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We stress the role of nonequivalent representations of the canonical commutation relations in quantum mechanics. Such representations occur when the space accessible to the system is not simply connected. An example is the magnetic Aharonov-Bohm effect, which can be interpreted in terms of nonequivalent representations, without explicitly invoking the vector potential in the region accessible to the electrons. [DOI: 10.1119/1.2360994]

The Boltzmann reservoir (BR) is a model constant-temperature environment that exhibits highly atypical thermodynamic behavior. Its microcanonical ensemble entropy is a linear, nonconcave function of its internal energy U, and its zero-work heat capacity is infinite. Its canonical partition function diverges because all possible energies are equally likely, so the microcanonical and canonical ensembles are not equivalent. If two BRs with the same temperature T(sub B) are put in thermal contact, either can have any fraction of the total energy; i.e., there is no unique equilibrium state. If two BRs with different temperatures are in thermal contact, the higher temperature BR gives all its energy to the other. A BR's temperature cannot be changed by a heat process but, in principle, can be altered by a work process. These and other properties that challenge conventional wisdom provide thought-provoking examples for thermal physics courses.

Brownian motion represents simple diffusion random walk processes. More complex random walk processes also can occur when probability distributions describing the random jump distances and times have infinite moments. We explore the manner in which these distributions can arise and how they underlie various scaling laws that play an important role in both random and deterministic systems.

The fluctuations in the number of particles of the ideal gas are calculated using the canonical and the grand canonical ensembles. The two results differ by a factor which accounts for the relative size of the total volume and the subvolume where the canonical ensemble fluctuations are calculated. This factor gives a simple example of explicit finite-size effects, because it arises from considering a fixed number of particles in the canonical ensemble. Our simulation results for the isothermal compressibility of the hard disk fluid with a finite number of particles are improved by applying this correction.

We describe a simple experiment, suitable for an undergraduate laboratory, in which the collector current in a transistor is measured as a function of the base-emitter voltage at various temperatures. The experiment gives a very convincing demonstration of the canonical distribution of statistical mechanics, in which the probability of occupancy of a state of energy E is proportional to e-(super E/kT).