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The instantaneous density of representative points can be characterized by a distribution function ρ (q1...qn, p1...pn, t) abbreviated for convenience ρ(x). To calculate thermodynamic properties of macroscopic systems, in accordance to ergodic hypothesis, it is necessary to determine explicitly the distribution function ρ (x) for the appropriate ensemble. The distribution function is time –independent, so 0 (3.9) t Hence, if the probability density distribution function in a particular ensemble is ρ, the average of a property, X, can be written as an integral X dX ()() (3.10) where are the ensemble variables, which will include the coordinates and the momenta of the particles, and d indicates the volume element for a multidimensional integration over these variables. Practically, there are two main methods to simulate statistical ensembles. The first is though molecular dynamics, which solves the equations of motion, and the second is Monte Carlo simulation, in which a statistical method is used. Although the total energy appearing above is the sum of the kinetic energy and potential energy, only the latter needs to be provided as an explicit function of coordinates through a suitable intermolecular potential energy model for Monte Carlo simulation. 3.3 MONTE CARLO SIMULATION 3.3.1 Principles Consider the integral I, of a function f(x) over a region [a,b]: The thermodynamic X ...dq1...dpn(q, p,t)X(q, p) properties estimating by the 1 ensembleaverageas andthetimeaverageas X( q,p) dtXq(t),p(t) 0 38PDF Image | MOLECULAR SIMULATION STUDIES IN THE SUPERCRITICAL REGION
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