Thermal mechanics: A quantum mechanical analogue of nonequilibrium statistical thermodynamics

A formal but not conventional equivalence between stochastic processes in nonequilibrium statistical thermodynamics and Schrödinger dynamics in quantum mechanics is shown. It is found, for each stochastic process described by a stochastic differential equation of Itô type, there exists a Schrödinger-like dynamics in which the absolute square of a wavefunction gives us the same probability distribution as the original stochastic process. In utilizing this equivalence between them, that is, rewriting the stochastic differential equation by an equivalent Schrödinger equation, it is possible to obtain the notion of deterministic limit of the stochastic process as a semi-classical limit of the “Schrödinger” equation. The deterministic limit thus obtained improves the conventional deterministic approximation in the sense of Onsager-Machlup. The present approach is valid for a general class of stochastic equations where local drifts and diffusion coefficients depend on the position. Two concrete examples are given. It should be noticed that the approach in the present form has nothing to do with the conventional one where only a formal similarity between the Fokker-Planck equation and the Schrödinger equation is considered.     

Aging in subdiffusion generated by a deterministic dynamical system

We investigate aging behavior in a simple dynamical system: a nonlinear map which generates subdiffusion deterministically. Asymptotic behaviors of the diffusion process are described using aging continuous time random walks. We show how these processes are described by an aging diffusion equation which is of fractional order. Our work demonstrates that aging behavior can be found in deterministic low dimensional dynamical systems.     

Some properties of the fundamental solution to the signalling problem for the fractional diffusion-wave equation

In this paper, the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order α, 1 ≤ α ≤ 2 and with constant coefficients is revisited. It is known that the diffusion and the wave equations behave quite differently regarding their response to a localized disturbance. Whereas the diffusion equation describes a process where a disturbance spreads infinitely fast, the propagation speed of the disturbance is a constant for the wave equation. We show that the time-fractional diffusion-wave equation interpolates between these two different responses and investigate the behavior of its fundamental solution for the signalling problem in detail. In particular, the maximum location, the maximum value, and the propagation velocity of the maximum point of the fundamental solution for the signalling problem are described analytically and calculated numerically.     

Nonstationarity induced by long-time noise correlations in the langevin equation

We solve the generalized Langevin equation driven by a stochastic force with a power-law autocorrelation function. A stationary Markov process has been applied as a model of the noise. However, the resulting velocity variance does not stabilize but diminishes with time. It is shown that algebraic distributions can induce such effects. Results are compared to those obtained with a deterministic random force. Consequences for the diffusion process are also discussed.     

On the diffusion of a fast molecule

We consider the motion of a heavy particle in interaction with an infinite ideal gas of slow atoms. We prove that the velocity of the heavy particle is, in a suitable limit, modeled by a deterministic process. We also treat the process of rescaled velocity fluctuations around a certain deterministic motion and show that this is appropriately modeled by a nonhomogeneous diffusion process.Supported in part by NSF grants PHY-8201708 and DMR81-14726.Supported in part by NSF grant PHY-8003298 and the Seton Hall University, Research and Faculty Development Council.     

Navier-Stokes Equation and Diffusions on the Group of Homeomorphisms of the Torus

A stochastic variational principle for the (two dimensional) Navier-Stokes equation is established. The velocity field can be considered as a generalized velocity of a diffusion process with values on the volume preserving diffeomorphism group of the underlying manifold. Navier-Stokes equation is reinterpreted as a perturbed equation of geodesics for the L ² norm. The method described here should hold as well in higher dimensions.     

Diffusion modeling of the Rayleigh piston

A Markov jump process in which a massive labeled particle undergoes random elastic collisions with a thermal bath is investigated. It is found that the behavior of the labeled particle can be divided into three distinct regimes depending on whether its velocity is (1) much less than, (2) on the order of, or (3) much greater than the mean speed of a bath particle. In each regime the jump process can be approximated by a particular continuous-path diffusion process. The first case corresponds to the Ornstein-Uhlenbeck process, while each of the latter can be modeled by a deterministic process with a nonlinear Langevin equation. In addition, in cases (2) and (3), the scaled deviation from the mean velocity can be modeled by a nonstationary diffusion. By scaling the time and letting the mass of the labeled particle become large, a continuous-path diffusion is constructed which approximates the jump process in each regime. Analytic solutions for the transition probability density are provided in each case, and numerical comparisons are made between the mean and variance of the diffusions and the original jump process.     

