Phase space density

A particularly convenient notation for trajectory bundle system can be introduced by using the classical Liouville equation which describes an ensemble of Hamiltonian trajectories by a phase space density / = f q, q, t). In textbooks of classical mechanics, e.g. [12], it is shown that Liouville s equation... 

A = A q, g) with respect to the phase-space density / is given by... 

We consider a rigid system of / mechanical degrees of freedom in thermal contact with a solvent. As in the discussion of equilibria, p q,p) is the phase space density and /( ) is the reduced distribution for the coordinates alone. Following BCAH, we also define a conditional average (A)p of an arbitrary dynamical variable A with respect to the rapid fluctuations of the momenta and solvent forces, at fixed values of the coordinates q, as... 

Evolution of Probability Densities The phase space density p q,p) evolves according to the Liouville equation... 

This formalism is the quantum-mechanical analog of the Klimontovich phase-space density formalism, and is particularly effective in the theory of long-range correlation and fluctuation in the kinetic theory of plasmas. 

Numerical calculations on Eq. (5.6) can be compared to classical mechanics by computing the classical dynamics of the phase-space density pc (x,y,px,pY), which is obtained as the solution to the Fokker-Planck equation ... 

It is an experimental fact that cross field transport in fusion edge plasmas cannot be realistically described by the classical Coulomb collision effects. Strictly then a term — q,a/maV (((f (f/Q)) resulting from turbulent fluctuations in the electric field SE and in the phase space density Sfa must be included on the right-hand side of (2.1), see [3]. 

Classical statistical mechanics is concerned with the probability distribution of phase points. In a classical microcanonical ensemble the phase space density is constant. Loosely speaking, aU phase points with the same energy are equally likely. In consequence the number of states of the classical system in a given energy range E to E + dE is proportional to the volume of the phase space shell defined by this energy range. 

Because it is not possible to reproduce on experiment with exactly the same configuration, we are not only not interested in the precise position of the atoms, we are not even interested in specific configurations, but only in characteristic ones. Although there may be differences on a microscopic scale, the behavior of a system on a macroscopic, and often also on a mesoscopic, scale will be the same. So we do not look at individual trajectories in phase space, but we average over all possible trajectories. This means that we have a phase space density p and a probabihty Pa of finding the system in configuration a. These cire related via... 

It follows trivially that dS/dt = 0 for all time in the equilibrium state. Thus, there is no purely microscopic mechanism that will give rise to entropy production, dS/dt > 0. The reason for this is that the phase space density / q accounts for all of the microstructure of phase space. In reality, we can never know this full microstructure, as was recognized by Ehrenfest, who suggested that one should work with a coarse-grained density, /eq, obtained by averaging over suitably small cells in phase space. Then one can define an entropy in analogy with Eq. [50], but in terms of /eq. In this way, entropy production can be realized microscopically, as discussed in detail in Ref. 23. Thus, the fine-grained entropy as defined in Eq. [50] always has a zero time derivative. 

Figure 12. Upper panels. Initial thermal ensemble, optimal pump and dump pulses, intermediate target. Lower panels. Snapshots of the dynamics oj obtained by propagating the ensemble corresponding to the intermediate target after the optimized pump-dump at 250 fs on the ground state showing the localization of the phase space density in the basin corresponding to isomer II [51]. See color insert.

The optimization of the pump pulse leads to a localization of the phase space density around the intermediate target. The intermediate target operator can be represented in the Wigner representation [Eq. (20)] by a minimum uncertainty wavepacket ... 

The dump pulse optimization leads to a spatial localization of the phase space density in the objective (isomer II). For this purpose, the intermediate target operator [Eq. (22)] can be propagated on the ground state, and the dump pulse is obtained from Eqs. (17) and (21). The largest eigenvalue (e.g., 0.78) can be obtained. This corresponds to 78% efficiency of localization of isomer II. The optimized dump pulse is very short ( 20 fs cf. the part of the signal after trf = 250 fs in Fig. 11). This implies that the time window around tj for... 

For the calculation using Variflex, the number of a variational transition q uantum s tates, N ej, w as given b y t he v ariationally d etermined minimum in Nej (R), as a function of the bond length along the reaction coordinate R, which was calculated by the method developed by Wardlaw-Marcus [6, 7] and Klippenstein [8]. The basis of their methods involves a separation of modes into conserved and transitional modes. With this separation, one can evaluate the number of states by Monte Carlo integration for the convolution of the sum of vibrational quantum states for the conserved modes with the classical phase space density of states for the transitional modes. 

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