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 W 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.

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A typical model for a gyrating engine consists of an inertial wheel powered by an energy source that generates an angle-dependent torque. Examples of such engines include a pendulum with an externally applied torque, Stirling engines, and the Brownian gyrating engine. Variations in the torque are averaged out by the inertia of the system to produce limit cycle oscillations. While torque generating mechanisms are also ubiquitous in the biological world, where they typically feed on chemical gradients, inertia is not a property that one naturally associates with such processes. In the present work, seeking ways to dispense of the need for inertial effects, we study an inertia-less concept where the combined effect of coupled torque-producing components averages out variations in the ambient potential and helps overcome dissipative forces to allow sustained operation for vanishingly small inertia. We exemplify this inertia-less concept through analysis of two of the aforementioned engines, the Stirling engine and the Brownian gyrating engine. An analogous principle may be sought in biomolecular processes as well as in modern-day technological engines, where for the latter, the coupled torque-producing components reduce vibrations that stem from the variability of the generated torque.