Ensembles and Arrangements

The approach we are following for systems with large numbers of particles is statistical rather than mechanical averaging. Thus, we need to consider the many different ways a system can be found to exist, rather than how it evolves from one single set of initial conditions. An ensemble is a hypothetical collection of many like systems that helps us do that. Our objective is to average over all the possible ways a system exists, and these ways are included in an ensemble. 

To work with ensembles, we have to distinguish between particles and systems, for we will be considering both. Let us start with a single molecule for which the electronic-vibrational- 

The i subscript specifies a state among the infinite number of eigenfunctions and eigenen-ergies. The classical Hamiltonian from which the quantum mechanical Hamiltonian is derived is a function of position and momentum coordinates for all the particles in the molecule. Let this number of coordinates be d. A system might consist of some number, N, of these molecules. We can formally write the Schrodinger equation for the system of N molecules as 

An ensemble for a system of N diatomic molecules might consist of M replicants of the system. Different types of ensembles are used to average under different sets of conditions. A microcanonical ensemble is one for which each replicant system in the ensemble has the same number of molecules, N, the same volume, V, and the same energy, E. That is, N, V, and E are fixed. These systems are identical from a thermodynamic perspective, but at the molecular level, they may be different. A canonical ensemble is one for which N, V, and the temperature, T, are the same fixed values for each replicant system in the ensemble. In a grand canonical ensemble, V, T, and the chemical potential are fixed. This allows N to change, as would occur in multiple-phase and reaction systems. One can construct other types of ensembles with other constraints. 

Although an ensemble is a hypothetical or virtual construct, we can appreciate the physical idea of at least a canonical ensemble. Taking M as a very large number, consider there to be M different impermeable vessels of volume V. In each are placed N molecules. The vessels are in contact, and the vessel walls conduct heat. The assembly of M vessels is placed in a constant-temperature heat bath to establish some temperature T. After that is achieved, the assembly is thermally isolated from its surroundings heat exchange 

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