## ABSTRACT

This chapter describes an approach of Meirovitch’s group for extracting the absolute entropy, S from a single sample obtained by Monte Carlo (MC), molecular dynamics (MD), or any other exact method. This approach is based on the recognition that two samples generated by different exact simulation techniques are equivalent in the sense that they lead to the same averages and fluctuations. Therefore, one can assume that a given MC sample of SAWs, for example, has rather been generated by the scanning method (SM) (Chapter 14), which allows reconstructing for each chain the transition probabilities (TPs) that hypothetically were used by SM to construct it. Since SM is approximate, one obtains an upper bound, S
^{A}(f), and the method is called “hypothetical scanning” (HS). One can define the stochastic HS, where each TP at step k is calculated by an additional MC simulation (of n-size sample) applied to the f future steps, or to the entire future of N – k + 1 steps, leading to HSMC and complete HSMC, respectively. One can define several lower bounds, S
^{B}
^{i}
, and the corresponding averages, S
^{M}
^{i}
= (S
^{A} + S
^{B}
^{i}
)/2, which are expected to be better approximations than S
^{A} and S
^{B}
^{i}
individually. Thus, increasing f and n, leads to converging results for S. This methodology has been applied to peptides, Ising models, a Lennard-Jones fluid, and TIP3P water. Combining HSMD with TI creates HSMD-TI, which allows calculating the free energy of a peptide and the water around. This approach (unlike the “quasi-harmonic approximation”) is applicable to any chain flexibility, ranging from harmonic microstates to the random coil state. Another method is the “local states” (LS) method, where the TPs are calculated from the frequencies of occurrence of certain local states. LS was applied to Ising models, peptides, and fluid dynamics.