Phase space distribution function

Absorption and photodissociation cross sections are calculated within the classical approach by running swarms of individual trajectories on the excited-state PES. Each trajectory contributes to the cross section with a particular weight PM (to) which represents the distribution of all coordinates and all momenta before the vertical transition from the ground to the excited electronic state. P (to) should be a state-specific, quantum mechanical distribution function which reflects, as closely as possible, the initial quantum state (indicated by the superscript i) of the parent molecule before the electronic excitation. The theory pursued in this chapter is actually a hybrid of quantum and classical mechanics the parent molecule in the electronic ground state is treated quantum mechanically while the dynamics in the dissociative state is described by classical mechanics. 

In quantum mechanics, the coordinate distribution is given by the square of the modulus of the wavefunction in coordinate-space, i(Ei) 2. 

Alternatively, we can work in momentum-space with the momentum distribution given by the square of the modulus of the momentum wavefunc-tion. However, because of Heisenberg s uncertainty relation it is impossible to specify uniquely the coordinates and the momenta simultaneously. Either the coordinates or the momenta can be defined without uncertainty. In classical mechanics, on the other hand, the coordinates as well as the momenta are simultaneously measurable at each instant. In particular, both the coordinates and the momenta must be specified at t — 0 in order to start the trajectory. Thus, we have the problem of defining a distribution function in the classical phase-space which simultaneously weights coordinates and momenta and which, at the same time, should mimic the quantum mechanical distributions as closely as possible. 

The quantum mechanical definition of a distribution function in the classical phase-space is an old theme in theoretical physics. Most frequently used is the so-called Wigner distribution function (Wigner 1932 Hillery, O Connell, Scully, and Wigner 1984). Let us consider a onedimensional system with coordinate R and corresponding classical momentum P. The Wigner distribution function is defined as 

For a harmonic oscillator in the nth vibrational state with mass m and frequency ujjo the integral in (5.10) can be evaluated analytically (e.g., 

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To be more precise, let us assume, as Boltzman first did in 1872 [boltz72], that we have N perfectly elastic billiard balls, or hard-spheres, inside a volume V, and that a complete statistical description of our system (be it a gas or fluid) at, or near, its equilibrium state is contained in the one-particle phase-space distribution function f x,v,t) ... 

Newton, the limit h —> 0 is singular. The symmetries underlying quantum and classical dynamics - unitarity and symplecticity, respectively - are fundamentally incompatible with the opposing theory s notion of a physical state quantum-mechanically, a positive semi-definite density matrix classically, a positive phase-space distribution function. 

Suppose we are given an arbitrary system Hamiltonian H(x, p) in terms of the dynamical variables x and p we will be more specific regarding the precise meaning of x and p later. The Hamiltonian is the generator of time evolution for the physical system state, provided there is no coupling to an environment or measurement device. In the classical case, we specify the initial state by a positive phase space distribution function fci(x,p) in the quantum case, by the (position-representation) positive... 

If one is interested in spectroscopy involving only the ground Born Oppenheimer surface of the liquid (which would correspond to IR and far-IR spectra), the simplest approximation involves replacing the quantum TCF by its classical counterpart. Thus pp becomes a classical variable, the trace becomes a phase-space integral, and the density operator becomes the phase-space distribution function. For light frequency co with ho > kT, this classical approximation will lead to substantial errors, and so it is important to multiply the result by a quantum correction factor the usual choice for this application is the harmonic quantum correction factor [79 84]. Thus we have... 

In C. George s method of projectors, the basic point is to distinguish between different pieces of the phase space distribution function... 

D. J. Tannor, unpublished (1984) D. Kohen, Phase-Space Distribution Function Approach to Molecular Dynamics in Solutions, Ph.D. Thesis, University of Notre Dame, 1995. 

Usually, a number of ions N is very large, and description of the system on the basis of this probability density is very inconvenient. Often, it is enough to restrict the consideration to lower order phase-space distribution functions defined as... 

