Equilibration problem for the generalized Langevin equation

We consider the problem of equilibration of a single oscillator system with dynamics given by the generalized classical Langevin equation. It is well-known that this dynamics can be obtained if one considers a model where the single oscillator is coupled to an infinite bath of harmonic oscillators which are initially in equilibrium. Using this equivalence we first determine the conditions necessary for equilibration for the case when the system potential is harmonic. We then give an example with a particular bath where we show that, even for parameter values where the harmonic case always equilibrates, with any finite amount of nonlinearity the system does not equilibrate for arbitrary initial conditions. We understand this as a consequence of the formation of nonlinear localized excitations similar to the discrete breather modes in nonlinear lattices. Introduction. – One of the simplest phenomenological ways of modeling the interactions of a system with a heat bath is through the Langevin equation which, for a single particle, of unit mass and moving in a one dimensional potential V (x), is given by ẍ = −dV (x)/dx − γẋ+ η(t) , (1) where η(t) is a Gaussian white noise which satisfies the fluctuation-dissipation (FD) relation 〈η(t)η(t)〉 = 2kBTγδ(t − t ) . Here T is the temperature of the heat bath. The dynamics in eq. (1) ensures that at long times the system reaches thermal equilibrium. Thus the phase space density P (x, p, t), where p = ẋ, converges in the limit t → ∞ to the Boltzmann distribution es/Z where Hs = p /2+V (x) and Z is the corresponding partition function. The proof for this uses the correspondence between the Langevin equation and the Fokker-Planck equation [1]. However the δ-correlated nature of the noise is unphysical and this has led to the study of the generalized Langevin equation [2, 3] ẍ = −dV (x)/dx − ∫ t −∞ dt′γ(t− t′)ẋ(t′) + η(t) , (2) where the noise is correlated and the the dissipative term involves a memory kernel. The noise is again Gaussian and is related to the dissipative term through the generalized FD relation 〈η(t)η(t′)〉 = kBTγ(t− t ′) . (3) A standard method of microscopically modeling a heat bath is to consider an infinite collection of harmonic oscillators, with a distribution of frequencies, coupled linearly to the system. In that case it can be shown [4, 5] that the effective equation of motion of the system is precisely given by eq. (2) where the dissipation kernel γ(t) depends on the bath oscillator frequencies and the coupling constants. Unlike the case with δ-correlated noise there exists no general proof that, for a general potential V (x), the system will reach thermal equilibrium at long times. The reason for this is that in this case the construction of a Fokker-Planck description is difficult and is known in few cases (e.g. harmonic oscillator case treated in Ref. [6]). For the special case of a harmonic potential and with certain restrictions on the form of γ(t) one can prove equilibration by a direct solution of the equations of motion [7]. In the quantum mechanical case the oscillator bath model has been widely used to model the effects of noise, dissipation and decoherence in quantum systems [8,9]. In this case the approach to equilibrium has been proved only for a special class of potentials for the cases where the system-reservoir coupling is weak [10]. A number of papers [11–13] have attempted to understand various aspects of the generalized Langevin equation such as anomalous diffusion, nonstationarity and ergodic-