What Is an Ensemble

So far in this chapter, we have considered only systems in which T,V,N) are constant. In statistical mechanics, such a system is called the Canonical ensemble. Ensemble has two meanings. First, it refers to the collection of all the possible microstates, or snapshots, of the arrangements of the system. We have counted the number of arrangements W of particles on a lattice or configurations of a model polymer chain. The word ensemble describes the complete set of all such configurations. Ensemble is also sometimes used to refer to the constraints, as in the T, V,N) ensemble. The (T, V,N) ensemble is so prominent in statistical mechanics that it is called canonical. A system constrained by (T, p,N), is called the isobaric-isothermal ensemble. 

When the constraints are (T, V, p), energy and particles can exchange across the boundary. This is called the Grand Canonical ensemble. It is important for processes of ligand binding, and is illustrated in Chapter 28. When (U,V,N) are held constant at the boundary, no extensive quantity exchanges across the boundary, so there is no bath, and maximum entropy 5([/, V,N) identifies the state of equilibrium (see Chapter 7). This is called the Microcanonical ensemble. 

The Microcanonical ensemble is qualitatively different from the Canonical and Grand Canonical ensembles because it involves no bath and no fluctuations of any extensive property across the boundary. 

The system has the same probability of being in any one microstate as any other. This prediction is the same as the prediction in Chapter 6 for die rolis in the absence of knowledge of an average score, or for dipole orientations if all directions are equivalent. 

The general definition of entropy is S/k = -ZiPilnpi. For the Micro-canonical ensemble U,V,N), the entropy is also given by the Boltzmann expression, Equation (6.1), S = k nW(U,V,N). You can see this by substituting Pj = lit = IIW into Equation (6.2), to get 

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What is an ensemble What is identical about members of the ensemble What might be examples... 

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