Hidden Nambu mechanics II: Quantum/semiclassical dynamics

Nambu mechanics is a generalized Hamiltonian dynamics characterized by an extended phase space and multiple Hamiltonians. In a previous paper [Prog. Theor. Exp. Phys. 2013, 073A01 (2013)] we revealed that the Nambu mechanical structure is hidden in Hamiltonian dynamics, that is, the classical time evolution of variables including redundant degrees of freedom can be formulated as Nambu mechanics. In the present paper we show that the Nambu mechanical structure is also hidden in some quantum or semiclassical dynamics, that is, in some cases the quantum or semiclassical time evolution of expectation values of quantum mechanical operators, including composite operators, can be formulated as Nambu mechanics. We present a procedure to find hidden Nambu structures in quantum/semiclassical systems of one degree of freedom, and give two examples: the exact quantum dynamics of a harmonic oscillator, and semiclassical wave packet dynamics. Our formalism can be extended to many-degrees-of-freedom systems; however, there is a serious difficulty in this case due to interactions between degrees of freedom. To illustrate our formalism we present two sets of numerical results on semiclassical dynamics: from a one-dimensional metastable potential model and a simplified Henon–Heiles model of two interacting oscillators.

Mean escape time of ballistically transporting Brownian particles due to thermal noise

The mean escape time of Brownian particles from a metastable potential well is investigated under the influence of a thermal time derivative Ornstein–Uhlenbeck noise, which can induce ballistic diffusion of force-free Brownian particles. Compared with the usual Ornstein–Uhlenbeck noise, some new characteristics are obtained. The Brownian particles escape fast remarkably for a larger correlation time of the Ornstein–Uhlenbeck noise. The mean escape time increases as the correlation time decreases, which is contrary to the Ornstein–Uhlenbeck noise case. The escape time for derivative Ornstein–Uhlenbeck noise is longer than that of Ornstein–Uhlenbeck noise for small strength of noise and small correlation time of Ornstein–Uhlenbeck noise. These features are explained by the spectrum features of derivative Ornstein–Uhlenbeck noise and the corresponding dynamical effects.

IMPLICATION OF BARRIER FLUCTUATIONS ON THE RATE OF WEAKLY ADIABATIC ELECTRON TRANSFER

The problem of escape of a Brownian particle in a cusp-shaped metastable potential is of special importance in nonadiabatic and weakly-adiabatic rate theory for electron transfer (ET) reactions. For the weakly-adiabatic reactions, the reaction follows an adiabaticity criterion in the presence of a sharp barrier. In contrast to the nonadiabatic case, the ET kinetics can be, however considerably influenced by the medium dynamics. In this paper, the problem of the escape time over a dichotomously fluctuating cusp barrier is discussed with its relevance to the high temperature ET reactions in condensed media.

THE KRAMERS TURNOVER PROBLEM: RECROSSING EFFECTS AND THE ENERGY DIFFUSION DESCRIPTION OF THERMAL ACTIVATION

Kramers' Brownian motion model for escape from a metastable potential well is reconsidered in terms of the particle's energy and the action variable near the peak of the barrier. The pertinent phase space density ρ(ε, s) is uniquely determined (i) by means of a spectral analysis and (ii) upon specifying the energy distribution of (re-)entering particles. The ensuing decay rate Γ goes to zero in the low as well as in the high friction limit according to Kramers' original formulae. The nature of the intermediate turnover regime is critically discussed — and a comparison with related recent work by Büttiker, Harris and Landauer, Mel'nikov and Meshkov, and Grabert is made — while a problem with the underlying density is pointed out.