Equilibrium mean square displacement

For thejth subunit in a weakly bending filament (Lbending modes have all relaxed to their equilibrium mean squared displacements, one has(109)... 

For the system in thennal equilibrium, one can compute the time-dependent mean square displacement (ICr)... 

The exponent Mk depends on the mean square displacement of the atom from its equilibrium position and hence upon temperature. It is linear with (kT/m Xsin / where k is the Boltzmann constant, T the absolute temperature, the scattering angle, the wavelength and m the atomic mass (for a monatomic material). In addition there are complicated expressions dependent upon the crystal symmetry. As an example, for silicon at room temperature the /, are reduced by approximately 6%. With this correction all the equations of dynamical theory still apply. 

In sharp contrast to the large number of experimental and computer simulation studies reported in literature, there have been relatively few analytical or model dependent studies on the dynamics of protein hydration layer. A simple phenomenological model, proposed earlier by Nandi and Bagchi [4] explains the observed slow relaxation in the hydration layer in terms of a dynamic equilibrium between the bound and the free states of water molecules within the layer. The slow time scale is the inverse of the rate of bound to free transition. In this model, the transition between the free and bound states occurs by rotation. Recently Mukherjee and Bagchi [14] have numerically solved the space dependent reaction-diffusion model to obtain the probability distribution and the time dependent mean-square displacement (MSD). The model predicts a transition from sub-diffusive to super-diffusive translational behaviour, before it attains a diffusive nature in the long time. However, a microscopic theory of hydration layer dynamics is yet to be fully developed. 

Atoms taking part in diffusive transport perform more or less random thermal motions superposed on a drift resulting from field forces (V//,-, Vrj VT, etc.). Since these forces are small on the atomic length scale, kinetic parameters established under equilibrium conditions (i.e., vanishing forces) can be used to describe the atomic drift and transport, The movements of atomic particles under equilibrium conditions are Brownian motions. We can measure them by mean square displacements of tagged atoms (often radioactive isotopes) which are chemically identical but different in mass. If this difference is relatively small, the kinetic behavior is... 

Figure 5. Mean squared displacement for the fractional (a = 1 /2, full line) and normal (dashed) Omstein-Uhlenbeck process. The Brownian process shows the typical proportionality to t for small times it approaches the saturation value much faster than its subdiffusive analogue, which starts off with the r /2 behavior and approaches the thermal equilibrium value by a power law, compare Eq. (61)...

Lattice vibrations are described as follows [9, 10]. If the deviation of an atom from its equilibrium position is u, then <(u2> is a measure for the average deviation of the atom (the symbol < ) represents the time average note that <(u> = 0). This so-called mean-squared displacement depends on the solid and the temperature, and is characteristic for the rigidity of a lattice. Lattice vibrations are a collective phenomenon they can be visualized as the modes of vibration... 

In Section VI, we consider a classical particle diffusing in an out-of-equilibrium environment. In this case, all the dynamical variables attached to the particle, even its velocity, are aging variables. We analyze how the drift and diffusion properties of the particle can be interpreted in terms of an effective temperature of the medium. From an experimental point of view, independent measurements of the mean-square displacement and of the mobility of a particle immersed in an aging medium such as a colloidal glass give access to an out-of-equilibrium generalized Stokes-Einstein relation, from which the effective temperature of the medium can eventually be deduced. 

Therefore, before describing the modification of the equilibrium FDT, we need to study in details the behavior of D(t). Note, however, that the integrated velocity correlation function [, Cvv(/) df takes on the meaning of a time-dependent diffusion coefficient only when the mean-square displacement increases without bounds (when the particle is localized, this quantity characterizes the relaxation of the mean square displacement Ax2 t) toward its finite limit Ax2(oo)). 

Here we show how the modified Kubo formula (187) for p(co) leads to a relation between the (Laplace transformed) mean-square displacement and the z-dependent mobility (z denotes the Laplace variable). This out-of-equilibrium generalized Stokes-Einstein relation makes explicit use of the function (go) involved in the modified Kubo formula (187), a quantity which is not identical to the effective temperature 7,eff(co) however re T (co) can be deduced from this using the identity (189). Interestingly, this way of obtaining the effective temperature is completely general (i.e., it is not restricted to large times and small frequencies). It is therefore well adapted to the analysis of the experimental results [12]. 

As displayed by the out-of-equilibrium generalized Stokes-Einstein relation (203), independent measurements of the particle mean-square displacement and frequency-dependent mobility in an aging medium give access, once Ax2(z) and p(z) = p(co = iz) are determined, to T (z) and to T (co) = T (z = — iffi). Then, the identity (189) yields the effective temperature ... 

