Diffusion in different models of active Brownian motion

Active Brownian particles (ABP) have served as phenomenological models of self-propelled motion in biology. We study the effective diffusion coefficient of two one-dimensional ABP models (simplified depot model and Rayleigh-Helmholtz model) differing in their nonlinear friction functions. Depending on the choice of the friction function the diffusion coefficient does or does not attain a minimum as a function of noise intensity. We furthermore discuss the case of an additional bias breaking the left-right symmetry of the system. We show that this bias induces a drift and that it generally reduces the diffusion coefficient. For a finite range of values of the bias, both models can exhibit a maximum in the diffusion coefficient vs. noise intensity.     

Diffusive model of protein folding dynamics with Kramers turnover in rate

We study the folding kinetics of a three-helix bundle protein using a coarse polymer model. The folding dynamics can be accurately represented by one-dimensional diffusion along a reaction coordinate selected to capture the transition state. By varying the solvent friction, we show that position-dependent diffusion coefficients are determined by microscopic transitions on a rough energy landscape. A maximum in the folding rate at intermediate friction is explained by "Kramers turnover" in these microscopic dynamics that modulates the rate via the diffusion coefficient; overall folding remains diffusive even close to zero friction. For water friction, we find that the "attempt frequency" (or "speed limit") in a Kramers model of folding is about 2 micros-1, with an activation barrier of about 2kBT, and a folding transition path duration of approximately equal to 100 ns, 2 orders of magnitude less than the folding time of approximately equal to 10 micros.     

Fokker–Planck Equations as Scaling Limits of Reversible Quantum Systems

We consider a quantum particle moving in a harmonic exterior potential and linearly coupled to a heat bath of quantum oscillators. Caldeira and Leggett derived the Fokker–Planck equation with friction for the Wigner distribution of the particle in the large-temperature limit; however, their (nonrigorous) derivation was not free of criticism, especially since the limiting equation is not of Lindblad form. In this paper we recover the correct form of their result in a rigorous way. We also point out that the source of the diffusion is physically restrictive under this scaling. We investigate the model at a fixed temperature and in the large-time limit, where the origin of the diffusion is a cumulative effect of many resonant collisions. We obtain a heat equation with a friction term for the radial process in phase space and we prove the Einstein relation in this case.     

Uphill anomalous transport in a deterministic system with speed-dependent friction coefficient

We investigate the transport of a deterministic Brownian particle theoretically, which moves in simple onedimensional, symmetric periodic potentials under the influence of both a time periodic and a static biasing force. The physical system employed contains a friction coefficient that is speed-dependent. Within the tailored parameter regime, the absolute negative mobility, in which a particle can travel in the direction opposite to a constant applied force, is observed.This behavior is robust and can be maximized at two regimes upon variation of the characteristic factor of friction coefficient. Further analysis reveals that this uphill motion is subdiffusion in terms of localization(diffusion coefficient with the form D(t) ~t~(-1) at long times). We also have observed the non-trivially anomalous subdiffusion which is significantly deviated from the localization; whereas most of the downhill motion evolves chaotically, with the normal diffusion.     

Noise-Induced Drift in Stochastic Differential Equations with Arbitrary Friction and Diffusion in the Smoluchowski-Kramers Limit

We consider the dynamics of systems with arbitrary friction and diffusion. These include, as a special case, systems for which friction and diffusion are connected by Einstein fluctuation-dissipation relation, e.g. Brownian motion. We study the limit where friction effects dominate the inertia, i.e. where the mass goes to zero (Smoluchowski-Kramers limit). Using the Itô stochastic integral convention, we show that the limiting effective Langevin equations has different drift fields depending on the relation between friction and diffusion. Alternatively, our results can be cast as different interpretations of stochastic integration in the limiting equation, which can be parametrized by α∈?. Interestingly, in addition to the classical Itô (α=0), Stratonovich (α=0.5) and anti-Itô (α=1) integrals, we show that position-dependent α=α(x), and even stochastic integrals with α?[0,1] arise. Our findings are supported by numerical simulations.     

碳在面心立方系合金钢与金屬中微扩散所引起的内耗峰

用扭摆作内耗测量,发现了几种面心立方系合金钢(18/8型不锈钢及高锰钢)中含碳可以引起内耗峰。当振动频率约为每秒1周时,峰的巅值温度在200-300℃之间。当钢中固溶体的碳量增多时,内耗峰升高而峰的位置移向低温,当钢中所含的碳因回火而发生沉淀时,内耗峰降低而峰的位置移向高温。把内耗方法所测得的激活能、弛豫时间和由此计算所得的扩散系数与资料上所载的碳在面心立方系的钢中宏观扩散的数据相比较,指出所观测的内耗峰确是由于碳在钢中的微扩散所引起来的。用同样的实验方法也发现了碳在镍铝合金及在纯镍中由于微扩散而引起的内耗峰。这些实验指出,碳在面心立方系晶体中微扩散而引起内耗峰这件事实,可能是一种普遍的现象。     

