Autocorrelation function current

The diffusion coefficient D is one-third of the time integral over the velocity autocorrelation function CvJJ). The second identity is the so-called Einstein relation, which relates the self-diffusion coefficient to the particle mean square displacement (i.e., the ensemble-averaged square of the distance between the particle position at time r and at time r + f). Similar relationships exist between conductivity and the current autocorrelation function, and between viscosity and the autocorrelation function of elements of the pressure tensor. 

Since j-c-v, the electrical current density autocorrelation function and the velocity autocorrelation function are proportional to each other. The latter function, however, can be expressed with the help of the time derivative of the decaying pro-... 

Using linear response theory and noting (according to the results at the end of Section 5.1.3) that the (complex) electrical conductivity a is the Fourier transform of the current density autocorrelation function, we obtain from Eqn. (5.75) (see the equivalent Eqn. (5.21))... 

Up to now, we have given a general theoretical development of the self-beat technique. As a practical illustration of the experimental apparatus used to detect autocorrelation functions in scattering experiments, the equipment currently used in our laboratory will now be described. While our treatment of the autocorrelation function has been in terms of an analog signal, the computer that measures this function is actually a digital device. This is based on the fact that it is also valid to count the scattered photons in order to calculate Ci(r) as the optical intensity signal is essentially determined by the number of photons that strike the photocathode per unit time. We have then... 

The expression of the transverse current autocorrelation function can also be derived from the linearized hydrodynamic equations. Because it is decoupled from all the longitudinal modes, the derivation is simple and the final expression in wavenumber and Laplace frequency plane can be written as... 

It is well known that the velocity autocorrelation function decays as f3/2 in the asymptotic limit due to the coupling between the tagged particle motion and the transverse current mode of the solvent [23, 56, 57]. The asymptotic limit of the Rn term can be calculated by assuming that Fs(q, t) and Ctt(q, t) have simple diffusive behavior. Thus the expression for Rn in this limit takes the following form ... 

In calculating the contribution from the current term given by Eq. (135), the required solvent dynamical variables are C (q, f), the current autocorrelation function of the solvent, and Ctio(q, t), the short-time part of the same. 

The inertial part of the the current autocorrelation function C Q(q, t) is given by... 

At high density the contribution from the transverse current terms becomes negligible since the transverse current autocorrelation function decays rapidly. 

To obtain an approximate expression for the density autocorrelation function, first we consider that the density fluctuation is coupled only to the longitudinal current fluctuation, and its coupling to the temperature fluctuation and other higher-order components are neglected. 

With this consideration die relaxation equation will give rise to a set of coupled equations involving the time autocorrelation function of the density and the longitudinal current fluctuation, and also there will be cross terms that involve the correlation between the density fluctuation and the longitudinal current fluctuation. This set of coupled equations can be written in matrix notation, which becomes identical to that derived by Gotze from the Liouvillian resolvent matrix [3]. 

One can now approximate the current autocorrelation function in the diffusive limit. When the time integration is performed, the above expression reduces to... 

The other dynamic variables required to calculate Rpp(t) and Rrr(t) are the dynamic structure factor of the solvent, F(q, t), the inertial part of the dynamic structure factor, Fo(q, t), the transverse current autocorrelation function of the solvent, C (q,t), the inertial part of the same, Ctf0(q, t), the self-dynamic structure factor of the solute, Fs(q, t), and the inertial part of the self-dynamic structure factor of the solute, Fs0(q,t). The expressions for all the above-mentioned dynamic quantities are similar to that given in Section IX but in two dimensions. 

The study of glass transition is an important subject in current research, and simulations may well be suited to help our understanding of the phenomenon. An example is the application of Monte Carlo techniques by Wittman, Kremer, and Binder.The authors employed a lattice method in two dimensions to model the system. The glass transition was determined by monitoring the free volume changes as well as isothermal compressibility. The glasslike behavior was determined by evaluating the bond autocorrelation function. The authors found that both the dynamic polymer structure factor and the orienta-... 

Equilibrium molecular dynamics uses the Green-Kubo relationship between the heat current autocorrelation function and the thermal conductivity [66] to obtain the thermal conductivity as ... 

Number of heat current autocorrelation function averages. 

In an "analog optical mixing experiment one measures either the time autocorrelation function of the photomultiplier output current or its corresponding spectral density. In the former case, the photomultiplier output is analyzed by an autocorrelator and in the latter, by a spectrum analyzer. 

This shows that the autocorrelation function C(t) of the photoelectron current is directly related to the second-order correlation function G r) of the light field. 

We illustrate how the present formalism leads to practical microscopic calculations by considering two applications. In Section 5 we study the density correlation function in the Boltzmann-Enskog approximation, and in Section 6 the recollision effects on the velocity autocorrelation function and the selfdiffusion coefficient are analyzed. In both cases, we assume, for simplicity that the fluid is a system of hard spheres. Although the calculations themselves are relatively crude and the numerical results have limited significance, these problems are of interest because they serve to indicate the level of complexity of current microscopic calculations and lead to a discussion of the areas where further calculations are needed. The chapter then closes with summary and discussions in Section 7. 

The methods used for expressing the data fall into two categories, time domain techniques and frequency domain techniques. The two methods are related because frequency and time are the reciprocals of each other. The analysis technique influences the data requirements. Reference 9 provides a brief overview of the various mathematical methods and a multitude of additional references. Specialized transforms (Fourier) can be used to transfer information between the two domains. Time domain measures include the normal statistical measures such as mean, variance, third moment, skewness, fourth moment, kurto-sis, standard deviation, coefficient of variance, and root mean squEire eis well as an additional parameter, the ratio of the standard deviation to the root mean square vtJue of the current (when measuring current noise) used in place of the coefficient of variance because the mean could be zero. An additional time domain measure that can describe the degree of randonmess is the autocorrelation function of the voltage or current signal. The main frequency domain... 

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