Phase-space integration issues

After the system has reached equilibrium, in order to compute the canonical ensemble average of any phase space function, one may continue integrating the trajectories using the constant temperature algorithm. Details about technical issues involved in this calculation can be found in the standard texts mentioned above. Here we briefly discuss the computation methodology of properties that are relevant to simulating interfacial systems. 

A probability distribution is assumed to be characterized by a measure on the state space of the problem which is the integral of a probability density function deflned on the phase space of the problem that assigns nonnegative real numbers to each state. We outline the technical setting here. To gain a fuller understanding of the mathematical issues involved in this chapter, it may be helpful to refer to an introductory text on analysis such as [26,317]. 

When considering stochastic differential equations, there is an additional issue, best brought out by directly considering the E-field as a fluctuating quantity. Even for classical systems - replacing the commutator by a Poisson bracket, and considering p as phase space density - there are several interpretations possible of the equations and how to integrate them [32]. 

In this section, a survey of the basic elements of the finite volume method, as applied to single phase flows, is provided [141, 201, 202, 49, 158]. The numerical issues considered are the approximations of surface and volume integrals, time discretizations, and space discretization of diffusive and convective (or advective) terms. 

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