We present the results of simulation of the chaotic dynamics of quantized vortices in the bulk of superfluid He II. Evolution of vortex lines is calculated on the base of the Biot - Savart law. The dissipative effects appeared from the interaction with the normal component, or/and from relaxation of the order parameter are taken into account. Chaotic dynamics appears in the system via a random forcing, e. i. we use the Langevin approach to the problem. In the present paper we require the correlator of the random force to satisfy the fluctuation-dissipation relation, which implies that thermodynamic equilibrium should be reached. In the paper we describe the numerical methods for integration of stochastic differential equation (including a new algorithm for reconnection processes), and we present the results of calculation of some characteristics of a vortex tangle such as the total length, distribution of loops in the space of their length, and the energy spectrum.

The numerical simulation of the diffusive-like decay of the vortex tangle without mutual friction was performed. The simulation was made with the use of the localized induction approximation. The early developed by authors algorithm, which is based on consideration of crossing lines, was used for vortex reconnection processes. We have determined the influence of different factors on decay of an inhomogeneous vortex tangle: a diffusion (large vortex loops break up to smaller ones which go away from initial volume), change of length owing to reconnection processes, the eliminations of small vortices below the space resolution, the insertion and removing of points to supply numerical algorithm stability. The obtained numerical results demonstrate that the vortex tangle, initially localized in the small region, is smearing into ambient space. The time evolution of vortex line density inside the initial domain satisfactory agrees with the ones, obtained from the solution of diffusion equation.

Квантова турбулентність - це сукупність взаємодіючих квантових вихрових петель (КВП) і ниток. Систематизовано інформацію про поточний стан досліджень КВП після реконнекції у надплинному гелії: геометрія, динаміка, властивості вихрових петель, енергетичний спектр до та після реконнекції . Обговорено можливу роль реконнекції у формуванні турбулентного спектра, а також основні методи та підходи до вивчення вихрових петель після реконнекції та квантової турбулентності.