ABSTRACT
Equilibrium thermodynamics may be developed as an autonomous macroscopic theory or it may be derived from microscopic statistical mechanics. The intrinsic beauty of the macroscopic approach is partially lost with the second treatment. Its beauty lies in its internal consistency. The advantage of the second treatment is that certain quantities are given explicit formulae in terms of fundamental constants, whereas the purely macroscopic approach must use measurements to determine these same quantities. The Stefan-Boltzmann constant is a prime example of this dichotomy. Using purely macroscopic thermodynamic arguments, Boltzmann showed that the energy density emitted per second from a unit surface of a black body is σT 4 where T is the temperature and σ is the Stefan-Boltzmann constant, but it takes statistical mechanics to produce the formula
σ = 2π 5k4
15c2h3
in which k is Boltzmann’s constant, c is the speed of light and h is Planck’s constant. This beautiful formula depends on three fundamental constants, an exhibition of the power of the microscopic viewpoint. Likewise, non-equilibrium thermodynamics may be developed as a purely autonomous macroscopic theory or it may be derived from microscopic kinetic theory, either classically or quantum mechanically. The separation between the macroscopic and microscopic approaches is a little less marked than for the equilibrium theory because the existence of the microscopic underpinning leads to the existence of fluctuations in the macroscopic picture, as well as to the celebrated Onsager reciprocal relations. On purely macroscopic grounds, the fluctuation-dissipation relation that connects the relaxation rates to the strengths of the fluctuations may be established, but it takes the full microscopic theory to compute their quantitative values, at least in principle. In practice, these computations are very difficult. This presentation is primarily about the macroscopic approach, although at the end the microscopic approach, based on linear response theory, is also reviewed.