位错芯区扩散引起的非线性滞弹性内耗峰

若干年来,我们对于出现在Al-Cu和Al-Mg系中的表现正常和反常振幅效应的坐落在室温附近的内耗峰进行了系统研究,测得的激活能接近于溶质原子在位错管道中扩散的激活能，从而认为内耗峰的基本过程是溶质原子在隹错芯内的扩散，并且提出了根据位错弯结模型的物理图像。在70年代，Windler-Gniewek等根据弦模型对于位错芯内的扩散进行了理论计算，推导出描述内耗行为的数学表达式与我们的实验结果有许多相似之处。本文对于弦模型和弯结模型进行了对比，分析了位错芯内的纵向扩散和横向扩散所引起的内耗的非线性表现以及内耗峰温和峰高随着应变振幅和测量温度而变化的情况，进一步了我们发现的室温非线性内耗峰（非线性滞弹性内耗）是由于溶质原子在位错芯内扩散所引起的。 关键词：     

Friction and diffusion coefficients in coordinate in nonequilibrium nuclear processes

The influence of different sets of friction and diffusion coefficients on the dynamics of a nuclear system is investigated. Taking as an example a dinuclear system we show in a “classic” investigation that with zero diffusion in the coordinate, the uncertainty relation can be violated during short initial times. Sets of diffusion coefficients are found for which the “classic” and quantum diffusion equations give close physical results. The tunneling through an energy barrier is sensitively influenced by the friction and diffusion coefficients in coordinate in the diffusion equation.     

模耦合理论计算纳米粒子在聚合物熔体中的含时扩散系数与常规扩散常数 (cited: 2)

分析和计算了纳米粒子在聚合物熔体中的含时扩散系数与常规扩散常数. 采用广义朗之万方程描述扩散动力学,并通过模耦合理论计算摩擦记忆内核.为简单起见,只考虑了来自两体碰撞和溶剂密度涨落耦合作用两类微观因素对摩擦记忆内核的贡献. 采用聚合物参考作用点模型以及Percus-Yevick闭合条件计算了聚合物-纳米粒子复合溶液的平衡态结构信息函数;详尽分析了纳米粒子的尺寸与聚合物链的尺寸对扩散动力学的影响. 揭示了结构函数、摩擦记忆内核以及扩散系数等随着纳米粒子半径和聚合物链长的变化关系. 结果表明,对于小尺寸的纳米粒子或者短链的聚合物,短时间的非马尔可夫扩散 动力学特征比较显著,含时扩散系数需要更长的时间弛豫到常规扩散常数. 微观因素对扩散常数的贡献随着纳米粒子尺寸的增加而减小,却随着聚合物链长的增加而增大. 此外,模耦合理论得到的扩散常数与Stokes-Einstein关系的预测值进行比较,发现对于小尺寸的纳米粒子或者长链的聚合物,微观因素对扩散常数的的贡献占主导地位. 相反,当纳米粒子较大或者聚合物链长较短时,流体力学的贡献会发挥重要作用.     

Kinetics of atoms in an elliptically polarized standing wave

We examine the kinetics of atoms with their ground and excited states being degenerate in the projection of angular momentum. The atoms are located in a standing wave with uniform elliptical polarization. Using the j _g =1/2→j _e =1/2 transition as an example, we show that the friction and diffusion of atoms strongly depend on the ellipticity of the field. For instance, in the low saturation limit the frictional force contains, in addition to the ordinary Doppler friction term, a term that can be interpreted as Sisyphean friction. Under certain conditions, the contributions reflecting the degeneracy of the ground state are dominant, with the result that the values of the friction and diffusion coefficients (and hence the rate of kinetic processes) may differ from the values predicted by the two-level atomic model by several orders of magnitude. Zh. éksp. Teor. Fiz. 115, 791–804 (March 1999)     

Surface diffusion anomaly near a substrate phase transition: H on W(100)

Using a linear optical diffraction method, we have experimentally studied the long predicted diffusion anomalous behavior for H/W(100) near the reconstructive phase transition of the W(100) substrate. This anomaly manifests itself in the form of a strong dip in the diffusion coefficient D at the transition temperature T(C). We interpret the strong reduction of D as a result of the diverging friction damping near the transition. The finite dip in D instead of a vanishing D at T(C) also demonstrates the importance of the non-Markovian (memory) deviation from the simple instantaneous damping picture.     

