Near-Equilibrium Measurements of Nonequilibrium Free Energy

A central endeavor of thermodynamics is the measurement of free energy changes. Regrettably, although we can measure the free energy of a system in thermodynamic equilibrium, typically all we can say about the free energy of a nonequilibrium ensemble is that it is larger than that of the same system at equilibrium. Herein, we derive a formally exact expression for the probability distribution of a driven system, which involves path ensemble averages of the work over trajectories of the time-reversed system. From this we find a simple near-equilibrium approximation for the free energy in terms of an excess mean time-reversed work, which can be experimentally measured on real systems. With analysis and computer simulation, we demonstrate the accuracy of our approximations for several simple models.

Figure 1

A simple driven system, amenable to numerical calculations. (a) Energy as a function of position. A single particle occupies a periodic, one-dimensional energy landscape. The position coordinate is discretized into $N_x$ uniformly spaced positions per period. Energy is also discretized, $E(x)=\lfloor N_e\mathbf{(}1+\sin (2\pi x/N_x)\mathbf{)}/2\rfloor /N_e$, for position $x$ and number $N_e$ of discrete energy bins. $N_x$ and $N_e$ are increased until results do not change appreciably with a finer discretization. (b) The system is initially in equilibrium with an external heat bath ($+$). At each discrete time step, the particle attempts to move one step left, one step right, or remain in the same location with equal probabilities, and the move is accepted according to the Metropolis criterion [17]. Every $1/v$ time steps, the energy surface shifts one position to the right. To ensure fully time-reversible dynamics, we simulate $1/2v$ time steps, shift the potential, and simulate another $1/2v$ time steps before examining the nonequilibrium properties of the system. All figures are drawn in the rest frame of the potential. Eventually the spatial distribution across a single periodic image converges to a nonequilibrium steady state ($\times$), approximated by Eq. (21) (□). Displayed results are for $v^{*}=24$, and reciprocal temperature $\beta =4$ reported in inverse units of the energy difference between top and bottom of the potential.

Figure 2

Equilibrium ($+$), steady-state ($\times$), and approximate steady-state (□) [Eq. (21)] probability distributions, for the system described in Fig. 1, at various driving rates and temperatures. Driving rates are reported in the dimensionless velocity $v^{*}\equiv v\ell /D$ for repeat length $\ell$ and diffusion coefficient $D$. The quality of our approximate distributions, including overall normalization, deteriorates at low temperature and high driving velocity. The dotted box highlights the conditions shown in Fig. 1.

Figure 3

The approximate steady-state free energy difference per periodic image, $\beta \Delta F_{\mathrm{approx}}$ [Eq. (19)], is very close to the exact steady-state free energy difference $\beta \Delta F_{\mathrm{exact}}=\beta (F_{{\lambda }_a;\Lambda }-F_{{\lambda }_b})$, as shown by the fractional error $1-\Delta F_{\mathrm{approx}}/\Delta F_{\mathrm{exact}}$ being much less than unity. Colors denote temperatures ranging from hot ($\beta =1/4$, red, bottom left) to cold ($\beta =32$, black, top right), with dimensionless velocity varying from $v^{*}=3$ (diamonds) to $v^{*}=48$ (crosses). Empirically for this system, $\beta \Delta F_{\mathrm{approx}}$ is always less than $\beta \Delta F_{\mathrm{exact}}$, and the fractional error shows a power-law dependence on exact free energy with exponent $\sim 1$ (dotted line plots $\frac{1}{5}\beta \Delta F_{\mathrm{exact}}$). Note that before convergence to steady state, fractional errors do not collapse onto a single curve even at low $\beta$.