Averages over Momentum Space

We shall often abbreviate integrals of h (r , p ), defined in the phase space of a single molecule, as follows  

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We can make the concept of the quality of a reaction coordinate q r) more precise by considering the so-called commitment probability, or conunittor. The conunit-tor pb (r) is defined as the probability that a trajectory started at configuration r with random momenta reaches state B before it reaches state A (see Fig. 10). (The commitment probability for state A is defined analogously.) The commitment probability was introduced as splitting probability already by Onsager, who used this concept to analyze ion pair recombination [262]. It has proven very useful in theoretical studies of protein folding, where the committor is known as pfou [263], and even in experimental work on liquid-solid nucleation [264]. Calculation of the probability pb r) involves a MaxweU-Boltzmann average over momentum space... 

In the Anderson picture the suppression of classical chaotic diffusion is understood as a destructive phase interference phenomenon that limits the spread of the rotor wave function over the available angular momentum space. The localization effect has no classical analogue. It is purely quantum mechanical in origin. The localization of the quantum rotor wave function in the angular momentum space can be demonstrated readily by plotting the absolute squares of the time averaged expansion amplitudes... 

The spectral function (11.9) in the weak-coupling approximation is proportional to the square of the one-hole structure amplitude (a a(q) 0), averaged over the angles of q. This amplitude is the momentum-space... 

The relation between the physical coordinate and momentum q, pq and the normal modes is known (cf. Eq. (41)), so one can express any function of the particle coordinate and momentum in terms of the time dependence of the normal modes and their initial conditions. It follows that the average of any time-dependent function of the coordinates and momenta, given initial conditions in the physical space (f[q(t), pq(t), q(0),, (0)])> can be determined by averaging over the initial conditions of the normal modes under suitable constraints. 

As a first example, we consider the momentum correlation function pq(t)pq(0)) thermally averaged over all initial configurations of the particle and the phase space of the bath. Since the correlation function is quadratic in all variables, it is convenient to... 

Furthermore, appropriate energy space averaging over Eqs. (12), derived through two-term approximation from the Boltzmann equation, yields the consistent macroscopic balance equations of the electrons. In particular, the particle and power balance can be derived from the first equation of system (12) and the momentum balance equation, normalized on the electron mass can be derived from the second equation of (12). These balance equations are... 

In Paper I, general imaginary-time correlation functions were expressed in terms of an averaging over the coordinate-space centroid density p (qj and the centroid-constrained imaginary-time-position correlation function Q(t, qj. This formalism was extended in Paper III to the phase-space centroid picture so that the momentum could be treated as an independent variable. The final result for a general imaginary-time correlation function is found to be given approximately by [5,59]... 

As mentioned previously, the constant o corresponds to some physically observable quantity such as position, momentum, kinetic energy, or total energy of the system, and it is called the expectation value. Since the expression in Equation 2-22 is being integrated over all space, the value obtained for the physically observable quantity corresponds to the average value of that quantity. This leads to the fourth postulate of quantum mechanics. 

Averaging the pore scale transport process over the REV and assigning the average properties to the centroid of the REV results in continuous functions in space of the hydrodynamic properties and state variables. As for the flow equation (1), differential calculus can be applied to establish mass and momentum balance equations for infinitesimal small soil volume and time increments. For the case of inert solute transport in a macroscopic homogeneous soil, the general continuity equation applies ... 

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