Calculating potentials of mean force and diffusion coefficients from nonequilibrium processes without Jarzynski’s equality

In general, the direct application of the Jarzynski equality (JE) to reconstruct potentials of mean force (PMFs) from a small number of nonequilibrium unidirectional steered molecular-dynamics (SMD) paths is hindered by the lack of sampling of extremely rare paths with negative dissipative work. Such trajectories that transiently violate the second law of thermodynamics are crucial for the validity of JE. As a solution to this daunting problem, we propose a simple and efficient method, referred to as the $FR$ method, for calculating simultaneously both the PMF $U(z)$ and the corresponding diffusion coefficient $D(z)$ along a reaction coordinate $z$ for a classical many-particle system by employing a small number of fast SMD pullings in both forward $(F)$ and time reverse $(R)$ directions, without invoking JE. By employing Crooks [Phys. Rev. E 61, 2361 (2000)] transient fluctuation theorem (that is more general than JE) and the stiff-spring approximation, we show that (i) the mean dissipative work $W¯d$ in the $F$ and $R$ pullings is the same, (ii) both $U(z)$ and $W¯d$ can be expressed in terms of the easily calculable mean work of the $F$ and $R$ processes, and (iii) $D(z)$ can be expressed in terms of the slope of $W¯d$⁠. To test its viability, the $FR$ method is applied to determine $U(z)$ and $D(z)$ of single-file water molecules in single-walled carbon nanotubes (SWNTs). The obtained $U(z)$ is found to be in very good agreement with the results from other PMF calculation methods, e.g., umbrella sampling. Finally, $U(z)$ and $D(z)$ are used as input in a stochastic model, based on the Fokker-Planck equation, for describing water transport through SWNTs on a mesoscopic time scale that in general is inaccessible to MD simulations.