Upside/Downside statistical mechanics of nonequilibrium Brownian motion. I. Distributions, moments, and correlation functions of a free particle

Abstract

Statistical properties of Brownian motion that arise by analyzing, separately, trajectories over which the system energy increases (upside) or decreases (downside) with respect to a threshold energy level are derived. This selective analysis is applied to examine transport properties of a nonequilibrium Brownian process that is coupled to multiple thermal sources characterized by different temperatures. Distributions, moments, and correlation functions of a free particle that occur during upside and downside events are investigated for energy activation and energy relaxation processes and also for positive and negative energy fluctuations from the average energy. The presented results are sufficiently general and can be applied without modification to the standard Brownian motion. This article focuses on the mathematical basis of this selective analysis. In subsequent articles in this series, we apply this general formalism to processes in which heat transfer between thermal reservoirs is mediated by activated rate processes that take place in a system bridging them.

On the definitions and simulations of vibrational heat transport in nanojunctions

“…These issues were observed with different integrators, RK4, BBK and the VEC. Insights on the poor performance of the interface currents could be achieved based on the upside/downside statistical analysis suggested in Refs 59,60 . While so far much effort had been placed on comparing the performance of different integrators, here we emphasize that using different working definitions for an observable may result in different convergence behavior.…”

Upside/Downside statistical mechanics of nonequilibrium Brownian motion. II. Heat transfer and energy partitioning of a free particle

“…(92) Figure 9 shows the dependence of these heat ratios on t. The variance in the simulation results shown in Fig. 9 for small t is a consequence of the underlying probability densities being independent of t in the t ′ → t limit, which means that over small time-intervals the probability density changes rapidly to go from the initial distribution ρ 0 at t ′ = 0 to the distribution at t ′ = t. 30 This leads to increased variance in the results from simulation. We have confirmed that simulation results converge to the analytical results with increased sampling.…”

“…which can be applied to construct expressions for the moments and time-correlation functions of a nonequilibrium Brownian process driven by N thermal sources. Because it is proportional to the energy E of the system, the second moment 30,59 v…”

“…Resolving these energy partitioning problems for Brownian motion is the focus of the current study. To this end, we apply the mathematical framework and statistical mechanics developed in the previous article in this series 30 in which analysis of energy activation and relaxation events is performed separately, as opposed to the typical case where these fluctuations are analyzed together. Considering the motion of the system under the influence of the different thermal baths, and focusing on the energy E(t) of the system, the fundamental step in the implementation of this formalism entails separating, at any time t, the full ensemble of trajectories into two groups: upside and downside.…”

“…In the previous article in this series, 30 an upside/downside mathematical framework for Brownian processes that are driven by multiple thermal sources was developed and applied to construct restricted dynamical properties of a free particle that are pertinent for thermal energy transfer. The focus of the present article is the application of those properties to examine heat cur-rents and energy partitioning between thermal reservoirs during energy activation and energy relaxation events, and also during positive and negative energy fluctuations from the average system energy.…”