Scaling laws and memory effects in the dynamics of liquids and proteins

Recent progress in the numerical calculation of memory functions from molecular dynamics simulations allowed the gaining of deeper insight into the relaxation dynamics of liquids and proteins. The concept of memory functions goes back to the work of R. Zwanzig on the generalized Langevin equation, and it was the basis for the development of various dynamical models for liquids. In this article we present briefly a method for the numerical calculation of memory functions, which is then applied to study their scaling behavior in normal and fractional Brownian dynamics. It has been shown recently that the model of fractional Brownian dynamics constitutes effectively a link between protein dynamics on the nanosecond time scale, which is accessible to molecular dynamics simulations and thermal neutron scattering, and the much longer time scale of functional protein dynamics, which can be studied by fluorescence correlation spectroscopy. The text was submitted by the authors in English. Affiliated with the University of Orléans.     

Non-equilibrium dynamics of simple spherical spin models

We investigate the non-equilibrium dynamics of spherical spin models with two-spin interactions. For the exactly solvable models of the d-dimensional spherical ferromagnet and the spherical Sherrington-Kirkpatrick (SK) model the asymptotic dynamics has for large times and large waiting times the same formal structure. In the limit of large waiting times we find in both models an intermediate time scale, scaling as a power of the waiting time with an exponent smaller than one, and thus separating the time-translation-invariant short-time dynamics from the aging regime. It is this time scale on which the fluctuation-dissipation theorem is violated. Aging in these models is similar to that observed in spin glasses at the level of correlation functions, but different at the level of response functions, and thus different at the level of experimentally accessible quantities like thermoremanent magnetization. Received 22 April 1999     

Dynamic fluctuations in optical bistability

We present results on the time correlation function of a system undergoing absorptive optical bistability. We first use the Zwanzig-Mori formalism to calculate time correlation functions both near marginal stability points and in the coexistence region. Near marginal stability the theory predicts large deviations from a single exponential form of the correlation function. The truncated continued fraction expansion is shown to become inapplicable close to the coexistence point. The difficulties are due to the presence of long time scales, viz. the very large mean first passage times between the two metastable steady states. When these scales are important we show that the memory kernel relaxation is no longer faster than that of a field fluctuation. An increase in the size of the system increases the disparity of the time scales and thus exacerbates the problems of the projection operator formalism as used here following reference [14].We next present an ansatz for the correlation function incorporating the four major time scales important near coexistence, the two single branch relaxation times and the two mean first passage times for transitions between the stable states. This form of the correlation function avoids the difficulties cited in connection with the use of the projection operator method.     

Relaxation time distributions for an anomalously diffusing particle

As well known, the generalized Langevin equation with a memory kernel decreasing at large times as an inverse power law of time describes the motion of an anomalously diffusing particle. Here, we focus attention on some new aspects of the dynamics, successively considering the memory kernel, the particle’s mean velocity, and the scattering function. All these quantities are studied from a unique angle, namely, the discussion of the possible existence of a distribution of relaxation times characterizing their time decay. Although a very popular concept, a relaxation time distribution cannot be associated with any time-decreasing quantity (from a mathematical point of view, the decay has to be described by a completely monotonic function).Technically, we use a memory kernel decaying as a Mittag-Leffler function (the Mittag-Leffler functions interpolate between stretched or compressed exponential behaviour at short times and inverse power law behaviour at large times). We show that, in the case of a subdiffusive motion, relaxation time distributions can be defined for the memory kernel and for the scattering function, but not for the particle’s mean velocity. The situation is opposite in the superdiffusive case.     

