Statistical mechanics microcanonical ensemble

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. 

The equations of motions can be formulated in the Newtonian, Hamiltonian, or Lagrangian approaches, all of which correspond to the microcanonical (NVE) statistical-mechanical ensemble in which the number of atoms, N, volume, V, and total energy, E, are conserved. Simulations in other statistical-mechanical ensembles can... 

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... 

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. 

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. 

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 use of non-Hamiltonian dynamical systems has a long history in mechanics [8] and they have recently been used to study a wide variety of problems in molecular dynamics (MD). In equilibrium molecular dynamics we can exploit non-Hamiltonian systems in order to generate statistical ensembles other than the standard microcanonical ensemble NVE) that is generated by traditional Hamiltonian dynamics. These ensembles, such as the canonical (NVT) and isothermal-isobaric (NPT) ensembles, are much better than the microcanonical ensemble for representing the actual conditions under which experiments are carried out. 

Another way to view MD simulation is as a technique to probe the atomic positions and momenta that are available to a molecular system under certain conditions. In other words, MD is a statistical mechanics method that can be used to obtain a set of configurations distributed according to a certain statistical ensemble. The natural ensemble for MD simulation is the microcanonical ensemble, where the total energy E, volume V, and amount of particles N (NVE) are constant. Modifications of the integration algorithm also allow for the sampling of other ensembles, such as the canonical ensemble (NVT) with constant temperature... 

From statistical mechanics the second law as a general statement of the inevitable approach to equilibrium in an isolated system appears next to impossible to obtain. There are so many different kinds of systems one might imagine, and each one needs to be treated differently by an extremely complicated nonequilibrium theory. The final equilibrium relations however involving the entropy are straightforward to obtain. This is not done from the microcanonical ensemble, which is virtually impossible to work with. Instead, the system is placed in thermal equilibrium with a heat bath at temperature T and represented by a canonical ensemble. The presence of the heat bath introduces the property of temperature, which is tricky in a microscopic discipline, and relaxes the restriction that all quantum states the system could be in must have the same energy. Fluctuations in energy become possible when a heat bath is connected to the... 

The partition function and the sum or density of states are functions which are to statistical mechanics what the wave function is to quantum mechanics. Once they are known, all of the thermodynamic quantities of interest can be calculated. It is instructive to compare these two functions because they are closely related. Both provide a measure of the number of states in a system. The partition function is a quantity that is appropriate for thermal systems at a given temperature (canonical ensemble), whereas the sum and density of states are equivalent functions for systems at constant energy (microcanonical ensemble). In order to lay the groundwork for an understanding of these two functions as well as a number of other topics in the theory of unimolecular reactions, it is essential to review some basic ideas from classical and quantum statistical mechanics. 

It is also interesting to consider the classical/quantal correspondence in the number of energized molecules versus time N(/, E), Eq. (8.22), for a microcanonical ensemble of chaotic trajectories. Because of the above zero-point energy effect and the improper treatment of resonances by chaotic classical trajectories, the classical and quantal I l( , t) are not expected to agree. For example, if the classical motion is sufficiently chaotic so that a microcanonical ensemble is maintained during the decomposition process, the classical N(/, E) will be exponential with a rate constant equal to the classical (not quantal) RRKM value. However, the quantal decay is expected to be statistical state specific, where the random 4i s give rise to statistical fluctuations in the k and a nonexponential N(r, E). This distinction between classical and quantum mechanics for Hamiltonians, with classical f (/, E) which agree with classical RRKM theory, is expected to be evident for numerous systems. 

Molecular dynamics simulations may be performed under a variety of conditions and constraints, corresponding to different ensembles in statistical mechanics. Most commonly the microcanonical (NVE) ensemble is used, i.e., the number of particles, the volume, and the total energy of the system remain constant during the simulation. The relationships in Equations (l)-(3) are valid for this case. 

Let us consider an ensemble of N molecules in a fixed volume V with a fixed total energy E. This is a microcanonical ensemble of classical statistical mechanics. Typical values for N used in these simulations of chemical interest is of the order of hundreds to a few thousands. In order to simulate an infinite system, periodic boundary conditions are invariably imposed. Thus a typical MD system would consist of N molecules enclosed in a cubic box with each side equal to length L. MD solves the equations of motion for a molecule i ... 

In normal classical statistical mechanics, it is assumed that all states which are fixed by the same external constraints, e.g., total volume V, average energy < ), average particle number N), are equally probable. All possible states of the system are generated and are assigned weight unity if they are consistent with these constraints and, zero otherwise. Thus in the case of an iV-particle system with classical Hamiltonian //j, the microcanonical ensemble entropy S E) is obtained from the total number of states ( ) via the definition... 

It is more realistic to treat an equilibrium state with the assumption the system is in thermal equilibrium with an external constant-temperature heat reservoir. The internal energy then fluctuates over time with extremely small deviations from the average value U, and the accessible microstates are the ones with energies close to this average value. In the language of statistical mechanics, the results for an isolated system are derived with a microcanonical ensemble, and for a system of constant temperature with a canonical ensemble. 

A standard MD computer simulation consists in the computation of the trajectory in the phase space of a system of N interacting bodies. The time evolution is determined by solving Newton s equations of motion of classical mechanics with finite difference methods. Such a model system corresponds to the microcanonical ensemble (NVE) of statistical mechanics with a constant number of particles N, volume V, and total energy E. In MD simulations the collective properties are then determined from the trajectory of all particles, i.e., from the time evolution of positions r = r, and momenta p = p,. The method relies on the assumption that stationary values of every average observable A can be defined as time integrals over the trajectory in the phase space ... 

Simulations generally involve a fixed number of particles (N). To model bulk systems with a finite number of particles, periodic boundary conditions are applied. The volume (V) of the periodically repeating simulation box may be kept fixed or varied. Extensions to the scheme in Eq. (2) allow coupling the system to a thermostat and/or barostat to control the temperature T and pressure P of the system [22-24]. Hence, depending on the chosen conditions, the distribution sampled during the MD simulation is representative of either the microcanonical (NVE), canonical (NVT), or isothermal-isobaric (NPT) statistical mechanics ensembles [19]. 

MD is readily applicable to a wide range of models, with and without constraints. It has been extended from the original microcanonical ensemble formulation to a variety of statistical mechanical ensembles. It is flexible and valuable for extracting dynamical information. The Achilles heel of MD is its high demand of computer time, as a result of which the longest times that can be simulated with MD fall short of the longest relaxation times of most real-life... 

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