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Register a new userPinball dynamics: unlimited energy growth in switching Hamiltonian systems
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A family of discontinuous symplectic maps on the cylinder is considered. This family arises naturally in the study of nonsmooth Hamiltonian dynamics and in switched Hamiltonian systems. The transformation depends on two parameters and is a canonical model for the study of bounded and unbounded behavior in discontinuous area-preserving mappings due to nonlinear resonances. This paper provides a general description of the map and points out its connection with another map considered earlier by Kesten. In one special case, an unbounded orbit is explicitly constructed.
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This paper studies homogenization of stochastic differential systems. The standard example of this phenomenon is the small mass limit of Hamiltonian systems. We consider this case first from the heuristic point of view, stressing the role of detailed balance and presenting the heuristics based on a multiscale expansion. This is used to propose a physical interpretation of recent results by the authors, as well as to motivate a new theorem proven here. Its main content is a sufficient condition, expressed in terms of solvability of an associated partial differential equation (the cell problem), under which the homogenization limit of an SDE is calculated explicitly. The general theorem is applied to a class of systems, satisfying a generalized detailed balance condition with a position-dependent temperature.
We present a procedure for averaging one-parameter random unitary groups and random self-adjoint groups. Central to this is a generalization of the notion of weak convergence of a sequence of measures and the corresponding generalization of the concept of convergence in distribution. The convergence is established in determination of the sequence of compositions of independent random transformations. When sequences of compositions of independent random transformations of the shift by the Euclidean vector in space, the results obtained coincide with the central limit theorem for the sums independent random vectors. The results are applied to the dynamics of quantum systems arising random quantization of the classical Hamiltonian system.
We study the dynamics of a class of Hamiltonian systems with dissipation, coupled to noise, in a singular (small mass) limit. We derive the homogenized equation for the position degrees of freedom in the limit, including the presence of a {em noise-induced drift} term. We prove convergence to the solution of the homogenized equation in probability and, under stronger assumptions, in an $L^p$-norm. Applications cover the overdamped limit of particle motion in a time-dependent electromagnetic field, on a manifold with time-dependent metric, and the dynamics of nuclear matter.
We consider quantum Hamiltonians of the form H(t)=H+V(t) where the spectrum of H is semibounded and discrete, and the eigenvalues behave as E_n~n^alpha, with 0<alpha<1. In particular, the gaps between successive eigenvalues decay as n^{alpha-1}. V(t) is supposed to be periodic, bounded, continuously differentiable in the strong sense and such that the matrix entries with respect to the spectral decomposition of H obey the estimate |V(t)_{m,n}|<=epsilon*|m-n|^{-p}max{m,n}^{-2gamma} for m!=n where epsilon>0, p>=1 and gamma=(1-alpha)/2. We show that the energy diffusion exponent can be arbitrarily small provided p is sufficiently large and epsilon is small enough. More precisely, for any initial condition Psiin Dom(H^{1/2}), the diffusion of energy is bounded from above as <H>_Psi(t)=O(t^sigma) where sigma=alpha/(2ceil{p-1}gamma-1/2). As an application we consider the Hamiltonian H(t)=|p|^alpha+epsilon*v(theta,t) on L^2(S^1,dtheta) which was discussed earlier in the literature by Howland.
The notion of monodromy was introduced by J. J. Duistermaat as the first obstruction to the existence of global action coordinates in integrable Hamiltonian systems. This invariant was extensively studied since then and was shown to be non-trivial in various concrete examples of finite-dimensional integrable systems. The goal of the present paper is to give a brief overview of monodromy and discuss some of its generalisations. In particular, we will discuss the monodromy around a focus-focus singularity and the notions of quantum, fractional and scattering monodromy. The exposition will be complemented with a number of examples and open problems.
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