Microcanonical-ensemble statistical

The above realization of the abstract mesoscopic equilibrium thermodynamics is called a Canonical-Ensemble Statistical Mechanics. We shall now briefly present also another realization, called a Microcanonical-Ensemble Statistical Mechanics since it offers a useful physical interpretation of entropy. 

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 / =... 

If all the resonance states which fomi a microcanonical ensemble have random i, and are thus intrinsically unassignable, a situation arises which is caWtA. statistical state-specific behaviour [95]. Since the wavefunction coefficients of the i / are Gaussian random variables when projected onto (]). basis fiinctions for any zero-order representation [96], the distribution of the state-specific rate constants will be as statistical as possible. If these within the energy interval E E+ AE fomi a conthuious distribution, Levine [97] has argued that the probability of a particular k is given by the Porter-Thomas [98] distribution... 

This section deals with the question of how to approximate the essential features of the flow for given energy E. Recall that the flow conserves energy, i.e., it maps the energy surface Pq E) = x e P H x) = E onto itself. In the language of statistical physics, we want to approximate the microcanonical ensemble. However, even for a symplectic discretization, the discrete flow / = (i/i ) does not conserve energy exactly, but only on... 

When g = 1 the extensivity of the entropy can be used to derive the Boltzmann entropy equation 5 = fc In W in the microcanonical ensemble. When g 1, it is the odd property that the generalization of the entropy Sq is not extensive that leads to the peculiar form of the probability distribution. The non-extensivity of Sq has led to speculation that Tsallis statistics may be applicable to gravitational systems where interaction length scales comparable to the system size violate the assumptions underlying Gibbs-Boltzmann statistics. [4]... 

Thus, in equilibrium the system spends on the average equal times in each of the Q. states. The calculation of the time average can therefore be replaced by averaging over the quantum statistical microcanonical ensemble. However, as in classical theory, equating time average with microcanonical ensemble average remains conjectural. 

In the case of an equilibrium system the Hamiltonian is the same as that of an ensemble of conservative systems in statistical equilibrium. If the energy of the system is measured to lie between Ek and EK + AE, then the representative ensemble is also restricted to the energy shell [AE K. From the hypotheses of equal a priori probabilities and random a priori phases it then follows that the diagonal elements of the density matrix lying inside [AE]k are all equal and that all others vanish. The density matrix of the quantum statistical microcanonical ensemble is thereby determined as... 

The microcanonical ensemble in quantum statistics describes a macroscopi-cally closed system in a state of thermodynamic equilibrium. It is assumed that the energy, number of particles and the extensive parameters are known. The Hamiltonian may be defined as... 

Of course, one is not really interested in classical mechanical calculations. Thus in normal practice the partition functions used in TST, as discussed in Chapter 4, are evaluated using quantum partition functions for harmonic frequencies (extension to anharmonicity is straightforward). On the other hand rotations and translations are handled classically both in TST and in VTST, which is a standard approximation except at very low temperatures. Later, by introducing canonical partition functions one can direct the discussion towards canonical variational transition state theory (CVTST) where the statistical mechanics involves ensembles defined in terms of temperature and volume. There is also a form of variational transition state theory based on microcanonical ensembles referred to by the symbol p,. Discussion of VTST based on microcanonical ensembles pVTST is beyond the scope of the discussion here. It is only mentioned that in pVTST the dividing surface is... 

One important property of these equations is that they conserve energy that is, E K U does not change as time advances. In the language of statistical mechanics, the atoms move within a microcanonical ensemble, that is, a set of possible states with fixed values of N, V, and E. Energy is not the only... 

If the statistical approximation were correct, one could estimate the rate constant for a microcanonical ensemble of reactant molecules by estimating the volume of... 

The RRKM theory is the most widely used of the microcanonical, statistical kinetic models It seeks to predict the rate constant with which a microcanonical ensemble of molecules, of energy E (which is greater than Eq, the energy of the barrier to reaction) will be converted to products. The theory explicitly invokes both the transition state hypothesis and the statistical approximation described above. Its result is summarized in Eq. 2... 

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 basic assumption in statistical theories is that the initially prepared state, in an indirect (true or apparent) unimolecular reaction A (E) —> products, prior to reaction has relaxed (via IVR) such that any distribution of the energy E over the internal degrees of freedom occurs with the same probability. This is illustrated in Fig. 7.3.1, where we have shown a constant energy surface in the phase space of a molecule. Note that the assumption is equivalent to the basic equal a priori probabilities postulate of statistical mechanics, for a microcanonical ensemble where every state within a narrow energy range is populated with the same probability. This uniform population of states describes the system regardless of where it is on the potential energy surface associated with the reaction. 

