  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,
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 Crook's
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 are equal, (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.
