ar X iv : c on d - m at / 0 20 10 34 v 1 3 J an 2 00 2 Thermal Fluctuations of a Liquid Helium Inside a Spherical Volume

The theory of the thermal hydrodynamic fluctuations in bounded systems has been developed only for non-superfluid liquids on the base of the linearized Navier-Stokes equations [1]. However, the study of effect of confined geometry on correlation functions of superfluid liquid is important for explanation of experiments in which capillaries, pores, dense suspensions and other objects are probed by radiation [2]. In this paper we present the correlation theory of the equilibrium thermal hydrody-namic fluctuations of compressible superfluid helium confined by the spherical cavity of radius R. The hydrodynamic fluctuations in the superfluid helium are described on the base of linearized two-liquids Khalatnikov's equations [3] with spontaneous heat fluxes and strains. For mass flow δ J and relative velocity δ w = v n − v s of normal-fluid and superfluid components these equations have form ∂δρ ∂t +∇·δ J = 0, ρ ∂δs ∂t +ρ s s∇·δ w− κ T ∆δT = − 1 T ∇· q, ∂δ J ∂t +∇δp− η ρ ∆δ J− 1 ρ (1 3 η+ζ 2)∇(∇·δ J)− ρ s η ρ ∆δ w+ ρ s ρ (ρζ 1 − 1 3 η−ζ 2)∇(∇·δ w) = − f , ∂δ J ∂t −ρ n ∂δ w ∂t +∇δp−ρs∇δT −ζ 1 ∇(∇·δ J)+ ρ s ρ (−ρζ 1 +ρ 2 ζ 3)∇(∇·δ w) = −ρ∇H, (1) where δp = ∂p ∂ρ T δρ + ∂p ∂T ρ δT, δs = ∂s ∂ρ T δρ + ∂s ∂T ρ δT. Here s is the entropy per unit mass, p is the pressure; κ, η, and ζ 1 , ζ 2 , ζ 3 are the heat conductivity, the " first " viscosity, and the " second " viscosity coefficients, respectively; ρ s , ρ n , and ρ are the superfluid, normal-fluid, and total densities. The system (1) is written with account of the Onsager symmetry principle for the kinetic coefficients. All kinetic coefficients and thermodynamic derivatives in (1) assumed to be independent of coordinates. By (1) the fluctuation fields of mass flow, mass density and other variables are linear functionals of random forces in the right parts of these equations.