Jarzynski equality (theorem)

Non-equilibrium work identity: the exponential work average equals the equilibrium free-energy difference for a system driven by an arbitrary protocol from an initial Gibbs state.

Jarzynski equality (theorem)

Theorem (work identity)

Consider a system with Hamiltonian $H(x;\lambda)$ depending on an external control parameter $\lambda$ . Assume the system starts in equilibrium at inverse temperature $\beta$ with parameter value $\lambda_0$ , i.e. with initial distribution given by the canonical ensemble .

Drive the system with an arbitrary protocol $\lambda_t$ from $t=0$ to $t=\tau$ , producing a random work value $W$ along each realization (trajectory). Then the Jarzynski equality states

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

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

This statement is the theorem-level form of Jarzynski equality .

Work along a protocol (common classical convention)

For Hamiltonian dynamics, the (inclusive) work performed on the system is often written as

W = \int_0^\tau \dot{\lambda}_t\,\partial_\lambda H(x_t;\lambda_t)\,dt,

where $x_t$ is the phase-space trajectory generated by the driven dynamics.

The distribution of $W$ is the work distribution .

Immediate corollary (second-law inequality)

By Jensen’s inequality applied to $\langle e^{-\beta W}\rangle$ ,

\langle W\rangle \ge \Delta F,

consistent with the second law in isothermal processes (temperature as in temperature ).

Conditions (typical sufficient assumptions)

Different derivations use different dynamics, but common sufficient conditions include:

Equilibrium start: initial state is Gibbs/canonical at $\lambda_0$ .
Microreversibility: underlying dynamics satisfy time-reversal invariance (Hamiltonian case) or, for stochastic dynamics, are consistent with detailed balance with respect to the instantaneous equilibrium family.
Well-defined work functional: depends on a clear convention for what counts as work vs. heat.

Connections

The Jarzynski identity is closely related to Crooks fluctuation theorem and implies it under standard assumptions.
It provides a practical route to compute free-energy differences from nonequilibrium experiments/simulations via exponential averaging.