Probability density, classical equilibrium

Such a method has recently been developed by Miller. et. al. (28). It uses short lengths of classical trajectory, calculated on an upside-down potential energy surface, to obtain a nonlocal correction to the classical (canonical) equilibrium probability density Peq(p, ) at each point then uses this corrected density to evaluate the rate constant via eq. 4. The method appears to handle the anharmonic tunneling in the reactions H+HH and D+HH fairly well (28), and can... 

Let us now briefly review the important physical concepts and quantities describing the static structure of classical liquid systems. For a system of N identical interacting particles in a volume V, in thermal equilibrium at the temperature T, we can introduce a set of configurational functions, namely, the -particle distribution functions, which provide a quantitative measure of the correlations between the positions of the particles. The normalized configuration probability density P(R) is given by... 

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. 

Fig. 2.3 Wavefunctions (A) and probability densities (B) of a barmonic oscillator. The dimensionless quantity plotted on the abscissa ( ) is the distance of the partiele from its equilibrium position divided by whoe m, is the reduced mass and v is the classical oscillation...

We consider only the equilibrium case so that the distribution of these points phase space is time-independent. In quantum statistical mechanics, we had a discrete list of possible states. In classical statistical mechanics, we have coordinates and momentum components that can range continuously. We denote the probability disttibution (probability density) for the ensemble by / and define the probability that the phase point of a randomly selected system of the ensemble will lie in the 6A -dimensional volume element d tNci pi to be... 

Classical mechanical formulas must agree with those obtained by taking the limit of quantum mechanical formulas as masses and energies become large (the correspondence limit). This limit does not affect the formula representing the equilibrium canonical probability density, so it must therefore be the same function of the energy as that of quantum statistical mechanics. For a one-component monatomic gas or liquid of N molecules without electronic excitation but with intermolecular forces, the classical energy (classical Hamiltonian function Jf) is expressed in terms of momentum components and coordinates ... 

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... 

The implimentation of quantum statistical ensemble theory applied to physically real systems presents the same problems as in the classical case. The fundamental questions of how to define macroscopic equilibrium and how to construct the density matrix remain. The ergodic theory and the hypothesis of equal a priori probabilities again serve to forge some link between the theory and working models. 

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