Crooks’ Fluctuation Theorem

guest post by Eric Downes

Christopher Jarzynski, Gavin Crooks, and some others have made a big splash by providing general equalities that relate free energy differences to non-equilibrium work values. The best place to start is the first two chapters of Gavin Crooks’ thesis:

• Gavin Crooks, Excursions in Statistical Dynamics, Ph.D. Thesis, Department of Chemistry, U.C. Berkeley, 1999.

Here is the ~1 kiloword summary:

If we consider the work W done on a system, Clausius’ Inequality states that

W \ge \Delta F

where \Delta F is the change in free energy. One must perform more work along a non-equilibrium path because of the second law of thermodynamics. The excess work

W- \Delta F

is dissipated as heat, and is basically the entropy change in the universe, measured in different units. But who knows how large the excess work will be…

One considers a small system for which we imagine there exists a distribution of thermodynamic work values W (more on that below) in moving a system through phase space. We start at a macrostate with free energy F_1, and (staying in touch with a thermal reservoir at inverse temperature \beta) move in finite time to a new non-equilibrium state. When this new state is allowed to equilibriate it will have free energy

F_1 + \Delta F

You can do this by changing the spin-spin coupling, compressing a gas, etc: you’re changing one of the parameters in the system’s Hamiltonian in a completely deterministic way, such that the structure of the Hamiltonian does not change, and the system still has well-defined microstates at all intervening times. Your total accumulated work values will follow

\displaystyle{ \langle \exp(-\beta W) \rangle = \exp(-\beta \Delta F)}

where the expectation value is over a distribution of all possible paths through the classical phase space. This is the Jarzynski Equality.

It has an analogue for quantum systems, which appears to be related to supersymmetry, somehow. But the proof for classical systems simply relies on a Markov chain that moves through state space and an appropriate definition for work (see below). I can dig up the reference if anyone wants.

This is actually a specific case of a more fundamental theorem discovered about a decade ago by Gavin Crooks: the Crooks fluctuation theorem:

\displaystyle{ \exp(-\beta(W- \Delta F)) = \frac{P_{\mathrm{fwd}}}{P_{\mathrm{rev}}} }

where P_{\mathrm{fwd}} is the probability of a particular forward path which requires work W, and P_{\mathrm{rev}} is the probability of its time-reversal dual (see Gavin Crooks’ thesis for more precise definitions).

How do we assign a thermodynamic work value to a path of microstates? At the risk of ruining it for you: It turns out that one can write a first law analog for a subsystem Hamiltonian. We start with:

H_{\mathrm{tot}} = H_{\mathrm{subsys}} + H_{\mathrm{environ}} + H_{\mathrm{interact}}

As with Gibbs’ derivation of the canonical ensemble, we never specify what H_{\mathrm{environ}} and H_{\mathrm{interact}} are, only that the number of degrees of freedom in H_{\mathrm{environ}} is very large, and H_{\mathrm{interact}} is a small coupling. You make the observation that work can be associated with changing the energy-levels of the microstates in H_{\mathrm{subsys}}, while heat is associated with the energy change when the (sub)system jumps from one microstate to another (due to H_{\mathrm{interact}}) with no change in the spectrum of available energies. This implies a rather deep connection between the Hamiltonian and thermodynamic work. The second figure in Gavin’s thesis explained everything for me, after that you can basically derive it yourself.

The only physical applications I am aware of are to Monte Carlo simulations and mesoscopic systems in nano- or molecular biophysics. In that regard John Baez’ recent relation between free energy and Rényi entropy is a nice potential competitor for the efficient calculation of free energy differences (which apparently normally requires multiple simulations at intervening temperaturess, calculating the specific heat at each.)

But the relation to Markov chains is much more interesting to me, because this is a very general mathematical object which can be related to a much broader class of problems. Heat ends up being associated with fluctuations in the system’s state, and the (phenomenological) energy values are kind of the “relative unlikelihood” of each state. The excess work turns out to be related to the Kullback-Leibler divergence between the forward and reverse path-probabilities.

For visual learners with a background in stat mech, this is all developed in a pedagogical talk I gave in Fall 2010 at U. Wisconsin-Madison’s Condensed Matter Theory Seminar; talk available here. I’m licensing it cc-by-sa-nc through the Creative Commons License. I’ve been sloppy with references, but I emphasize that this is not original work; it is my presentation of Crooks’ and Jarzynski’s. Nonetheless, any errors you find are my own. Hokay, have a nice day!

3 Responses to Crooks’ Fluctuation Theorem

  1. Blake Stacey says:

    It has an analogue for quantum systems, which appears to be related to supersymmetry, somehow.

    Mallick et al.‘s relation between Jarzynski’s equality and supersymmetry is actually set up in classical terms, using the “response field” formalism for writing a field theory with stochastic dynamics and Grassmann number-valued fields to bring in the idea of supersymmetry.

    (I wish I knew more about what they actually did, but that’s a bit of background on the tools they used.)

  2. Baez on Higher Categories for Concurrency, Crooks’ Fluctuation Theorem, Category-Theoretic Characterizations of Entropy, Networks and Population Biology

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