Equilibrium statistical mechanics canonical ensemble

Obviously Eq. (11) can also be derived directly from equilibrium thermodynamic considerations involving equilibrium constants. A statistical-mechanical derivation4 of Eq. (11), utilizing an equilibrium (actually grand canonical) ensemble, points up the analogy between this transition and a special case of helix-coil transition, the latter being most usually treated with equilibrium ensembles. 

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. 

Before we introduce our restricted partition function formalism, we need to review some salient aspects of the equilibrium statistical mechanics formalism. This will also help with the clarity and continuity of presentation. We will restrict our discussion to the canonical ensemble, but the extension to other ensembles is straightforward and trivial. 

In statistical mechanics the properties of a system in equilibrium are calculated from the partition function, which depending on the choice for the ensemble considered involves a sum over different states of the system. In the very popular canonical ensemble, that implies a constant number of particles N, volume V, and temperature T conditions, the quasiclassical partition function Q is... 

There is considerable interest in the use of discretized path-integral simulations to calculate free energy differences or potentials of mean force using quantum statistical mechanics for many-body systems [140], The reader has already become familiar with this approach to simulating with classical systems in Chap. 7. The theoretical basis of such methods is the Feynmann path-integral representation [141], from which is derived the isomorphism between the equilibrium canonical ensemble of a... 

In the following, we first describe (Section 13.3.1) a statistical mechanical formulation of Mayer and co-workers that anticipated certain features of thermodynamic geometry. We then outline (Section 13.3.2) the standard quantum statistical thermodynamic treatment of chemical equilibrium in the Gibbs canonical ensemble in order to trace the statistical origins of metric geometry in Boltzmann s probabilistic assumptions. In the concluding two sections, we illustrate how modem ab initio molecular calculations can be enlisted to predict thermodynamic properties of chemical reaction (Sections 13.3.3) and cluster equilibrium mixtures (Section 13.3.4), thereby relating chemical and phase thermodynamics to a modem ab initio electronic stmcture picture of molecular and supramolecular interactions. 

We have derived a formula for the molecular partition function by considering a system containing many molecules at equilibrium with a heat bath. We can generalize our statistical mechanics by a gedanken experiment of considering a large number of identical systems, each with volume V and number of particles N at equilibrium with the heat bath at temperature T. Such a supersystem is called a canonical ensemble. Our derivation is the same the fraction of systems that are in a state with energy Et is... 

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. 

In the past few years, development of new theories have led to completely new ways of determining free energy changes. Traditionally, the difference in the free energy of two equilibrium state is (AFi 2) and the free energy change of a process can be obtained directly from the statistical mechanical definition of the free energy, F, in terms of the partition function. For the canonical ensemble F = —k T In J = —ksTln Z, where ka is Boltzmann s constant, //(F) is the phase... 

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... 

Under very general conditions, it follows from classical statistical mechanics that the equilibrium behavior of our fluid system is adequately described % the behavior of a Gibbskn ensemble of systems characterized by a canonical distribution (in energy) in phase space. This has two immediate consequences. First it specifies the spatial distribution of our N molecule system. The simultaneous probability that some first molecule center hes in the volume element dr whose center is at and etc., and the Nih molecule center lies in the volume element dr f whose center is at is... 

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. 

The MC technique is a stochastic simulation method designed to generate a long sequence, or Markov chain of configurations that asymptotically sample the probability density of an equilibrium ensemble of statistical mechanics [105, 116]. For example, a MC simulation in the canonical (NVT) ensemble, carried out under the macroscopic constraints of a prescribed number of molecules N, total volume V and temperature T, samples configurations rp with probability proportional to, with, k being the Boltzmann constant and T the... 

The present article presents an introduction to the path integral formulation of quantum dynamics and quantum statistical mechanics along with numerical procedures useful in these areas and in electronic structure theory. Section 2 describes the path integral formulation of the quantum mechanical propagator and its relation to the more conventional Schrddinger description. That section also derives the classical limit and discusses the connection with equilibrium properties in the canonical ensemble, Numerical techniques are described in Section 3. Selective chemical applications of the path integral approach are presented in Section 4 and Section 5 concludes. 

In order to theoretically understand the Monte Carlo method, we need to use the concepts of statistical mechanics. These concepts are developed in detail in books of fundamental statistical mechanics [16]. This formalism guides us in the task of calculating various quantities and/or properties that describe the behavior of the system. This is accomplished from the partition function that specifies the statistical properties of the system in thermodynamic equilibrium. There are different classes of partition functions, each corresponding to a determined statistical ensemble. The canonical ensemble is applied for a system of N particles in a volume V and temperature T. These magnitudes have fixed values and heat exchange is allowed. For this ensemble the corresponding partition function is ... 

The energy and entropy, which are related in this equation to the molecular partition function, are statistical entities, defined by particles which do not interact except to maintain the equilibrium conditions. To obtain similar relationships for real systems, it is necessary to apply statistical mechanics to the calculation of the thermodynamic entities, which correspond to the molar quantities of particles, or that is N approaches 1 this treatment it is convenient to use the canonical ensemble already discussed and presented in Figure n.l. This ensemble consists of a very large number of systems, N, each containing 1 mol of molecules and separated from the others by diathermic walls, which allow heat conduction but do not allow particles to pass. The set of all the systans is isolated from the outside and has a fixed energy E, which is the energy of the canonical ensemble. 

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. 

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