Dynamics in condensed phase for systems involving phase functions obeying Gaussian statistics

We have carried out molecular dynamic simulations over a range of densities for two-dimensional fluids consisting of hard, soft, and Lennard-Jones disks. For comparison we have also carried out simulations for the corresponding systems in which all but one particle are frozen in position. We have studied the velocity autocorrelation functions and the closely related velocity-sign autocorrelation functions, and have examined the probabilities per unit time that a particle will undergo a first velocity sign reversal after an elapsed time t measured alternately from the last velocity reversal or from a given arbitrary time. At all densities studied, the first of these probabilities per unit time is zero at t=0 and rises to a maximum at a later time, but as the hardness of the disks is increased, the maximum moves in toward t→0. This maximum can be correlated with the ‘‘negative’’ dip observed in the velocity correlation functions when plotted versus time. Our conclusion is that all these phenomena can be explained qualitatively on the basis of a model where memory does not extend back beyond the last velocity reversal. However, at high density, the velocity-sign-autocorrelation function not only shows a negative dip (which is explained by the model) but also a second ‘‘oscillation’’ which is not described, even qualitatively, by the model. We conclude that the first dip in the velocity and velocity-sign correlation functions can occur even if there are no correlated or coherent librations, but the existence of a ‘‘second’’ oscillation is a better indication of such correlations.

Consider two interacting sub-systems with nonoverlapping charge distributions. How can the properties of the combined system be expressed in terms of properties of the individual sub-systems alone? This general problem appears in a wide variety of physical, chemical, and biological systems [Joester 2003, Stone 1996, Moore 2001]. In this chapter we provide a prescription for addressing this issue by the computation of (i) response functions and (ii) correlation functions of spontaneous fluctuations of relevant degrees of freedom in the individual sub-systems. The computation of response and correlation functions is greatly simplified by using the density matrix in Liouville space [Mukamel 1995]. Hilbert and Liouville spaces offer very different languages for the description of nonlinear response. Computing dynamical observables in terms of the wavefunction in Hilbert space requires both forward and backward propagations in time. In contrast, the density matrix calculated in Liouville space should only be propagated forward. The choice is between following only the ket, moving it both forward and backward, or following the joint forward dynamics of the ket and the bra. Artificial time variables (Keldysh loops) commonly used in many-body theory [Haug 1996] are connected with the wavefunction. The density matrix which uses the real laboratory time throughout the calculation offers a more intuitive picture. Wavefunction-based theories calculate transition amplitudes, which by themselves are not observable. The density matrix, on the other hand directly calculates physical observables. Moreover, dephasing processes (damping of off-diagonal elements of the density matrix caused by phase fluctuations) which are naturally included into the Liouville space formulation may not be described in terms of the wavefunction. The “causality paradox” of TDDFT [Gross 1996] can be clearly resolved in Liouville space [Mukamel 2005]. In this chapter we present a method for expressing the joint response of two interacting sub-systems in terms of their correlation and response functions. This factorization appears quite naturally in Liouville space. The p order response of the individual systems is a linear combination of 2 distinct (p + 1)-point correlation functions known as Liouville space pathways [Mukamel 1995], which differ by whether the interaction at each time

The evolution of density correlations in an expanding universe can be described by the BBGKY equations. This approach has been the subject of several previous studies, but always under the assumption of small-amplitude fluctuations, where the hierarchy of equations has a natural truncation. Reslts of these studies cannot be compared to the present universe because the galaxy two-point correlation function xi (r) is much greater than unity at r9 or approx. =1h/sup -1/ Mpc, and the three-point function zeta is on the order of xi (r)/sup 2/. In this strongly nonlinear situation the hierarchy is dominated by terms ignored in the linear analysis. Our method of truncating the hierarchy is based on the empirical result that zeta can be represented to good accuracy as a simple function of xi. We solve the equations via the velocity-moment method, and we truncate the resulting velocity-moment hierarchy for the two-point function by assuming that the distribution in the relative velocity of particle pairs has zero skewness about the mean. The second equation in this velocity-moment hierarchy is our main equation for xi. It involves the three-point spatial correlation function zeta, which we write as a function of xi following the empirical result. Themore » third equation involves the first velocity moment of the three-point position and velocity correlation function. We model this term in a way consistent with our model for zeta and with a constraint equation that expresses conservation of triplets.The equations admit a similarity transformation if (1) the effects of the discreteness of particles can be ignored, (2) the initial spectrum of density perturbations assumes a power law shape, and (3) the universe is described by an Einstein-de Sitter model (..cap omega..approx. =1). The numerical results presented here are based on this similarity solution.« less

