Free-energy difference from nonequilibrium work

Equilibrium ΔF expressed via nonequilibrium work statistics: ΔF = -β^{-1} ln ⟨e^{-βW}⟩ and related inference using Crooks’ theorem.

Free-energy difference from nonequilibrium work

Equilibrium free-energy difference

For a system at fixed inverse temperature $\beta=1/(k_B T)$ (with $T$ the temperature ) and control parameter $\lambda$ , the equilibrium Helmholtz free energy is

F(\lambda) \;=\; -\beta^{-1}\ln Z(\lambda),

where $Z(\lambda)$ is the canonical partition function (compare Helmholtz free energy and statistical free energy ).

The equilibrium free-energy difference between endpoints of a protocol is

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

Nonequilibrium identification via work statistics

Even if the protocol $\lambda_t$ is fast and drives the system out of equilibrium, $\Delta F$ is determined by nonequilibrium work fluctuations:

Jarzynski estimator

If realizations start in equilibrium at $\lambda_0$ and work $W$ is measured (see work distribution ), then

\Delta F \;=\; -\beta^{-1}\ln\left\langle e^{-\beta W}\right\rangle,

which is exactly Jarzynski’s equality rewritten as a formula for $\Delta F$ .

Crooks crossing / likelihood methods

If both forward and reverse experiments are available, the Crooks fluctuation theorem

\frac{P_F(W)}{P_R(-W)} \;=\; e^{\beta(W-\Delta F)}

implies:

Crossing method: solve $P_F(W)=P_R(-W)$ to get $W=\Delta F$ .
Likelihood-based estimation: infer $\Delta F$ from the full set of observed forward/reverse work samples (statistically efficient in practice).

Dissipation and information

Define dissipated work $W_{\mathrm{diss}}=W-\Delta F$ . Then $\langle W_{\mathrm{diss}}\rangle\ge 0$ (by Jensen) is a nonequilibrium refinement of the second law and can be related to a relative entropy between forward and reverse path ensembles (a path-space irreversibility measure).

Practical cautions

The Jarzynski exponential average can be dominated by rare low-work events, leading to slow convergence for finite samples.
Using reverse data (Crooks-based inference) often reduces variance compared to forward-only estimation.