Calculation of microcanonical entropy differences from configurational averages

A simple expression is derived, enabling the calculation of the entropy difference between two microcanonical equilibrium states at different energies in atomistic computer simulations. This expression only requires potential energy samples from molecular dynamics or Monte Carlo simulations at the relevant energies. This presents an alternative to switching methods such as thermodynamic integration or nonequilibrium work relations, as well as flat-histogram random walks, all of which involve sampling in between the relevant states. The method is especially suited for small (nanoscopic) systems such as clusters and proteins, and is applicable to first-principles data directly.

Figure 1

Entropy difference $\Delta S$ vs internal energy $E$ for a highly densified 108-atom Lennard-Jones system. Circles were calculated using the $\sigma$ method presented in this work, and the solid line using integration of the inverse temperature. From left to right, the arrows indicate the energies required to obtain solid and liquid at the melting temperature $T_m$, and the dashed lines indicate the entropies measured with the $\sigma$ method at those energies.

Figure 2

Probability density functions for $-\ln \sigma (\mathbf{r};E_0)$ at (a) $E=150$ eV, $E_0=110$ eV and (b) $E=330$ eV, $E_0=210$ eV. In each case, the solid blue rectangle indicates the 95$\%$ confidence interval, while the solid red line indicates the expectation value $-\langle \ln \sigma \rangle$.

Figure 3

Estimated entropy difference $\Delta S$ between $E=170$ eV and $E^{\prime }=180$ eV, as a function of the reference energy $E_0$ employed.

Figure 4

Probability density functions (PDFs) for the potential energy $\Phi$ at $E=170$ eV and $E^{\prime }=180$ eV. Vertical dashed lines represent the lower (1.02 eV/atom) and upper (1.18 eV/atom) limits for optimal averages in Fig. 3.