Fluctuation-dissipation theorems from the generalised Langevin equation

Abstract

The generalised Langevin equation (GLE), originally developed in the context of Brownian motion, yields a convenient representation for the mobility (generalised susceptibility) in terms of a frequency-dependent friction (memory function). Kubo has shown how two deep consistency conditions, or fluctuation-dissipation theorems, follow from the GLE. The first relates the mobility to the velocity auto-correlation in equilibrium, as is also derivable from linear response theory. The second is a generalised Nyquist theorem, relating the memory function to the auto-correlation of the random force driving the velocity fluctuations. Certain subtle points in the proofs of these theorems have not been dealt with sufficiently carefully hitherto. We discuss the input information required to make the GLE description a complete one, and present concise, systematic proofs starting from the GLE. Care is taken to settle the points of ambiguity in the original version of these proofs. The causality condition imposed is clarified, and Felderhof's recent criticism of Kubo's derivation is commented upon. Finally, we demonstrate how the 'persistence' of equilibrium can be used to evaluate easily the equilibrium auto-correlation of the 'driven' variable (e.g., the velocity) from the transient solution of the corresponding stochastic equation.