Fluctuation theorem

Remarks

The proof pairs each trajectory segment with its time reverse $M\Gamma(t)$, which has dissipation $-\Omega_t$. The ratio of the probabilities of the two segments is $e^{\Omega_t}$, by the change of variables formula along the flow, whose Jacobian is $\exp \int_0^t \Lambda\, ds$. This form is the transient fluctuation theorem of Evans and Searles; the steady-state form of Gallavotti and Cohen holds as $t \to \infty$ under extra chaoticity hypotheses. It implies the integral form $\mathbb{E}[e^{-\Omega_t}] = 1$ and, by Jensen's inequality, the second-law inequality $\mathbb{E}[\Omega_t] \ge 0$, where $\mathbb{E}$ is the expected value. Example: let a system start in the canonical distribution at temperature $T$ and then be driven by a field. Then $\Omega_t$ is the dissipative work done by the field divided by $k_B T$, and negative values have exponentially small probability.

Going deeper

For a student

The theorem gives the second law as a statement about probabilities, valid far from equilibrium and for small systems. It has been tested in experiments with colloidal particles in optical traps. The entropy production measured in units of $k_B$ plays the role of $\Omega_t$ in many models. The Jarzynski equality and the Crooks fluctuation theorem are relatives for the work done on a system driven between two equilibrium states.