Interaction quench in the Hubbard model

Motivated by recent experiments in ultracold atomic gases that explore the nonequilibrium dynamics of interacting quantum many-body systems, we investigate the opposite limit of Landau's Fermi-liquid paradigm: We study a Hubbard model with a sudden interaction quench, that is, the interaction is switched on at time t=0. Using the flow equation method, we are able to study the real time dynamics for weak interaction U in a systematic expansion and find three clearly separated time regimes: (i) An initial buildup of correlations where the quasiparticles are formed. (ii) An intermediate quasi-steady regime resembling a zero temperature Fermi liquid with a nonequilibrium quasiparticle distribution function. (iii) The long-time limit described by a quantum Boltzmann equation leading to thermalization of the momentum distribution function with a temperature T proportional, variantU.     

A comparison of projection operator formalisms for the study of self-diffusion

The utility of projection operator formalisms for describing the dynamics of many-body systems is studied, and the compatibility of these formalisms with certain approximation schemes is evaluated in the light of known behavior of such systems. For simplicity the investigation is limited to the study of Brownian motion. Specifically, a memory kernel formalism and a kinetic equation formalism are compared for the calculation of the time evolution of the momentum autocorrelation function. Both perturbation expansions and averaged propagator approximations are investigated. The results from these studies suggest that the long-time behavior of the momentum autocorrelation function is sensitive to the long-range nature of the interparticle potential.This work was supported in part by the National Science Foundation under Grant GK-19360X.     

Non-Markovian weak coupling limit of quantum Brownian motion

We derive and solve analytically the non-Markovian master equation for harmonic quantum Brownian motion proving that, for weak system-reservoir couplings and high temperatures, it can be recast in the form of the master equation for a harmonic oscillator interacting with a squeezed thermal bath. This equivalence guarantees preservation of positivity of the density operator during the time evolution and allows one to establish a connection between the dynamics of Schrödinger cat states in squeezed environments and environment-induced decoherence in quantum Brownian motion.

     

长时相干效应与分形布朗运动

 我们研究了阻尼布朗粒子，在具有幂律长时相干C(t)~t^-β（0<β<1，1<β<2）的无规涨落力作用下的运动情况。我们发现它是作分形布朗运动，而不是作普通的布朗运动，而且，找出了分形布朗运动的有效Fokker-Planck方程，以及相应的精确解。于是第一次把长时相干效应和分形布朗运动建立了定量的联系。     

An Enskog repeated-ring kinetic equation; Long-time tails and the Brownian limit

A kinetic equation for the motion of a particle of arbitrary size and mass through a moderately dense gas is derived and discussed. The “long-time tail” of the velocity correlation function is calculated and found to agree with existing results. For a Brownian particle, the theory gives the Stokes-Einstein law for the self-diffusion coefficient, with the shear viscosity given by its Enskog value.     

Nonergodic Brownian Motion

The long-time behavior of a system is suggested to confirm nonergodicity of non-Markovian Brownian dynamics, namely, whether the stationary probability density function （PDF） of the system characterized mainly by low moments of variables depends on the initial preparation. Thus we classify nonergodic Brownian motion into two classes： the class-I is that the PDF of a force-free particle depends on the initial velocity and the equilibration can be recovered through a bounded potential; while the PDF in the class-H depends on the initial coordinate and the equilibration can not be approached by introducing any potential. We also compare our result with the conditions of three kinds for ergodicity.     

Kinetic equations without “memory” for the time-displaced correlation functions

The relations between the kinetic equations with and without convolution in time are discussed on the basis of the kinetic equation for the Van Hove self-correlation function. Formal equivalence of both the equations is shown, and approximate scattering operators for the dilute-gas case and for the Brownian particle are considered.     

Brownian particles in random and quasicrystalline potentials: How they approach the equilibrium

We study the Brownian motion of an ensemble of single colloidal particles in a random square and a quasicrystalline potential when they start from non-equlibrium. For both potentials, Brownian dynamics simulations reveal a widespread subdiffusive regime before the diffusive long-time limit is reached in thermal equilibrium. We develop a random trap model based on a distribution for the depths of trapping sites that reproduces the results of the simulations in detail. Especially, it gives analytic formulas for the long-time diffusion constant and the relaxation time into the diffusive regime. Aside from detailed differences, our work demonstrates that quasicrystalline potentials can be used to mimic aspects of random potentials.     

Friction tensor for a pair of Brownian particles: Spurious finite-size effects and molecular dynamics estimates

In this work, we show thatin any finite system, the binary friction tensor for two Brownian particlescannot be directly estimated from an evaluation of the microscopic Green-Kubo formula, involving the time integral of force-force autocorrelation functions. This pitfall is associated with a subtle inversion of the thermodynamic and long-time limits and leads to spurious results for the estimates of the friction matrix based on molecular dynamics simulations. Starting from a careful analysis of the coupled Langevin equations for two interacting Brownian particles, we derive a method to circumvent these effects and extract the binary friction tensor from the correlation function matrix of the instantaneous forces exerted by the bath particles on the fixed Brownian particles, and from the relaxation of the total momentum of the bath in afinite system. The general methodology is applied to the case of two hard or soft Brownian spheres in a bath of light particles. Numerical estimates of the relevant correlation functions and of the resulting self and mutual components of the matrix of friction tensors are obtained by molecular dynamics simulations for various spacings between the Brownian particles. This paper is dedicated to B. Jancovici on the occassion of his 65th birthday.     