Boltzmann6 proposed that at the temperature T = 0, all thermal motion stops (except for zero-point vibration), and the entropy function S can be evaluated by a statistical function W, called the thermodynamic probability W (or, as we will learn in Section 5.2, the partition function Q for a microcanonical ensemble) ... 

We want to find out all about one system, with a macroscopic number (say, Avogadro s number Na) of constituents. Let us construct a "microcanonical ensemble" (juCE), described above (see Fig. 5.1). Within this juCE, we seek the statistically most likely distributions. Since we assume that the number of particles N and the energy U of the system are limited, we state that the restraints, or constraint equations, are... 

In order to remove the need for explicit trajectory analysis, one makes the statistical approximation. This approximation can be formulated in a number of equivalent ways. In the microcanonical ensemble, all states are equally probable. Another formulation is that the lifetime of reactant (or intermediate) is random and follows an exponential decay rate. But perhaps the simplest statement is that intramolecular vibrational energy redistribution (IVR) is faster than the reaction rate. IVR implies that if a reactant is prepared with some excited vibrational mode or modes, this excess energy will randomize into all of the vibrational modes prior to converting to product. 

RRKM theory assumes both the statistical approximation and the existence of the TS. It assumes a microcanonical ensemble, where all the molecules have equivalent energy E. This energy exceeds the energy of the TS (Eq), thanks to vibration, rotation, and/or translation energy. Invoking an equilibrium between the TS (the activated complex) and reactant, the rate of reaction is... 

Classical statistical mechanics is concerned with the probability distribution of phase points. In a classical microcanonical ensemble the phase space density is constant. Loosely speaking, aU phase points with the same energy are equally likely. In consequence the number of states of the classical system in a given energy range E to E + dE is proportional to the volume of the phase space shell defined by this energy range. 

The assumption of strict adiabaticity is lifted by assuming that channels may be chosen statistically, according to a microcanonical ensemble. The rate coefficient is then expressed by a formula that is very similar to the RRKM result. 

Elementary processes in chemical dynamics are universally important, besides their own virtues, in that they can link statistical mechanics to deterministic dynamics based on quantum and classical mechanics. The linear surprisal is one of the most outstanding discoveries in this aspect (we only refer to review articles [2-7]), the theoretical foundation of which is not yet well established. In view of our findings in the previous section, it is worth studying a possible origin of the linear surprisal theory in terms of variational statistical theory for microcanonical ensemble. 

The general mathematical formulation of the equilibrium statistical mechanics based on the generalized statistical entropy for the first and second thermodynamic potentials was given. The Tsallis and Boltzmann-Gibbs statistical entropies in the canonical and microcanonical ensembles were investigated as an example. It was shown that the statistical mechanics based on the Tsallis statistical entropy satisfies the requirements of equilibrium thermodynamics in the thermodynamic limit if the entropic index z=l/(q-l) is an extensive variable of state of the system. 

The entropy S for the Tsallis and Boltzmann-Gibbs statistics in the microcanonical ensemble can be written as... 

Tet us consider the nonrelativistic ideal gas of N identical particles governed by the classical Maxwell-Boltzmann statistics in the framework of the Tsallis and Boltzmann-Gibbs statistics in the microcanonical ensemble. For this special model, the statistical weight (111) can be written as (see [6] and reference therein)... 

Thus, the principle of additivity (Eqs. (21), (24), and (25)) is totally satisfied by the Tsallis statistics in the microcanonical ensemble. Equation (140) proves the zero law of thermodynamics for the microcanonical ensemble [6]. 

Substituting Eq. (144) into Eq. (130) for the entropy of the microcanonical ensemble, we obtain the entropy of the canonical ensemble (Eq. (90)). Equation (134) for the pressure of the microcanonical ensemble is identical to Eq. (92) for the pressure of the canonical ensemble. Substituting Eqs. (144) and (86) into Eq. (135) for the chemical potential of the microcanonical ensemble, we obtain Eq. (94) for the chemical potential of the canonical ensemble. Moreover, substituting Eqs. (144) and (86) into Eq. (136) for the variable E of the microcanonical ensemble, we obtain Eq. (96) for the variable E of the canonical ensemble. Thus, for the Tsallis statistics, the canonical and microcanonical ensembles are equivalent in the thermodynamic limit when the entropic parameter z is considered to be an extensive variable of state. 

Parvan A S. Microcanonical ensemble extensive thermodynamics of Tsallis statistics. Phys. Lett. A. 2006 350(5-6) 331-338. 

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