Ensembles

The objects whose properties are calculated by statistical mechanics are known as systems. Conceptually, to define a system one takes the world, or a model of the world, and draws a closed boundary around a part of the world. ``Closed'' is here meant in the geometric sense, the boundary serving to divide the world into two discontinuous pieces. The part of the world inside the boundary is the system. The open, infinite region outside the boundary is the bath. Baths are used extensively in thermodynamics; see later Lectures. The detailed properties of baths are not central to much work in statistical mechanics. The boundary need not be solid; it may be possible for molecules or energy to pass through the boundary. There are no fundamental constraints on where or how the boundary between the system and the bath is drawn. However, some choices of boundary are much more interesting than others. A system's behavior can be modified by the boundary's properties.

A fundamental requirement of statistical mechanics is: any system that can be treated by statistical mechanics can given a complete, microscopic description. In a classical system, a complete microscopic description consists of the Hamiltonian coordinates and their canonically conjugate momenta. For a single point gas atom, a complete microscopic description could be the position coordinates x,y, z and the corresponding momenta . (We will not treat systems with non-holonomic constraints.) In a quantum system, the microscopic description consists of the amplitude and phase for each basis vector. In general, a complete microscopic description of a system is a list of variables. This list has the property that it is complete, so that any mechanical variable can be written as a function of the variables specified by the complete microscopic description. Note that the system is required to have a complete description, but a complete microscopic description of the bath is not required.

To discuss statistical mechanics one also needs the concept of a state of the system. To specify a state, one specifies values for every variable of the complete microscopic description. Specifying values for all of the microscopic variables is sufficient to specify the state of the system. Each state of the system corresponds to a unique set of values of the microscopic variables, and vice versa: any allowed set of values for the microscopic variables correspond to a state of the system.

An ensemble is a complete, non-repeating list of all of the allowed states (or phases[3]) of a system, together with the statistical weight to be associated with each state. An allowed state of the system is known as an element of the ensemble. There are a variety of different ensembles, some more important than others. This list of allowed states of an ensemble can usually be described by specifying constraints. By constraining a variable, one limits which of the system's conceivable states are ``allowed states of the system''. For example, one might constrain the number of molecules N in the system to have a specific value, in which case states in which N molecules were not present would not be in the ensemble.

A function whose value can be calculated from the microscopic description of a single element of an ensemble is known as a mechanical variable. For example, the kinetic energy K of a gas molecule is a mechanical variable, since K can be obtained from the gas molecule's momentum p. The phrase 'calculated from' includes the identity operation, is a mechanical variable via . There are important variables which are not mechanical; these are the ensemble variables. The value of an ensemble variable can be obtained by examining the entire ensemble, but can not be calculated from o complete microscopic description of a single element of the ensemble. The temperature T and entropy S are important ensemble variables.

The great innovations of J. Willard Gibbs in statistical mechanics were the mechanical validation of the ensemble concept, the identification of the fundamental thermodynamic ensemble as the Canonical Ensemble, and the correct treatment of classical systems in which the number of molecules is not fixed.

In Gibbs' formulation of statistical mechanics, the average value of a mechanical variable A is obtained by determining A in each element of an ensemble, and then taking a correctly weighted average of A over the whole ensemble. While Gibbs' statistical mechanics primarily gives average values, a clever choice of the function being averaged can yield detailed information about a system. Lecture 6 will illustrate computations of the average pressure of a gas. One might also be interested in pressure fluctuations. By computing such averages as or , being the Dirac delta function, one can characterize the spectrum of the pressure fluctuations, or determine how often a particular pressure is observed.

Simple images of operational time and ensemble averages will sometimes prove useful. To obtain a time average of some parameter A in a real system, one could imagine making a motion picture of the system, using a special camera which captures the full microscopic description of the system, including all particle positions, momenta ...By analyzing each frame of film, A(t) could be determined for each frame and averaged over all frames. A theoretical time average may also be envisioned as a motion picture. The initial conditions of the system are the first frame of film. By numerically integrating the equations of motion, successive frames of film are digitally generated, thereby creating a numerical motion-picture of an evolving system.[2] The integration step is cumbersome, though feasible for small systems. The associated computer methods are known as molecular dynamics and Monte Carlo simulations, as treated in later Lectures.

Time integration can be avoided by making an ensemble average. To continue the film analogy, in an ensemble average one generates all possible pictures of the system. is obtained by computing A in every possible picture of the system, and taking a correctly-weighted average over A from each possible picture. By ``all possible'' I mean not only all the pictures that would be generated by integrating in time from a single starting point, but all pictures which are compatible with the laws of nature governing the system.While it may seem more tedious to look at all possible pictures than to look only at a single reel of film, it turns out to be much simpler to generate all possible pictures than it is to work out which picture follows which (the latter being required for a theoretical time average), so it is easier to compute an ensemble average than to compute a time average.