The Kramers theory for the escape rate of a Brownian particle from a potential well is extended to the full damping range. It is shown that the most adequate description of the underdamped Brownian motion in a deep potential well is provided by a Green function of the Fokker–Planck equation in the energy-position variables. The problem of lifetime of a particle in a single potential well is reduced to an integral equation in energy variable, with the Green function being the kernel of this equation. The straightforward solution by the Wiener–Hopf method yields an explicit expression for the lifetime, which describes the crossover from the extremely underdamped regime to that of a moderate damping. With the use of Kramer’s result for moderate-to-large damping an expression for the lifetime is presented, which holds at arbitrary damping. The problem of the rate of transitions between the two minima of a double-well potential is reduced to a system of two integral equations, which is also solved by the Wiener–Hopf method. An explicit expression for the relaxation time of nonequilibrium populations of the two minima is given.

We study the properties of a two-dimensional spinless particle moving in a random magnetic field. This problem arises in the context of a modern theory of strongly correlated systems as well as in the theory of vortex-lines dynamics in high-T, materials. The problem is investigated with a variety of methods including direct perturbation theory, quasiclassical approximation, the method of an optimal fluctuation, and Monte Carlo simulations. We obtain a shape of the density of states near the unrenormalized lower boundary of the spectrum, a particle mobility, and its diamagnetic orbital susceptibility.

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The low-temperature behaviour of the thermodynamic quantities in the one-dimensional 1D antiferromagnet with a Haldane ground state was analysed using a concept of non-interacting quasiparticles as a starting point. The proposed description appeared to be in good agreement with the results of the Monte Carlo study of a 128-site spin chain and experimental data obtained in NENP (Ni(C2H8N2)2NO2ClO4) provided that the elementary excitations constituting the Haldane triplet obey Fermi statistics.

The fast algorithm of the Maximum Entropy (MaxEnt) numerical solution of the linear inverse problem is described. The minimization of a general functional intrinsic to the MaxEnt approach is reduced to an iteration procedure with each step being a constrained least-squares problem (minimization of a quadratic functional with linear inequality constraints). The algorithm is structurally simple and can be assembled from blocks available in standard program libraries. The algorithm is tested on “toy” tasks with exponential kernel, as well as on practical problems of the recovery of the spectral density of strongly correlated quantum systems from the imaginary time Green’s functions obtained by Monte Carlo.