Fluctuations in large classical systems

Abstract

Based on statistical regularities, this paper describes fluctuations in a multi-particle system of the Gibbs ensemble that is in thermal equilibrium with a thermostat. In particular, using Gibbs' canonical distributions, we calculate energy fluctuations during random thermal exchanges, the variance of particle number fluctuations, and the mean square of random energy. According to the results of these calculations, a formula for the relative energy fluctuations of a monoatomic ideal gas was also obtained. To calculate the fluctuations of any physical quantity, it is necessary to express the entropy variation in terms of that quantity. Based on this widely used thermodynamic principle, and taking into account the relationship between entropy probabilities and non-equilibrium states, a generalized formula for the variance of thermodynamic quantity fluctuations was derived, along with a formula for calculating the probability density of a non-equilibrium state. Using the resulting formula, it was shown that the probability density of a non-equilibrium state is a function of small random deviations of thermodynamic quantities from their equilibrium values. Based on these results, expressions were derived for the variances of temperature, volume, and density fluctuations in a monoatomic ideal gas, as well as for the relative fluctuations of these quantities. Additionally, internal energy fluctuation was calculated. Using a method that considers the transition of a system from an equilibrium state to a fluctuating one, along with thermodynamic parameter fluctuations, a criterion was presented that enables the calculation of fluctuations in mechanical quantities that describe the state of a macroscopic body in a given medium. In particular, the transition of a resting body to a fluctuating state due to molecular impacts from a fluid (or gas) was considered, and the probability density and variance of the fluctuating velocity of the body were calculated.