When the environment of the particle is itself out-of-equilibrium, as is the case for a particle evolving in an aging medium, we showed how the study of both the mobility and the diffusion of the particle allows one to obtain the effective temperature of the medium. We derived an out-of-equilibrium generalized Stokes-Einstein relation finking the Laplace transform of the mean-square displacement and the z-dependent mobility. This relation provides an efficient way of deducing the effective temperature from the experimental results. 

When a particle moves in brownian motion, the chance that it will ever return to its initial position is negligibly small. Thus, there will be a net displacement with time of any single particle, even though the average displacement for all particles is zero. For example, during a short time interval one particle may move a distance sls another a distance s2 and so on. Some of these displacements will be positive, others negative some up, others down but with equilibrium conditions the sum of the displacements will be zero. It is possible to estimate the displacement of any particle in terms of its root-mean-square displacement. 

Thus, during dynamics, the particles could leave the central box. At equilibrium, it is not necessary for the particles to be brought back into the central box. However, when this must be done, the PBC procedure, which is similar to minimum imaging, can be performed. In this procedure, the particle coordinates q,- are converted to scaled coordinates s,-. These are then brought into the ntral cubic box by means of the dnint operation, and then unsealed using h to give back q, in the central cell. Because unshifted particle coordinates along the trajectory are often required (to calculate, e.g., mean-squared displacements), it is not necessary to perform PBC under equilibrium conditions. 

Temperature factor An exponential expression by which the scattering of an atom is reduced as a consequence of vibration (or a simulated vibration resulting from static disorder). For isotropic motion the exponential factor is exp(—5iso sin 0/A ), where Biso is the isotropic temperature factor. It equals 87r (ti ), where (ti ) is the mean-square displacement of the atom from its equilibrium position. For anisotropic motion the exponential expression usually contains six parameters, the anisotropic vibration or displacement parameters, which describe ellipsoidal rather than isotropic (spherically symmetrical) motion or average static displacements. 

Simulations—isoergic and isothermal, by molecular dynamics and Monte Carlo—as well as analytic theory have been used to study this process. The diagnostics that have been used include study of mean nearest interparticle distances, kinetic energy distributions, pair distribution functions, angular distribution functions, mean square displacements and diffusion coefficients, velocity autocorrelation functions and their Fourier transforms, caloric curves, and snapshots. From the simulations it seems that some clusters, such as Ar, 3 and Ar, 9, exhibit the double-valued equation of state and bimodal kinetic energy distributions characteristic of the phase change just described, but others do not. Another kind of behavior seems to occur with Arss, which exhibits a heterogeneous equilibrium, with part of the cluster liquid and part solid. 

Latora et al. [18] discussed a relation between the process of relaxation to equilibrium and anomalous diffusion in the HMF model by comparing the time series of the temperature and of the mean-squared displacement of the phases of the rotators. They showed that anomalous diffusion changes to a normal diffusion after a crossover time, and they also showed that the crossover time coincides with the time when the canonical temperature is reached. They also claim that anomalous diffusion occurs in the quasi-stationary states. 

Figure 12. Time series of the mean-square displacement of the phases aj t). N = 100,1000, and 10,000 from top to bottom. The vertical axis is the original scale only for N = 10,000, and is multiplied by 103 and 106 for N = 1000 and 100, respectively, just for a graphical reason. In Stage I where the system is quasi-stationary, the numerical results are approximated by solid curves that are obtained from Eq. (12) using functions in Eq. (19). After the system reaches equilibrium, diffusion becomes normal. Anomaly in diffusion is observed only in Stage II. The two short vertical lines on each curve show the end of Stages I and II, which correspond to the ones found in Fig. 2. [Reproduced with permission from Y. Y. Yamaguchi, Phys. Rev. E 68, 066210 (2003). Copyright 2004 by the American Physical Society.]...

To obtain the diffusion constant, D, we consider two alternative equilibrium time correlation function approaches. First, D can be obtained from the long time limit of the slope of the time-dependent mean square displacement of the electron from its starting position. The quantum expression for this estimator is... 

At the moment the noise in the mean squared displacement versus time curves does not permit easy observation of this type of time-dependent behavior, and it remains a matter for future investigation. Also of interest is the question of whether, under the highly diffusive conditions of computer simulations, the systems very near equilibrium relax exponentially ( single relaxation time ) or otherwise, as for laboratory glasses. ... 

From Equation 4 and by analogy with the normal diffusion equation, H(Am) = 7r2/t to reach 99% equilibrium. Equation 16 then reduces to the familiar relationship of the diffusion coefficient to the mean square displacement distance,... 

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