Exact rate calculations by trajectory parallelization and twisting

Introduction. The main objective of this work is to revisit and extend in scope the nonequilibrium sampling procedure to compute steady state probability distributions proposed in Refs. 1,2. Specifically, we show how this procedure can be modified to calculate exactly certain dynamical quantities such as the rate of reactions occurring in arbitrary equilibrium or nonequilibrium systems at statistical steady state. This is done by exploiting results of transition path theory (TPT) 3,4,5 which indicate how to twist the dynamics of a given system to calculate the reaction rate between a given reactant and product state. The method proposed in this note can be seen as a generalization to arbitrary nonequilibrium processes at statistical steady state of the milestoning procedure with Voronoi tessellation proposed in Ref. 6 by building upon the original works in Refs. 7,8,9. This generalization makes the procedure more expensive computationally, but it permits to relax completely the assumptions made in milestoning -these assumptions were discussed in detail in Ref. 10. Our method can also be viewed as a generalization of the transition interface sampling (TIS) 11,12,13 and forward flux sampling (FFS) 14,15 methods in which arbitrary sets of interfaces can be used that do not have to be placed in monotone succession.

The remainder of this note is organized as follows. First we revisit from an original perspective the nonequilibrium sampling procedure of Refs. 1,2. Next, we show how this procedure can be modified to calculate reaction rates exactly using TPT. We then compare our procedure with the Markovian milestoning method proposed in Ref. 6 and with TIS 11,12,13 and FFS 14,15 . Finally we illustrate our procedure on a simple example. In terms of notations and assumptions, we will denote by z the location of the system in its state-space Ω ⊂ R d (e.g. it could be the positions and velocities of all the atoms in a molecular system, in which case z = (x, v) and d = 6n if n is the number of atoms). The specifics of the dynamics of the system are not important except that we assume that (i) its evolution is Markovian and (ii) it is ergodic with respect to a probability density function which we denote by ̺(z). Notice that we do not require detailed balance, i.e. ̺(z) is associated with a nonequilibrium statistical steady state in general.

Restricted sampling with flux matching. The methods in Refs. 1,2,6 are based on a factorization of the dynamics in which one artificially constrains the system to evolve in a set of cells partitioning state-space in a way that (i) does not bias the dynamics inside the cells and (ii) is consistent with the exact probability fluxes in and out of these cells. In other words, the procedure guarantees that a true unconstrained trajectory of the system can be reconstructed exactly by patching together in some appropriate way the pieces computed in the cells. These pieces can be calculated in parallel in each cell, hence the name trajectory parallelization. The method in Ref. 6 is restricted to equilibrium systems and exploits the timereversibility of the dynamics. The method in Refs. 1,2 is more costly but it works also for nonequilibrium systems. To explain how the latter method works, we first recall how to construct the cells using the Voronoi tessellation associated with a given set of generating points or centers 6,16 . Denoting these centers by z α ∈ Ω ⊂ R d , with α = 1, . . . , Λ, the Voronoi cell B α associated to z α contains all the points that are closer to z α than to any other center, i.e. (see Fig. 1 for an illustration)

where • is some appropriate norm (e.g. the Euclidean norm in which case z 2 = d i=1 z 2 i ). If we were to generate an infinitely long trajectory of the system, z(t) with t > 0, this trajectory would keep going in and out of the cells B α by crossing the edges between these cells. Out of this trajectory we could therefore generate an ensemble of exit-entry points, i.e. those points on the edges of the cells at which the trajectory goes from a cell B α into a neighboring cell B β . By the Markovian assumption, these points are all we need to generate an exact sample of trajectories inside each of the cells B α simply by starting trajectories at the points leading into that cell and running these trajectories for- ward in time until they exit the cell. The procedure in Refs. 1,2 is a way to generate these pieces of trajectories inside the cells without having to compute the entry points beforehand from a long unbiased trajectory but rather by generating them on-the-fly. To understand how this is done, imagine that we associate an independent copy, or replica, of the system to each cell B α . Let us denote the instantaneous position of these replicas by z α (t) ∈ B α , α = 1, . . . , Λ. Even if we start z α (t) inside B α , sooner or later this trajectory will try to exit B α and go to another cell. When this happens, we store the exit point on the bounda

…(Full text truncated)…

This content is AI-processed based on ArXiv data.