Phase-space conditional probability density

Equation [16] is known as the Liouville equation and is, in fact, a statement of the conservation of the phase space probability density. Indeed, it can be seen that the Liouville equation takes the form of a continuity equation for a flow field on the phase space satisfying the incompressibility condition dIdT F = 0. Thus, given an initial phase space distribution function /(F, 0) and some appropriate boundary conditions on the phase space satisfied by f, Eq. [16] can be used to determine /(F, t) at any time t later. 

Considering an ensemble of initial conditions each representing a possible state of the system, we express the probability of a given ensemble or density distribution of system points in phase space T by a continuous function... 

The starting point of the classical description of motion is the Newton equations that yield a phase space trajectory (r (f), p (f)) for a given initial condition (r (0), p (0)). Alternatively one may describe classical motion in the framework of the Liouville equation (Section (1,2,2)) that describes the time evolution of the phase space probability density p f). For a closed system fully described in terms of a well specified initial condition, the two descriptions are completely equivalent. Probabilistic treatment becomes essential in reduced descriptions that focus on parts of an overall system, as was demonstrated in Sections 5.1-5.3 for equilibrium systems, and in Chapters 7 and 8 that focus on the time evolution of classical systems that interact with their thermal environments. 

The conditions are such that the particle is originally in a potential hole, but it may escape in the course of time by passing over a potential barrier. The analytical problem is to calculate the escape probability as a function of the temperature and of the viscosity of the medium, and then to compare the values so found with the ones of the activated state method. For sake of simplicity, Kramers studied only the one-dimensional model, and the calculation rests on the equation of diffusion obeyed by a density distribution of particles in the. phase space. Definite results can be obtained in the limiting cases of small and large viscosity, and in both cases there is a close analogy with the Cristiansen treatment of chemical reactions as a diffusion problem. When the potential barrier corresponds to a rather smooth maximum, a reliable solution is obtained for any value of the viscosity, and, within a large range of values of the viscosity, the escape probability happens to be practically equal to that computed by the activated state method. 

Imagine a distribution po(X) which we may take to be an initial macroscopic state of a stochastic differential equation system. This might be a smooth probability density such as a Gaussian, the indicator function for a small disk D in the phase space, or, in the extreme case a Dirac delta distribution (indicating that all initial conditions are clustered at a single point in phase space). The density evolves according to the partial differential equation... 

The remainder of this section is devoted to the derivation of Eq.[54]. Besides the mathematics we also define the range of applicability of simulations based on the Nernst-Planck equation. The starting point for deriving the Nernst-Planck equation is Langevin s equation (Eq. [45]). A solution of this stochastic differential equation can be obtained by finding the probability that the solution in phase space is r, v at time t, starting from an initial condition ro, Vo at time = 0. This probability is described by the probability density function p r, v, t). The basic idea is to find the phase-space probability density function that is a solution to the appropriate partial differential equation, rather than to track the individual Brownian trajectories in phase space. This last point is important, because it defines the difference between particle-based and flux-based simulation strategies. 

Figure 2. Schematic representation of the four conceptually different paths (the heavy lines) one may utilize to attack the phase-coexistence problem. Each figure depicts a configuration space spanned by two macroscopic properties (such as energy, density. ..) the contours link macrostates of equal probability, for some given conditions c (such as temperature, pressure. ..). The two mountaintops locate the equilibrium macro states associated with the two competing phases, under these conditions. They are separated by a probability ravine (free-energy barrier). In case (a) the path comprises two disjoint sections confined to each of the two phases and terminating in appropriate reference macrostates. In (b) the path skirts the ravine. In (c) it passes through the ravine. In (d) it leaps the ravine.

As an example, let us consider suspension polymerization. In suspension polymerization, the monomers from the monomer droplets transfer into the polymer particles as long as monomer droplets exist. Thus, regarding the change of monomer concentration in the space of polymerization, that is, polymer particles, this reaction can be considered a semibatch reactor from the viewpoint of the change of monomer concentration. Figure 10 depicts the calculated results of crosslink density distribution of suspension polymerization and homogeneous phase polymerization under Flory s simplified conditions [48, 49]. In Fig. 10, the abscissa indicates the reactivity (0) at ttie time of each primary polymer formation and the ordinate shows the probability of crosslink density of each primary polymer. In the figure, for example, the line with l/ = 0.6... 

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