3 - The canonical ensemble

Abstract

When two systems with fixed total energy are thermally coupled, the systems exchange energy until the joint system approaches the equilibrium state which is characterized by maximum total entropy and a common temperature T. If a system exchanges energy with a much larger reservoir, then the probability distribution of energy E in the smaller system is proportional to the Boltzmann factor exp⁡(−βE), where β=1/kT and k is Boltzmann's constant. The canonical partition function describes a system coupled to a thermal reservoir, with the Helmholtz free energy being proportional to the logarithm of the partition function. Energy fluctuations are small compared to the total energy, with the variance proportional to the heat capacity. The equipartition theorem gives 12kT as the average value of any squared microscopic variable in the Hamiltonian. We use the canonical ensemble to determine the thermodynamic behavior of classical ideal gases, systems of harmonic oscillators, and paramagnets.