Maximizing free energy gain

Maximizing the amount of free energy that a system extracts from its environment is important for a wide variety of physical, biological and technological processes, from energy harvesting processes such as photosynthesis to energy storage systems such as fuels and batteries. We extend recent results from non-equilibrium thermodynamics to derive closed-form expressions for the maximum amount of free energy that a system can extract from its environment over the course of a fixed process. We also analyze how our bounds on extractable free energy vary with the initial distribution of the states of the system. Simple equations allow us to compare the amount of free energy that can be extracted under the optimal initial distribution with that for a sub-optimal initial distribution. We show that the problem of finding that optimal initial distribution is convex and solvable via gradient descent. We demonstrate our results by analyzing how the amount of extractable free energy varies with the initial distribution of a simple Szilard engine.