Time correlation functions stationary systems

Time correlation functions are our main tools for conveying this information in stationary systems. These are systems at thermodynamic equilibrium or at steady state with steady fluxes present. In such systems macroscopic observables do not evolve in time and there is no time origin that specifies the beginning of a process. However, it is meaningful to consider conditional probabilities such as P(B, t2 I A, Ix dB—the probability that a dynamical variable B will have a value in the range (B,... .B + dB) at time Z2 if another dynamical variable A had the value A at time Zi, and the joint probability P B, t2, A, tfjdBdA that A will be in the range (A,..., A + dA) at time t = ti and B will be in (5,..., 5 + d/jf) at time Z2 These two probabilities are connected by the usual relation (cf. Eq. (1.188))... 

In the previous sections, we derived general correlation function expressions for the nonlinear response function that allow us to calculate any 4WM process. The final results were recast as a product of Liouville space operators [Eqs. (49) and (53)], or in terms of the four-time correlation function of the dipole operator [Eq. (57)]. We then developed the factorization approximation [Eqs. (60) and (63)], which simplifies these expressions considerably. In this section, we shall consider the problem of spontaneous Raman and fluorescence spectroscopy. General formal expressions analogous to those obtained for 4WM will be derived. This will enable us to treat both experiments in a similar fashion and compare their information content. We shall start with the ordinary absorption lineshape. Consider our system interacting with a stationary monochromatic electromagnetic field with frequency w. The total initial density matrix is given by... 

The parallel-replica method [5] is perhaps the least glamorous of the AMD methods, but is, in many cases, the most powerful. It is also the most accurate AMD method, assuming only first-order kinetics (exponential decay) i.e., for any trajectory that has been in a state long enough to have lost its memory of how it entered the state (longer than the correlation time icorr, the time after which the system is effectively sampling a stationary distribution restricted to the current state), the probability distribution function for the time of the next escape from that state is given by... 

We notice that the correlation function defined by Eq. (147) is stationary. Thus, it fits the Onsager principle [101], which establishes that the regression to equilibrium of an infinitely aged system is described by the unperturbed correlation function. The authors of Ref. 102 have successfully addressed this issue, using the following arguments. According to an earlier work [96] the GME of infinite age has the same time convoluted structure as Eq. (59), with the memory kernel T(t) replaced by (1>,XJ (f). They proved that the Laplace transform of Too is... 

Figures 7.2 and 7.3 show the relevant correlation functions. In three dimensions at the dimensionless time pvot = 10 the steady-state is already nearly achieved (the deviation from the unity is seen only at r < lOro). Since the correlation length at large t is finite, microscopic defect segregation takes place for d = 3. Quite contrary, for low (J < 2) dimensions the correlation functions are no longer stationary. Similarly to the recombination decay kinetics treated in [14], the accumulation kinetics demonstrates also an infinite increase in time of the correlation length (defined by a coordinate where X (r, f) 1 or F(r, t)

The system seems quasi-stationary in Stage I. The existence of quasi-stationary states for a sufficiently long time has been questioned in Ref. 36. We will positively answer to the question by observing dependence on t of the correlation function Cp(t xj in Section VI.B. 

The Redfield tensor S is defined in terms of stationary correlation functions of the system-bath coupling operator, V, evolving under the bath Hamiltonian, Thus the dynamics of the bath are retained in Eq. (9), the only assumptions being that the bath is in thermal equilibrium and that its dynamics are independent of the state of the system beyond some correlation time, t, short compared to the rate of change of cr. The tensor element R,, / can be written [26, 42]... 

We discussed some aspects of the responses of chemical systems, linear or nonlinear, to perturbations on several earlier occasions. The first was the responses of the chemical species in a reaction mechanism (a network) in a nonequilibrium stable stationary state to a pulse in concentration of one species. We referred to this approach as the pulse method (see chapter 5 for theory and chapter 6 for experiments). Second, we studied the time series of the responses of concentrations to repeated random perturbations, the formulation of correlation functions from such measurements, and the construction of the correlation metric (see chapter 7 for theory and chapter 8 for experiments). Third, in the investigation of oscillatory chemical reactions we showed that the responses of a chemical system in a stable stationary state close to a Hopf bifurcation are related to the category of the oscillatory reaction and to the role of the essential species in the system (see chapter 11 for theory and experiments). In each of these cases the responses yield important information about the reaction pathway and the reaction mechanism. 

The time distribution of the fluorescence photons emitted by a single dye molecule reflects its intra- and intermolecular dynamics. One example are the quantum jumps just discussed which lead to stochastic fluctuations of the fluorescence emission caused by singlet-triplet quantum transitions. This effect, however, can only be observed directly in a simple fluorescence counting experiment when a system with suitable photophysical transition rates is available. By recording the fluorescence intensity autocorrelation function, i.e. by measuring the correlation between fluorescence photons at different instants of time, a more versatile and powerful technique is available which allows the determination of dynamical processes of a single molecule from nanoseconds up to hundreds of seconds. It is important to mention that any reliable measurement with this technique requires the dynamics of the system to be stationary for the recording time of the correlation function. 

The starting point of the proof is two sum-rules of Jhon et oL Ihese are identities, obtained from different ways of evaluating time derivatives, which are consequences of the stationary values of ensemUe averages in a system at equilihrium. Thus if the average of the time derivative of the correlation function (l(ii t)J(r2 t)) is to he zero at all times, then... 

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