Solvable Small Thermal Engines

With the advent of modern microscopic and nanoscopic techniques it became possible to understand the workings and constraints of very small systems, such as biological ones.

It has become clearer in the last few years that the physics of those systems is quite different from the macroscopic ones that obey the laws of thermodynamics and equilibrium statistical mechanics.

Microscopic systems are driven by fluctuations, and are mostly in non-equilibrium states. Eventual violations of the 2nd Law of Thermodynamics become possible for a single realization of a thermodynamic process, but the 2nd Law is still valid for averages of many realizations.

In order to study these interesting problems, here we propose a minimal model consisting of a massive damped Brownian particle subjected to an harmonic and quadratic ($k_3 x^4/4$) potentials. The particle is enclosed in a box of size $L(t)$ and in contact with a heat bath of temperature $T(t)$. Having full control of the expressions for $T(t)$ and $L(t)$ allows us to write a wide range of cycles, from equilibrium to non-equilibrium.

Our goal is to obtain the exact analytical expansions in orders of $k_3$ for the averages of functionals such as work and absorbed heat. From there we can calculate the efficiency of our engines for different cycles.