As it was already written above, we would like to study structural changes in the charge distribution between macroscopic objects, that is caused by the image forces, and depends on the wall-to-wall distance. To obtain direct structural information about the system, we will introduce a configurational analogue of the phase-space distribution function. At equilibrium, the definition of an fth order distribution function given by Eq. (12) can be applied to the equilibrium probability density [Eq. (13)], and the integration with respect to impulses can easily be carried out. We write for the rth order local 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. 

The microcanonical ensemble is characterized by fixed values of the thermodynamic variables N, the total particle number, V, the volume of the system, and E, the total energy. The phase space distribution function for the micro-canonical ensemble is... 

Recall from the section on non-Hamiltonian dynamics that the phase space distribution function satisfies a generalized Liouville equation ... 

Equation [95] is capable of predicting the phase space distribution function of a system on a general manifold by incorporating a dynamical metric Vg(r, t). Recall also from prior sections that Eq. [95] must be supplemented with the equation of motion for the metric... 

An immediate use for this conserved quantity is obvious it can (and should) be used to check the NEMD code for algorithmic and programming errors. It is also possible to use the conserved energy in obtaining a knowledge of the phase space. The approach proceeds in the same fashion as presented in the section on equilibrium molecular dynamics. Let F denote the full phase space of the variables, p , q,, ri,, I. We now make the assumption of equal a priori probability for each of the microstates F with energy H. This assumption has traditionally been applied to equilibrium systems only. In the isolated system we consider, this assumption is the most obvious one to make. Thus, one can write the phase space distribution function /(F) as... 

Although we have assumed in Eq. [209] that the velocity profile in the confined fluid is linear, it is not immediately obvious that this is technically possible in the absence of moving boundary conditions. A parallel to this situation is the comparison between Nose-Hoover (NH) thermostats and Nose-Hoover chain (NHC) thermostats. Although the Nose-Hoover equations of motion can be shown to generate the canonical phase space distribution function, for a pedagogical problem like the simple harmonic oscillator (SHO), the trajectory obtained from the NH equations of motion has been found not to fill up the phase space, whereas the NHC ones do. The SHO is a stiff system and thus to make it ergodic, one needs additional degrees of freedom in the form of an NHC.2 ... 

There are two distinct contributions to the flux. The initial 3-correlated contribution, which gives rise to the transition state rate, and a retarded backflow j t) associated with third-body collisions. The temporal characteristics of the flux can be determined from the phase space distribution function R, t R 0), R(0)) which, for the inverted parabolic potential, is ... 

Let s now consider the impact of these results on the evolution of an ensemble of systems that evolve according to the equations of motion, (58), that is described by a phase space distribution function f x,t). Recall that, for a Hamiltonian system, the distribution function /(x, t) will satisfy the Liouville equation ... 

It is important to note that it is possible to construct a distribution function that satisfies the general Liouville equation, (65), while not generating the particular phase space distribution function corresponding to the given d3mamical system. To illustrate, consider a distribution function that is constructed from a product of an arbitrary subset of the -functions of the conservation laws. This distribution function would also satisfy the generalized Liouville equation and be of the form ... 

It is convenient to define the phase space distribution function... 

Phase Space, Distribution Function, Means and Moments... 

The quantum analogs of the phase space distribution function and the Lionville equation discussed in Section 1.2.2 are the density operator and the quantum Lionville equation discussed in Chapter 10. Here we mention for future reference the particularly simple results obtained for equilibrium systems of identical noninteracting particles. If the particles are distinguishable, for example, atoms attached to their lattice sites, then the canonical partitions function is, for a system of N particles... 

Here we describe an alternative derivation of the Liouville equation (1.104) for the time evolution of the phase space distribution function f (r, p t). The derivation below is based on two observations First, a change inf reflects only the change in positions and momenta of particles in the system, that is, of motion of phase points in phase space, and second, that phase points are conserved, neither created nor destroyed. 

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