Weakly driven anomalous diffusion in non-ergodic regime: an analytical solution

We derive the probability density of a diffusion process generated by nonergodic velocity fluctuations in presence of a weak potential, using the Liouville equation approach. The velocity of the diffusing particle undergoes dichotomic fluctuations with a given distribution ψ(τ) of residence times in each velocity state. We obtain analytical solutions for the diffusion process in a generic external potential and for a generic statistics of residence times, including the non-ergodic regime in which the mean residence time diverges. We show that these analytical solutions are in agreement with numerical simulations.     

Three-dimensional Lorentz model in a magnetic field: Exact and Chapman–Enskog solutions

We derive the exact solution of the Boltzmann kinetic equation for the three-dimensional Lorentz model in the presence of a constant and uniform magnetic field. The velocity distribution of the electrons reduces exponentially fast to its spherically symmetric component. In the long time hydrodynamic limit there remains only the diffusion process governed by an anisotropic diffusion tensor. The systematic way of building the Chapman–Enskog solutions is described.     

Lorentz lattice gases: Basic theory

We present several ballistic models of the Lorentz gas in two-dimensional lattices with deterministic and stochastic deflection rules, and their corresponding Liouville equations. Boltzmann-level-equation results are obtained for the diffusion coefficient and velocity autocorrelation function for models with stochastic deflection rules. The long-time behavior of the mean square displacement is briefly discussed and the possibility of abnormal diffusion indicated. Even if the diffusion coefficient exists, its low-density limit may not be given correctly by the Boltzmann equation.     

Diffusion dynamics in branched spherical structure

Diffusion on a spherical surface with trapping is a common phenomenon in cell biology and porous systems. In this paper, we study the diffusion dynamics in a branched spherical structure and explore the influence of the geometry of the structure on the diffusion process. The process is a spherical movement that occurs only for a fixed radius and is interspersed with a radial motion inward and outward the sphere. Two scenarios govern the transport process in the spherical cavity: free diffusion and diffusion under external velocity. The diffusion dynamics is described by using the concepts of probability density function (PDF) and mean square displacement (MSD) by Fokker-Planck equation in a spherical coordinate system. The effects of dead ends, sphere curvature, and velocity on PDF and MSD are analyzed numerically in detail. We find a transient non-Gaussian distribution and sub-diffusion regime governing the angular dynamics. The results show that the diffusion dynamics strengthens as the curvature of the spherical surface increases and an external force is exerted in the same direction of the motion.     

Anomalous diffusion with absorption: exact time-dependent solutions

Recently, analytical solutions of a nonlinear Fokker-Planck equation describing anomalous diffusion with an external linear force were found using a nonextensive thermostatistical Ansatz. We have extended these solutions to the case when an homogeneous absorption process is also present. Some peculiar aspects of the interrelation between the deterministic force, the nonlinear diffusion, and the absorption process are discussed.     

QUALITATIVE AND QUANTITATIVE ANALYSIS OF STOCHASTIC PROCESSES BASED ON MEASURED DATA, I: THEORY AND APPLICATIONS TO SYNTHETIC DATA

Analysis of stochastic processes governed by the Langevin equation is discussed. The analysis is based on a general method for non-parametric estimation of deterministic and random terms of the Langevin equation directly from given data. Separate estimation of the terms corresponds to decomposition of process dynamics into deterministic and random components. Such decomposition provides a basis for qualitative and quantitative analysis of process dynamics. In Part I, the following analysis possibilities are described and illustrated using various synthetic datasets: (1) qualitative inspection of the estimated terms presented as fields, (2) reconstruction of the deterministic and stochastic evolution of the process and (3) approximation of the deterministic term by an analytical function and quantitative treatment of the equations obtained. In Part II, these analysis possibilities are applied to experimental datasets from metal cutting and laser-beam welding.     

