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

The local states and hypothetical scanning methods enable one to define a series of lower bound approximations for the free energy, $F^A$ from a sample of configurations simulated by any exact method. $F^A$ is expected to anticorrelate with its fluctuation ${\sigma }^A,$ i.e., the better (i.e., larger) is $F^A$ the smaller is ${\sigma }^A,$ where ${\sigma }^A$ 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 $F^A{\mathrm{=F}}^{\mathrm{extp}}{\mathrm{+C[\sigma }}^A{]}^{\alpha }$ where C, and α are parameters to be optimized, and $F^{\mathrm{extp}}$ is the extrapolated value of the free energy. If this function is also convex (concave down), one can obtain an upper bound denoted $F^{\mathrm{up}}.$ This is the intersection of the tangent to the function at the lowest ${\sigma }^A$ measured with the vertical axis at ${\sigma }^A\text{=0.}$ 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 $F^{\mathrm{extp}}$ and $F^{\mathrm{up}}$ are stable. In particular, extremely accurate results for the free energy and the entropy have been obtained for the Ising model.