烧结过程中Ni-Al金属间化合物形成的内耗

Ni-Al金属间化合物是一类重要的高温结构材料,在多种领域具有明确的目标需求.粉末冶金技术是制备Ni-Al金属间化合物的一种重要选择.探索烧结过程中Ni-Al金属间化合物形成和转变过程,明确固相扩散反应发生温度和金属间化合物种类对调控烧结工艺和优化产品质量至关重要.本文采用内耗技术系统研究了Ni-Al粉末混合物压坯烧结过程的内耗行为.在内耗-温度谱上观察到一个显著内耗峰,随测量频率的增大而降低,但峰温无明显频率依赖性.同时,内耗峰随升温速率的增大向高温方向移动且峰值增加.分析认为,该峰与升温过程中金属间化合物NiAl3和Ni2Al3的形成有关,属于典型的相变内耗峰.此外,机械球磨可调控Ni-Al粉末混合物的微观结构,内耗峰随球磨时间增加向低温方向移动且峰值降低,表明固相扩散反应可在低温区域以较低速率进行.这与球磨过程中粉末颗粒的细化、粉末混合物的片层化、固溶度和表面能的提高以及缩短的原子扩散路径有关.同时也表明机械球磨可有效降低固相扩散反应起始温度进而降低烧结温度.     

Adiabatic elimination for systems of Brownian particles with nonconstant damping coefficients

We discuss the problem of eliminating the momentum variable in the phase space Langevin equations for a system of Brownian particles in two related situations: (i) position-dependent damping and (ii) existence of hydrodynamic interactions. We discuss the problems associated with the conventional elimination and we develop an alternative elimination procedure, in the Lagevin framework, which leads to the correct Smoluchowski equation. We give a heuristic argument on the basis of stochastic differential equations for the Smoluchowski limit and establish rigorously the limit for the general case of position-dependent friction and diffusion coefficents.     

用内耗方法研究碳在γ-铁中的扩散

在含锰1.7％的γ-铁中观察到因含碳而引起的内耗峰。当振动频率约为2周/秒时,内耗峰出现在240℃左右。这个内耗峰由于含碳量的增加而升高,由于在较高温度保温而降低。根据内耗测量所得的激活能是34,000±2000卡/克分子,与碳在γ-铁中的激活能(宏观扩散实验的数据)相合,由此可知所观测的内耗峰是由于碳在γ-铁(含锰1.7％)中的应力感生微扩散所引起来的。根据内耗测量的数据求出了碳在200—250℃的温度范围内在γ-铁中扩散的扩散系数。在logD—l/T坐标上,这些数据与宏观扩散实验所测得的在1000—1200℃的扩散系数坐落在一条直线上。所观测的内耗峰高度似乎与含碳量的二次方成正比,这表示在应力感生扩散过程中直接参加元跳动的乃是两个碳原子。由此可以提出一个产生内耗峰的初步模型:由于点阵中有空穴存在,两个碳原子很容易构成一个在各方向引起不同畸变的原子对,由于这个原子对在应力作用下的转动而产生内耗。     

A theoretical model for the collective motion of proteins by means of principal component analysis

A coarse grained model in the frame work of principal component analysis is presented. We used a bath of harmonic oscillators approach, based on classical mechanics, to derive the generalized Langevin equations of motion for the collective coordinates. The dynamics of the protein collective coordinates derived from molecular dynamics simulations have been studied for the Bovine Pancreatic Trypsin Inhibitor. We analyzed the stability of the method by studying structural fluctuations of the C _a atoms obtained from a 20 ns molecular dynamics simulation. Subsequently, the dynamics of the collective coordinates of protein were characterized by calculating the dynamical friction coefficient and diffusion coefficients along with time-dependent correlation functions of collective coordinates. A dual diffusion behavior was observed with a fast relaxation time of short diffusion regime 0.2–0.4 ps and slow relaxation time of long diffusion about 1–2 ps. In addition, we observed a power law decay of dynamical friction coefficient with exponent for the first five collective coordinates varying from −0.746 to −0.938 for the real part and from −0.528 to −0.665 for its magnitude. It was found that only the first ten collective coordinates are responsible for configuration transitions occurring on time scale longer than 50 ps.     