Reorientational motion of acetone in solutions

Dipolar correlation functions of acetone in various solvents are obtained by Fourier transform of infrared spectra. The analysis of the shapes of correlation functions shows that the reorientational process is diffusional in the time scale longer than about 0.3 × 10^?12 sec. In nonpolar solvents the ratios of correlation times τ_A : τ_B : τ_C do not vary in different solvents, and the reorientation around b axis is more restricted than around the other axis. The length of correlation time is dependent on the density of solvent rather than the viscosity. These observations are unexpected if only the electrostatic interaction is considered. In polar solvents correlation times are longer than in nonpolar solvents and the tendency is especially remarkable in τ_B. An electrostatic interaction between polar molecules seems to be responsible for these observations. The solvent effect on the frequency shifts of carbonyl stretching vibration is consistent with the result of correlation times.     

Velocity correlation functions for finite one-dimensional systems

Certain systems consisting of a one-dimensional gas of a finite number of point particles interacting with a “hard-core” potential are considered.We use the technique developed by Jepsen to calculate exactly the velocity correlation functions for these systems. We discover that after a slow decay for times of the order of the relaxation time, there is a “fast” decay to the equilibrium value on a macroscopic time scale characterized by L/v_TH (L is the length of the container and v_TH the thermal velocity).The dependence of the velocity correlation functions on the initial position of the specified particle is also considered. In particular, the behaviour approaching the boundary of the container is analyzed. These considerations are generalized to systems of higher spatial dimension.     

A new method for measuring the outer scale of atmospheric turbulence

A method, based on the correlation functions of the wandering (lateral fluctuations) of two separated thin parallel laser beams propagating through the atmosphere, is proposed for measuring the outer scale of atmospheric turbulence. The method utilizes the ratio between the correlation functions of the wandering in two perpendicular planes. A simple relationship to obtain the outer scale from the measured correlation functions is established for a particular model of turbulence, the modified von Karman model. The method is validated by experimental data both in the laboratory and in the open air. To this purpose special measurements, which were conducted during an outdoor measurements campaign in 2004, are analysed. An instantaneous value of the outer scale can be obtained accurately. The results show that the size of outer scale of atmospheric turbulence changes very rapidly with time and that the proposed method can be applied in real atmospheric conditions.     

Time correlation functions in classical dense fluids

The velocity autocorrelation functions and memory functions of dense classical fluids may be directly obtained from the static radial distribution function g(r) in an approximate way. Following the Mori projection operator formalism, the memory functions may be related to the fluctuating force correlation. At low densities, these functions may be evaluated by following the trajectories of particle pairs in the interatomic potential. At higher densities, the force correlation functions can be evaluated approximately from particle pair trajectories via the potential of the mean force. The contributions to the memory function come mainly from particle pairs with rather specific and rather short interatomic distances. At higher temperatures, this specific distance is even shorter, hence the memory function decays quickly with time. At lower temperatures, a negative region of the memory function may develop. On the other hand, there is relatively little density dependence of the normalized memory function. The results for argon fluids at various densities and temperatures agree satisfactorily with the molecular dynamics and the Enskog values. The decrease of the diffusion coefficient with density is partly due to the nature of g(r) which reflects the stronger clustering of atoms at higher densities.     

A statistical-mechanical theory of self-diffusion in glass-forming liquids

A statistical-mechanical theory of self-diffusion in glass-forming liquids is presented. A non-Markov linear Langevin equation is derived from a Newton equation by employing the Tokuyama-Mori projection operator method. The memory function is explicitly written in terms of the force-force correlation functions. The equations for the mean-square displacement, the mean-fourth displacement, and the non-Gaussian parameter are then formally derived. The present theory is applied to the glass transitions in the glass-forming liquids to discuss the crossover phenomena in the dynamics of a single particle from a short-time ballistic motion to a long-time self-diffusion process via a β (caging) stage. The effects of the renormalized friction coefficient on self-diffusion are thus explored with the aid of analyses of the simulation results by the mean-field theory proposed recently by the present author. It is thus shown that the relaxation time of the renormalized memory function is given by the β-relaxation time. It is also shown that for times longer than the β-relaxation time the dynamics of a single particle is identical to that discussed in the suspensions.     

