Jarzynski equality

Nonequilibrium work identity: ⟨e^{-βW}⟩ = e^{-βΔF} for protocols starting in equilibrium, yielding ΔF from nonequilibrium experiments and implying ⟨W⟩≥ΔF.

Jarzynski equality

Statement

Let a system at inverse temperature $\beta=1/(k_B T)$ (with $T$ the temperature ) start in equilibrium at control parameter $\lambda_0$ , typically the canonical ensemble .

Drive the system with an arbitrary (possibly fast) protocol $\lambda_t$ over $t\in[0,\tau]$ . For each realization, measure the work $W$ performed on the system (see work distribution ). Let

\Delta F \;=\; F(\lambda_\tau)-F(\lambda_0)

be the equilibrium free-energy difference at the same $\beta$ (see nonequilibrium free-energy difference ).

Then the Jarzynski equality is

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

where $\langle\cdot\rangle$ denotes the ensemble average over repeated realizations (an expectation with respect to the induced work distribution).

Conditions (minimal checklist)

Initial state is equilibrium at $\lambda_0$ (canonical, or appropriate equilibrium ensemble).
Dynamics are microreversible (time-reversal symmetric Hamiltonian dynamics, or suitable stochastic dynamics coupled to a heat bath).
Work is defined consistently with the driven parameter (protocol work).

These assumptions are also those used for the Crooks fluctuation theorem , from which Jarzynski can be derived.

Consequences

Second-law bound (via Jensen)

Since $\exp$ is convex,

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

hence

\langle W\rangle \;\ge\; \Delta F,

consistent with the second law .

Dissipated work

Define the dissipated work

W_{\mathrm{diss}} \;=\; W-\Delta F.

Jarzynski becomes $\langle e^{-\beta W_{\mathrm{diss}}}\rangle = 1$ , emphasizing that fluctuations with unusually small $W_{\mathrm{diss}}$ control the exponential average.

Near-equilibrium expansion

If $W$ has small fluctuations around its mean, cumulant expansion gives

\Delta F \approx \langle W\rangle - \frac{\beta}{2}\,\mathrm{Var}(W) + \cdots,

linking nonequilibrium work fluctuations to equilibrium free-energy differences (compare with linear-response identities such as fluctuation–dissipation ).

Practical note (convergence)

Estimating $\Delta F$ via $-\beta^{-1}\ln\langle e^{-\beta W}\rangle$ can be statistically challenging because rare low-work events dominate the exponential average. Using both forward and reverse processes with Crooks often improves efficiency.