Equilibrium free energy differences from nonequilibrium measurements: a master equation approach

TL;DR

The paper proves an exact Jarzynski-style identity within a master equation framework: for a system driven by a time-dependent parameter $\lambda(t)$ between $0$ and $1$, the nonequilibrium work ensemble satisfies $\overline{e^{-\beta W}}=e^{-\beta\Delta F}$ for any switching time $t_s$, under Markovian dynamics and detailed balance. The derivation introduces a weighted density $g(\mathbf z,t)=f(\mathbf z,t)Q(\mathbf z,t)$ and shows its evolution collapses to the instantaneous canonical form, yielding the identity without requiring Hamiltonian Liouville dynamics. The authors demonstrate the result across Hamiltonian, Langevin, isothermal MD, and Monte Carlo switching, and discuss practical implications for free-energy computations, including adiabatic switching and estimators like $W^x=-\beta^{-1}\ln\Big(\frac{1}{N_s}\sum_i e^{-\beta W_i}\Big)$. Numerical results corroborate the theory, showing $W^a$ overestimates $\Delta F$ while $W^x$ accurately recovers it for finite switching times, and highlighting the method’s potential to extract equilibrium information from nonequilibrium processes. The work broadens the toolkit for free-energy estimation by embedding nonequilibrium work within a rigorous master-equation context, with implications for simulations and potential micro- or mesoscopic experiments.

Abstract

It has recently been shown that the Helmholtz free energy difference between two equilibrium configurations of a system may be obtained from an ensemble of finite-time (nonequilibrium) measurements of the work performed in switching an external parameter of the system. Here this result is established, as an identity, within the master equation formalism. Examples are discussed and numerical illustrations provided.

Figures (9)

• Figure 1: Distribution of values of work, $\rho(W,t_s)$, performed during an ensemble of independent switching measurements at a given switching time $t_s$. The vertical line represents a delta function at $W=\Delta F$, and corresponds to $t_s\rightarrow\infty$; in that limit, the work performed during a single switching process is exactly equal to $\Delta F$. The smooth distribution represent $\rho(W,t_s)$ for a finite value of $t_s$. In this case the ensemble average work exceeds the free energy difference, $\overline{W}>\Delta F$, since energy is dissipated in a finite-time (irreversible) process.
• Figure 2: Simulations of an isolated harmonic oscillator whose natural frequency is switched from $\omega_0=1.0$ to $\omega_1=2.0$ over a switching time $t_s$. At each of five values of $t_s$, $10^5$ simulations were carried out. The upper and lower sets of points show the ordinary averages ($W^a$) and the exponential averages ($W^x$) of the work, respectively. The dashed line is at $W=1.5$, the dotted line at $W=\Delta F=1.0397$.
• Figure 3: Same as Fig.\ref{['fig:ham']}, except that the harmonic oscillator is now subject to a frictional and a stochastic force, as per Eq.\ref{['eq:langevin']}. The dashed line gives the free energy difference, $\Delta F = 1.0397$.
• Figure 4: In these simulations, the harmonic oscillator is thermostatted with the IMD scheme described in the text. $10^5$ simulations at $t_s=1.0$ were performed, and the dots in this figure show the final locations in phase space of these trajectories.
• Figure 5: Contour plot of the distribution $f(x,p,t_s)$, constructed from the data shown in Fig.\ref{['fig:imd1']}, with Gaussian smoothing.