Rare events and the convergence of exponentially averaged work values

Christopher Jarzynski

Phys. Rev. E 73, 046105 – Published 5 April 2006

Abstract

Equilibrium free energy differences are given by exponential averages of nonequilibrium work values; such averages, however, often converge poorly, as they are dominated by rare realizations. I show that there is a simple and intuitively appealing description of these rare but dominant realizations. This description is expressed as a duality between “forward” and “reverse” processes, and provides both heuristic insights and quantitative estimates regarding the number of realizations needed for convergence of the exponential average. Analogous results apply to the equilibrium perturbation method of estimating free energy differences. The pedagogical example of a piston and gas [R.C. Lua and A.Y. Grosberg, J. Phys. Chem. B 109, 6805 (2005)] is used to illustrate the general discussion.

Received 23 December 2005

DOI:https://doi.org/10.1103/PhysRevE.73.046105

Authors & Affiliations

Christopher Jarzynski^*

Theoretical Division, T-13, MS B213, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

^*Electronic address: chrisj@lanl.gov

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Vol. 73, Iss. 4 — April 2006

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Images

Figure 1
During most realizations of the process, we observe work values near the peak of $ρ (W)$ . However, the average of $e^{- β W}$ is dominated by realizations for which the work is observed to be in the region around the peak of $g (W) = ρ (W) e^{- β W}$ .Reuse & Permissions
Figure 2
A conjugate pair of trajectories. The horizontal axis $(q)$ represents the complete set of configurational coordinates (e.g., particle positions), while the vertical axis $(p)$ represents the set of associated momenta. $Γ$ denotes a point in this many-dimensional phase space; and $Γ_{t}^{F ∕ R}$ denotes a realization of the forward or reverse process, with time running from $t = 0$ to $t = τ$ . The two trajectories are related by time-reversal: $Γ_{t}^{R} = Γ_{τ - t}^{F *}$ , where $(q, p)^{*} \equiv (q, - p)$ . In the notation of Sec. 3, the upper curve is the trajectory $γ^{F}$ , the lower curve is the trajectory $γ^{R}$ .Reuse & Permissions
Figure 3
(a) A typical realization of the forward process. Due to the great speed of the piston [Eq. (10)], most particles remain in the left half the box for the duration of the process. (b) The conjugate twin of the realization depicted in (a). Almost all particles begin in the left half of the box, and the piston then moves rapidly through a largely empty region. (The vertical gray arrows specify the direction of increasing time.)Reuse & Permissions
Figure 4
(a) A typical realization of the reverse process. As the piston moves rapidly into the gas, the particles with which it collides gain large components of velocity $(\sim 2 u)$ along the direction of motion of the piston. The particles with the attached arrows are meant to represent these fast particles, while the unadorned ones are characterized by thermal velocities. (b) The conjugate twin of the realization depicted in (a). Half the particles are initially moving at great speeds $(\sim 2 u)$ in the direction of the piston. By the end of the process, after each of these has collided with the piston, the velocity distribution of the entire gas is thermal.Reuse & Permissions
Figure 5
The upper box represents the space of all forward trajectories, the lower box the space of reverse trajectories, and in this visual depiction conjugate pairing is indicated by reflection about a horizontal line between the two boxes (e.g., the two crosses represent a pair of conjugate twins). The grey regions $ζ_{typ}^{F}$ and $ζ_{typ}^{R}$ denote the peaks of the probability distributions $P^{F}$ and $P^{R}$ , while the vertically striped regions $ζ_{dom}^{F}$ and $ζ_{dom}^{R}$ are the peaks of the functions $Q^{F}$ and $Q^{R}$ . The dashed arrows indicate the conjugate pairing that is the central result of this paper [Eq. (27)].Reuse & Permissions
Figure 6
Distributions of work values when ${\bar{W}}_{d}^{F} > {\bar{W}}_{d}^{R}$ . Since the two distributions cross at $W = Δ F$ , $ρ^{F}$ is wider than $ρ^{R}$ . Thus work values near $- {\bar{W}}^{R}$ are more frequently sampled from $ρ^{F} (W)$ , than those near ${\bar{W}}^{F}$ are sampled from $ρ^{R} (- W)$ .Reuse & Permissions

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