A new approach to visualizing spectral density functions and deriving motional correlation time distributions: applications to an alpha-helix-forming peptide and to a well-folded protein

A new approach to visualizing spectral densities and analyzing NMR relaxation data has been developed. By plotting the spectral density function, J(omega), as F(omega)=2 omega J(omega) on the log-log scale, the distribution of motional correlation times can be easily visualized. F(omega) is calculated from experimental data using a multi-Lorentzian expansion that is insensitive to the number of Lorentzians used and allows contributions from overall tumbling and internal motions to be separated without explicitly determining values for correlation times and their weighting coefficients. To demonstrate the approach, (15)N and (13)C NMR relaxation data have been analyzed for backbone NH and C(alpha)H groups in an alpha-helix-forming peptide 17mer and in a well-folded 138-residue protein, and the functions F(omega) have been calculated and deconvoluted for contributions from overall tumbling and internal motions. Overall tumbling correlation time distribution maxima yield essentially the same overall correlation times obtained using the Lipari-Szabo model and other standard NMR relaxation data analyses. Internal motional correlational times for NH and C(alpha)H bond motions fall in the range from 100 ps to about 1 ns. Slower overall molecular tumbling leads to better separation of internal motional correlation time distributions from those of overall tumbling. The usefulness of the approach rests in its ability to visualize spectral densities and to define and separate frequency distributions for molecular motions.     

Continuous-Time Random Walk with multi-step memory: an application to market dynamics

An extended version of the Continuous-Time Random Walk (CTRW) model with memory is herein developed. This memory involves the dependence between arbitrary number of successive jumps of the process while waiting times between jumps are considered as i.i.d. random variables. This dependence was established analyzing empirical histograms for the stochastic process of a single share price on a market within the high frequency time scale. Then, it was justified theoretically by considering bid-ask bounce mechanism containing some delay characteristic for any double-auction market. Our model appeared exactly analytically solvable. Therefore, it enables a direct comparison of its predictions with their empirical counterparts, for instance, with empirical velocity autocorrelation function. Thus, the present research significantly extends capabilities of the CTRW formalism.     

Correlations in the adiabatic response of chaotic systems

Adiabatic variation of the parameters of a chaotic system results in a fluctuating reaction force. The quantum analog of a classical dissipative force, proportional to the time integral of the force-force correlation function, vanishes. We study this quantum-classical crossover for random matrix models. For the Gaussian unitary ensemble the crossover is found to take place on the Heisenberg time scale and the finite time integral practically vanishes for longer times. For the Gaussian orthogonal case, there is no such time scale and the integral falls off inversely proportional to time.     

On the material frame indifference controversy: Some results from group theory and computer simulation.

Group theory and computer simulation are used to show that molecular and atomic scale time cross correlation functions have different symmetry properties in different frames of reference. Constitutive equations, written in any frame of reference, are approximations to classical molecular dynamics, and must therefore reproduce the results from group theory applied to ensemble averages such as time correlation functions, and corroborated by computer simulation. This may help to resolve some of the contemporary literature claims regarding material frame indifference and the implementation of “co- rotating” frames.     

A memory function model for the velocity autocorrelation function and the self-diffusion coefficient in simple dense fluids

A memory-function model is used to compute the velocity autocorrelation function and the self-diffusion coefficient of a dense Lennard-Jones fluid from the zero-time correlation functions of the molecular velocity and its first two time derivatives. It is shown that these zero-time correlation functions can be evaluated in terms of the radial distribution function and the pair potential only, i.e. without considering higher order correlation functions. Since molecular dynamics results are available for the radial distribution function as well as the velocity autocorrelation function and the self-diffusion coefficient, a rigorous test of the chosen memory function is possible. The agreement is reasonable, although generally not within the error bands of the molecular dynamics results.     

