We present the results of a simulation of the chaotic dynamics of quantized vortices in the bulk of superfluid He II. The evolution of vortex lines is calculated on the basis of the Biot—Savart law. The dissipative effects arising 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, i.e., 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. We describe numerical methods for integration of the stochastic differential equation (including a new algorithm for reconnection processes), and we present the results of a calculation of some characteristics of a vortex tangle such as the total length, the length distribution of the loops, and the energy spectrum.