  We present the theoretical study on non-equilibrium (NEQ) fluctuations for
diffusion dynamics in high dimensions driven by a linear drift force. We
consider a general situation in which NEQ is caused by two conditions: (i)
drift force not derivable from a potential function and (ii) diffusion matrix
not proportional to the unit matrix, implying non-identical and correlated
multi-dimensional noise. The former is a well-known NEQ source and the latter
can be realized in the presence of multiple heat reservoirs or multiple noise
sources. We develop a statistical mechanical theory based on generalized
thermodynamic quantities such as energy, work, and heat. The NEQ fluctuation
theorems are reproduced successfully. We also find the time-dependent
probability distribution function exactly as well as the NEQ work production
distribution $P({\mathcal W})$ in terms of solutions of nonlinear differential
equations. In addition, we compute low-order cumulants of the NEQ work
production explicitly. In two dimensions, we carry out numerical simulations to
check out our analytic results and also to get $P({\mathcal W})$. We find an
interesting dynamic phase transition in the exponential tail shape of
$P({\mathcal W})$, associated with a singularity found in solutions of the
nonlinear differential equation. Finally, we discuss possible realizations in
experiments.
