We consider the classical dynamics of a particle in a one-dimensional space-periodic potential U(X) = U(X+2pi) under the influence of a time-periodic space-homogeneous external field E(t) = E(t+T). If E(t) is neither a symmetric function of t nor antisymmetric under time shifts E(t+/-T/2) not equal-E(t), an ensemble of trajectories with zero current at t = 0 yields a nonzero finite current as t-->infinity. We explain this effect using symmetry considerations and perturbation theory. Finally we add dissipation (friction) and demonstrate that the resulting set of attractors keeps the broken symmetry property in the basins of attraction and leads to directed currents as well.

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We study rectification of a current of particles moving in a spatially pe- The model Hamiltonian of a particle of unit mass and its dynamical equation of motion can be written as:1

We consider low-dimensional dynamical systems exposed to a heat bath and to additional ac fields. The presence of these ac fields may lead to a breaking of certain spatial or temporal symmetries which in turn cause nonzero averages of relevant observables. Nonlinear (non)adiabatic response is employed to explain the effect. We consider a case of a particle in a periodic potential as an example and discuss the relevant symmetry breakings and the mechanisms of rectification of the current in such a system.

We have developed a general method for the description of separatrix chaos, based on the analysis of the separatrix map dynamics. Matching it with the resonant Hamiltonian analysis, we show that, for a given amplitude of perturbation, the maximum width of the chaotic layer in energy may be much larger than it was assumed before. We use the above method to explain the drastic facilitation of global chaos onset in time-periodically perturbed Hamiltonian systems possessing two or more separatrices, previously discovered [S. M. Soskin, O. M. Yevtushenko, and R. Mannella, Phys. Rev. Lett. 90, 174101 (2003)]. The theory well agrees with simulations. We also discuss generalizations and applications. The method may be generalized for single-separatrix cases. The facilitation of global chaos onset may be relevant to a variety of systems, e.g., optical lattices, magnetic and semiconductor superlattices, meandering flows in the ocean, and spinning pendulums. Apart from dynamical transport, it may facilitate noise-induced transitions and the stochastic web formation.

We develop a supersymmetric field theoretical description of the Gaussian ensemble of the almost diagonal Hermitian Random Matrices. The matrices have independent random entries H i≥j with parametrically small off-diagonal elementsWe derive a regular virial expansion of correlation functions in the number of "interacting" supermatrices associated with different sites in the real space and demonstrate that the perturbation theory constructed in this way is controlled by a small parameter B. General form of the integral expression for the m-th virial coefficient governed by the "interaction" of m supermatrices is presented and calculated explicitly in the cases of 2-and 3-matrix "interaction". The suggested technique allows us to calculate both the spectral correlations and the correlations of the eigenfunctions taken at different energies and in different space points.

Energy level statistics of Hermitian random matricesĤ with Gaussian independent random entries H i≥j is studied for a generic ensemble of almost diagonal random matrices with |H ii | 2 ∼ 1 and |H i =j | 2 = b F(|i − j|) ≪ 1. We perform a regular expansion of the spectral form-factorpowers of b ≪ 1 with the coefficients K m (τ ) that take into account interaction of (m + 1) energy levels. To calculate K m (τ ), we develop a diagrammatic technique which is based on the Trotter formula and on the combinatorial problem of graph edges coloring with (m + 1) colors. Expressions for K 1 (τ ) and K 2 (τ ) in terms of infinite series are found for a generic function F(|i − j|) in the Gaussian Orthogonal Ensemble (GOE), the Gaussian Unitary Ensemble (GUE) and in the crossover between them (the almost unitary Gaussian ensemble). The Rosenzweig-Porter and power-law banded matrix ensembles are considered as examples.

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