Phase space cell

WIson K G 1971 Renormalization group and critical phenomena. II. Phase space cell analysis of critical behaviour P/rys. Rev. B 4 3184-205... 

Exercise. The neutrons in a nuclear reactor behave as free particles until they are absorbed, scatter, or cause fission and thereby produce more neutrons. The master equation for the joint probability distribution of the occupation numbers nk of the phase space cells X is... 

Fig. A.2.1 An illustration of the 2s-dimensional phase space of a system with s degrees of freedom. The solid path describes the motion of the system according to Hamilton s equations of motion. The cube illustrates a phase-space cell of volume ha that contains one state.

Substituting (2.110,2.111) into (2.109), we obtain for the spatial part of the phase space cell... 

Figure 2.34 Coherence volume and phase space cell...

Covering of Energy Cells Assume that the energy cells under consideration are compact sets and the stepsize r is fixed. We want to construct a collection B of boxes in phase space such that the union Q of these subsets is a covering of the energy cell we focus on. To this end, consider... 

Labeling each cell of the one-dimensional lattice by i G Z, where Z is the set of integers, the collection of all configurations S defines the CA phase space, and is denoted by T =. ... 

We restrict our attention for the moment to elementary one-dimensional systems, so that each cell of the lattice is indexed by an integer i G Z and takes on one phase space, L 3 is a compact metric space under the metric... 

Starting with the partition function of translation, consider a particle of mass m moving in one dimension x over a line of length I with velocity v. Its momentum Px = mVx and its kinetic energy = Pxllm. The coordinates available for the particle X, px in phase space can be divided into small cells each of size h, which is Planck s constant. Since the division is so incredibly small we can replace the sum with integration over phase space in x and Px, and so calculate the partition function. By normalizing with the size of the cell h the expression becomes... 

When thermal energy is negligible, cells in phase space are uniformly occupied up to the Fermi momentum pp, given by... 

The elements A and B therefore have equal areas. Liouville s theorem states that an element in phase space is conserved, which means that the element within which a system can be found is constant. Further, if the range Ae in the phase space is divided into equal elements, the system spends equal times passing through these elements, or cells. 

The prior distribution is often not what is observed, and there can be extreme deviations from it. [1, 3, 23] By our terminology this means that the dynamics do impose constraints on what can happen. How can one explicitly impose such constraints without a full dynamical computation At this point I appeal again to the reproducibility of the results of interest as ensured by the Monte Carlo theorem. The very reproducibility implies that much of the computed detail is not relevant to the results of interest. What one therefore seeks is the crudest possible division into cells in phase space that is consistent with the given values of the constraints. This distribution is known as one of maximal entropy". 

This function is normahzed to take the unit value for 0 = 2n. For vanishing wavenumber, the cumulative function is equal to Fk Q) = 0/(2ti), which is the cumulative function of the microcanonical uniform distribution in phase space. For nonvanishing wavenumbers, the cumulative function becomes complex. These cumulative functions typically form fractal curves in the complex plane (ReF, ImF ). Their Hausdorff dimension Du can be calculated as follows. We can decompose the phase space into cells labeled by co and represent the trajectories by the sequence m = ( o i 2 n-i of cells visited at regular time interval 0, x, 2x,..., (n — l)x. The integral over the phase-space curve in Eq. (60) can be discretized into a sum over the paths a>. The weight of each path to is... 

The phase-space region Af/ corresponding to the lattice vector / is partitioned into cells A. The probability that the system is found in the cell A at time t is given by... 

The phase space is partitioned into cells co of diameter 5. In the limit of an arbitrarily fine partition, the entropy per unit time tends to the Kolmogorov-Sinai entropy per unit time which is equal to the sum of positive Lyapunov exponents by Pesin theorem [16] ... 

For much of the discussion in this chapter, the BOA is assumed valid so that the bond making/breaking is simply described by motion of nuclei on a multidimensional ground state PES. For example, dissociation of a molecule from the gas phase is described as motion on the PES from a region of phase space where the molecule is far from the surface to one with the adsorbed atoms on the surface. Conversely, the time-reversed process of associative desorption is described as motion on the PES from a region of phase space with the adsorbed atoms on the surface to one where the intact molecule is far from the surface. For diatomic dissociation/associative desorption, this PES is given as V(Z, R, X, Y, ft, cp, < ), where Z is the distance of the diatomic to the surface, R is the distance between atoms in the molecule, X and Y are the location of the center of mass of the molecule within the surface unit cell, ft and cp are the orientation of the diatomic relative to the surface normal and represent the thermal distortions of the hh metal lattice atom... 

As an example take a gas in a cylindrical vessel. In addition to the energy there is one other constant of the motion the angular momentum around the cylinder axis. The 6A/-dimensional phase space is thereby reduced to subshells of 6N-2 dimensions. Consider a small sub volume in the vessel and let Y(t) be the number of molecules in it. According to III.2 Y(t) is a stochastic function, with range n = 0,1,2,. .., N. Each value Y = n delineates a phase cell ) one expects that Y(t) is a Markov process if the gas is sufficiently dilute and that pi is approximately a Poisson distribution if the subvolume is much smaller than the vessel. 

So far we only considered transport of particles by diffusion. As mentioned in 1 the continuous description was not strictly necessary, because diffusion can be described as jumps between cells and therefore incorporated in the multivariate master equation. Now consider particles that move freely and should therefore be described by their velocity v as well as by their position r. The cells A are six-dimensional cells in the one-particle phase space. As long as no reaction occurs v is constant but r changes continuously. As a result the probability distribution varies in a way which cannot be described as a succession of jumps but only in terms of a differential operator. Hence the continuous description is indispensable, but the method of compounding moments can again be used. 

The development of quantum theory, particularly of quantum mechanics, forced certain changes in statistical mechanics. In the development of the resulting quantum statistics, the phase space is divided into cells of volume hf. where h is the Planck constant and / is the number of degrees of freedom. In considering the permutations of the molecules, it is recognized that the interchange of two identical particles does not lead to a new state. With these two new ideas, one arrives at the Bose-Einstein statistics. These statistics must be further modified for particles, such as electrons, to which the Pauli exclusion principle applies, and the Fermi-Dirac statistics follow. 

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