Phase-space probability

The problems with the adiabatic Yamada-Kawasaki distribution and its thermostatted versions can be avoided by developing a nonequilibrium phase space probability distribution for the present case of mechanical work that is analogous to the one developed in Section IVA for thermodynamic fluxes due to imposed thermodynamic gradients. The odd work is required. To obtain this, one extends the work path into the future by making it even about t ... 

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

The availability of a phase space probability distribution for the steady state means that it is possible to develop a Monte Carlo algorithm for the computer simulation of nonequilibrium systems. The Monte Carlo algorithm that has been developed and applied to heat flow [5] is outlined in this section, following a brief description of the system geometry and atomic potential. 

For nonequilibrium statistical mechanics, the present development of a phase space probability distribution that properly accounts for exchange with a reservoir, thermal or otherwise, is a significant advance. In the linear limit the probability distribution yielded the Green-Kubo theory. From the computational point of view, the nonequilibrium phase space probability distribution provided the basis for the first nonequilibrium Monte Carlo algorithm, and this proved to be not just feasible but actually efficient. Monte Carlo procedures are inherently more mathematically flexible than molecular dynamics, and the development of such a nonequilibrium algorithm opens up many, previously intractable, systems for study. The transition probabilities that form part of the theory likewise include the influence of the reservoir, and they should provide a fecund basis for future theoretical research. The application of the theory to molecular-level problems answers one of the two questions posed in the first paragraph of this conclusion the nonequilibrium Second Law does indeed provide a quantitative basis for the detailed analysis of nonequilibrium problems. 

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... 

For = 380 V/cm Fig. 7.5(c) shows that the starting action / = 36 is in the middle of a chaotic sea. Practically all the classical phase-space probability started out at / = 36 is now free to diffuse to the continuum, resulting in large ionization. This result, again, is consistent with the experimental data. 

Fig. 8.3. Powerlaw decay of the phase-space probability for the kicked hydrogen atom.

Prom the physical point of view the absence of stable islands means that all the phase-space probability eventually ionizes. Since all the atomic physics systems investigated to date possess a mixed phase space that shows regular islands embedded in a chaotic sea, the absence of stable islands in the kicked hydrogen atom is a very unique property. 

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... 

These normal modes evolve independently of each other. Their classical equations of motion are Uk = —co Uk, whose general solution is given by Eqs (6.81). This bath is assumed to remain in thermal equilibrimn at all times, implying the phase space probability distribution (6.77), the thermal averages (6.78), and equilibrium time correlation functions such as (6.82). The quantum analogs of these relationships were discussed in Section 6.5.3. 

How do we calculate the probability of a fluctuation about an equilibrium state Consider a system characterized by a classical Hamiltonian H r, p ) where p and denote the momenta and positions of all particles. The phase space probability distribution isf (r, p ) = Q exp(—/i22(r, p )), where Q is the canonical partition function. 

The need to include quantum mechanical effects in reaction rate constants was realized early in the development of rate theories. Wigner [8] considered the lowest order terms in an -expansion of the phase-space probability distribution function around the saddle point, resulting in a separable approximation, in which bound modes are quantized and a correction is included for quantum motion along the reaction coordinate - the so-called Wigner tunneling correction. This separable approximation was adopted in the standard ad hoc procedure for quan-... 

As a consequence, within statistical mechanics a macroscopic state F (f) is characterized by its phase space probability distribution Pr (o( 5P) defined so that PT (t) phase points near q, p. 

To start, we note that in analogy to Eq. (A.2), the phase space probability distribution for state Fp is given by... 

Statistical mechanics forms the foundation of the methodological developments of the free energy difference techniques, providing the link between macroscopic, measurable quantities of chemical systems, and the detailed, microscopic description of the molecular system. The thermodynamic quantities of interest are expressed in terms of ensemble averages, phase space probabilities or partition functions, all of which eventually are determined by the system Hamiltonian. The main difficulty in practical calculations does not lie in... 

The normalized phase space probability Tt(p, q ) gives the probability of finding the system at a particular point in phase space defined by the coordinates q and momenta p. This probability is proportional to the Boltzmann factor and determines the distribution of states of the system at equilibrium. 

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. 

W is one quantum analogue of a classical phase space probability distribution function. One can, in fact, formulate quantum mechanics in terms of W instead of /. Now W(R,P,t=0) shown in Fig. 2b appears as a two dimensional Gaussian centred on R=6.75 a.u. and P = 0 a.u. It contains little information about the underlying classical phase space structure (Fig. 2a) except for the fact it is peaked where, classically, one has a significant area of quasibound motion. One sees no effects of the resonance islands, for example, because the phase space area occupied by these islands is less than Planck s constant h = 2tc a.u.. 

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. 

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