一般阻尼随机微分方程的拟局部振荡算法

提出了一种模拟随机微分方程的拟局部振荡算法,即利用算符劈裂方法和势函数的泰勒展开,对噪声作用下耗散粒子的时间演化算符进行分解,得到了对应涨落行为的扩散算符和对应确定轨迹的漂移算符.其中局部简谐势场的涨落过程可获得解析解,而剩余的确定项则利用简单的Euler算法积分.应用到几个算例并与常用的两种算法相比较,结果表明:本算法随时间步长最稳定,可使用较大的时间步长.     

Dynamics of a creep-slip model of earthquake faults

Starting off from the relationship between time-dependent friction and velocity softening we present a generalization of the continuous, one-dimensional homogeneous Burridge–Knopoff (BK) model by allowing for displacements by plastic creep and rigid sliding. The evolution equations describe the coupled dynamics of an order parameter-like field variable (the sliding rate) and a control parameter field (the driving force). In addition to the velocity-softening instability and deterministic chaos known from the BK model, the model exhibits a velocity-strengthening regime at low displacement rates which is characterized by anomalous diffusion and which may be interpreted as a continuum analogue of self-organized criticality (SOC). The governing evolution equations for both regimes (a generalized time-dependent Ginzburg–Landau equation and a non-linear diffusion equation, respectively) are derived and implications with regard to fault dynamics and power-law scaling of event-size distributions are discussed. Since the model accounts for memory friction and since it combines features of deterministic chaos and SOC it displays interesting implications as to (i) material aspects of fault friction, (ii) the origin of scaling, (iii) questions related to precursor events, aftershocks and afterslip, and (iv) the problem of earthquake predictability. Moreover, by appropriate re-interpretation of the dynamical variables the model applies to other SOC systems, e.g. sandpiles.     

Hydrodynamics in two dimensions and vortex theory

We consider the Navier-Stokes equation for a viscous and incompressible fluid inR ². We show that such an equation may be interpreted as a mean field equation (Vlasov-like limit) for a system of particles, called vortices, interacting via a logarithmic potential, on which, in addition, a stochastic perturbation is acting. More precisely we prove that the solutions of the Navier-Stokes equation may be approximated, in a suitable way, by finite dimensional diffusion processes with the diffusion constant related to the viscosity. As a particular case, when the diffusion constant is zero, the finite dimensional theory reduces to the usual deterministic vortex theory, and the limiting equation reduces to the Euler equation.Partially supported by Italian CNR     

Aging properties of an anomalously diffusing particule

We report new results about the two-time dynamics of an anomalously diffusing classical particle, as described by the generalized Langevin equation with a frequency-dependent noise and the associated friction. The noise is defined by its spectral density proportional to ω^δ−1 at low frequencies, with 0<δ<1 (subdiffusion) or 1<δ<2 (superdiffusion). Using Laplace analysis, we derive analytic expressions in terms of Mittag–Leffler functions for the correlation functions of the velocity and of the displacement. While the velocity thermalizes at large times (slowly, in contrast to the standard Brownian motion case δ=1), the displacement never attains equilibrium: it ages. We thus show that this feature of normal diffusion is shared by a subdiffusive or superdiffusive motion. We provide a closed form analytic expression for the fluctuation–dissipation ratio characterizing aging.     

Microscopic derivation of a Markovian Master equation in a deterministic model of chemical reaction

We consider a (deterministic, conservative) one-dimensional system of colored hard points, changing color each time they hit one another with a relative velocity above a threshold. In the limit of rare reactions, theN-particle color distribution follows a Markovian birth-and-death process. Using the reaction rate as an intrinsic time scale, we also obtain the reaction-diffusion equation for a test particle in this hydrodynamic limit. Explicit results are given for a discrete and a Maxwellian velocity distribution.