Relaxation Processes and Fractional Differential Equations

Relaxation properties of different media (dielectrics, semiconductors, ferromagnetics, and so on) are normally expressed in terms of response function f(t) or of real and imaginary components of its Fourier transform dependent on the frequency . It had been recently recognized that most of real materials show deviation from classical Debye process. There exist a few empirical approximations of non-Debye response functions. One of them is the two-power approximation containing and , where and belong to the interval (0, 1). This formula gives the basis for introducing of fractional differential equation considered in this paper. A stochastic interpretation of this equation is offered; its solution is found and investigated. The results are in agreement with experimental data.     

Multiscale Analysis for Interacting Particles: Relaxation Systems and Scalar Conservation Laws

We investigate the derivation of semilinear relaxation systems and scalar conservation laws from a class of stochastic interacting particle systems. These systems are Markov jump processes set on a lattice, they satisfy detailed mass balance (but not detailed balance of momentum), and are equipped with multiple scalings. Using a combination of correlation function methods with compactness and convergence properties of semidiscrete relaxation schemes we prove that, at a mesoscopic scale, the interacting particle system gives rise to a semilinear hyperbolic system of relaxation type, while at a macroscopic scale it yields a scalar conservation law. Rates of convergence are obtained in both scalings.     

Algorithmical Unsolvability of the Ergodicity Problem for Locally Interacting Processes with Continuous Time

We prove algorithmical unsolvability of the ergodicity problem for a class of one-dimensional translation-invariant random processes with local interaction with continuous time, also known as interacting particle systems. The set of states of every component is finite, the interaction occurs only between nearest neighbors, only one particle can change its state at a time and all rates are 0 or 1.     

非球体填充的组合球模型及松弛算法

现有的松弛算法由于仅用于球填充而只考虑颗粒的平动,故提出考虑非球体转动的改进松弛算法并采用组合球模型,使其能够模拟任意形状非球体的随机填充以及多种非球体的混合填充.用多个球体的外包络面近似一个非球体外形的组合球模型,将非球体之间的接触转化为球体之间的接触,从而简化并统一非球体接触判断算法.通过引入非球体的转矩和转角松弛机制,使改进松弛算法克服了"自锁"现象,并能生成非球体的随机密填充.算例表明,填充结果与现有的数值模拟及实验结果相符.     

岩心孔隙介质中流体的核磁共振弛豫

弛豫时间是核磁共振研究中的一个重要参数,岩心孔隙介质流体的弛豫过程是自由流体弛豫机制、表面弛豫机制和流体的扩散弛豫机制共同作用的结果,它包含了丰富的孔隙和流体本身的信息. 弛豫时间和自扩散系数的测量及对弛豫时间的分析是核磁共振技术应用于岩心分析和石油勘测的重要内容.     

Relaxation effects in the quantification of fat using gradient echo imaging

Quantification of fat has been investigated using images acquired from multiple gradient echoes. The evolution of the signal with echo time and flip angle was measured in phantoms of known fat and water composition and in 21 research subjects with fatty liver. Data were compared to different models of the signal equation, in which each model makes different assumptions about the T1 and/or T2* relaxation effects. A range of T1, T2*, fat fraction and number of echoes was investigated to cover situations of relevance to clinical imaging. Results indicate that quantification is most accurate at low flip angles (to minimize T1 effects) with a small number of echoes (to minimize spectral broadening effects). At short echo times, the spectral broadening effects manifest as a short apparent T2 for the fat component.     

Relaxation behaviour of a biased two-level system,in metals in the weak damping limit

The dynamic properties of a biased two-level system in contact with a dissipative bath are studied in the weak coupling limit using a resolvent expansion method. The theory yields consistent results at low temperatures, a regime in which the widely used dilute bounce gas approximation (DBGA) to an underlying functional integral expression breaks down. The present results are however equivalent to a recently adapted functional integral technique that goes beyond the DBGA. The calculated expressions are relevant for analyzing the neutron scattering data on tunneling of light interstitials, e.g., hydrogen, in metals, at very low temperatures.     

Decay to equilibrium in random spin systems on a lattice. II

We study the continuous spin systems on ad3-dimensional lattice with random ferromagnetic interactions of finite range. We show that, if the temperature is sufficiently high and the probability of interaction to be large is small enough, the almost sure decay to equilibrium has a subexponential upper bound.     

