The theories of Chapters 12 and 13 are developed close to equilibrium. Definition of the close to equilibrium regime is based on the local-equilibrium hypothesis, which states: The system can be split into small cells, small enough that changes in thermodynamic properties within a cell (that is, gradients) can be neglected. However, the cells are sufficiently large (containing a sufficiently large number of particles) that microscopic fluctuations can be neglected as well. Moreover, all thermodynamic quantities (that is, T, S and Gibbs equation) are rigorously defined at each cell as in equilibrium. These quantities remain uniform at each cell, but slightly differ from cell to cell, and are changed in time; the basic unit of time, Δt, is much shorter than any macroscopic time evolution, but is the largest for which thermodynamic properties in a cell remain constant. Thus, the relaxation of a system to equilibrium is not very fast.

Derivation of the two Fick’s laws and the diffusion equation is based on the local-equilibrium hypothesis. The close to equilibrium condition is used in Einstein’s derivation of the Brownian motion, and in the solution of the Langevin equation, which leads to the fluctuation-dissipation theorem. The probability distributions of the velocity in the Langevin equation are calculated by both the central limit theorem and the Fokker-Planck equation. The Langevin equation with a charge in an electric field, the Langevin equation with an external force in the strong damping velocity, the Smoluchowski equation, and the Fokker-Planck equation for a full Langevin equation with a force, are all treated. Finally, the stochastic dynamics method is discussed and compared to molecular dynamics.