Quantum–Classical Correspondence Principle for Heat Distribution in Quantum Brownian Motion

Quantum Brownian motion, described by the Caldeira–Leggett model, brings insights to the understanding of phenomena and essence of quantum thermodynamics, especially the quantum work and heat associated with their classical counterparts. By employing the phase-space formulation approach, we study the heat distribution of a relaxation process in the quantum Brownian motion model. The analytical result of the characteristic function of heat is obtained at any relaxation time with an arbitrary friction coefficient. By taking the classical limit, such a result approaches the heat distribution of the classical Brownian motion described by the Langevin equation, indicating the quantum–classical correspondence principle for heat distribution. We also demonstrate that the fluctuating heat at any relaxation time satisfies the exchange fluctuation theorem of heat and its long-time limit reflects the complete thermalization of the system. Our research study justifies the definition of the quantum fluctuating heat via two-point measurements.     

Stochastic elastohydrodynamics of a microcantilever oscillating near a wall

We consider the thermally driven motion of a microcantilever in a fluid environment near a wall, a configuration characteristic of the atomic force microscope. A theoretical model is presented which accounts for hydrodynamic interactions between the cantilever and wall over a wide range of frequencies and which exploits the fluctuation-dissipation theorem to capture the Brownian dynamics of the coupled fluid-cantilever system. Model predictions are tested against experimental thermal spectra for a cantilever in air and water. The model shows how, in a liquid environment, the effects of non-delta-correlated Brownian forcing appear in the power spectrum, particularly at low frequencies. The model also predicts accurately changes in the spectrum in liquid arising through hydrodynamic wall effects, which we show are strongly mediated by the angle at which the cantilever is tilted relative to the wall.     

Curvilinear All-Atom Multiscale (CAM) Theory of Macromolecular Dynamics

A method is introduced for simulating long timescale macromolecular structural fluctuations and transitions with atomic-scale detail. The N-atom Liouville equation for the macromolecule/host medium system provides the starting point for the analysis. Order parameters characterizing overall macromolecular architecture are demonstrated to be slowly evolving. For single-stranded macromolecules, a curvilinear coordinate provides a way to introduce the order parameters. Using a multiscale approach, Fokker–Planck equations are derived. A nanocanonical method for constructing the lowest order solution to the Liouville equation and the equivalence of long-time and ensemble averages avoid the tedious bookkeeping needed to preserve the number of degrees of freedom (required in earlier methods). The method overcomes the large energy barriers that plague other approaches for estimating rates of transition between macromolecular conformations. A reduced dynamics is derived for the friction dominated limit. New experimental methods for observing macromolecular dynamics and medical sciences applications are discussed.     

On the persistence of breathers at deep water

The long-time behavior of a perturbation to a uniform wavetrain of the compact Zakharov equation is studied near the modulational instability threshold. A multiple-scale analysis reveals that the perturbation evolves in accord with a focusing nonlinear Schrodinger equation for values of wave steepness μ < μ₁ ≈ 0.274. The long-time dynamics is characterized by interacting breathers, homoclinic orbits to an unstable wavetrain. The associated Benjamin-Feir index is a decreasing function of μ, and it vanishes at μ₁. Above this threshold, the perturbation dynamics is of defocusing type and breathers are suppressed. Thus, homoclinic orbits persist only for small values of wave steepness μ ? μ₁, in agreement with recent experimental and numerical observations of breathers.     

From Ballistic to Diffusive Behavior in Periodic Potentials

The long-time/large-scale, small-friction asymptotic for the one dimensional Langevin equation with a periodic potential is studied in this paper. It is shown that the Freidlin-Wentzell and central limit theorem (homogenization) limits commute. We prove that, in the combined small friction, long-time/large-scale limit the particle position converges weakly to a Brownian motion with a singular diffusion coefficient which we compute explicitly. We show that the same result is valid for a whole one parameter family of space/time rescalings. The proofs of our main results are based on some novel estimates on the resolvent of a hypoelliptic operator.     

双层生物膜的动力学性质

采用傅里叶空间中的布朗运动方程研究了具有双层结构的生物膜的动力学性质,给出了生物膜在空间热库作用下随机运动的三维图像.研究表明,双层生物膜之间的相对滑动是一个非常重要的动力学过程,对高度-高度相关函数有非常显著的影响. 关键词： 双层细胞膜 相关函数     

Dynamical properties of strongly interacting Brownian particles : II. Self-diffusion

The self-diffusion process of Brownian particles is theoretically investigated for concentrated systems in the presence of strong potential interactions between particles. Starting from an N-particle diffusion equation, a formalism is developed to calculate the self-diffusion coefficient and the velocity autocorrelation function on the basis of the superposition approximation for the three-particle distribution function of non-equilibrium states. Explicit calculations are carried out for model systems of hard spheres with a screened Coulomb potential. Calculated time-dependent self-diffusion coefficients are compared with available data of the Brownian dynamics. Without introducing any phenomenological or adjustable parameters, quantitative agreement is achieved.     

Master equation for a quantum particle in a gas

The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts nonperturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.