How Does Pressure Fluctuate in Equilibrium?

Abstract

We study fluctuations of pressure in equilibrium for classical particle systems. In equilibrium statistical mechanics, pressure for a microscopic state is defined by the derivative of a thermodynamic function or, more mechanically, through the momentum current. We show that although the two expectation values converge to the same equilibrium value in the thermodynamic limit, the variance of the mechanical pressure is in general greater than that of the pressure defined through the thermodynamic relation. We also present a condition for experimentally detecting the difference between them in an idealized measurement of momentum transfer.

$$\mu PT$$ statistical ensemble: systems with fluctuating energy, particle number, and volume

“…Fluctuations of the volume are related to fluctuations of the pressure, while fluctuations of the particle number are related to fluctuations of the chemical potential, through uncertainty relations that hold for statistically conjugated variables 25 – 28 . Even though pressure and chemical potential are fixed parameters, they can be seen as external fields subject to noise, e.g.…”

Ensemble transformation in the fluctuation theory

“…However, intensive variables cannot be statistical variables within the quasi-thermodynamic formalism at the thermodynamic limit. This can be shown straightforwardly and intuitively via the order-ofmagnitude analysis within our formalism, thereby shedding light onto the much-debated topic [31,32,35,42,43] despite its entry into standard textbooks [21,44].…”

“…Our goal is to facilitate thermodynamic variable transformation in the fluctuation solution theory from a geometrical perspective. The geometric basis of fluctuation will become apparent when one reformulates the Kirkwood-Buff solution theory [6,8,11] based on the quasi-thermodynamic formalism of fluctuation by von Smoluchowski and Einstein [36][37][38][39][40], which has been refined since then [21,[29][30][31][32][33][34][35]45]. Let us consider an isochoric system, which consists of a small, open isochoric (constant volume) subsystem and a reservoir (denoted as 𝑟).…”

“…To clarify the geometric foundation of fluctuation, the quasi-thermodynamic formalism [21,[29][30][31][32][33][34][35] of von Smoluchowski [36][37][38] and Einstein [38][39][40] will provide a useful perspective complementary to the ensemble-based formalism of Gibbs [41] which is used commonly as the foundation of the Kirkwood-Buff theory [6,8,17]. The quasi-thermodynamic formalism employs the minimum work associated with moving a molecule from the reservoir into the system, which is expressed in a quadratic form involving the Hessian matrix and the deviation vector of thermodynamic variables [25,27].…”

$μPT$ statistical ensemble: systems with fluctuating energy, particle number, and volume

“…Fluctuations of the volume are related to fluctuations of the pressure, while fluctuations of the particle number are related to fluctuations of the chemical potential, through uncertainty relations that hold for statistically conjugated variables [24][25][26][27]. Even though pressure and chemical potential are fixed parameters, they can be seen as external fields subject to noise, e.g.…”