Phase space probability density

From Eq. (152), the static or time-reversible phase space probability density is... 

The state of the entire system at time t is described by the /V-particle phase space probability density function, P(x/V, t). In MPC dynamics the time evolution of this function is given by the Markov chain,... 

Multiparticle collisions are carried out at time intervals x as described earlier. We can write the equation of motion for the phase space probability density function as a simple generalization of Eq. (15) by replacing the free-streaming operator with streaming in the intermolecular potential. We find... 

For a rather diluted system, the interaction potential U can be represented as a combination of the one-body and pair potentials given by Eqs. (8) and (9). Further, we introduce a phase-space probability density /( r, p v 0- This function describes the probability, with which the system acquires an ensemble configuration, where each ion occupies a point given by the unique combination (r, p,- in the phase-space. Knowing the ensemble configuration, the average value of any function of coordinates and impulses can be calculated [13,14]. 

If the system under consideration is at the equilibrium, the phase-space probability densities (12) become then independent of time, and can be factorized as... 

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. 

The time evolution of this phase space probability density is governed by the Liouville equation expressed as... 

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. 

This chapter deals with the analogous quantum mechanical problem. Within the limitations imposed by its nature as expressed, for example, by Heisenberg-type uncertainty principles, the Schrodinger equation is deterministic. Obviously it describes a deterministic evolution of the quantum mechanical wavefunction. The analog of the phase space probability density f (r, p f) is now the quantum mechanical density operator (often referred to as the density matrix ), whose time evolution is detennined by the quantum Liouville equation. Again, when the system is fully described in terms of a well specified initial wavefunction, the two descriptions are equivalent. The density operator formalism can, however, be carried over to situations where the initial state of the system is not well characterized and/or... 

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. 

Let us try to derive an equation for the evolution in time of the phase space probability density p(p,r, t) for a random process generated by... 

The method proposed by Berendsen is much simpler and easier to program than that proposed by Nose and Hoover. It suffers, however, from the fact that the phase-space probability density it defines does not conform to a specific statistical ensemble (e.g., NVT, NPT). Consequently, there exists no Hamiltonian that should be conserved during the MD simulation. 

From Eq. (96), the real-system phase-space probability density for the Nosd thermostat (virtual-time sampling) can be written... 

Using these results and the definition of T, it is easily shown that dp/dt = 0 in Eq. (Ill) provided that g = Ndf. This shows that the extended-system phase-space density pe,r(i , p, y) is a stationary (equilibrium) solution of Eq. (111) corresponding to the Nose-Hoover equations of motion. Integrating out the y variable leads to the real-system phase-space probability density for the Nosd-Hoover thermostat (realtime sampling)... 

All results presented so far can be put in one overarching formalism, that of the Fokker-Planck equation. This equation governs the probability of finding a system in a particular area of phase space. The phase space probability density is denoted by p( r), [p], t), which is a function of the positions (r) and momenta (p) of the particles (or modes) involved in the barrier transition, and of the particles (modes) coupled to this barrier transition process. The equation describes classical Newtonian dynamics, but also incorporates damped, or decaying, motion to a final equilibrium state. It is based on a mesoscopic picture of the environment... 

We define the phase space probability density p(, writing... 

The question we will answer is the following how are these systems distributed in phase space More precisely, we will determine the phase space probability density, pnve(K)-... 

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