A statistical-mechanical theory of slow dynamics near the glass transition

A statistical-mechanical theory of slow dynamics near the glass transition in two kinds of glass-forming systems, (M) molecular systems and (S) suspensions of colloids, is presented from a unified point of view based on the Tokuyama-Mori projection operator method. The exact diffusion equations for the coherent- and the incoherent-intermediate scattering functions are first derived, whose memory functions are convolutionless in time and contain the correlation effects due to the hydrodynamic interactions in (S). The analytic expressions of the memory functions are then calculated within the mode-coupling theory (MCT) approximation and are shown to coincide with the conventional ones obtained by MCT. Alternative mode-coupling equations are thus obtained in (M) and (S) separately. Self-diffusion is also discussed. Alternative equations for the mean-square displacement and the non-Gaussian parameter are also derived within MCT approximation. All results in both the systems are compared with those obtained by MCT.     

On the propagation of hydrodynamic interactions

We present explicit expressions in real space and time for the equilibrium correlation functions of the density and velocity fields in a compressible fluid. This allows to discuss the time scales involved in the propagation of hydrodynamic interactions and its relevance on the dynamics of suspended colloidal particles. We show that hydrodynamic interactions develop in a sonic time scale, not in a vorticity time scale.     

聚酯聚氨酯分子动力学的核磁共振研究

测量了对聚酯聚氨酯在浓溶液状态下质子自旋-晶格弛豫时间随温度的变化,对其中有代表性的基团进行了详细的研究,根据BPP理论用各向同性模型计算分子局部运动的相关时间,从计算结果来看,聚酯聚氨酯的局部分子运动是符合BPP理论的,硬链段的局部分子运动比软链段的局部分子运动要慢得多,两者相差至少一个数量级以上;质子的自旋-晶格弛豫时间主要是偶极-偶极作用的影响;强烈的氢键有相互作用使得硬链段基因的局部运动相关时间接近;通过形成氢键不同的NH峰的相关时间和活化能的观察可以研究软、硬段间的相互作用。     

35MeV/u 36Ar+112,124Sn反应中中等质量碎片关联函数的入射道依赖

³⁶Ar+^112,124Sn反应中小角关联出射的中等质量碎片(IMF)约化速度关联函数. 结果表明³⁶Ar+¹²⁴Sn反应系统中的约化速度关联函数在小约化速度处的反关联程度比³⁶Ar+¹¹²Sn反应系统中的强, 表现出明显的入射道依赖性. 考察出射粒子对的单核子总动量时, 发现这种差异主要来自于高动量粒子对的贡献. 用三体弹道理论模型MENEKA分别计算了两个系统的IMF发射时标, 在³⁶Ar+¹¹²Sn反应系统中约为150fm/c, 而在³⁶Ar+¹²⁴Sn反应系统中, 约为120fm/c. 同位旋相关的量子分子动力学计算表明, ³⁶Ar+¹²⁴Sn系统中IMF的发射时间谱比³⁶Ar+¹¹²Sn系统略有前移, 相应地, 其中心密度从最高点随时间的下降亦比³⁶Ar+¹¹²Sn系统略快.     

Fluctuations in a fluid under a stationary heat flux II. Slow part of the correlation matrix

The general formulas, derived in a previous paper, are used to calculate the correlation functions of the hydrodynamic variables in the Rayleigh-Bénard system. The behavior of the correlation functions on a time scale slow compared to that of sound propagation is determined, using systematically nonequilibrium hydrodynamic eigenmodes. These (slow) eigenmodes of the linearized Boussinesq equations in the presence of gravity and a temperature gradient are the viscous and the visco-heat modes. They are determined for ideal heat-conducting plates with stick boundary conditions. The visco-heat modes are found to behave qualitatively different from those obtained with slip boundary conditions. Using these eigenmodes, the slow part of the correlation functions can be determined explicitly. On a small length scale, as probed by light scattering, we recover the same expression for the Rayleigh line as quoted in the literature. On larger length scales, as probed by microwaves, the coupling of gravity to the temperature gradient gives rise to a convective instability (heating form below) or to propagating visco-heat modes (heating from above). The corresponding correlation functions and the Rayleigh line are calculated and discussed.