Homogenization for Generalized Langevin Equations with Applications to Anomalous Diffusion

Lim, Soon Hoe; Wehr, Jan; Lewenstein, Maciej

doi:10.1007/s00023-020-00889-2

Cited by 12 publications

(11 citation statements)

References 70 publications

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“…The following theorem follows from a straightforward application of theorem B.1. The last statement in the theorem follows from the proof of theorem B.1 (see (Lim et al 2020) for details).…”

Section: S H Limmentioning

confidence: 89%

“…We consider a class of non-Markovian Langevin equations, whose coefficients are possibly state-dependent, describing the dynamics of a particle moving in a force field and interacting with the environment. The evolution of the particle's position, x t ∈ R d , t 0, is given by the solution to the following stochastic integro-differential equation (SIDE) (Lim et al 2020):…”

Section: Introductionmentioning

confidence: 99%

“…The results obtained in this paper not only recover our earlier results in (Bo et al 2019), but also give new results and uncover interesting insights in more general settings. We emphasize that the present paper has a rather applied flavor, i.e., it focuses on applying the general homogenization theorems obtained in our earlier works (Lim et al 2020, Lim and to shed light on the above issue in the field of stochastic thermodynamics, rather than presenting novel mathematical techniques and proofs for homogenization. Moreover, the notion of stochastic Lévy areas is, for the first time, connected to thermodynamic quantities such as work done on physical systems.…”

Section: Introductionmentioning

confidence: 99%

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Anomalous thermodynamics in homogenized generalized Langevin systems

Lim

2021

J. Phys. A: Math. Theor.

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We study functionals, such as heat and work, along trajectories of a class of multi-dimensional generalized Langevin systems in various limiting situations that correspond to different level of homogenization. These are the situations where one or more of the inertial time scale(s), the memory time scale(s) and the noise correlation time scale(s) of the systems are taken to zero. We find that, unless one restricts to special situations that do not break symmetry of the Onsager matrix associated with the fast dynamics, it is generally not possible to express the effective evolution of these functionals solely in terms of trajectory of the homogenized process describing the system dynamics via the widely adopted Stratonovich convention. In fact, an anomalous term is often needed for a complete description, implying that convergence of these functionals needs more information than simply the limit of the dynamical process. We trace the origin of such impossibility to area anomaly, thereby linking the symmetry breaking and area anomaly. This hold important consequences for many nonequilibrium systems that can be modeled by generalized Langevin equations. Our convergence results hold in a strong pathwise sense.

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Section: S H Limmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Anomalous thermodynamics in homogenized generalized Langevin systems

Lim

2021

J. Phys. A: Math. Theor.

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“…In both cases, the coupling term in the initial Hamiltonian was nonlinear, which would lead to quantum stochastic equations with inhomogeneous damping and multiplicative noise. This is not always tractable in the case of non-Ohmic spectral densities [78][79][80], presenting a challenge for mathematical physics. The recipe used in both cases was to linearize this coupling term and subsequently to establish the regimes of validity in which this assumption holds.…”

Section: Introductionmentioning

confidence: 99%

Quantum dynamics of a Bose polaron in a d -dimensional Bose-Einstein condensate

et al. 2021

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We study the quantum motion of an impurity atom immersed in a Bose-Einstein condensate in arbitrary dimensions. It was shown, for all dimensions, that the Bogoliubov excitations of the Bose-Einstein condensate act as a bosonic bath for the impurity, where linear coupling is possible for a certain regime of validity, which was assessed only in one dimension. Here we present the detailed derivation of the d-dimensional Langevin equations that describe the quantum dynamics of the system, and of the associated generalized tensor that describes the spectral density in the full generality, and assesses the linear assumption in all dimensions. As results, we obtain, when the impurity is not trapped, the mean square displacement in all dimensions, showing that the motion is superdiffusive. We obtain also explicit expressions for the superdiffusive coefficient in the small and large temperature limits. We find that, in the latter case, the maximal value of this coefficient is the same in all dimensions, but is only reachable in one dimension, within the validity of the assumptions. We study also the behavior of the average energy and compare the results for various dimensions. In the trapped case, we study squeezing and find that the stronger position squeezing can be obtained in lower dimensions. We quantify the non-Markovianity of the particle's motion and find that it increases with dimensionality.

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“…A generalized homogenization theorem for Langevin systems was proved in [13]. Lim et al [14] discussed generalized Langevin equation for non-Markovian anomalous diffusions. We point out that most existing works mentioned above are for Gaussian noise.…”

Section: Introductionmentioning

confidence: 99%

Homogenization of Dissipative Hamiltonian Systems under Lévy Fluctuations

Wang¹,

Lv²,

Duan³

2021

Preprint

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This work is devoted to deriving small mass limiting equation for a class of Hamiltonian systems with multiplicative Lévy noise. Derivation of the limiting equation depends on the structure of the stochastic Hamiltonian systems, in which a noise-induced drift term arises. We prove convergence to the limiting equation in probability under appropriate assumptions on smoothness and boundedness. Furthermore, we demonstrate convergence in moment under stronger assumptions. A Lévy type Smoluchowski-Kramers approximation result is presented as an illustrative example.

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Homogenization for Generalized Langevin Equations with Applications to Anomalous Diffusion

Cited by 12 publications

References 70 publications

Anomalous thermodynamics in homogenized generalized Langevin systems

Anomalous thermodynamics in homogenized generalized Langevin systems

Quantum dynamics of a Bose polaron in a d -dimensional Bose-Einstein condensate

Homogenization of Dissipative Hamiltonian Systems under Lévy Fluctuations

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