In order to study the microdynamics of translational and orientational motion of ions in liquid electrolytes, a molecular dynamics simulation of molten LiNO3, which includes all degrees of freedom of vibrating nitrate ions, has been performed. Simple Coulomb pair potentials with Born-type repulsions have been adopted in the simulation. The calculated properties include: translational velocity and force correlation functions, rotational correlation functions, and angular momentum and torque correlation functions. The nature of the ionic motion is analyzed in terms of quasioscillation and quasilibration. Local structures surrounding nitrate ions, which can be characterized by, e.g., the coordination number, are found to persist for a long time (∼1 ps). The coupling between the environmental structure and the translational and orientational behaviors of a given nitrate ion has been examined through selectively sampled correlation functions. We found that the change of coordination number enhances the translational diffusion and the rotational diffusion is slow if the coordination number remains low. This shows that the dynamics are very sensitive to the exact shape of the potential and the dynamics itself. Results of the simulation are compared with the experimental diffusion constants and dynamical behaviors revealed by Raman spectroscopy. While the interionic interaction assumed has been found to describe quite well the static structure of molten LiNO3, simulated translational diffusion and rotational relaxation seem to be somewhat too fast, suggesting too shallow an attractive well in the potential we have assumed.

Velocity correlation functions of charged particles derived from the Fokker–Planck equation

The scaling behaviour of the correlation functions of the velocity and temperature fields in convective turbulence at moderately high Rayleigh number is studied through direct numerical integration. An extended self-similarity analysis is applied. Neat evidence for Bolgiano scaling of the spatial correlation functions is obtained.

We consider stochastic vibrations of road vehicles excited by random road surfaces. The vehicle is modeled as a multibody system consisting of rigid bodies coupled by springs and dampers. Characteristics of springs and dampers are approximated by polynomials. The resulting mathematical model is a system of nonlinear ordinary differential equations of second order with random input functions. Its solution is found by using a perturbation method. Approximations of correlation and spectral density functions of motions, velocities and accelerations of the rigid bodies are calculated using limit theorems of weakly correlated random functions. Numerical results are presented.

Stochastic Liouville equations are used to study the relaxation behavior of position and orientation correlation functions of a fluid particle, if the second-order velocity or angular-velocity correlation is given. Two assumptions for reducing higher-order (angular) velocity correlation functions to two-point correlation functions are compared, the Gaussian assumption and ’’the assumption of vanishing higher-order G-cumulants.’’ In the case of translation both assumptions predict the same diffusion constant; only the latter one yields a Burnett coefficient that diverges in accordance with microscopic results. For rotational diffusion the second assumption is studied and compared with results obtained by Pomeau and Weber on the basis of the first assumption.

On the brownian movement of a rotator in a cosine potential

The responses of gel electrolytes to stimuli make them useful in applications such as sensors and actuators. However, few studies have explored their transport properties from a molecular viewpoint. We studied the transport coefficients of gel electrolytes based on perfluorinated sulfonic acid using molecular dynamics simulations. The transport coefficients for electric and pressure fields, namely, the ionic conductivity, Darcy permeability, and cross coupling constant, were calculated based on Kubo's linear response theory from the corresponding velocity correlation functions and mean square displacements. The effects of the water content of the gel electrolyte and those of the monovalent cationic species were also analyzed. The calculated transport coefficients qualitatively agree with the reported experimental results. The role of the cross coupling constants in determining the functional efficiency of gel electrolytes as pressure sensors or electroactive actuators is discussed.

