Cells in phase space

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

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". 

We must now perform the averaging in Eq. (1.4). We note that there are two sorts of terms first, those for which i = j secondly, those for which i j. We shall now show that the terms of the second sort average to zero. The reason is the statistical independence of two molecules i and j in the Boltzmann distribution. To find the average of such a term, we multiply by the fraction of all systems of the assembly in which the ith and jth molecules have the particular energies and ej, where k% and k3 are indices referring to particular cells in phase space, and sum over all states of the assembly. From Eq. (1.2) of Chap. IV, giving the fraction of all systems of the assembly in which each particular molecule, as the ith, is in a particular state, as the th, we see that this average is... 

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. 

However, electrons in solids obey Fermi statistics. Now, Boltzmann statistics makes no assertions regarding the number of particles which are permitted in the cells in phase space, but rather, it examines the probability of finding a given number of particles in the individual cells. Fermi statistics, on the other hand, asserts from the start that, because of the Pauli principle, only two particles are permitted in any cell of phase space [7]. In order to show how Fermi statistics affects the formulation of the chemical potentials of electronic defects in crystals, and how the mass action laws are thereby affected, let us take the compound silver sulfide Aga+ S as an example [15]. 

First, we must decide on a logical choice for the size of the cell in phase space. We know from Heisenberg s imcertainfy principle that we cannot specify position and momentum of a particle to a precision greater than h, AyAp h, and AzAv h therefore, the... 

The total volume of a cell in phase space is Ax Ap = h, where Ax = V, the total volume of the system. Therefore, the number of cells of phase space between E and E + dE is... 

The density of states is foimd as before, except that we have made the cells in phase space half as small to assure no more than single occupancy. Thus Equation 15.18 becomes... 

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... 

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... 

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. 

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... 

As already remarked, the idea underlying the Thomas-Fermi (TF) statistical theory is to treat the electrons around a point r in the electron cloud as though they were a completely degenerate electron gas. Then the lowest states in momentum space are all doubly occupied by electrons with opposed spins, out to the Fermi sphere radius corresponding to a maximum or Fermi momentum pt(r) at this position r. Therefore if we consider a volume dr of configuration space around r, the volume of occupied phase space is simply the product dr 47ipf(r)/3. However, we know that two electrons can occupy each cell of phase space of volume h3 and hence we may write for the number of electrons per unit volume at r,... 

Figure 22, to be compared with Figure 17, shows the probability of the state (X, y) in the central cell obtained by time averaging of the trajectories in phase space. The result of the simulation of macroscopic equations (a) and of the master equation (b) are close although the noise associated to the fluctuations tends to broaden the distribution, the most probably states (dark regions) are preserved. This conclusion is also confirmed by the histograms of P versus the variable X and corroborates further the results of Section 4.1 on the robustness of the statistical properties of chaotic systems toward thermodynamic fluctuations. 

In view of Heisenberg s uncertainty principle, there is a minimum volume /z in phase space that may be associated with a single particle. This minimum volume is called a phase cell. In the phase space of N particles, the volume of a phase cell is. Thus, is a natural unit of volume... 

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