  In this work, we establish formally exact stochastic equations of motion
(SEOM) theory to describe the dissipative dynamics of fermionic open systems.
The construction of the SEOM is based on a stochastic decoupling of the
dissipative interaction between the system and fermionic environment, and the
influence of environmental fluctuations on the reduced system dynamics is
characterized by stochastic Grassmann fields. Meanwhile, numerical realization
of the time-dependent Grassmann fields has remained a long-standing challenge.
To solve this problem, we propose a minimal auxiliary space (MAS) mapping
scheme, with which the stochastic Grassmann fields are represented by
conventional c-number fields along with a set of pseudo-levels. This eventually
leads to a numerically feasible MAS-SEOM method. The important properties of
the MAS-SEOM are analyzed by making connection to the well-established
time-dependent perturbation theory and the hierarchical equations of motion
(HEOM) theory. The MAS-SEOM method provides a potentially promising approach
for accurate and efficient simulation of fermionic open systems at ultra-low
temperatures.

Statistical Mechanics