Time correlation functions definition

The basic definition of a time correlation function is the equilibrium ensemble average (at t = 0)... 

If the symmetry is different, then of course iL /, > can be nonzero. In this article we assume that 0t,..., VN have definite albeit different time reversal symmetries. The properties can be represented by vectors t/j >... t/jy>... in Hilbert space with scalar product defined above. It is a simple matter to demonstrate that L is Hermitian in this Hilbert Space. Define the time correlation function... 

It is possible to derive an equation which describes the time evolution of the time-correlation function Cn(t) where C stands for different autocorrelation functions depending on the definition of the scalar product (i), (ii), or (iii) of Eq. (73) adopted. 

There are also situations when one is not in the classical limit, and so Equation (13) would not seem applicable, and instead one would like to approximate one of the quantum mechanical expressions for Ti by relating the relevant quantum time-correlation function to its classical analog. For the sake of definiteness, let us consider the case where the oscillator is harmonic and the oscillator-bath coupling is linear in q, as discussed above. In this case k 0 can be written as... 

This equality implies that p". >neq (y, t2 — t ) is not positive definite, a price that we have to pay to ensure the equivalence between the density and trajectory picture in the non-Poisson case. Thus, the two-time correlation function is evaluated using only density prescriptions, and the result turns out to be identical to Eq. (148), which is known to correspond to the prescription of renewal theory [see Eq. (147)]. In the Poisson case the equilibrium distribution is flat. Thus, the contributionp s " cq(y, t2 — t ) vanishes. 

The two-point one time correlation functions, in the form presented in the preceding discussion, are not suitable for analyzing motions at different scales and specifically they are not suitable for understanding relations between movements of fluid characterized by different length and time scales. That is why it is better to use the 3D Fourier transforms of two-point correlations and to decompose them into waves of different frequencies or wave numbers. Turbulence has by definition a 3D character so it is obvious that the spectrum has to be 3D as well, to characterize turbulence properly. The ID spectrum of Taylor (see, e.g., [66]) oversimplifies the observed features of turbulence and may give misleading interpretations of the 3D held (see also, [113], p. 18). The differences and consequences of ID and 3D spectrum analysis are discussed by Hinze ([66], sects. 1-12 and 3-4) and Pope ([121], sect. 6.5). 

The phase space trajectory r (Z), p (Z) is uniquely determined by the initial conditions r (Z = 0) = r p (Z = 0) = p. There are therefore no probabilistic issues in the time evolution from Z = 0 to Z. The only uncertainty stems from the fact that our knowledge of the initial condition is probabilistic in nature. The phase space definition of the equilibrium time correlation function is therefore. 

How do the definitions (6.3) and (6.6) relate to each other While a formal connection can be made, it is more important at this stage to understand their range of applicability. The definition (6.6) involves the detailed time evolution of all particles in the system. Equation (6.3) becomes useful in reduced descriptions of the system of interest. In the present case, if we are interested only in the mutual dynamics of the observables A and B we may seek a description in the subspace of these variables and include the effect of the huge number of all other microscopic variables only to the extent that it affects the dynamics of interest. This leads to a reduced space dynamics that is probabilistic in nature, where the functions P(B,t2, A,t].) and P(B, Z2 A, Zi) emerge. We will dwell more on these issues in Chapter 1. Common procedures for evaluating time correlation functions are discussed in Section 7.4.1. 

H. Kono, Y. Nomura, and Y. Fujimura, Adv. Chent. Phys. 80, 403 (1991). Here, the time-correlation function formalism is used to classify the RSR into the fluorescencelike, the Raman-like and the interference-like components. This classification corresponds to the nomenclature II in [6] and the interference-like component is not positive definite in general. 

The basic formulation of this problem was given by Van Hove [25] in the form of his space-time correlation functions, G ir, t) and G(r, t). He showed that the scattering functions, as defined above, for a diffusing system are given by the Fourier transformation of these correlation functions in time and space. Incoherent scattering is linked to the self-correlation function, Gs(r, t) which provides a full definition of tracer diffusion while coherent scattering is the double Fourier transform of the full correlation function which is similarly related to chemical or Fick s law diffusion. Formally the equations can be written ... 

Compared to crystalline materials, the production and handling of amorphous substances are subject to serious complexities. Whereas the formation of crystalline materials can be described in terms of the phase rule, and solid-solid transformations (polymorphism) are well characterised in terms of pressure and temperature, this is not the case for glassy preparations that, in terms of phase behaviour, are classified as unstable . Their apparent stability derives from their very slow relaxations towards equilibrium states. Furthermore, where crystal structures are described by atomic or ionic coordinates in space, that which is not possible for amorphous materials, by definition, lack long-range order. Structurally, therefore, positions and orientations of molecules in a glass can only be described in terms of atomic or molecular distribution functions, which change over time the rates of such changes are defined by time correlation functions (relaxation times). 

