×

zbMATH — the first resource for mathematics

Diffusion, mobility and the Einstein relation. (English) Zbl 0559.60065
Statistical physics and dynamical systems. Rigorous results, Pap. 2nd Colloq. Workshop, Köszeg/Hung. 1984, Prog. Phys. 10, 405-441 (1985).
[For the entire collection see Zbl 0557.00018.]
We investigate the Einstein relation \(\sigma =\beta D\) between the diffusion constant D and the ”mobility”, of a ”test particle” interacting with its environment: \(\beta^{-1}\) is the temperature of the system where D is measured and \(\sigma\) E is the drift in a constant external field E. The relation is found to be satisfied for all model systems in which we can find a unique stationary non-equilibrium state of the environment, as seen from the test particle in the presence of the field. For some systems, e.g. infinite systems of hard rods in one dimension, we find non-unique stationary states which do not satisfy the Einstein relation. For some models in a periodic box the Einstein relation is the most direct way of obtaining D. A precise macroscopic formulation of the Einstein relation which makes it mathematically very plausible is given.

MSC:
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
81P20 Stochastic mechanics (including stochastic electrodynamics)
60K35 Interacting random processes; statistical mechanics type models; percolation theory