Ensemble, canonical

In any case, statistical methods work better if there is a variation in the total energy. This is the main reason to use ensembles other than the microcanonical ensemble. An ensemble consists of copies of the same system, called members of the ensemble. In the canonical ensemble, the volume is the same and all members are in contact with the same heat bath at a given temperature T. However, the total energy of all the members of the ensemble has to be the same. In this way, it is not necessary to assume that the members have almost the same energy as is done in the microcanonical ensemble. 

The canonical ensemble prescribes a constant number of particles, N. Alternatively, the number of particles may also be allowed to vary. This is done in the grand canonical ensemble. The new condition is that the total number of particles in all ensembles has to be constant. The purpose of ensembles is to resemble a natnral system as much as possible, for example, a piece of metal or a certain volnme. 

All subsystems are considered to be in the same heat bath, and the temperature (T) is therefore a constant. The collection of aUIV subsystems, with constant N but with varying E, are considered to represent aU aspects of the original system. This collection is called the canonical ensemble. 

When the Boltzmann distribution was derived in Section 5.2, we assumed a constant number of particles (N) and a given volume (V). We also assumed that the energy was constant. In the canonical ensanble, we imagine that the subsystems (members of the canonical ensemble) may exchange energy but not particles with each other. The ensemble that we are supposed to use is the one where the assumptions agree with the experimental conditions. Since the total energy is constant in the microcanonical ensemble, the definition of such a simple concept as temperature has to be done indirectly. 

The different subsystems have the same energy levels since the external conditions are identical. The total number of particles is constant. We assume that there are n, subsystems with energy E U2 subsystems with energy Ej, etc. The set of numbers Uj is called n. Evidently, we have 

Consider a system of N particles in volume V. Assume that the temperature T of the system is constant. The ensemble of microscopic points in phase space that correspond to this NVT macroscopic state is called the ATT or canonical ensemble. The canonical ensemble is called that simply because it was the first ensemble tackled by Gibbs. Indeed, ensemble theory as described in Gibbs seminal text (see Further reading) was formulated entirely for the constant temperature ensemble. The reason, as will become apparent in this chapter, is the ease in determining the canonical partition function. 

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The canonical ensemble is a set of systems each having the same number of molecules N, the same volume V and the same temperature T. This corresponds to putting the systems in a thennostatic bath or, since the number of systems is essentially infinite, simply separating them by diathennic walls and letting them equilibrate. In such an ensemble, the probability of finding the system in a particular quantum state / is proportional to where UfN, V) is tire energy of the /th quantum state and /c, as before, is the Boltzmaim... 

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... 

Consider two systems in thennal contact as discussed above. Let the system II (with volume and particles N ) correspond to a reservoir R which is much larger than the system I (with volume F and particles N) of interest. In order to find the canonical ensemble distribution one needs to obtain the probability that the system I is in a specific microstate v which has an energy E, . When the system is in this microstate, the reservoir will have the energy E = Ej.- E due to the constraint that the total energy of the isolated composite system H-II is fixed and denoted by Ej, but the reservoir can be in any one of the R( r possible states that the mechanics within the reservoir dictates. Given that the microstate of the system of... 

The above derivation leads to the identification of the canonical ensemble density distribution. More generally, consider a system with volume V andA particles of type A, particles of type B, etc., such that N = Nj + Ag +. . ., and let the system be in themial equilibrium with a much larger heat reservoir at temperature T. Then if fis tlie system Hamiltonian, the canonical distribution is (quantum mechanically)... 

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. 

In a canonical ensemble, the system is held at fixed (V, T, N). In a grand canonical ensemble the (V, T p) of the system are fixed. The change from to p as an independent variable is made by a Legendre transfomiation in which the dependent variable, the Flelmlioltz free energy, is replaced by the grand potential... 

The coimection between the grand canonical ensemble and thennodynamics of fixed (V, T, p) systems is provided by the identification... 

In the grand canonical ensemble, the number of particles flucPiates. By differentiating log E, equation (A2.2.121) with respect to Pp at fixed V and p, one obtains... 

Thennodynamics of ideal quantum gases is typically obtained using a grand canonical ensemble. In principle this can also be done using a canonical ensemble partition function, Q =. exp(-p E ). For the photon and... 

This result is identical to that obtained from a canonical ensemble approach in the thennodynamic limit, where the fluctuations in N vanish and (N) = N. The single-particle expression for the canonical partition fiinction = (-" can be evaluated using ih r rV i f<,2M) or a particle in a cubical box of volume V. 

This is the same as that in the canonical ensemble. All the thennodynamic results for a classical ideal gas tlien follow, as in section A2.2.4.4. In particular, since from equation (A2.2.158) the chemical potential is related to which was obtained m equation (A2.2.88). one obtains... 

A direct and transparent derivation of the second virial coefficient follows from the canonical ensemble. To make the notation and argument simpler, we first assume pairwise additivity of the total potential with no angular contribution. The extension to angularly-mdependent non-pairwise additive potentials is straightforward. The total potential... 

The correlation functions provide an alternate route to the equilibrium properties of classical fluids. In particular, the two-particle correlation fimction of a system with a pairwise additive potential detemrines all of its themiodynamic properties. It also detemrines the compressibility of systems witir even more complex tliree-body and higher-order interactions. The pair correlation fiinctions are easier to approximate than the PFs to which they are related they can also be obtained, in principle, from x-ray or neutron diffraction experiments. This provides a useful perspective of fluid stmcture, and enables Hamiltonian models and approximations for the equilibrium stmcture of fluids and solutions to be tested by direct comparison with the experimentally detennined correlation fiinctions. We discuss the basic relations for the correlation fiinctions in the canonical and grand canonical ensembles before considering applications to model systems. 

The grand canonical ensemble is a collection of open systems of given chemical potential p, volume V and temperature T, in which the number of particles or the density in each system can fluctuate. It leads to an important expression for the compressibility Kj, of a one-component fluid ... 

It was shown in section A2.3.3.2 that the grand canonical ensemble (GCE) PF is a generating fiinction for the canonical ensemble PF, from which it follows that correlation fiinctions in the GCF are just averages of the fluctuating numbers N and N - 1... 

Orkoulas G and Panagiotopoulos A Z 1999 Phase behavior of the restricted primitive model and square-well fluids from Monte Carlo simulations in the grand canonical ensemble J. Chem. Phys. 110 1581... 

Hiroike K 1972 Long-range correlations of the distribution functions in the canonical ensemble J. Phys. Soc. Japan 32 904... 

No system is exactly unifomi even a crystal lattice will have fluctuations in density, and even the Ising model must pemiit fluctuations in the configuration of spins around a given spin. Moreover, even the classical treatment allows for fluctuations the statistical mechanics of the grand canonical ensemble yields an exact relation between the isothemial compressibility K j,and the number of molecules Ain volume V ... 

Day P N and Truhlar D G 1991 Benchmark calculations of thermal reaction rates. II. Direct calculation of the flux autocorrelation function for a canonical ensemble J. Chem. Phys. 94 2045-56... 

The canonical ensemble corresponds to a system of fixed and V, able to exchange energy with a thennal bath at temperature T, which represents the effects of the surroundings. The thennodynamic potential is the Helmholtz free energy, and it is related to the partition fiinction follows ... 

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