Microcanonical distribution

RRKM theory assumes a microcanonical ensemble of A vibrational/rotational states within the energy interval E E + dE, so that each of these states is populated statistically with an equal probability [4]. This assumption of a microcanonical distribution means that the unimolecular rate constant for A only depends on energy, and not on the maimer in which A is energized. If N(0) is the number of A molecules excited at / =... 

Figure A3.13.9. Probability density of a microcanonical distribution of the CH cliromophore in CHF within the multiplet with cliromophore quantum nmnber V= 6 (A. g = V+ 1 = 7). Representations in configuration space of stretching and bending (Q coordinates (see text following (equation (A3.13.62)1 and figure A3.13.10). Left-hand side typical member of the microcanonical ensemble of the multiplet with V= 6...

For a microcanonical distribution of initial conditions all initial conditions have the same energy and the acceptance probability becomes... 

For Hamiltonian dynamics with a canonical or microcanonical distribution of initial conditions the acceptance probability for pathways generated with the shifting algorithm is particularly simple. Provided forward and backward shifting moves are carried out with the same probability the acceptance probability from (7.23) reduces to... 

For Newtonian dynamics and a canonical distributions of initial conditions one can reject or accept the new path before even generating the trajectory. This can be done because Newtonian dynamics conserves the energy and the canonical phase-space distribution is a function of the energy only. Therefore, the ratio plz ]/p z at time 0 is equal to the ratio p[.tj,n ]/p z ° at the shooting time and the new trajectory needs to be calculated only if accepted. For a microcanonical distribution of initial conditions all phase-space points on the energy shell have the same weight and therefore all new pathways are accepted. The same is true for Langevin dynamics with a canonical distribution of initial conditions. 

Consider a gas whose phase density in T space is represented by a microcanonical ensemble. Let it consist of molecules with //-spaces pi with probability distributions gt. Denote the element of extension in pi by fa. Since energy exchanges may occur between the molecules, pi cannot be represented by a microcanonical distribution. There must be a finite density corresponding to points of the ensemble that do not satisfy the requirement of constant energy. Nevertheless, the simultaneous probability that molecule 1 be within element di of its p-space, molecule 2 within dfa of its //-space, etc., equals the probability that the whole gas be in the element... 

It is necessary that this functional equation should be satisfied for every value of the total energy H = 2 Ht, although the constant g(H) for any given H is still described by the microcanonical distribution. 

For a system with many degrees of freedom the canonical distribution of energy closely approximates the microcanonical distribution about the most... 

Exercise. Generalize this ring distribution to r variables evenly distributed on a hypersphere in r dimensions, i.e., the microcanonical distribution of an ideal gas. Find the marginal distribution for xx. Show that it becomes Gaussian in the limit r-> oo, provided that the radius of the sphere also grows, proportionally to y/r. 

Thirdly it is easy to see that the condition that the X are independent is important. If one takes for all r variables one and the same X the result cannot be true. On the other hand, a sufficiently weak dependence does not harm. This is apparent from the calculation of the Maxwell velocity distribution from the microcanonical ensemble for an ideal gas, see the Exercise in 3. The microcanonical distribution in phase space is a joint distribution that does not factorize, but in the limit r -> oo the velocity distribution of each molecule is Gaussian. The equivalence of the various ensembles in statistical mechanics is based on this fact. 

The adaptation is such as to permit the equilibrium microcanonical distribution for the slow coordinate X to be a solution (2.3) when k(X) = 0. The SU(X) in Eq. (2.3) is the vibrational entropy change needed to reach X from... 

The introduction of certain special stationary density distributions in T-space (canonical and microcanonical distributions)... 

A) The microcanonical distribution p is everywhere zero except between the two energy surfaces E=Eo and E=E0+BE0) where BE0 is very small. Within this shell p has a constant value. This distribution of volume density p(q, p) for BE0=0 is obviously equivalent to the ergodic surface distribution (Eq. 31) of Section 10b. 

The theory then assumes that the rate of reaction is proportional to this probability, and that the microcanonical distribution is not depleted by reaction. 

This set of approximations is essentially similar to that for the more familiar canonical transition state theory, apart from the final assumption, that the state is described by a microcanonical distribution (i.e. fixed energy) rather than the Boltzmann distribution (thermal equilibrium) of canonical transition state theory. 

Rosenblum, E.l.Dashevskaya, E.E.Nikitin, and I.Oref, On the sampling of microcanonical distributions for rotating harmonic triatomic molecules, Chem. Phys. 213, 243 (1996)... 

We will first consider the fluctuation-dissipation theorem for the microcanonical distribution. The microcanonical equilibrium distribution is given as... 

The microcanonical distribution is formally written in the superstatistical form... 

However, p is a complex variable so that we cannot treat the microcanonical distribution as the superstatistical distribution. 

These phenomena lead us to a rather complicated situation. The first phenomenon reminds us of ergodicity, the realization of microcanonical distribution in systems with many degrees of freedom and the validity of statistical mechanics. We know that KAM tori cannot divide the phase space (or energy surface) for systems with many degrees of freedom, and the first phenomenon tells us that two neighborhoods in different parts of the phase space are connected not only topologically but also dynamically. In this sense the phenomenon can be considered as an elementary process of relaxation in systems with many degrees of freedom. 

As a result, a trajectory generated by the dynamics of (58) will not sample the entire phase space, but instead will sample a subspace of the entire phase space surface determined by the intersection of the hypersurfaces ylfc(x) = Cfc, where Ck is a set of constants. The microcanonical distribution function, that is generated by these systems, can be constructed from a product of 5-functions that represent these conservation laws ... 

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