The Molecular Phase Space

The Molecular Phase Space.—In the last chapter, we pointed out that for a gas of N identical molecules, each of n degrees of freedom, the phase space of 2Nn dimensions could be subdivided into N subspaces, each of 2n dimensions. We shall now consider a different way of describing our assembly. We take simply a 2a-dimensional space, like one of our previous subspaces, and call it a molecular phase space, since a point in it gives information about a single molecule. This molecular phase space will be divided, according to the quantum theory, into cells of volume hn. A given quantum state, or complexion, of the whole gas of N 

Ns = 0, meaning that one molecule is in the first cell, two in the second, and none in the third. Then, as we see in Table V-l, there are three apparently different complexions leading to this same set of Ni s. In complexion o, molecule 1 is in cell 1, and 2 and 3 are in cell 2 etc. We 

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A situation that arises from the intramolecular dynamics of A and completely distinct from apparent non-RRKM behaviour is intrinsic non-RRKM behaviour [9], By this, it is meant that A has a non-random P(t) even if the internal vibrational states of A are prepared randomly. This situation arises when transitions between individual molecular vibrational/rotational states are slower than transitions leading to products. As a result, the vibrational states do not have equal dissociation probabilities. In tenns of classical phase space dynamics, slow transitions between the states occur when the reactant phase space is metrically decomposable [13,14] on the timescale of the imimolecular reaction and there is at least one bottleneck [9] in the molecular phase space other than the one defining the transition state. An intrinsic non-RRKM molecule decays non-exponentially with a time-dependent unimolecular rate constant or exponentially with a rate constant different from that of RRKM theory. 

Assemblies in the Molecular Phase Space.—When we describe a system by giving the Ni s, the numbers of molecules in each cell of the molecular phase space, we automatically avoid the difficulties described in the last section relating to the identity of molecules. We now meet immediately the distinction between the Fcrmi-Dirac, the Einstein-Bose, and the classical or Boltzmann statistics. In the Einstein-Bose statistics, the simplest form in theory wa art. up a complexion bv giving a set of Nj s, and we say that any possible set of Ni s, subject only to the obvious restriction... 

Equation (2.5) is of an interesting form being a quantity independent of G, raised to the G power, we may interpret it as a product of terms, one for each cell of the molecular phase space. Now to get the whole number of complexions for the system, we should multiply quantities like (2.5), for each group of G cells in the whole molecular phase space. Plainly this will give us something independent of the exact way we divide up the cells into groups, or independent of G, and we find... 

This holds for every transition for which Aft j 0 that is, for every collision satisfying the laws of conservation of energy and momentum. Using the notation of Sec. 3 in Chap. IV, we can let the average number of molecules in an element dx dy dz dpx dpy dpz of the molecular phase space be/m dx dy dz dpz dpy dpz. According to Chap. Ill, Sec. 3, the volume of molecular phase space associated with one cell is A3. Then we have the relation... 

As a second illustration of the use of the general formula (2.14), we take a perfect gas and consider the fluctuations of the number of molecules in a group of G cells in the molecular phase space. Two important physical problems are special cases of this. In the first place, the G cells may include all those, irrespective of momentum, which lie in a certain region of coordinate space. Then the fluctuation is that of the number of molecules in a certain volume, leading immediately to the fluctuation in density. Or in the second place, we may be considering the number of molecules striking a certain surface per second and the fluctuation of this number. In this case, the G cells include all those whose molecules will strike the surface in a second, as for example the cells contained in prisms similar to those shown in Fig. IV-2. Such a fluctuation is important in the theory of the shot effect, or the fluctuation of the number of electrons emitted thermionically from an element of surface of a heated conductor, per second we assume that the number emitted can be computed from the number striking the surface from inside the metal. 

Using Eq. (3.10), we can rewrite T In Zx and its temperature derivatives in terms of C%. First, we consider the behavior of T In Z% and its derivatives at the absolute zero. Let there be g0 cells of the lowest energy in the molecular phase space, gy of the next higher, and so on. It is customary to call these g s the a priori probabilities of the various energy levels, meaning merely the number of elementary colls which happen to have the same energy. Then we have, from Eq. (3.3),... 

The methods of Nonlinear Dynamics(8) can be applied to gain new theoretical insight into the underlying dynamics in terms of the molecular phase space structure. In particular, the existence of low order Fermi resonances between vibrationally anharmonic local (or normal) modes or between bending vibrations and rotations, can cause dramatic changes in phase space structure, manifest in the breakdown... 

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