Temperature and equilibrium

Abstract

Statistical mechanics explains the comprehensible behavior of microscopically complex systems by using the weird geometry of high-dimensional spaces, and by relying only on the known conserved quantity: the energy. Particle velocities and density fluctuations are determined by the geometry of spheres and cubes in dimensions with twenty three digits. Temperature, pressure, and chemical potential are defined and derived in terms of the volume of the high-dimensional energy shell, as quantified by the entropy. In particular, temperature is the inverse of the cost of buying energy from the rest of the world, and entropy is the currency being paid. Exercises discuss the weird geometry of high dimensions, how taste and smell measure chemical potentials, equilibrium fluctuations, and classic thermodynamic relations.