Analysis of Trajectory Entropy for Continuous Stochastic Processes at Equilibrium

Abstract

The analytical expression for the trajectory entropy of the overdamped Langevin equation is derived via two approaches. The first route goes through the Fokker−Planck equation that governs the propagation of the conditional probability density, while the second method goes through the path integral of the Onsager− Machlup action. The agreement of these two approaches in the continuum limit underscores the equivalence between the partial differential equation and the path integral formulations for stochastic processes in the context of trajectory entropy. The values obtained using the analytical expression are also compared with those calculated with numerical solutions for arbitrary time resolutions of the trajectory. Quantitative agreement is clearly observed consistently across different models as the time interval between snapshots in the trajectories decreases. Furthermore, analysis of different scenarios illustrates how the deterministic and stochastic forces in the Langevin equation contribute to the variation in dynamics measured by the trajectory entropy. ■ INTRODUCTIONConsider very generally a system that interacts with its environment and evolves over time. Let x be some continuous order parameter of interest (or an experimental or computational observable) that characterizes the spatial configuration and represents the state of the system. For example, x could be a molecule-scale observable in single-molecule experiments, such as the orientation-averaged dipole−dipole distance in a single-molecule Forster-type resonance energy transfer experiment, 1 the constant-force extension in a laser tweezers/atomic force pulling experiment, 2 or a collective coordinate from molecular dynamics simulations, 3 even a state variable in a quantum-control experiment. 4 In many problems of this type, including the examples given above, the time evolution of x can be described by the overdamped Langevin equation, 5where the subscript in x t is time. In the Langevin equation, D is the diffusion coefficient specifying the magnitude of the stochastic forces modeled by the Wiener process, dW t , satisfying ⟨dW t dW t′ ⟩ = δ(t − t′) dt. On the other hand, the deterministic component of the Langevin equation comes from the potential of mean force (PMF), V(x), and the mean force is F(x) = −dV(x)/dx. Here, we consider the PMF as nondimensionalized by k B T, where k B is the Boltzmann constant and T is the temperature. Suppose in one realization that the system is at an initial configuration x 0 at time zero. Then propagating the Langevin equation will trace out a trajectory X(t), a continuous but nondifferentiable function that gives a value of x t at time t, with t varying between 0 and the finite period of observation, t obs . For stationary processes, the system reaches equilibrium and the probability density of obtaining a particular value of X(t) = x along the trajectory is given by p eq (x) = exp(−V(x))/Z eq , where Z eq = ∫ dx exp(−V(x)) is the equilibrium partition function. 6 In this work, we consider an ense...

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