Entropic anomaly and maximal efficiency of microscopic heat engines

The efficiency of microscopic heat engines in a thermally heterogenous environment is considered. We show that, as a consequence of the recently discovered entropic anomaly, quasi-static engines, whose efficiency is maximal in a fluid at uniform temperature, have in fact vanishing efficiency in presence of temperature gradients. For slow cycles the efficiency falls off as the inverse of the period. The maximum efficiency is reached at a finite value of the cycle period that is inversely proportional to the square root of the gradient intensity. The relative loss in maximal efficiency with respect to the thermally homogeneous case grows as the square root of the gradient. As an illustration of these general results, we construct an explicit, analytically solvable example of a Carnot stochastic engine. In this thought experiment, a Brownian particle is confined by a harmonic trap and immersed in a fluid with a linear temperature profile. This example may serve as a template for the design of real experiments in which the effect of the entropic anomaly can be measured.