Fluctuations of the critical Casimir force

TL;DR

This paper investigates the fluctuations of the critical Casimir force, analyzing its static and dynamic correlations, and explores how these fluctuations influence observable quantities like film thickness and colloidal positions.

Contribution

It provides a rigorous analysis of the static and dynamic correlations of the CCF within the Gaussian approximation, revealing cutoff independence in dynamics and effects on film thickness fluctuations.

Findings

Dynamic correlation function decays algebraically in time.

Dynamic exponent depends only on universality class.

Fluctuations induce non-Markovian noise affecting film thickness.

Abstract

The critical Casimir force (CCF) arises from confining fluctuations in a critical fluid and thus it is a fluctuating quantity itself. While the mean CCF is universal, its (static) variance has previously been found to depend on the microscopic details of the system which effectively set a large-momentum cutoff in the underlying field theory, rendering it potentially large. This raises the question how the properties of the force variance are reflected in experimentally observable quantities, such as the thickness of a wetting film or the position of a suspended colloidal particle. Here, based on a rigorous definition of the instantaneous force, we analyze static and dynamic correlations of the CCF for a conserved fluid in film geometry for various boundary conditions within the Gaussian approximation. We find that the dynamic correlation function of the CCF is independent of the momentum cutoff and decays algebraically in time. Within the Gaussian approximation, the associated exponent depends only on the dynamic universality class but not on the boundary conditions. We furthermore consider a fluid film, the thickness of which can fluctuate under the influence of the time-dependent CCF. The latter gives rise to an effective non-Markovian noise in the equation of motion of the film boundary and induces a distinct contribution to the position variance. Within the approximations used here, at short times, this contribution grows algebraically in time whereas, at long times, it saturates and contributes to the steady-state variance of the film thickness.