Classical canonical partition function

The grand canonical ensemble is a set of systems each with the same volume V, the same temperature T and the same chemical potential p (or if there is more than one substance present, the same set of p. s). This corresponds to a set of systems separated by diathennic and penneable walls and allowed to equilibrate. In classical thennodynamics, the appropriate fimction for fixed p, V, and Tis the productpV(see equation (A2.1.3 7)1 and statistical mechanics relates pV directly to the grand canonical partition function... 

The path-integral (PI) representation of the quantum canonical partition function Qqm for a quantized particle can be written in terms of the effective centroid potential IT as a classical configuration integral ... 

With neglect of the quantum effects that arise from the exchange of identical particles [147], (8.66) gives the exact quantum partition function in the limit P — oo. For finite P, Qp((3) is the canonical partition function of a classical system composed of ring polymers. Each quantum particle corresponds to a ring polymer of P beads in which neighboring beads are connected by harmonic springs with force... 

For a classical system of N point particles enclosed in a volume V,at a temperature T, the canonical partition function can be decomposed in two factors. The first one (Qt) comes from the integration over the space of momenta of the kinetic term of the classical Hamiltonian, which represents the free motion of noninteracting particles. The second one, which introduces the interactions between the particles and involves integration over the positions, is the configuration integral. This way, equation (30)... 

In 1933, J.G. Kirkwood explicitly showed that the canonical partition function Q for a system of N monatomic particles reduces to an integral over phase space in the limit of high temperature (Equation 4.81). The result corresponds to classical mechanics (i.e. the spacing between energy levels is small compared to kT)... 

Here H is the Hamiltonian function for one N atomic molecule (i) and s is its symmetry number. One might have expected this result immediately from the Kirkwood formulation for the classical canonical partition function. H is a function of the 3N Cartesian momenta and the 3N Cartesian coordinates of molecule i. 

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... 

For a classical system at equilibrium, the canonical partition function is written... 

There are two basic approaches to the computer simulation of liquid crystals, the Monte Carlo method and the method known as molecular dynamics. We will first discuss the basis of the Monte Carlo method. As is the case with both these methods, a small number (of the order hundreds) of molecules is considered and the difficulties introduced by this restriction are, at least in part, removed by the use of artful boundary conditions which will be discussed below. This relatively small assembly of molecules is treated by a method based on the canonical partition function approach. That is to say, the energy which appears in the Boltzman factor is the total energy of the assembly and such factors are assumed summed over an ensemble of assemblies. The summation ranges over all the coordinates and momenta which describe the assemblies. As a classical approach is taken to the problem, the summation is replaced by an integration over all these coordinates though, in the final computation, a return to a summation has to be made. If one wishes to find the probable value of some particular physical quantity, A, which is a function of the coordinates just referred to, then statistical mechanics teaches that this quantity is given by... 

So much for old memories from classical thermodynamics. Now let us get more serious. The canonical partition function, using the degeneracies directly, is... 

Generally, the (classical) canonical partition function Q N, V, T) can be related... 

The denominator in Eq. (5.2) is related to the classical canonical partition function. Using Eq. (5.1) this distribution function can be written as a product of... 

How do we calculate the probability of a fluctuation about an equilibrium state Consider a system characterized by a classical Hamiltonian H r, p ) where p and denote the momenta and positions of all particles. The phase space probability distribution isf (r, p ) = Q exp(—/i22(r, p )), where Q is the canonical partition function. 

It is useful to keep the classical way of expressing the partition function and extend its application to more complex situations. First of all, one may write an expression for the canonical partition function of an ideal gas in terms of the partition function for each molecule. On the basis of the total Hamiltonian for each molecule (equation (2.2.18)), Hj, the canonical partition function is... 

In section 1.2, we introduced the quantum mechanical partition function in the T, V, N ensemble. In most applications of statistical thermodynamics to problems in chemistry and biochemistry, the classical limit of the quantum mechanical partition function is used. In this section, we present the so-called classical canonical partition function. 

While elegant in its simplicity, such an approach is extremely problematic when computing properties of physical systems. To appreciate this, we need to introduce some fundamental relationships of classical statistical mechanics, The volume of phase space that can be accessed by the N hard disks in the example above is called the canonical partition function, Q ... 

The free energy difference methods reviewed in this chapter, unless specified otherwise, are discussed for conditions of constant volume and constant temperature (NVT). The extension to ensembles of other types is straightforward.The classical canonical partition function is determined by the classical Hamiltonian 3 6(p, q ), describing the interactions of all N particles in the system in terms of the set of generalized coordinates and conjugated momenta p. For a system with N particles at temperature T, the canonical partition function can be written as... 

The thennodynamic properties of a classical hard-sphere system are derivable from the Gibbs canonical partition function... 

For simplicity we discuss a classical fluid system of N equal particles of mass m contained in a box of volume V and interacting by a two-body potential that is velocity independent (e.g., a 6-12 Lennard-Jones potential). The system is in equilibrium with a reservoir at temperature T [i.e., an (NVT) ensemble]. A configuration of the N particles is defined by their Cartesian coordinates and is denoted by the 3N vector x the ensemble of these vectors defines the configurational space ft of volume V. The momenta of the particles are denoted by the 3N vector p and the corresponding space by ft. Because the forces do not depend on the velocities, the contributions of the kinetic energy, pyim, and the interaction energy, (x), to the canonical partition function Q are separated,... 

However, since the canonical PF requires the knowledge of all energy levels of a system, and this is impossible to calculate at present (except for highly simplified models), we resort to the classical analog of the canonical partition function Qclass 111 making this transition from the quantum mechanical to the classical PF, we actually make a few assumptions and approximations. 

A, the Helmholtz energy, is a thermodynamic potential for the canonical ensemble. Qnvt, often symbolized simply as Q, is the canonical partition function. The last of Eqs. (46) defines a fundamental equation in the Helmholtz energy representation by expressing as a function of N, V, T. Often in classical systems it is possible to separate the energy contributions that depend on the momenta only (kinetic energy K) from the potential energy V, which depends only on the coordinates. When Cartesian coordinates are used as degrees of freedom, for example, the partition function can be factorized as ... 

Using (1.6.1) in the classical canonical partition function (1.5.9), we immediately obtain... 

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