Hydrodynamic fields in fluctuating environment: a model for isochoric heat capacity of simple liquids

TL;DR

The paper develops a fluctuating-hydrodynamics framework in which simple liquids at low temperature interact with a delta-correlated noise field to model high-energy non-hydrodynamic effects. Through a configurational zeta-function method, it derives a functional-series generating functional that reveals both phonon-like excitations and gapped-momentum (k-gap) excitations in the liquid, and uses spectral-zeta regularization of functional determinants to compute the isochoric heat capacity. The results show that the phase space for vibrational modes in liquids is not fixed but grows with the inclusion of noise-induced excitations, yielding a temperature-dependent $C_V$ consistent with a reduced low-frequency density of states relative to solids. This framework connects hydrodynamic fields to quantum noise effects and provides a route to quantify how non-hydrodynamic degrees of freedom shape liquid thermodynamics, with potential links to recent topological and viscoelastic approaches. Overall, it offers a coherent, field-theoretic account of emergent phononic and $k$-gap excitations and their thermodynamic consequences in simple liquids.

Abstract

Using functional methods, we investigate the sound quanta arising from quantized hydrodynamic fields in simple liquids at low temperatures, under the influence of high-energy processes, coming from non-hydrodynamic degrees of freedom. To model these effects on the hydrodynamic fields, we assume that the quantum fields are coupled to an additive, delta-correlated (in space and time) quantum noise field. Thus, the hydrodynamic fields are defined in a fluctuating environment. After defining the generating functional of connected correlation functions in the presence of the noise field, we perform a functional integral over all noise field configurations. This is done using a formal object inspired by the distributional zeta-function method, called the configurational zeta-function. We obtain a new generating functional written in terms of an analytically tractable functional series. This formalism allow us to obtain the excitation spectra of liquids. In the liquid, each term of the series describes the emergent non-interacting elementary excitations with the usual phonon-like dispersion relation and additional excitations with dispersion relations with gaps in pseudo-momentum space. Finally, the behavior of the constant volume specific heat as a function of temperature is obtained for different simple liquids.

Figures (1)

• Figure 1: Plot of heat capacity, $C_V$, as a function of normalized temperature $T/T_D$ for different values of $\sigma/T_D$. Here we consider $V T^3_D = 1$ and ${\epsilon_2}/T_D = 0.001$. We assume $\epsilon_2$ is a regularization factor introduced in the calculation of the heat capacity and $T_D$ is the Debye temperature of the liquid. To generate the curves, we consider a summation of $k$ from 1 to 100.