Autocorrelation times

A typical noisy light based CRS experiment involves the splitting of a noisy beam (short autocorrelation time, broadband) into identical twin beams, B and B, tlnough the use of a Michelson interferometer. One ami of the interferometer is computer controlled to introduce a relative delay, x, between B and B. The twin beams exit the interferometer and are joined by a narrowband field, M, to produce the CRS-type third order polarization in the sample ([Pg.1209]

Exercise 4.5. Evaluate the autocorrelation function and the autocorrelation time for the function exp(-f) cos(2f). 

From the various autocorrelation times which characterized macromolecular fluctuations, those associated with the fluctuation of the electrostatic field from the protein on its reacting fragments are probably the most important (see Ref. 8). These autocorrelation times define the dielectric relaxation times for different protein sites and can be used to estimate dynamical effects on biological reactions (see Chapter 9 for more details). 

FIGURE 9.4. The autocorrelation function of the time-dependent energy gap Q(t) = (e3(t) — 2(0) for the nucleophilic attack step in the catalytic reaction of subtilisin (heavy line) and for the corresponding reference reaction in solution (dotted line). These autocorrelation functions contain the dynamic effects on the rate constant. The similarity of the curves indicates that dynamic effects are not responsible for the large observed change in rate constant. The autocorrelation times, tq, obtained from this figure are 0.05 ps and 0.07ps, respectively, for the reaction in subtilisin and in water. 

Equation (10.6) for the mobility in the two-state model implicitly assumes that the electron lifetime in the quasi-free state is much greater than the velocity relaxation (or autocorrelation) time, so that a stationary drift velocity can occur in the quasi-free state in the presence of an external field. This point was first raised by Schmidt (1977), but no modification of the two-state model was proposed until recently. Mozumder (1993) introduced the quasi-ballistic model to correct for the competition between trapping and velocity randomization in the quasi-free state. 

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). 

The dispersion of this waiting time distribution, i.e., its second central moment, is a measure that we can use to define a homogenization time scale on which the dispersion is equal to that of a homogeneous (Poisson) system on a time scale given by the torsional autocorrelation time. The homogenization time scale shows a clear non-Arrhenius temperature dependence and is comparable with the time scale for dielectric relaxation at low temperatures.156... 

In order to see whether the development in time of a given situation could be follcwed, autocorrelation functions of all relevant features were constructed. Frcm these functions it was observed that, provided the weather was not too unstable, an autocorrelation time of about four hours was encountered. This autocorrelation was best defined for S02 concentrations that are measured hourly at various... 

For reference, we provide brief definitions and discussions of basic statistical quantities the mean, variance, autocorrelation function, and autocorrelation time. 

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. 

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. 

When the solute molecules are significantly larger than the solvent molecules, the detailed motion of individual solvent molecules around a solute molecule matters less the solvent may be better approximated as a hydrodynamic continuum. Both the velocity relaxation time Tiei = mj Girrja and the velocity autocorrelation time rc are larger for larger solute... 

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. 

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... 

The oldest and best known example of a Markov process in physics is the Brownian motion.510 A heavy particle is immersed in a fluid of light molecules, which collide with it in a random fashion. As a consequence the velocity of the heavy particle varies by a large number of small, and supposedly uncorrelated jumps. To facilitate the discussion we treat the motion as if it were one-dimensional. When the velocity has a certain value V, there will be on the average more collisions in front than from behind. Hence the probability for a certain change AV of the velocity in the next At depends on V, but not on earlier values of the velocity. Thus the velocity of the heavy particle is a Markov process. When the whole system is in equilibrium the process is stationary and its autocorrelation time is the time in which an initial velocity is damped out. This process is studied in detail in VIII.4. 

Yet it turned out that this picture did not lead to agreement with the measurements of Brownian motion. The breakthrough came when Einstein and Smoluchowski realized that it is not this motion which is observed experimentally. Rather, between two successive observations of the position of the Brownian particle the velocity has grown and decayed many times the interval between two observations is much larger than the autocorrelation time of the velocity. What is observed is the net displacement resulting after many variations of the velocity. 

This example exhibits several features that are of general validity. First it is clear that the Markov property holds only approximately. If the previous displacement Xk — Xk-x happened to be a large one, then the chances are slightly in favor of a large velocity at the time when Xk is observed. This velocity will survive for a short time of the order of the autocorrelation time of the velocity, and thereby favor a large value of Xk + 1 — Xk. Thus the fact that the autocorrelation time of the velocity is not strictly zero gives rise to some correlation between two successive displacements. This effect is small, provided that the time between two observations is much longer than the autocorrelation time of the velocity. 

Exercise. Delta functions do not occur in nature. In any physical application L(t) has an autocorrelation time tc > 0 for a Brownian particle tc is at least as large as the duration of an individual collision. It is therefore more physical to write instead of the delta function in (1.3) some sharply peaked function (j>(t — t ) of width tc. Show that this leads to the same results provided that 1. 

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. 

Exercise. Apply the result to the harmonic oscillator (1.3) with frequency co2(t) = cog l H- a (t), where t) is a stationary random process with zero mean and autocorrelation time tc. The answer is... 

E(x, t) is random with zero mean, stationary in time and space, and has an autocorrelation time tc. The density of an ensemble of such particles obeys (5.2),... 

A third class of equations, which permit to study the effect of autocorrelation times of arbitrary length, has been encountered in IX.7. This class consists of equations (1.1) in which Y(t) is a Markov process. We write 77 for its transition probability density 77(y, t y0, t0) and... 

The time of return can itself be made the criterion of success, by forgetting about the B region and counting two consecutive crossings of S as independent transitions if and only if they are separated by a time interval greater than some characteristic time To, e.g. the autocorrelation time of the velocity normal to... 

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