Simulation of a free energy upper bound, based on the anticorrelation between an approximate free energy functional and its fluctuation

Abstract

The local states and hypothetical scanning methods enable one to define a series of lower bound approximations for the free energy, FA from a sample of configurations simulated by any exact method. FA is expected to anticorrelate with its fluctuation σA, i.e., the better (i.e., larger) is FA the smaller is σA, where σA becomes zero for the exact F. Relying on ideas proposed by Meirovitch and Alexandrowicz [J. Stat. Phys. 15, 123 (1976)] we best-fit such results to the function FA=Fextp+C[σA]α where C, and α are parameters to be optimized, and Fextp is the extrapolated value of the free energy. If this function is also convex (concave down), one can obtain an upper bound denoted Fup. This is the intersection of the tangent to the function at the lowest σA measured with the vertical axis at σA=0. We analyze such simulation data for the square Ising lattice and four polymer chain models for which the correct F values have been calculated with high precision by exact methods. For all models we have found that the expected concavity always exists and that the results for Fextp and Fup are stable. In particular, extremely accurate results for the free energy and the entropy have been obtained for the Ising model.