Thermodynamical Approach to the Longest Common Subsequence Problem

We introduce an interacting particle model in a random media and show that this particle process is equivalent to the Longest Common Subsequence (LCS) problem of two binary sequences. We derive a differential equation which links the mean LCS-curve to the average speed of the particles given their density and prove that the average speed of the particles and density converges uniformly on every scale which is somewhat larger than  . All three authors are grateful to SFB 701. M.V. is grateful to CNPq (304561/2006–1 and 471925/2006–3) and FAPESP (thematic grant 04/07276–2) for partial support.     

Relaxation Property and Stability Analysis of the Quasispecies Models

The relaxation property of both Eigen model and Crow-Kimura model with a single peak fitness landscape is studied from phase transition point of view. We first analyze the eigenvalue spectra of the replication mutation matrices. For sufficiently long sequences, the almost crossing point between the largest and seeond-largest eigenvalues locates the error threshold at which critical slowing down behavior appears. We calculate the critical exponent in the limit of infinite sequence lengths and compare it with the result from numerical curve fittings at sufficiently long sequences. We find that for both models the relaxation time diverges with exponent 1 at the error （mutation） threshold point. Results obtained from both methods agree quite well. From the unlimited correlation length feature, the first order phase transition is further confirmed. Finally with linear stability theory, we show that the two model systems are stable for all ranges of mutation rate. The Igigen model is asymptotically stable in terms of mutant classes, and the Crow-Kimura model is completely stable.     

准分子介质中的弛豫振荡(英文)

本文描述了稀有气体卤化物准分子介质中光强的弛豫振荡,使用的高压混合气体由稀有气体.卤素和相应地缓冲(稀有)气体所组成.借助氩离子激光束(514.5nm)探测激活介质,测得三原子准分子Xe_2Cl的弛豫振荡周期值为4nm左右.系统用相对论强电子束进行泵浦.在对准分子介质的光学增益观测中,发现了光场强度弛豫振荡的有趣现象.这种振荡表明了光强与被激励介质间的相互作用.本文首次描述了准分子介质中的这种振荡,其物理学机制可以认为是:光强增加导致受激发射速率增加使得粒子数反转下降,这就引起光学增益减小,而光学增益的减小反过来又导致光强的减弱.我们假设,高压混合气体被电子束泵浦后形成均匀加宽的四能级系统,而激光下能级是空的.利用微扰方法,建立起了描述腔内光子数密度变化R_1的微分方程如下:(?)_1+B(?)_1+Cn_1=D.进一步,类比无线电技术中RLC线路理论,可以得到弛豫振荡的周期T为T=2α[t.τ/(p-1)]~(1/2).其中,p——是泵浦函数,t——是腔内光子寿命,τ——是除受激发射之外,其他一切因素造成的激发态寿命.由这一公式计算出的周期T理论值和实验中观测到的值基本上是一致的.这样,反过来就可由测量弛豫振荡周期,去确定泵浦函数或原子寿命,并和其他方法相佐证.     

Relaxation Property and Stability Analysis of the Quasispecies Models

The relaxation property of both Eigen model and Crow-Kimura model with a single peak fitness landscape is studied from phase transition point of view. We first analyze the eigenvalue spectra of the replication mutation matrices. For sufficiently long sequences, the almost crossing point between the largest and second-largest eigenvalues locates the error threshold at whichcritical slowing down behavior appears. We calculate the critical exponent in the limit of infinite sequence lengths and compare it with the result from numerical curve fittings at sufficiently long sequences. We find that for both models the relaxation time diverges with exponent 1 at the error (mutation) threshold point. Results obtained from both methods agree quite well. From the unlimited correlation length feature, the first order phase transition isfurther confirmed. Finally with linear stability theory, we show that the two model systems are stable for all ranges of mutation rate. The Eigen model is asymptotically stable in terms of mutant classes, and the Crow-Kimura model is completely stable.     

Fluorescence Study of the Relaxation Process in Excited Hetarylthiazole Cations

The absorption and fluorescence spectra of five cations protonated at the quinolyl nitrogen atom (IH⁺–VH⁺) and one ethylated (IEt⁺) cation were investigated. For these compounds (except VH⁺) both an anomalously large fluorescence Stokes shift (up to 238 nm) and a large short-wavelength fluorescence shift (up to 145 nm) at decreasing temperatures (down to 77 K) were observed. This is not the case for unprotonated molecules. The ground-state conjugation between quinolyl and another molecular fragment was found for II, IH⁺, IIH⁺, and IEt⁺. The relaxation process of excited cations is medium viscosity and temperature dependent. The experimental results are explained in terms of excited-state structural relaxation.     

