Finite autocorrelation time

Suppose Y(t) is a stationary, zero-average process with a finite autocorrelation time tc then <7(r)2> is independent of time and one has... 

Exercise. The same equations can be solved for arbitrary (/). Express the characteristic functional of x in that of . Find that x(t 2) t3 whenever has a finite autocorrelation time. 

Here (g)T = (e/m)Tf2/(r( + Tt) is called the ballistic mobility and (/t)H = + Tt) is the usual trap-controlled mobility. (q)F is the applicable mobility when the velocity autocorrelation time ( 1) is much less than the trapping time scale in the quasi-free state (fTf l). In the converse limit, (jj)t applies, that is—trapping effectively controls the mobility and a finite mobility results due to random trapping and detrapping even if the quasi-free mobility is infinite (see Eq. 10.8). 

Here, 8(f — f ) represents a sharply peaked but finite and differentiable autocorrelation function with a small but nonzero autocorrelation time, which is assumed to be an even function of t — t. The Stratonovich interpretation is obtained in the limit of vanishing autocorrelation time. 

As remarked in II.3, strictly stationary processes do not exist in nature, let alone in the laboratory, but they may be approximately realized when a process lasts much longer than the phenomena one is interested in. One condition is that it lasts much longer than the autocorrelation time. Processes without a finite tc never forget that they have been switched on in the past and can therefore not be treated as approximately stationary. 

Since Eq. (49) takes into account only the term of order Dt, the term of order in Eq. (51) is meaningless and the term linear in t in vanishes exactly. For T = 0, our result equals the well-known Smoluchowski rate. The main conclusion we can draw is that the activation rates for non-Markovian processes like Eq. (44) decrease as t increases the exact result of ref. 44 can thus be extended to the case of Gaussian random forces of finite correlation time as well. However, if we take Eq. (50) seriously, we obtain an Arrhenius factor, exp(A /Z)), of T(x) which does not exhibit a dependence on T. This is in contrast to the result found for telegr hic noises, where the Arrhenius factor increases with increasing autocorrelation time r (see ref. 44). The result of a numerical simulation for J(x) based on the bi-... 

Due to the finite propagation time T of the wavepackets, the Fourier transformation causes artifacts known as the Gibbs phenomenon [122]. In order to reduce this effect, the autocorrelation function is first multiplied by a damping function cos jtt/IT) [81,123]. Furthermore, to simulate the experimental line broadening, the autocorrelation functions will be damped by an additional multiplication with a Gaussian function exp — t/xd)% where zj is the damping parameter. This multiplication is equivalent to a convolution of the spectrum with a Gaussian with a full width at half maximum (FWHM) of /xd- The convolution thus simulates... 

There are many experiments which determine only specific frequency components of the power spectra. For example, a measurement of the diffusion coefficient yields the zero frequency component of the power spectrum of the velocity autocorrelation function. Likewise, all other static coefficients are related to autocorrelation functions through the zero frequency component of the corresponding power spectra. On the other hand, measurements or relaxation times of molecular internal degrees of freedom provide information about finite frequency components of power spectra. For example, vibrational and nuclear spin relaxation times yield finite frequency components of power spectra which in the former case is the vibrational resonance frequency,28,29 and in the latter case is the Larmour precessional frequency.8 Experiments which probe a range of frequencies contribute much more to our understanding of the dynamics and structure of the liquid state than those which probe single frequency components. 

Optic-like collective excitations are not a unique feature of binary mixture of charged particles. Such modes can also be found in binary mixtures of neutral particles. However, the behavior of mode contributions to time correlation functions in small k range in these two cases is quite different. In particular, amplitude of optic-like modes to the mass concentration autocorrelation function tends to zero for the latter case, whereas for the former one these modes produce the finite contribution even in the hydrodynamic limit. 

Successful first-principles molecular dynamics simulations in the Car-Paxrinello framework requires low temperature for the annealed electronic parameters while maintaining approximate energy conservation of the nuclear motion, all without resorting to unduly small time steps. The most desirable situation is a finite gap between the frequency spectrum of the nuclear coordinates, as measured, say, by the velocity-velocity autocorrelation function. 

In conclusion let us note that there is an error due to the finite time average [cf. Eq. (2.4.7)] in the determination of the full autocorrelation functions. If T is the averaging time and rc is the correlation tme, then there are effectively (Tj2xc) independent samplings contributing to the correlation function. The relative error is proportional to the reciprocal of the square root of thenumberor (2tc/T)112. Aformulaoften used for an estimate of this error is... 

The time-resolved monitoring and evaluation of the autocorrelation function of the emission intensity fluctuations yield information on the diffusion rate of fluorescent species. However, the experimental measurement is affected by a number of parasite side effects. First, the intensity of the focused beam is very high, which promotes photobleaching (mainly the transition to the triplet state). Hence, only a few very stable and resistant fluorophores (rhodamine dyes or BODIPY) can be employed and still an appropriate correction has to be used when evaluating the diffusion coefficients. Second, the multiple labeling of particles of finite size can generate additional problems. We will focus on some of these complications in Sects. 3.5 and 4.3. 

In our recent smdies, we focused on several complicating factors arising in studies of nanoparticles of a non-negligible size (e.g., polymeric micelles, vesicles) that can carry several fluorescent labels. When the dimensions of such particles become comparable to the typical dimensions of the effective volume (coi, (O2), the correlated motion of the fluorophores located on a single particle affects the shape of the autocorrelation function. Recently, an approximate expression for the FCS autocorrelation function of diffusing particles of finite size has been derived by Wu et al. [85]. They have shown that the autocorrelation function of uniformly labeled spherical particles can be expressed in a form similar to (12) where the diffusion time, concentration, and dimensions of the active volume are replaced by corresponding apparent quantities that depend on the particle size. Qualitatively, the same results were obtained in our computer simulations, which are discussed later (see Sect. 4.3). 

Space does not permit us to give here a detailed discussion of the effects of finite-system size on molecular dynamic calculations of time correlations functions. We have given elsewhere a discussion of such effects on the velocity autocorrelation function from a hydrodynamic point of view. This reference can also be consulted for a more extensive discussion of results for both the velocity autocorrelation function and the super-Burnett self-diffusion coefficient, including comparisons with theoretical predictions. 

This concept of an autocorrelation function is central to the understanding of polymer motions. A homely analogy may make it more accessible. A telephone directory is (in theory) an accurate list of numbers and addresses at the moment its editing ceases. In other words, it correlates precisely with the truth, which makes G(t) = 1 at t = 0. As t, the age of the directory, increases, this autocorrelation decays, with a half-life of a few years, and must eventually dwindle to near zero because of the finite lifespan of both humans and institutions. Exactly the same would be true in reverse if some time-traveller were able to obtain a copy of a future directory. 

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