  Anisotropy in temperature, chemical potential, or ion concentration, provides
the fuel that feeds dynamical processes that sustain life. At the same time,
anisotropy is a root cause of incurred losses manifested as entropy production.
In this work we consider a rudimentary model of an overdamped stochastic
thermodynamic system in an anisotropic temperature heat bath, and study minimum
entropy production when driving the system between thermodynamic states in
finite time. While entropy production in isotropic temperature environments can
be expressed in terms of the length (in the Wasserstein-2 metric) traversed by
the thermodynamic state of the system, anisotropy complicates substantially the
mechanism of entropy production since, besides dissipation, seepage of energy
between ambient anisotropic heat sources by way of the system dynamics is often
a major contributing factor. A key result of the paper is to show that in the
presence of anisotropy, minimization of entropy production can once again be
expressed via a modified Optimal Mass Transport (OMT) problem. However, in
contrast to the isotropic situation that leads to a classical OMT problem and a
Wasserstein length, entropy production may not be identically zero when the
thermodynamic state remains unchanged (unless one has control over
non-conservative forces); this is due to the fact that maintaining a
Non-Equilibrium Steady-State (NESS) incurs an intrinsic entropic cost that can
be traced back to a seepage of heat between heat baths. As alluded to, NESSs
represent hallmarks of life, since living matter by necessity operates far from
equilibrium. Therefore, the question studied herein, to characterize minimal
entropy production in anisotropic environments, appears of central importance
in biological processes and on how such processes may have evolved to optimize
for available usage of resources.
