The Canonical Ensemble

The canonical ensemble is considered in the next Section, while for the grand canonical ensemble the reader is referred to more comprehensive texts on the subject (Reed and Gubbins, for example). 

It is of course desirable to combine thermodynamics with our knowledge of the structure of matter. In particular we want to calculate Thermodynamie quantities on the basis of microscopic interactions between atoms and molecules or even subatomic particles. 

We assume a large completely isolated system containing a by comparison extremely small subsystem. This subsystem is allowed to exchange heat with its surroundings and we have 

Here E is the total internal energy of the isolated system, whereas is one particular value which the internal energy of the subsystem may assume. The difference between these internal energies is Eenv, i.e. the internal energy of the subsystem s environment. 

Hentschke, Thermodynamics, Undergraduate Lecture Notes in Physics, 

Having said this we may now continue by studying U Eenv) = E — Ey), the number of microstates of the environment all possessing the same energy E — Ey. Progress requires two additional and important assumptions (i) all microstates are equally probable (ii) the probability that our subsystem has the energy Ey, py, is proportional to the number of microstates available to the environment under this constraint, i.e. 

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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... 

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 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... 

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... 

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

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 ... 

Since H=K. + V, the canonical ensemble partition fiinction factorizes into ideal gas and excess parts, and as a consequence most averages of interest may be split into corresponding ideal and excess components, which sum to give the total. In MC simulations, we frequently calculate just the excess or configurational parts in this case, y consists just of the atomic coordinates, not the momenta, and the appropriate expressions are obtained from equation b3.3.2 by replacing fby the potential energy V. The ideal gas contributions are usually easily calculated from exact... 

It is a standard result in the canonical ensemble that energy fluctuations are related to the heat capacity Cy=... 

It is instructive to see this in temis of the canonical ensemble probability distribution function for the energy, NVT - Referring to equation B3.3.1 and equation (B3.3.2I. it is relatively easy to see that... 

Statistical mechanics may be used to derive practical microscopic fomuilae for themiodynamic quantities. A well-known example is tire virial expression for the pressure, easily derived by scaling the atomic coordinates in the canonical ensemble partition fiinction... 

Finally, by considering increasing the number of particles by one in the canonical ensemble (looking at the excess, non-ideal, part), it is easy to derive the Widom [34] test-particle fomuila... 

Consider simulating a system m the canonical ensemble, close to a first-order phase transition. In one phase, is essentially a Gaussian centred around a value j, while in the other phase tlie peak is around Ejj. 

The coordinate representation of the density matrix, in the canonical ensemble, may be written... 

Nos e S 1984 A molecular dynamics method for simulations In the canonical ensemble Mol. Phys. 52 255-68... 

Martyna G J, Klein M L and Tuckerman M 1992 Nos e-Hoover chains the canonical ensemble via continuous dynamics J. Chem. Phys. 972635—43... 

Tobias D J, Martyna G J and Klein M L 1993 Molecular dynamics simulations of a protein In the canonical ensemble J. Phys. Chem. 9712959-66... 

Nose, S. A molecular dynamics method for simulations in the canonical ensemble. Mol. Phys. 52 (1984) 255-268 ibid. A unified formulation of the constant temperature molecular dynamics method. J. Chem. Phys. 81 (1984) 511-519. 

The canonical ensemble partition function is the phase space integral... 

Since the averaging operator is not normalized and in general (1), 1 for g 7 1, it is necessary to compute Zq to determine the average. To avoid this difficulty, we employ a different generalization of the canonical ensemble average... 

The canonical ensemble is the name given to an ensemble for constant temperature, number of particles and volume. For our purposes Jf can be considered the same as the total energy, (p r ), which equals the sum of the kinetic energy (jT(p )) of the system, which depends upon the momenta of the particles, and the potential energy (T (r )), which depends upon tlie positions. The factor N arises from the indistinguishability of the particles and the factor is required to ensure that the partition function is equal to the quantum mechanical result for a particle in a box. A short discussion of some of the key results of statistical mechanics is provided in Appendix 6.1 and further details can be found in standard textbooks. 

The criterion used to accept or reject a new configuration is slightly different for the is thermal-isobaric simulation than for a simulation in the canonical ensemble. The followi] quantity is used ... 

The excess chemiccil potential is thus determined from the average of exp[—lT (r )/fe In ensembles other than the canonical ensemble the expressions for the excess chem potential are slightly different. The ghost particle does not remain in the system and the system is unaffected by the procedure. To achieve statistically significant results m Widom insertion moves may be required. However, practical difficulties are encounte when applying the Widom insertion method to dense fluids and/or to systems contain molecules, because the proportion of insertions that give rise to low values of y f, dramatically. This is because it is difficult to find a hole of the appropriate size and sha... 

Note This method of temperature regulation does not give all properties of the canonical ensemble. In particular, you cannot calculate Cy, heat capacity at constant volume. 

Eor many systems the ensemble that is used in an MC simulation refers to the canonical ensemble, (N, F/ T). This ensemble permits a rise and fall in the pressure of the system, P, because the temperature and volume are held constant. Thus, the probabiUty that any system of N particles, in a volume H at temperature Tis found in a configuration x is proportional to the Boltzmann weighted energy at that state, E, and it is given by... 

In this expression. Ait is the size of the integration time step, Xj is a characteristic relaxation time, and T is the instantaneous temperature. In the simulation of water, they found a relaxation time of Xj = 0.4 ps to be appropriate. However, this method does not correspond exactly to the canonical ensemble. 

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