Chaos and Energy Spreading for Time-Dependent Hamiltonians, and the Various Regimes in the Theory of Quantum Dissipation

Abstract

We make the first steps toward a generic theory for energy spreading and quantum dissipation. The Wall formula for the calculation of friction in nuclear physics and the Drude formula for the calculation of conductivity in mesoscopic physics can be regarded as two special results of the general formulation. We assume a time-dependent Hamiltonian H(Q, P; x(t)) with x(t)=Vt, where V is slow in a classical sense. The rate-of-change V is not necessarily slow in the quantum-mechanical sense. The dynamical variables (Q, P) may represent some``bath'' which is being parametrically driven by x. This bath may consist of just a few degrees of freedom, but it is assumed to be classically chaotic. In the case of either the Wall or Drude formula, the dynamical variables (Q, P) may represent a single particle. In any case, dissipation means an irreversible systematic growth of the (average) energy. It is associated with the stochastic spreading of energy across levels. The latter can be characterized by a transition probability kernel P t (n | m), where n and m are level indices. This kernel is the main object of the present study. In the classical limit, due to the (assumed) chaotic nature of the dynamics, the second moment of P t (n | m) exhibits a crossover from ballistic to diffusive behavior. In order to capture this crossover within quantum mechanics, a proper theory for the quantal P t (n | m) should be constructed. We define the V regimes where either perturbation theory or semiclassical considerations are applicable in order to establish this crossover. In the limit Ä 0 perturbation theory does not apply but semiclassical considerations can be used in order to argue that there is detailed correspondence, during the crossover time, between the quantal and the classical P t (n | m). In the perturbative regime there is a lack of such correspondence. Namely, P t (n | m) is characterized by a perturbative core-tail structure that persists during the crossover time. In spite of this lack of (detailed) correspondence there may be still a restricted correspondence as far as the second moment is concerned. Such restricted correspondence is essential in order to establish the universal fluctuation-dissipation relation.

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

“…Step (i): Our presentation is loosely based on Cohen [20]; we emphasize that this is not a mathematical proof, rather one form of a heuristic common in physics literature (cf. [54,39,31,24]).…”

Stadium billiard with moving walls

“…(11), then perturbation theory becomes useless [3,12]. In such case P t (n|m) becomes purely non-perturbative.…”

“…If we adhere to the present (more physical) approach of using x-dependent basis, an equivalent method [12] is to determine the sudden time t sdn . This is the time to resolve the expanding "core".…”

“…The quantum mechanical derivation of this formula is much more subtle [3,10,11,12] and leads to the distinction between "adiabatic" and "(extended) perturbative" and "non-perturbative" regimes. The semiclassical limit is contained in the latter regime.…”

“…The semiclassical limit is contained in the latter regime. It is known that the LRT formula does not hold in the adiabatic regime [20], but it is valid in the (extended) perturbative regime [10,12]. It is not necessarily valid in the non-perturbative regime [10,11], but if the system has a classical limit, it must be valid again in the semiclassical regime.…”