An artificial random force is introduced into Burgers' model equation for turbulence. This forced model equation is solved numerically as an initial-value problem. Both the driving force and initial velocity field are assumed Gaussian and are generated by a white noise process. Many statistical properties of this model for turbulence are studied. By adjusting the external force, the turbulence can reach an equilibrium state. The velocity correlation function and energy spectrum are calculated for the equilibrium turbulence. It is found that the energy spectrum falls off as the inverse second power of the wavenumber. The velocity correlation function is similar to the result obtained in real turbulence experiments. With Gaussian random driving force and Gaussian initial velocity field, it is found that the velocity field remains very nearly Gaussian by comparing the fourth-order velocity correlation with the quasinormal assumption. Although the process remains very nearly Gaussian, it is found that the projection of the process on the initial white noise process becomes smaller and smaller. This is to be expected, since the dynamic system is escaping from the original random base.

We present a new approach to the modelling of sound radiated from vibrating structures that will provide a platform for simulation of larger problems by the method of Dynamical Energy Analysis (DEA), which characterises sound using densities in a ray-dynamical phase space. We consider in particular the radiated acoustic power from plates with diffuse bending vibrations, which we characterise using a two-point correlation function of normal velocities. Such field-field correlation functions are transformed to ray-dynamical phase space densities using a Wigner transformation, which allows insertion into ray-based methods such as DEA. Using correlation functions to characterise plate vibrations also allows the method to cater for stochastic, noisy driving of such systems. The results for the radiation efficiency of a plate are presented in an asymptotic form, with leading contributions from the plate interior and its boundary and with corner corrections also being given for particular boundary conditions and right-angled corners. A notable feature of this analysis is that the bulk contribution vanishes below a critical frequency, and the asymptotic estimate of radiated power then leads with a boundary contribution. This is shown to agree well with a more traditional calculation based on modal analysis in the special case of a rectangular plate.

The translational dynamics of supercooled and normal liquid water is investigated via a specific correlation function DeltaB with the aim of explaining the behavior of the centers of mass velocity correlation function (VCF). DeltaB is divided into diffusive and nondiffusive parts that yield separated contributions to the VCF, namely an Enskog-type diffusive one, modeled by an exponential function, and a nondiffusive one, made up by damped oscillations of a vanishing time integral. In the translational density of states (DOS), the oscillations yield the bands at omega(1) congruent with 50 cm(-1), omega(3) congruent with 240 cm(-1) (the two well-known experimental bands of the Raman spectra) and omega(2) congruent with 160 cm(-1) (the Einstein frequency of the liquid). It is shown that the chief negative lobe of the VCF is mainly due to the DOS component at the lowest frequency omega(1). The study of the relative pair dynamics shows that this lobe is due to the transverse dynamics, while the longitudinal one determines the fast DOS component at omega(3). The presence of a negative tail is highlighted. Its contribution extends beyond the region of the fast dynamics (t<0.7 ps) up to about 1.5 ps and is due to a low-frequency oscillating mode that produces a low-frequency DOS band centered at about omega(0)=20 cm(-1).

Lagrangian velocity and acceleration statistics of fluid and inertial particles measured in pipe flow with 3D particle tracking velocimetry

The memory functions for the velocity, angular-momentum, and dipolar autocorrelation functions from a series of molecular-dynamics studies of liquid carbon monoxide are examined. The velocity and angular-momentum memory functions decay initially almost to zero in a Gaussian fashion. However, their long-time behavior has a much slower time dependence. The dipolar memory function from a simulation using a strong noncentral potential is approximately this system's angular-momentum autocorrelation function. Approximate velocity and angular-momentum correlation functions are generated from approximate memory functions and the results are compared to experiment. Gaussian memories based on the second and fourth moments of the corresponding autocorrelation functions give the best agreement with experiment. However, none of the approximate memories examined adequately represents the long-time behavior of the experimental memories. The static atomic radial distribution functions are given and are shown to depend upon the strength of the orientational parts of the pair potential used in the dynamics calculations. The non-Gaussian characteristics of the Van Hove self-correlation functions are examined and shown to depend on the potential and number of particles used in the dynamics calculations. The intermediate scattering function and its memory are also examined.