such a system.

I know this is a rather broad subject, and any pointers to the existing body of knowledge will be much appreciated. It would be lovely if an example is discussed for illustration purposes.

## 3 Answers

You might want to have a look at the work of Gavin Crook (http://threeplusone.com/gec/), especially the first two chapters of his PhD thesis (to be found on his website) are quite revealing. I'll quickly summarize his main result:

Assume a system is what he calls microscopically reversible, that is, the probability of a trajectory through phase space is related to the probability of the system taking the reverse trajectory by a simple function of the heat (Eq. 1.10 in his thesis). It is initially in equilibrium. Then you drive it out of equilibrium by some (time-reversible) protocol. Now for an arbitrary function $F$ depending on the path of the system through phase space, it holds that $$\langle F \rangle_{\mathrm{F}} = \langle \hat F \exp(-\beta W_\mathrm{d})\rangle_{\mathrm R}$$ where $\langle \ldots \rangle$ denotes an average over all possible paths the system can take through phase space and F / R denotes the forward / reverse non-equilibrium process. $W_{\mathrm{d}}=W_{\mathrm{tot}} - W_{\mathrm{r}}$ is what he calls dissipative work; it's just the total work minus the minimum amount of work required (reversible work, that is, the free energy difference). $\hat F$ is the time reversal of $F$. And now it comes: This holds regardless of the strength of the perturbation!

By choosing $F=1$ (or any other constant), one obtains the Jarzynski equality $$\langle \exp(-\beta W) \rangle = \langle \exp(-\beta \Delta F) \rangle$$ ($\langle \ldots \rangle$ again denotes an average over all possible realizations of the non-equilibrium process) which relates the work performed during a non-equilibrium process to the free energy difference by an equality (!) instead of the inequality resulting from the second law. With more sophisticated $F$'s one also obtains other relations like the transient fluctuation theorem, the Kawasaki response (which gives you a probability distribution for a non-equilibrium ensemble).

There's a lot more literature; I also recommend the 1997 papers of Christopher Jarzynski (sadly no free access). At the moment, I'm learning about all these things myself, so the above might not be 100% waterproof explained, but I hope, one gets the idea.

• Interesting, Let me get back to you once I digest what you've written. Also please tell me the names of the papers of Jarynski, I have access to journals. Commented May 31, 2013 at 18:36
• Sure :) Nonequilibrium equality for free energy differences is the paper in which he first derives his equality and Equilibrium free-energy differences from nonequilibrium measurements: A master-equation approach takes a more general approach and includes really nice and simple examples to test the equality numerically. Commented May 31, 2013 at 23:20
• Can you clarify what you mean by a protocol in this context? Thanks for pointing to this work, this approach looks intriguing. Commented Jun 3, 2013 at 6:08
• By "protocol" I mean some fixed prescription of how exactly the system is driven out of equilibrium. In all the realisations of the non-equilibrium process (which are obtained by starting from canonically distributed initial conditions), this prescription is always the same. An (actually not that) trivial example: a harmonic oscillator with time-dependent force constant, $H(q,p;t)=\frac{p^2}{2}+\frac{1}{2}k(t) q^2$. Now as the protocol I would denote the exact dependence of $k$ on $t$, including the initial and final value of $t$. Commented Jun 3, 2013 at 6:47
• In the literature, you often don't find a direct dependence on $t$, but rather on a "switching parameter" $\lambda(t)$, which is being switched from $0$ to $1$ during the non-equilibrium process. Commented Jun 3, 2013 at 6:47

What you are asking for is how to treat out-of-equilibrium fenomena, and as such, if it is still possible to use traditional ensemble formalism to it.

I don't know about using ensembles to out-of-equilibrium thermodynamics (macro-state via micro-state counting), but I do know that you can approach these problems via kinectic theory (Boltzmann Equation and the like) or Stochastic Methods (Fokker-Planck equation, SDEs...).

If you can define (locally) temperature is these systems, it isn't always clear, but at least when you try to apply hydrodynamics to a system, it's exactly that you are doing: Local (Quasi-)Equilibrium Thermodynamics

Out-of-Equilibrium Physics is not only a very broad subject, but one with very active research, since it's of crucial importance for many pure and applied areas of knowledge, but also it have lot's of very important (fundamental) problems.

• Hi, User23873, Thankyou for replying, How do you define Temprature locally, especially when it is changing according to a microscopic law. As microscopic evolution does not enforce the system to locally have a thermal distribution. Commented May 31, 2013 at 16:48

The standard conception is that you cannot define a macrostate, that is a system with macroscopic properties, such as its temperature and pressure, out of equilibrium.

See this video too