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

## TL;DR

This paper develops a mathematical framework to analyze the statistical properties of nonequilibrium Brownian motion, focusing on energy fluctuations during upward and downward events, applicable to systems coupled to multiple thermal sources.

## Contribution

It introduces a novel selective analysis method for trajectories based on energy thresholds, enhancing understanding of nonequilibrium transport phenomena.

## Key findings

- Derived distributions, moments, and correlation functions for upside/downside events.
- Applicable to standard Brownian motion without modifications.
- Provides a general formalism for future studies of heat transfer in systems with thermal reservoirs.

## 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 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.