谐势中的广义 Langevin 方程：反常扩散与能量均分

Anomalous Diffusion and the Generalized Langevin Equation

The Generalized Langevin Equation (GLE) is a Stochastic Integro-Differential Equation that is commonly used to describe the velocity of microparticles that move randomly in viscoelastic fluids. Such particles commonly exhibit what is known as anomalous subdiffusion, which is to say that their position Mean-Squared Displacement (MSD) scales sublinearly with time. While it is common in the literature to observe that there is a relationship between the MSD and the memory structure of the GLE, and there exist special cases where explicit solutions exist, this connection has never been fully charac…

Homogenization for Generalized Langevin Equations with Applications to Anomalous Diffusion

We study homogenization for a class of generalized Langevin equations (GLEs) with state-dependent coefficients and exhibiting multiple time scales. In addition to the small mass limit, we focus on homogenization limits, which involve taking to zero the inertial time scale and, possibly, some of the memory time scales and noise correlation time scales. The latter are meaningful limits for a class of GLEs modeling anomalous diffusion. We find that, in general, the limiting stochastic differential equations (SDEs) for the slow degrees of freedom contain non-trivial drift correction terms and are …

Anomalous Thermodynamics in Homogenized Generalized Langevin Systems

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 fun…

Asymptotic analysis of the mean squared displacement under fractional memory kernels

The generalized Langevin equation (GLE) is a universal model for particle velocity in a viscoelastic medium. In this paper, we consider the GLE family with fractional memory kernels. We show that, in the critical regime where the memory kernel decays like $1/t$ for large $t$, the mean squared displacement (MSD) of particle motion grows linearly in time up to a slowly varying (logarithm) term. Moreover, we establish the well-posedness of the GLE in this regime. This solves an open question from [Mckinley 2018 Anomalous] and completes the answer to the conjecture put forward in [Morgado 2002 Rel…

The Fractional Langevin Equation: Brownian Motion Revisited

We have revisited the Brownian motion on the basis of the fractional Langevin equation which turns out to be a particular case of the generalized Langevin equation introduced by Kubo on 1966. The importance of our approach is to model the Brownian motion more realistically than the usual one based on the classical Langevin equation, in that it takes into account also the retarding effects due to hydrodynamic backflow, i.e. the added mass and the Basset memory drag. On the basis of the two fluctuation-dissipation theorems and of the techniques of the Fractional Calculus we have provided the ana…

分数布朗运动的动力学基础及相关具有随机扩散系数和时变随机Hurst参数的反常扩散模型的极限定理

Limit Theorems for the Dynamical Foundation of the Fractional Brownian Motion and Related Models of Anomalous Diffusion with Random Diffusion Coefficient and Time-Dependent Random Hurst parameter

本文通过由欠阻尼Langevin方程描述的多质量布朗粒子环境，在N→∞极限下为带随机扩散系数和时变随机Hurst参数的分数布朗运动及更一般反常扩散模型建立了动力学基础，并给出相应广义Kolmogorov–Fokker–Planck方程。