鋼鐵中碳、氮的擴散、脱溶和沉澱

一般认为钢的变脆例如蓝脆和回火脆都与钢中的扩散和沉淀有关系。本文尝试用内耗测量的方法研究碳与氮在α-铁及碳素钢中的扩散、脱溶和沉淀,从而进一步地了解钢的变脆的机构。实验的结果指出,碳在α-铁中的扩散不受自身浓度、合金元素和沉淀历史的影响。氮的情形与碳显著不同,合金元素使氮的扩散变慢,在沉淀初期使氮的扩散加速。较有系统地研究了碳、氮在加工后的α-铁中脱溶时所引起的内耗峰(当振动频率约为每秒1周时出现在250℃左右),发现了高温淬火在含碳或氮较多的试样中所产生的内应力也可以引起这个内耗峰。实验指出:①这个脱溶内耗峰的出现条件与钢的蓝脆的出现条件相同,都是一种应变时效或淬火时效的现象;②与这个内耗峰有关的碳、氮是处于原子的状态。这些联系使我们认为钢的蓝脆是一种应变脱溶或淬火脱溶的过程,是由于原子状态的碳或氮聚集于钢中的内应力区域或晶体缺陷(例如原子脱节)中所引起来的。关于这方面的深入研究正在继续进行中。根据本实验及以前关于钢铁中碳氮的扩散、脱溶和沉淀所得的结果,对于钢的回火脆的机构也提出了一种初步的看法。回火脆是一种回火沉淀的过程,引起回火脆的沉淀似乎是氮化物而不是碳化物。     

Inhomogeneous quantum diffusion and decay of a meta-stable state

We consider the quantum stochastic dynamics of a system whose interaction with the reservoir is considered to be linear in bath co-ordinates but nonlinear in system co-ordinates. The role of the space-dependent friction and diffusion has been examined in the decay rate of a particle from a meta-stable well. We show how the decay rate can be hindered by inhomogeneous dissipation due to nonlinear system–bath coupling strength.     

Hamiltonian and Brownian systems with long-range interactions: IV. General kinetic equations from the quasilinear theory

We develop the kinetic theory of Hamiltonian systems with weak long-range interactions. Starting from the Klimontovich equation and using a quasilinear theory, we obtain a general kinetic equation that can be applied to spatially inhomogeneous systems and that takes into account memory effects. This equation is valid at order 1/N in a proper thermodynamic limit and it coincides with the kinetic equation obtained from the BBGKY hierarchy. For N→+∞, it reduces to the Vlasov equation governing collisionless systems. We describe the process of phase mixing and violent relaxation leading to the formation of a quasistationary state (QSS) on the coarse-grained scale. We interpret the physical nature of the QSS in relation to Lynden-Bell’s statistical theory and discuss the problem of incomplete relaxation. In the second part of the paper, we consider the relaxation of a test particle in a thermal bath. We derive a Fokker-Planck equation by directly calculating the diffusion tensor and the friction force from the Klimontovich equation. We give general expressions of these quantities that are valid for possibly spatially inhomogeneous systems with long correlation time. We show that the diffusion and friction terms have a very similar structure given by a sort of generalized Kubo formula. We also obtain non-Markovian kinetic equations that can be relevant when the auto-correlation function of the force decreases slowly with time. An interesting factor in our approach is the development of a formalism that remains in physical space (instead of Fourier space) and that can deal with spatially inhomogeneous systems.     

Some New Features of Linear Theory for Describing Non-linear Diffusion

The non-linear flux equation, the non-linear Fokker-Planck equation (or Smoluchowski equation), and the non-linear Langiven equation are the basicequations for describing particle diffusion in non-ideal system subjected totime-dependent external fields. Nevertheless, the exact solution of thoseequations is still a challenge because of their inherent complexity of thenon-linear mathematics. Li et al. found that, based on the defined apparentvariables, the non-linear Fokker-Planck equation and the non-linear flux equation could be transformed to linear forms under the condition of strong friction limit or local equilibrium assumption. In this paper, some new features of the theory were found: (i) The linear flux equation for describing non-linear diffusion can be obtained from the irreversible thermodynamic theory; (ii) The linear non-steady state diffusion equation for describing non-linear diffusion of the non-steady state, which was described by the non-linear Fokker-Planck equation, can be derived more consistently from the microscopic molecular statistical theory; (iii) In the theory, thenon-linear Langiven equation also bears a linear form; (iv) For some special cases, e.g. diffusion in a periodic total potential system, the local equilibrium assumption or the strong friction limit is not required in establishing the linear theory for describing non-linear diffusion, so the linear theory may be important in the study of Brown motor.     

Effects of charge-density-wave depinning on the low frequency shear compliance of NbSe3

We report on measurements of the voltage dependence of the complex shear compliance of two crystals of the charge-density-wave (CDW) conductor NbSe₃, for torsional frequencies between 1 Hz and 71 Hz. For both samples, there is a frequency independent, ∼1% increase of the magnitude of the compliance when the voltage exceeds the threshold for CDW depinning, but the internal friction has a striking sample dependence which we do not understand. For one sample, there is a frequency independent decrease in internal friction with CDW depinning, suggesting that the elastic changes are not relaxational and might reflect changes in the screening of the crystal strain by the CDW. For the second sample, the sign and magnitude of the change in internal friction is strongly frequency dependent, which we associate with a change in screening due to the finite electron diffusion time. The second sample also exhibits a frequency dependent peak in internal friction near threshold that may reflect relaxation of the CDW phase. Received 14 August 2001 and Received in final form 3 October 2001