Methods for Calculating the Entropy from MC and MD Samples

Direct Methods for Calculating the Entropy from MC and MD Samples 

Two methods are described here the hypothetical scanning method and the local states (LS) method. Both were developed originally for spin systems and are discussed in a later section, but for simplicity we illustrate how they have been applied to a model of SAWs on a square lattice. These methods enable one to extract the approximate entropy from a sample simulated by any technique, in particular by the MC or the MD procedures. They are based on the concept that two samples in equilibrium generated by different simulation methods are equivalent in the sense that they lead to the same estimates (within statistical error) of average properties, such as the entropy, energy, and their fluctuations.  

S f) s an upper bound for the exact entropy S (see Ref. 38) and can be improved (i.e., decreased) systematically by increasing f. Another criterion for the improvement of S (f) is the decrease of its fluctuation r[S (/ )] as f is increased. For SAWs, the entropy S plays the role of the free energy in a system with finite interactions therefore the fluctuation of the exact S is zero (see discussion following Eq. [12]), which is also demonstrated by the fact that Pf is equal for all the SAWs. On the other hand, r(S ) is larger than zero but is expected to decrease as the approximation improves, which means that the sample size required for estimating with a given precision decreases as well. Indeed, such a decrease of a[S (/ )] has been found in all systems studied thus far.38,265,266 estimated from the given sample by (see Eq. [29]), 

One can define another approximation for the absolute entropy denoted S (/ ) (for details see Ref. 38), estimated by S (/ )  

Estimation of S is more difficult than that of S also, it can be shown rigorously that S S. However, for all the systems studied, where S could be estimated reliably, it has been found that S S, and the absolute value of the deviations of and S from S are approximately equal. This is an important result because the average value of and S , denoted by becomes a better approximation than either of them individually. 

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