By definition, the centroid variable occupies a central role in the behavior of the centroid-constrained imaginary-time correlation function in Eq. (2.1). However, it is even more interesting to analyze the role of the centroid variable in the real-time quantum position correlation function [4, 8]. This information can in principle be extracted from the exact centroid-constrained correlation function C (t, q ) through the analytic continuation t— if. Such a procedure, however, is generally not tractable unless there is some prior simplification of the problem. One such simplification is achieved [4, 8] through use of the optimized reference quadratic action functional, given by [3, 21-23]... 

Water residence times on the protein surface are not directly measurable experimentally, bnt can be defined as the relaxation time of time correlation functions of the popnlation of the hydration shell [5], Due to differences in the definition from one investigation to another, the values reported in the literature exhibit considerable variability. Nonetheless, heterogeneity in water dynamics near the protein surface is clearly manifested in distributions of water residence times. The distribution we have constrncted for the N state of HocLA in solution from the residence time of water next to each residue is plotted up to lOOps in Figure 16.1d. The distribution is very broad, ranging from 2.6 to 241 ps, but highly skewed toward shorter residence times. The mean residence time of 23 ps is about 2.5 times longer than the rotational correlation time for hydration water. 

A direct way to calculate the time correlation function from the values saved during the simulation run is just to literally implement its definition. Suppose we have M+ values of. 4(0 and B t), obtained at the regular time intervals mSt, where m is an integer running from 1 to M, stored in the... 

The dynamic behavior of macromolecules can be explained by adopting the time correlation functions of the equilibrium fluctuations of the chain end-to-end vectors, CRR(t) (for a precise definition of correlation... 

In applications of LLS, we often need to understand theories for both scattering techniques and properties of complex fluids, partly because scattering is an observation not in real space but in Fourier space. Therefore, the results do not yield direa images of stmctures and dynamics. Furthermore, soft matter often does not have definite stmctures and/or simple dynamics, but exhibits stochastic behavior described theoretically in terms of the space-time correlation function. In this sense, scattering methods are suitable and irreplaceable tools for investigating soft condensed matter. 

In order to extract some more information from the csa contribution to relaxation times, the next step is to switch to a molecular frame (x,y,z) where the shielding tensor is diagonal (x, y, z is called the Principal Axis System i.e., PAS). Owing to the properties reported in (44), the relevant calculations include the transformation of gzz into g x, yy, and g z involving, for the calculation of spectral densities, the correlation function of squares of trigonometric functions such as cos20(t)cos20(O) (see the previous section and more importantly Eq. (29) for the definition of the normalized spectral density J((d)). They yield for an isotropic reorientation (the molecule is supposed to behave as a sphere)... 

This estimate should be made more precise. To do it, let us use some results of the numerical solution of a set of the kinetic equations derived in the superposition approximation. The definition of the correlation length o in the linear approximation was based on an analysis of the time development of the correlation function Y(r,t) as it is noted in Section 5.1. Its solution is obtained neglecting the indirect mechanism of spatial correlation formation in a system of interacting particles, i.e., omitting integral terms in equations (5.1.14) to (5.1.16). Taking now into account such indirect interaction mechanism, the dissimilar correlation function, obtained as a solution of the complete set of equations in the superposition approximation... 

To simplify mathematical manipulations, let us consider now the case of equal diffusion coefficients, Da = D, in which case the similar correlation functions just coincide, Xv r),T) = X(t),t). Taking into account the definition of correlation length Id = VDt, where D = Da + D = 2D a, as well as time-dependence of new variables r) and r, one gets from (5.1.2) to (5.1.4) a set of equations... 

The origin of this unusual behaviour is partly clarified from Fig. 6.34(a) where the relevant curves 2 demonstrate the same kind of the non-monotonous behaviour as the critical exponents above. Since, according to its definition, equation (4.1.19), the reaction rate is a functional of the joint correlation function, this non-monotonicity of curve 2 arises due to the spatial re-arrangements in defect structure. It is confirmed by the correlation functions shown in Fig. 6.34(a). The distribution of BB pairs is quasi-stationary, XB(r,t) X°(r) = exp[(re/r)3], which describes their dynamic aggregation. (The only curve is plotted for XB in Fig. 6.35(a) for t = 102 (the dotted line) since for other time values XB changes not more than by 10 per cent.) This quasi-steady spatial particle distribution is formed quite rapidly already at t 10° it reaches the maximum value of XB(r, t) 103. The effect of the statistical aggregation practically is not observed here, probably, due to the diffusion separation of mobile B particles. 

This function describes the correlation between 0,(0) and Oj(t) as a function of the time. Corresponding to each definition of the scalar product (Eq. (73)) there is a different correlation function. 

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