Nonergodic Brownian oscillator

Ergodicity breaking in well-behaved generalized Langevin equations

By means of the concepts of dissipative stability and stochastic realizability, the phenomenon of ergodicity breaking observed in Generalized Langevin Equations (GLEs) in the presence of nonvanishing friction factors [Phys. Rev. E 83, 062102 (2011)1539-375510.1103/PhysRevE.83.062102] can be properly explained: it occurs at the boundary of the region of dissipative stability; in those cases, this region coincides with that of stochastic realizability. This is the case of the Plyukhin model that considers a generalized Debye kernel. In the presence of dissipative kernels characterized by real-valued relaxation rates corresponding to the rheological behavior of viscoelastic fluids, since the domain of stochastic realizability is strictly contained within the region of dissipative stability, this phenomenon cannot be observed if Kubo's theory of fluctuation-dissipation holds. The hydromechanic theory of GLEs also provides a physical interpretation of the ergodicity breaking reported by [Chin. Phys. Lett. 22, 1845 (2005)0256-307X10.1088/0256-307X/22/8/006] that stems from a dissipationless ''fluid-inertial" effect, leading to superdiffusion.

Dynamic fluctuation-dissipation theory for generalized Langevin equations: Constructive constraints, stability, and realizability

Using the initial-value formulation, a dynamic theory for systems evolving according to a generalized Langevin equation is developed, providing conditions on the existence of equilibrium behavior and its fluctuation-dissipation implications. For systems fulfilling the property of local realizability, that for all the practical purposes corresponds to the postulate of the existence of a Markovian embedding, physical constraints, expressed in the form of dissipative stability and stochastic realizability are derived. If these two properties are met, Kubo theory is constructively recovered, while if one of these conditions is violated a thermodynamic equilibrium behavior does not exist (and this occurs also for "well-behaved dissipative systems"), with implications in the linear response theory.

Nonergodic Brownian oscillator: High-frequency response

We consider a Brownian oscillator whose coupling to the environment may lead to the formation of a localized normal mode. For lower values of the oscillator's natural frequency ω≤ω_{c}, the localized mode is absent and the unperturbed oscillator reaches thermal equilibrium. For higher values of ω>ω_{c} when the localized mode is formed, the unperturbed oscillator does not thermalize but rather evolves into a nonequilibrium cyclostationary state. We consider the response of such an oscillator to an external periodic force. Despite the coupling to the environment, the oscillator shows the unbounded resonance (with the response linearly increasing with time) when the frequency of the external force coincides with the frequency of the localized mode. An unusual resonance ("quasiresonance") occurs for the oscillator with the critical value of the natural frequency ω=ω_{c}, which separates thermalizing (ergodic) and nonthermalizing (nonergodic) configurations. In that case, the resonance response increases with time sublinearly, which can be interpreted as a resonance between the external force and the incipient localized mode.