Temperature Dependence of Structural Relaxation in Glass-Forming Liquids and Polymers

Understanding the microscopic mechanism of the transition of glass remains one of the most challenging topics in Condensed Matter Physics. What controls the sharp slowing down of molecular motion upon approaching the glass transition temperature T_g, whether there is an underlying thermodynamic transition at some finite temperature below T_g, what the role of cooperativity and heterogeneity are, and many other questions continue to be topics of active discussions. This review focuses on the mechanisms that control the steepness of the temperature dependence of structural relaxation (fragility) in glass-forming liquids. We present a brief overview of the basic theoretical models and their experimental tests, analyzing their predictions for fragility and emphasizing the successes and failures of the models. Special attention is focused on the connection of fast dynamics on picosecond time scales to the behavior of structural relaxation on much longer time scales. A separate section discusses the specific case of polymeric glass-forming liquids, which usually have extremely high fragility. We emphasize the apparent difference between the glass transitions in polymers and small molecules. We also discuss the possible role of quantum effects in the glass transition of light molecules and highlight the recent discovery of the unusually low fragility of water. At the end, we formulate the major challenges and questions remaining in this field.     

Relaxation in statistical many-agent economy models

We review some statistical many-agent models of economic and social systems inspired by microscopic molecular models and discuss their stochastic interpretation. We apply these models to wealth exchange in economics and study how the relaxation process depends on the parameters of the system, in particular on the saving propensities that define and diversify the agent profiles.     

Relaxation in the wealth exchange models

We examine the concept of relaxation in the wealth exchange models that are recently proposed in econophysics to interpret wealth distributions. To quantify and characterize the process of relaxation, we define an appropriate quantity and evaluate that numerically for the systems of many agents. Also, heuristic arguments are provided in support of some numerical results.     

Relaxation times and line widths of isotopically-substituted nitroxides in aqueous solution at X-band

Optimization of nitroxides as probes for EPR imaging requires detailed understanding of spectral properties. Spin lattice relaxation times, spin packet line widths, nuclear hyperfine splitting, and overall lineshapes were characterized for six low molecular weight nitroxides in dilute deoxygenated aqueous solution at X-band. The nitroxides included 6-member, unsaturated 5-member, or saturated 5-member rings, most of which were isotopically labeled. The spectra are near the fast tumbling limit with T₁∼T₂ in the range of 0.50–1.1 μs at ambient temperature. Both spin–lattice relaxation T₁ and spin–spin relaxation T₂ are longer for ¹⁵N- than for ¹⁴N-nitroxides. The dominant contributions to T₁ are modulation of nitrogen hyperfine anisotropy and spin rotation. Dependence of T₁ on nitrogen nuclear spin state m_I was observed for both ¹⁴N and ¹⁵N. Unresolved hydrogen/deuterium hyperfine couplings dominate overall line widths. Lineshapes were simulated by including all nuclear hyperfine couplings and spin packet line widths that agreed with values obtained by electron spin echo. Line widths and relaxation times are predicted to be about the same at 250 MHz as at X-band.     

取代基对卟啉三阶光学非线性及弛豫过程的影响

研究了取代基对卟啉三阶光学非线性和激发态弛豫过程的影响，发现在同一取代位置，推电子基团取代使得卟啉三阶光学非线性极化纱数Ｘ值相对影响，而吸电子基团取代使得卟啉Ｘ值相对减少，对同一取代基在不同取代装置，发现推电子基团在邻位取代的Ｘ值大于其在对位取代卟啉的Ｘ值，而吸电子基团在邻位取代的Ｘ值小于其在对位取代卟啉的Ｘ值，另外，还发现推电子基团取代的卟啉弛豫过程比吸电子基团取代的卟啉弛豫过程快７倍。     

Freezing Transitions in Non-Fellerian Particle Systems

Non-Fellerian particle systems are characterized by nonlocal interactions, somewhat analogous to non-Gibbsian distributions. They exhibit new phenomena that are unseen in standard interacting particle systems. We consider freezing transitions in one-dimensional non-Fellerian processes which are built from the abelian sandpile additions to which in one case, spin flips are added, and in another case, so called anti-sandpile subtractions. In the first case and as a function of the sandpile addition rate, there is a sharp transition from a non-trivial invariant measure to the trivial invariant measure of the sandpile process. For the combination sandpile plus anti-sandpile, there is a sharp transition from one frozen state to the other anti-state.