On Statistical Mechanics of Vortex Lines

An ensemble of a large number of vortex lines in cylindrical bounded domain is being considered. A system of equations determining the statistical characteristics of vortex line motion is derived from the assumption that this motion is ergodic and vortex lines are sufficiently smooth. The smoothness of vortex lines is characterized by vortex diffusivity. Both cases of finite and infinite vortex diffusivity are discussed. The continuum limit equations are obtained. Introduction. Ideal fluid is a Hamiltonian system, and it is tempting to determine the statistical properties of fluid motion from the assumption that this Hamiltonian system is ergodic. If the fluid is incompresssible, then the carriers of independent degrees of freedom, the ”particles” of the system, are vortex lines. The problem is in developing statistical mechanics of vortex lines. An attempt to derive the relationships of statistical mechanics of vortex lines from the maximum entropy principle was undertaken in [1],[2]. It was considered an ensemble of a large number, N , of vortex lines with vanishing intensities γs, s = 1, ..., N ; γs = σs/N , σs remain finite as N → ∞. Probability of sth vortex line to pass through a small vicinity of a given contour Γ was found to be