Bogoliubov-Born-Green-Kirkwood-Yvon chain and kinetic equations for the level dynamics in an externally perturbed quantum system

Theoretical description and simulation of large quantum coherent systems out of equilibrium remains a daunting task. Here we are developing an approach to it based on the Pechukas-Yukawa formalism, which is especially convenient in the case of an adiabatically slow external perturbation, though it is not restricted to adiabatic systems. In this formalism the dynamics of energy levels in an externally perturbed quantum system as a function of the perturbation parameter is mapped on that of a fictitious one-dimensional classical gas of particles with cubic repulsion. Equilibrium statistical mechanics of this Pechukas gas allows us to reproduce the random matrix theory of energy levels. In the present work, we develop the nonequilibrium statistical mechanics of the Pechukas gas, starting with the derivation of the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) chain of equations for the appropriate generalized distribution functions. Sets of approximate kinetic equations can be consistently obtained by breaking this chain at a particular point (i.e., approximating all higher-order distribution functions by the products of the lower-order ones). When complemented by the equations for the level occupation numbers and interlevel transition amplitudes, they allow us to describe the nonequilibrium evolution of the quantum state of the system, which can describe better a large quantum coherent system than the currently used approaches. In particular, we find that corrections to the factorized approximation of the distribution function scale as $1/N$, where $N$ is the number of the “Pechukas gas particles” (i.e., energy levels in the system).

Figure 1

Evolution of eigenvalues: all the eigenvalues of Hamiltonian (18) for 100 simulations with random initial conditions obtained from the different values of $J$. These eigenvalues are of the form $J+\lambda H_n$, they are Gaussian distributed as $J$ is Gaussian distributed with mean 0 and standard deviation 1, through their evolution in $\lambda$ from 0 to 1 in steps of 0.1. When the perturbation is much weaker than the interaction $J$, the system stays close to its ground state. When the perturbation is of the same order as $J$, the eigenvalues deviate from an initially Gaussian distribution, evolving into four distinct peaks.

Figure 2

One-particle distributions: the evolution of $F_{1,0}\left(x_1,v_1\right)$ as $\lambda$ increases. Initially Gaussian distributed about a single peak, as $\lambda$ increases, it settles into four equally distributed peaks due to the large perturbation. The velocities are deterministic so the distribution is centered around the four velocity points.