Chapter 3
Ensembles: Systems of Particles
This chapter discusses the nature of microscopic configurations and states for
systems of particles, accessibility of microscopic states, introduces the Fundamental
Postulate of statistical mechanics and illustrates it explicitly for small model
systems. The ensemble concept is applied to an isolated system of two subsystems,
one small (but still macroscopic), the other much larger (the reservoir). A canonical
ensemble describes a small system that can exchange energy, but not matter, with
the reservoir, while a grand ensemble describes a small system that can exchange
both energy and matter with the reservoir. The canonical partition function that
determines the probability for the particle to be in a specific energy state is
obtained for a single-particle that can access various types of energy, and is then
generalized to many particles and to mixtures. The grand partition function is
similarly developed for a system that can exchange both energy and matter with
the reservoir.
3.1 Microscopic Configurations
The microscopic arrangements of particles in a container are often called configurations: they are also often called states (most appropriately when quantum
mechanics is employed). Since most modern students are often more familiar with
quantum mechanical considerations than with classical mechanical ones, it will be
appropriate to utilize the language of quantum mechanics in much of the following.
Each quantum state of an isolated system may be characterized by the energy
associated with that quantum state. Two or more states characterized by the same
energy are said to be degenerate. How do we specify the possible quantum states
of a system? In principle we could simply list the values belonging to all relevant
quantum numbers (essentially one for each degree of freedom possessed by the
system as a whole). This would represent a truly enormous number. Indeed, a
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9_3
129
Ensembles: Systems of Particles
This chapter discusses the nature of microscopic configurations and states for
systems of particles, accessibility of microscopic states, introduces the Fundamental
Postulate of statistical mechanics and illustrates it explicitly for small model
systems. The ensemble concept is applied to an isolated system of two subsystems,
one small (but still macroscopic), the other much larger (the reservoir). A canonical
ensemble describes a small system that can exchange energy, but not matter, with
the reservoir, while a grand ensemble describes a small system that can exchange
both energy and matter with the reservoir. The canonical partition function that
determines the probability for the particle to be in a specific energy state is
obtained for a single-particle that can access various types of energy, and is then
generalized to many particles and to mixtures. The grand partition function is
similarly developed for a system that can exchange both energy and matter with
the reservoir.
3.1 Microscopic Configurations
The microscopic arrangements of particles in a container are often called configurations: they are also often called states (most appropriately when quantum
mechanics is employed). Since most modern students are often more familiar with
quantum mechanical considerations than with classical mechanical ones, it will be
appropriate to utilize the language of quantum mechanics in much of the following.
Each quantum state of an isolated system may be characterized by the energy
associated with that quantum state. Two or more states characterized by the same
energy are said to be degenerate. How do we specify the possible quantum states
of a system? In principle we could simply list the values belonging to all relevant
quantum numbers (essentially one for each degree of freedom possessed by the
system as a whole). This would represent a truly enormous number. Indeed, a
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9_3
129