Crooks fluctuation theorem

Setup (forward vs. reverse driving)

Consider a system coupled to a heat bath at temperature $T$ (inverse temperature $\beta = 1/(k_B T)$), initially prepared in equilibrium for a control parameter value $\lambda_0$. The system is then driven by an externally prescribed protocol $\lambda_t$ from $t=0$ to $t=\tau$ (the forward process). Define the time-reversed protocol $\tilde\lambda_t := \lambda_{\tau-t}$ (the reverse process), with the reverse experiment initialized in equilibrium at $\lambda_\tau$.

• Equilibrium initialization and free energy: canonical ensemble , canonical partition function , Helmholtz free energy , temperature .
• Nonequilibrium work viewpoint: work distribution , free-energy difference from nonequilibrium work .

Let $W$ denote the work performed on the system during a single realization of the forward protocol, and let $P_F(W)$ be its probability density. Let $P_R(W)$ be the work density in the reverse protocol (defined with the same sign convention for work on the system).

Theorem (Crooks fluctuation theorem)

Assume microreversibility (time-reversal symmetry of the underlying dynamics) and consistent thermal contact with the bath (e.g., Markovian dynamics satisfying local detailed balance; see detailed balance ). Then the forward and reverse work distributions satisfy

where $\Delta F := F(\lambda_\tau)-F(\lambda_0)$ is the equilibrium Helmholtz free-energy difference.

Equivalently, in terms of dissipated work $W_{\mathrm{diss}} := W-\Delta F$,

Key consequences

1) Jarzynski equality (corollary)

Integrating the Crooks relation over $W$ yields

which is the Jarzynski equality (see also Jarzynski equality (concept) ).

2) Crossing criterion for estimating $\Delta F$

The forward and reverse distributions cross at the reversible work value:

This gives an operational way to extract $\Delta F$ from nonequilibrium sampling.

3) Second-law inequality from convexity

By Jensen’s inequality applied to the Jarzynski equality,

consistent with the second law and the role of dissipation.

Proof idea (path probabilities; sketch)

A standard route compares the probability of a forward trajectory $\Gamma$ to its time-reversed counterpart $\tilde\Gamma$ under the reverse protocol:

1. Use microreversibility plus local detailed balance (or Hamiltonian time-reversal symmetry for system+bath) to relate the ratio of path probabilities to entropy flow into the bath.
2. Identify the bath entropy flow with $\beta Q$ (heat $Q$ absorbed by the bath), and combine with the first-law form $W = \Delta E + Q$.
3. Use equilibrium initial distributions to produce the boundary term $-\beta\Delta F$, yielding a relation of the form $\frac{\mathbb P_F(\Gamma)}{\mathbb P_R(\tilde\Gamma)} = e^{\beta(W(\Gamma)-\Delta F)}.$
4. Push forward from trajectories to work values to obtain the work-distribution identity above.

Information-theoretic refinements often express average dissipation as a relative entropy between forward and reverse path measures (see relative entropy (KL divergence) ).

Prerequisites (minimal)

• Equilibrium ensembles and free energy: canonical ensemble , Helmholtz free energy .
• Nonequilibrium work notions: work distribution .
• Microscopic reversibility / balance conditions: detailed balance .
• Basic probability language: probability measure , expectation .