  Theoretical description and simulation of large quantum coherent systems out
of equilibrium remains a daunting task. Here we are developing a new approach
to it based on the Pechukas-Yukawa formalism, which is especially convenient in
case of an adiabatically slow external perturbation. 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 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 inter-level transition
amplitudes, they allow 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).
