Canonical partition functions

While studying the lattice models for exact calculations, we will use a discrete version of the configurational partition function (10.5) in the canonical ensemble  

It should be stressed that the nonnegativity of the heat capacity and the maximization principle only require W E) 0 and not W E) 1. Thus, both principles remain valid even if the entropy becomes negative, which is most certainly the case with the ideal gas. The principles refer to two independent aspects. What we will see below is that it is the condition W E) 1 that tends to be violated by the low-temperature SMSs, when extrapolated to T = 0. There is no such violation for low-temperature equilibrium states (CR). The latter is related to the absence of any energy gap for equilibrium states. 

9) It is well known that in classical statistical mechanics, the entropy in continuum space can become negative. This is true of the ideal gas at low temperatures. From the exact solution ofthe classical Tonks gas of rods in one dimension, one also finds that the entropy becomes negative at high coverage. Thus, W E) need not always represent the number of configurations, and care is needed to interpret W E) in all cases. 

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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 T-P partition ftmction can also be written in temis of the canonical partition function Qj as ... 

In an ideal gas there are no interactions between the particles and so the potential ener function, 1 ), equals zero. exp(- f (r )/fcBT) is therefore equal to 1 for every gas partic in the system. The integral of 1 over the coordinates of each atom is equal to the volume, ai so for N ideal gas particles the configurational integral is given by (V = volume). T1 leads to the following result for the canonical partition function of an ideal gas ... 

To introduce the transfer matrix method we repeat some well-known facts for a 1-D lattice gas of sites with nearest neighbor interactions [31]. Its grand canonical partition function is given by... 

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 ... 

The additional factor of Qi(V, T) in Eq. (21) makes the leading term in the sum unity, as suggested by the usual expression for the cluster expansion in terms of the grand canonical partition function. Note that i in the summand of Eq. (20) is not explicitly written in Eq. (21). It has been absorbed in the n , but its presense is reflected in the fact that the population is enhanced by one in the partition function numerator that appears in the summand. Equation (21) adopts precisely the form of a grand canonical average if we discover a factor of (9(n, V, T) in the summand for the population weight. Thus... 

As suggested previously, the density of states has a direct connection to the entropy, and, hence, to thermodynamics, via Boltzmann s equation. Alternately, we can consider the free energy analogue, using the Laplace transform of the density of states - the canonical partition function ... 

The excess density of states figures straightforwardly into the canonical partition function. Substituting the convolution in (1.4) into (1.3) and making the substitution S" = S — S", it follows that... 

In the second line, we have carried the integral over the ideal gas part, which results in the temperature-dependent de Broglie wavelength, A. The final expression is similar to the familiar casting of the canonical partition function,... 

In the canonical partition function of (5.1), we have for simplicity ignored combinatorial prefactors. Free energy perturbation theory [12] relies on evaluating effectively the ratio of the partition functions to obtain the free energy difference between the initial and final states corresponding to coupling parameters A = 1 and 0 (see also Chap. 2),... 

In the so-called primitive representation of the discretized path-integral approach [141], the canonical partition function for finite P has the form... 

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... 

A first step toward quantum mechanical approximations for free energy calculations was made by Wigner and Kirkwood. A clear derivation of their method is given by Landau and Lifshitz [43]. They employ a plane-wave expansion to compute approximate canonical partition functions which then generate free energy models. The method produces an expansion of the free energy in powers of h. Here we just quote several of the results of their derivation. 

The function Q is the canonical partition function or sum over states. The eigenvalues En are difficult, if not impossible, to obtain. It is significant,... 

From the normalization condition the relationship between Helmholtz free energy and the canonical partition function... 

In order to fix ideas, we will consider in this section a canonical ensemble [38], In this ensemble, the Helmholtz free energy is straightforwardly related with the canonical partition function through... 

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)... 

Statistical mechanics enables one to express the chemical potential i, for an ideal gas phase system in terms of the spectroscopic properties of individual gas phase molecules. The reader is referred to standard statistical mechanics texts (e.g. D. A. McQuarrie Statistical Mechanics , reading list) for the development of the relationship between the system Helmholtz free energy, A , and the corresponding canonical partition function Qi... 

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)... 

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