Ensemble Dependence of Fluctuations with Application to Machine Computations

The standard theory of fluctuations in thermodynamic variables in various ensembles is generalized to nonthermodynamic variables: e.g., the mean-square fluctuations of the kinetic energy $K$ in a classical microcanonical ensemble at fixed energy $E$ is given, for large systems, by $\frac{〈{\left(\delta K\right)}^2〉}{〈K〉=T\left[\frac{1-3}{2C}\right)}$, where $T$ is the temperature (corresponding to the energy $E$) and $C$ is the specific heat per particle (in units of Boltzmann's constant). The general results may be expressed in terms of the asymptotic behavior of the Ursell functions in various ensembles. Applications are made to molecular dynamic computations where time averages correspond (via ergodicity) to phase averages in an ensemble with fixed energy and momentum. The results are also useful for time-dependent correlations.