Integrated autocorrelation time

The additional Yukawa interactions also lead to a fluctuating internal energy E(t) that makes it possible to determine the rate at which the large (and slower) particles decorrelate. We consider the integrated autocorrelation time r obtained from the energy autocorrelation function [28],... 

Figure 3. Temperature dependence of several meEtsures of the local orientational mobility of the chain. The lower set of curves pertaining to the right abscissa shows the mean time between torsional transitions for the three relevant torsional angles along the chain. The upper two sets of curves give the integrated autocorrelation time for the second Legendre polynomial of the CH vector orientation, rcH, and the integrated autocorrelation time for the torsion angle autocorrelation function, rroa-...

Thus, Texp is the relaxation time of the slowest mode in the system. (If the state space is infinite, Texp might be +oo )i On the other hand, for a given observable A we define the integrated autocorrelation time... 

The factor of is purely a matter of convention it is inserted so that Tmt,A Texp,A if pAA t) With T 1.) The integrated autocorrelation time controls the statistical error in Monte Carlo estimates of ( A). More precisely, the sample mean... 

The present analysis follows the approach taken by aU three of these authors, in which SDEs are constructed by choosing the drift and diffusivity coefficients so as to yield a desired diffusion equation. Peters [13] has pioneered an alternative approach, in which expressions for the drift and diffusivity are derived from a direct, but rather subtle, analysis of the underlying inertial equations of motion, in which (for rigid systems) he integrates the instantaneous equations of motion over time intervals much greater than the autocorrelation time of the particle velocities. Peters has expressed his results both as standard Ito SDEs and in a nonstandard interpretation that he describes heuristically as a mixture of Stratonovich and Ito interpretations. Peters mixed Ito—Stratonovich interpretation is equivalent to the kinetic interpretation discussed here. Here, we recover several of Peters results, but do not imitate his method. 

On multiplying eqn. (284) by u0, integrating over time, and taking the ensemble average, the velocity autocorrelation function is obtained [271]. 

As A x was supposed stationary the integral is independent of time. The effect of the fluctuations is therefore to renormalize A0 by adding a constant term of order a2 to it. The added term is the integrated autocorrelation function of At. In particular, if one has a non-dissipative system described by A0, this additional term due to the fluctuations is usually dissipative. This relation between dissipation and the autocorrelation function of fluctuations is analogous to the Green-Kubo relation in many-body systems 510 but not identical to it, because there the fluctuations are internal, rather than added as a separate term as in (2.1). 

Figure 8 The integrated stress-stress autocorrelation function as described in Eqs. [121] for SPC/E water at 303.15 K as described in Ref. 42. Note the convergence of the integral over time deteriorates owing to insufficient data sampling. The experimental value of the shear viscosity is 7.97 x 10 Pa s, whereas the calculated value from this curve 6.6 0.8 x 10 Pa s.

Another approach to calculate thermal conductivity is equilibrium molecular dynamics (EMD) [125] that uses the Green-Kubo relation derived from linear response theory to extract thermal conductivity from heat current correlation functions. The thermal conductivity X is calculated by integrating the time autocorrelation function of the heat flux vector and is given by... 

The function h(t ) i(t + r)dt is often referred to as the autocorrelation function of the Amotion h(t) however, the reader should be careful to note the difference between the autocorrelation function of h(t)—an integrable function—and the autocorrelation function of Y(t)—a function that is not integrable because it does not die out in time. With this distinction in mind, Campbell s theorem can be expressed by saying that the autocovariance function of a shot noise process is n times the autocorrelation function of the function h(t). 

From this expression we see that the friction cannot be determined from the infinite-time integral of the unprojected force correlation function but only from its plateau value if there is time scale separation between the force and momentum correlation functions decay times. The friction may also be estimated from the extrapolation of the long-time decay of the force autocorrelation function to t = 0, or from the decay rates of the momentum or force autocorrelation functions using the above formulas. 

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. 

The time-dependent quantity in the integrand of Eq. (4.24), (,(r, R, f = 0) ,(r, R, f)), is called the autocorrelation function. It is the integral over all space of the product of the initial wavepacket with the wavepacket at time t. Rama Krishna and Coalson [86] have shown that the Fourier transform over time in Eq. (4.24) can be replaced by twice the half-Fourier transform where the time integral mns from f = 0 to f = oo. Using this result we obtain the final expression ... 

Cross-correlation measurements with time delay do not record these simultaneous diagonal contributions, which give rise to LO shot noise in autocorrelation measurements. In our schemes, these contributions have the form of a single directional integral N f Because of the even symmetry of (u ), this... 

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