Microcanonical system

The hypersurface fomied from variations in the system s coordinates and momenta at//(p, q) = /Tis the microcanonical system s phase space, which, for a Hamiltonian with 3n coordinates, has a dimension of 6n -1. The assumption that the system s states are populated statistically means that the population density over the whole surface of the phase space is unifomi. Thus, the ratio of molecules at the dividing surface to the total molecules [dA(qi, p )/A]... 

In this chapter we consider the problem of reaction rates in clusters (micro-canonical) modified by solvent dynamics. The field is a relatively new one, both experimentally and theoretically, and stems from recent work on well-defined clusters [1, 2]. We first review some theories and results for the solvent dynamics of reactions in constant-temperature condensed-phase systems and then describe two papers from our recent work on the adaptation to microcanonical systems. In the process we comment on a number of questions in the constant-temperature studies and consider the relation of those studies to corresponding future studies of clusters. 

In the next section we summarize two treatments of microcanonical systems [2, 44], one of the steady-state Kramers one-coordinate type and one including vibrational assistance. An earlier approach to the problem was given by Troe [45]. 

Over the last several years much effort has gone into the extension of MD to describe other than microcanonical systems (Nose and references therein). Nose, in particular, has provided a method that is capable in principle of producing canonical averages from time averages of static physical properties along continuous, deterministic trajectories. This method can also be used for the calculation of dynamic propierties that are inaccessible from MC simulations. However, the interpretation of time-dejjendent properties calculated from this and similar methods is not straightforward, particularly for very small systems. ... 

The PC analysis (PCA) originally developed in the statistical problem can provide us with not only collective coordinates but also the conjugate momenta for any microcanonical system Let a set of the original coordinates and the conjugate momenta be (p, q) and the new collective coordinates Q(Qi = S C/A/ j and the conjugate momenta P. By assuming a generating function F(q, P), we obtain... 

This assumption is not bad at all for sufficiently large microcanonical systems and long trajectories. It is consistent with the numerical observation of the similarity between canonical and microcanonical ensembles for systems with a few hundred coupled degrees of freedom. 

The hypersurface formed from variations in the system s coordinates and momenta ati/(p, q) = is the microcanonical system s phase space, which, for a Hamiltonian with 3n coordinates, has a dimension of 6n -... 

We began this chapter by considering the phase space volume and surface area which are related to the sum and density of states for a system at a given total energy E, that is, a microcanonical system. This is the system of major interest in this book. However, during the discussion of several topics it will be necessary to make use of the partition function, which is appropriate for constant-temperature, or canonical, systems. Because the partition functions for translations, rotations, and vibrations are derived in all undergraduate physical chemistry texts, we will not derive them here, but simply summarize the results. 

A unimolecular reaction can be viewed as a reaction flux in phase space. It is best to have in mind a potential energy surface with a real barrier in the product channel, that is, a saddle point. Figure 6.4 shows both the reaction coordinate and a picture of the phase space associated with the molecule and the transition state. Recall, that a molecule of several atoms having a total of m internal degrees of freedom can be fully described by the motion of m positions (q) and m momenta (p). At any instant in time, the system is thus fully described by 2m coordinates. A constant energy molecule (a microcanonical system) has its phase space limited to a surface in which the Hamiltonian H = E. Thus, the dimensionality of this hypersurface is reduced to 2m — 1. 

In this equation, E is the activation energy and e, is the translational energy associated with the momentum p in the reaction coordinate. Both of these energies must be subtracted from the total energy at the saddle point because these energies are not available for the — 1 momenta, p , and n — 1 coordinates, We can think of this expression as an equilibrium constant for a microcanonical system. 

The cornerstone of statistical mechanical theory is the concept of entropy, which may be taken as the number of states available to a discrete system (e.g. a quantum mechanical system), or, in the case of a classical system, the volume of the accessible phase space. In terms of a microcanonical system, the entropy is enshrined in the famous formula that appears on Boltzmann s tombstone ... 

The canonical temperature T of the closed system is identical with the microcanonical heat bath temperature, i.e., T = However, this does not mean that also the canonical and microcanonical system temperatures coincide actually = Tis only valid in the thermodynamic limit, where the energetic fluctuations... 

Microcanonical system of fixed values of mass, volume, and energy ... 

The microcanonical ensemble is a set of systems each having the same number of molecules N, the same volume V and the same energy U. In such an ensemble of isolated systems, any allowed quantum state is equally probable. In classical thennodynamics at equilibrium at constant n (or equivalently, N), V, and U, it is the entropy S that is a maximum. For the microcanonical ensemble, the entropy is directly related to the number of allowed quantum states C1(N,V,U) ... 

An explicit example of an equilibrium ensemble is the microcanonical ensemble, which describes closed systems with adiabatic walls. Such systems have constraints of fixed N, V and E < W< E + E. E is very small compared to E, and corresponds to the assumed very weak interaction of the isolated system with the surroundings. E has to be chosen such that it is larger than (Si )... 

The microcanonical ensemble is a certain model for the repetition of experiments in every repetition, the system has exactly the same energy, Wand F but otherwise there is no experimental control over its microstate. Because the microcanonical ensemble distribution depends only on the total energy, which is a constant of motion, it is time independent and mean values calculated with it are also time independent. This is as it should be for an equilibrium system. Besides the ensemble average value (il), another coimnonly used average is the most probable value, which is the value of tS(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e. if... 

In the last subsection, the microcanonical ensemble was fomuilated as an ensemble from which the equilibrium, properties of a dynamical system can be detennined by its energy alone. We used the postulate of... 

For a microcanonical ensemble, p = [F( )] for each of the allowed F( ) microstates. Thns for an isolated system in eqnilibrinm, represented by a microcanonical ensemble. 

This behaviour is characteristic of thennodynamic fluctuations. This behaviour also implies the equivalence of various ensembles in the thermodynamic limit. Specifically, as A —> oo tire energy fluctuations vanish, the partition of energy between the system and the reservoir becomes uniquely defined and the thennodynamic properties m microcanonical and canonical ensembles become identical. 

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]... 

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