Anisotropy in temperature fields, chemical potentials and ion concentration gradients provide 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 highlight the trade-off between energy extraction and entropic losses by considering a rudimentary model of an overdamped stochastic thermodynamic system in an anisotropic temperature heat bath. We analyze central problems in Stochastic Thermodynamics such as that of maximizing work output or that of minimizing entropy production, while driving the system between thermodynamic states in finite time.

Specifically, we first focus our interest on maximizing work output while driving the overdamped system over a cycle by periodic potential control. We show that path-lengths traversed in the manifold of thermodynamic states, measured in a suitableRiemannian metric (the Wasserstein-2 metric), represent dissipative energy losses, while area integrals of a work-density quantify work being extracted. Thus, the maximal amount of work that can be extracted relates to an isoperimetric problem, trading off area against length of an encircling path. We also derive an isoperimetric inequality that provides a universal bound on the efficiency of all cyclic operating protocols, and a bound on how fast a closed path can be traversed before it becomes impossible to extract positive work.

While entropy production in \emph{isotropic} temperature environments can also 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. We 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 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.

Finally, inspired by nature's ability to harvest energy from fluctuations and anisotropic chemical concentrations in conjunction with varying electrochemical potentials, we introduce an ``engine concept'', based on our previous results, that \emph{autonomously} extracts energy from an anisotropic temperature field. We explore the coupling between (fast) thermal fluctuations and a (slow) inertial mechanical component in a way that allows generation of mechanical power, and explain the mechanisms responsible for this energy transfer. We show that one can dispense with this inertia, often uncharacteristic of biological systems, by coupling multiple subsystems, while still ensuring stability of limit cycles.

As alluded to, anisotropic environments represent hallmarks of life, since living matter necessitates consuming fuel to operate far from equilibrium. Therefore, the questions addressed herein, aimed at characterizing maximal work output and minimal entropy production in anisotropic environments, appear of central importance in biological processes and on how such processes may have evolved to optimize for available usage of resources.