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particles. This is essentially equivalent to the system’s total energy, kinetic plus potential, and can be written as N12 H(p,r) p V(r)KUE ii (3.4) ii i1 2mi Note that the Hamiltonian is a function of 6N independent variables, the 3N particle momenta and the 3N particle positions. Integration of these equations of motion yields the trajectory xt =xt (x0 ) (3.5) where x0 specifies the state of the system at time t=0. Since the laws of classical mechanics are deterministic, the subsequent trajectory from a given point is uniquely determined. Therefore, a trajectory in phase space cannot intersect itself. 3.2.2 The concept of the ensemble The variables N, V and E are sufficient to specify a thermodynamic system macroscopically, but insufficient to determine the 2n-time dependent variables in order to specify the microscopic state. Moreover, the microscopic mechanical variables can be modified continuously in such way as to leave the macroscopic state unaltered. It must exist an infinite number of microscopic states, which are compatible with the macroscopic specification of a thermodynamic system. Gibbs (McQuarrie and Simon, 1999) denoted as an ‘ensemble’ a sufficiently representative set of microscopic states corresponding to a specified macroscopic state. For conceptual purposes an ensemble can be defined as a very large number (sometimes taken to the infinity) of systems, each being a replica on a macroscopic scale of a given thermodynamic system. It is useful to discuss a simple example before discuss further the concept of the ensemble. Consider using a thermocouple (i.e. a digital thermometer) to measure 33PDF Image | MOLECULAR SIMULATION STUDIES IN THE SUPERCRITICAL REGION
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