  An approach to non-adiabatic dynamics of atoms in molecular and condensed
matter systems under general non-equilibrium conditions is proposed. In this
method interaction between nuclei and electrons is considered explicitly up to
the second order in atomic displacements defined with respect to the mean
atomic trajectory. This method enables one to consider movement of atoms beyond
their simple vibrations. Both electrons and nuclei are treated fully
quantum-mechanically using a combination of path integrals applied to nuclei
and non-equilibrium Green's functions (NEGF) to elections. Our method is
partition-less: initially, the entire system is coupled and assumed to be at
thermal equilibrium. Then, the exact application of the Hubbard-Stratanovich
transformation in mixed real and imaginary times enables us to obtain, without
doing any additional approximations, an exact expression for the reduced
density matrix for nuclei and hence an effective quantum Liouville equation for
them, both containing Gaussian noises. It is shown that the time evolution of
the expectation values for atomic positions is described by an infinite
hierarchy of stochastic differential equations for atomic positions and momenta
and their various fluctuations. The actual dynamics is obtained by sampling all
stochastic trajectories. It is expected that applications of the method may
include photo-induced chemical reactions (e.g. dissociation), electromigration,
atomic manipulation in scanning tunneling microscopy, to name just a few.
