Why Is It So Difficult To Simulate Entropies, Free Energies, and Their Differences?

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Abstract
The classical 19th century thermodynamic inequalities of Clausius and Helmholtz are applied to the calculation of entropy and free energy changes by computer simulation. The irreversibility of finite-time thermodynamic paths is exploited to obtain upper and lower bounds on these quantities. Schrödinger's microscopic interpretation of heat and work provides the basis for a literal implementation of the key historical concepts on the computer using the Monte Carlo algorithm of Metropolis. Coupling schemes, paths, and reference states are variationally optimized to improve the convergence of the simulated properties, and a newly introduced variational flexibility, metric scaling, is overviewed. Reasons for expecting limiting power laws for the convergence are outlined.