Crooks fluctuation theorem

Setup (forward and reverse protocols)

Consider a system with control parameter $\lambda$ (e.g. a trap position, field, volume) driven by a time-dependent protocol $\lambda_t$ for $t\in[0,\tau]$.

• Forward (F): start in equilibrium at $\lambda_0$ with inverse temperature $\beta=1/(k_B T)$ (with $T$ the temperature ), typically the canonical ensemble .
• Reverse (R): start in equilibrium at $\lambda_\tau$ and drive with the time-reversed protocol $\tilde\lambda_t=\lambda_{\tau-t}$.

For each realization, define the work $W$ (see work distribution ). Let $P_F(W)$ and $P_R(W)$ be the corresponding work probability densities (with respect to the relevant probability measure ).

Let $\Delta F = F(\lambda_\tau)-F(\lambda_0)$ denote the equilibrium free-energy difference at the same $\beta$ (see nonequilibrium free-energy difference and Helmholtz free energy ).

Theorem (Crooks)

Under microreversible dynamics (time-reversal symmetric Hamiltonian dynamics, or Markov dynamics satisfying microscopic reversibility such as detailed balance with a heat bath), the work distributions satisfy

Immediate consequences

1. Crossing point identifies $\Delta F$.
   The value $W^\star$ where $P_F(W^\star)=P_R(-W^\star)$ satisfies $W^\star=\Delta F$.

2. Jarzynski follows by integration.
   Multiply both sides by $P_R(-W)e^{-\beta W}$ and integrate over $W$ to obtain

   $$\left\langle e^{-\beta W}\right\rangle_F \;=\; e^{-\beta \Delta F},$$

   i.e. Jarzynski’s equality .

3. Second-law inequality.
   Using Jensen’s inequality,

   $$\langle W\rangle_F \;\ge\; \Delta F,$$

   consistent with the second law .

Information-theoretic form (dissipation as relative entropy)

At the level of trajectories $\omega$ (paths), Crooks’ theorem is equivalent to a path-space likelihood ratio

so the mean dissipated work $\langle W-\Delta F\rangle_F$ is proportional to a relative entropy between forward and reverse path measures.

Practical note

Crooks’ relation underlies statistically efficient estimators of $\Delta F$ (e.g. Bennett-type methods) built from both $P_F$ and $P_R$; see free-energy difference from nonequilibrium work .