Correlation functions averages

A complete and detailed analysis of the formal properties of the QCL approach [5] has revealed that while this scheme is internally consistent, inconsistencies arise in the formulation of a quantum-classical statistical mechanics within such a framework. In particular, the fact that time translation invariance and the Kubo identity are only valid to O(h) have implications for the calculation of quantum-classical correlation functions. Such an analysis has not yet been conducted for the ILDM approach. In this chapter we adopt an alternative prescription [6,7]. This alternative approach supposes that we start with the full quantum statistical mechanical structure of time correlation functions, average values, or, in general, the time dependent density, and develop independent approximations to both the quantum evolution, and to the equilibrium density. Such an approach has proven particularly useful in many applications [8,9]. As was pointed out in the earlier publications [6,7], the consistency between the quantum equilibrium structure and the approximate... 

This is the time-correlation function averaged over a finite time T. 

Z Cco) Zr( >) Sample impedance Reference impedance < .. > Correlation function Averaged quantities... 

In general, it is diflfieult to quantify stnietural properties of disordered matter via experimental probes as with x-ray or neutron seattering. Sueh probes measure statistieally averaged properties like the pair-correlation function, also ealled the radial distribution function. The pair-eorrelation fiinetion measures the average distribution of atoms from a partieular site. 

Unlike the solid state, the liquid state cannot be characterized by a static description. In a liquid, bonds break and refomi continuously as a fiinction of time. The quantum states in the liquid are similar to those in amorphous solids in the sense that the system is also disordered. The liquid state can be quantified only by considering some ensemble averaging and using statistical measures. For example, consider an elemental liquid. Just as for amorphous solids, one can ask what is the distribution of atoms at a given distance from a reference atom on average, i.e. the radial distribution function or the pair correlation function can also be defined for a liquid. In scattering experiments on liquids, a structure factor is measured. The radial distribution fiinction, g r), is related to the stnicture factor, S q), by... 

I quantities x and y are different, then the correlation function js sometimes referred to ross-correlation function. When x and y are the same then the function is usually called an orrelation function. An autocorrelation function indicates the extent to which the system IS a memory of its previous values (or, conversely, how long it takes the system to its memory). A simple example is the velocity autocorrelation coefficient whose indicates how closely the velocity at a time t is correlated with the velocity at time me correlation functions can be averaged over all the particles in the system (as can elocity autocorrelation function) whereas other functions are a property of the entire m (e.g. the dipole moment of the sample). The value of the velocity autocorrelation icient can be calculated by averaging over the N atoms in the simulation ... 

The first-order El "golden-rule" expression for the rates of photon-induced transitions can be recast into a form in which certain specific physical models are easily introduced and insights are easily gained. Moreover, by using so-called equilibrium averaged time correlation functions, it is possible to obtain rate expressions appropriate to a... 

In this form, one says that the time dependenee has been reduee to that of an equilibrium averaged (n.b., the Si pi I)i ) time correlation function involving the eomponent of the dipole operator along the external eleetrie field att = 0(Eo p) and this eomponent at a different time t (Eo p (t)). 

In effect, i is replaced by the vibrationally averaged electronic dipole moment iave,iv for each initial vibrational state that can be involved, and the time correlation function thus becomes ... 

All of these time correlation functions contain time dependences that arise from rotational motion of a dipole-related vector (i.e., the vibrationally averaged dipole P-avejv (t), the vibrational transition dipole itrans (t) or the electronic transition dipole ii f(Re,t)) and the latter two also contain oscillatory time dependences (i.e., exp(icofv,ivt) or exp(icOfvjvt + iAEi ft/h)) that arise from vibrational or electronic-vibrational energy level differences. In the treatments of the following sections, consideration is given to the rotational contributions under circumstances that characterize, for example, dilute gaseous samples where the collision frequency is low and liquid-phase samples where rotational motion is better described in terms of diffusional motion. 

If the Bath relaxation constant, t, is greater than O.I ps, you should be able to calculate dynamic properties, like time correlation functions and diffusion constants, from data in the SNP and/or CSV files (see Collecting Averages from Simulations on page 85). 

This function measures the correlation of the property A (it) to itself at two different times separated by the time interval t, averaged over the whole trajectory. The auto-correlation function is reversible in time [i.e., CaU) = it)], and it is stationary (i.e., (A(it + t)A(t))... 

Here is the position operator of atom j, or, if the correlation function is calculated classically as in an MD simulation, is a position vector N is the number of scatterers (i.e., H atoms) and the angular brackets denote an ensemble average. Note that in Eq. (3) we left out a factor equal to the square of the scattering length. This is convenient in the case of a single dominant scatterer because it gives 7(Q, 0) = 1 and 6 u,c(Q, CO) normalized to unity. 

In order to develop integral equations for the correlation functions, we consider the system composed of N polydisperse spheres. The average density of particles with diameter <7, is given by... 

The structure of the chapter is as follows. First, we start with a brief introduction of the important theoretical developments and relevant interesting experimental observations. In Sec. 2 we present fundamental relations of the liquid-state replica methodology. These include the definitions of the partition function and averaged grand thermodynamic potential, the fluctuations in the system and the correlation functions. In the second part of... 

To define the correlation functions of partly quenched systems requires one to consider fluctuations. There are two types of fluctuations thermal fluctuations for a given configuration of matrix species, and fluctuations induced by disorder. We characterize the average over disorder of thermal fluctuations by the variance... 

In close relation to the fluctuations, one may introduce the correlation functions. The pair density distribution function for fluid particles (ri, r2) is defined as the average over all realizations of the matrix structure of the... 

Then, performing a disorder average in Eq. (19), and using Eq. (18) we can obtain the following two relations for the connected and blocking correlation functions... 

In the microemulsion the role of A is played by the period of damped oscillations of the correlation functions (Eq. (7)). The surface-averaged Gaussian curvature Ky, = 2t x/ S is the topological invariant per unit surface area. Therefore the comparison between Ryyi = Kyy / in the disordered microemulsion and in the ordered periodic phases is justified. We calculate here R= Since K differs for diffused films from cor-... 

A. Ciach. Four-point correlation functions and average Gaussian curvature in microemuisions. Phys Rev E 55 1954-1964, 1997. 

Notice that in this general case, correlation functions cannot be solved for directly instead, there is an entire hierarchy of lower-order correlations expressed as functions of higher-order correlations. For example if we take an average of equation 7.79 over all space-time histories, and assume that we have a steady-state so... 

Here u is a unit vector oriented along the rotational symmetry axis, while in a spherical molecule it is an arbitrary vector rigidly connected to the molecular frame. The scalar product u(t) (0) is cos 0(t) in classical theory, where 6(t) is the angle of u reorientation with respect to its initial position. It can be easily seen that both orientational correlation functions are the average values of the corresponding Legendre polynomials ... 

It must be stressed that every Un(t,J) brings dq(0) to a different coordinate system. Consequently, the averaged operator (A7.13) is actually a weighted sum of the quantities in differently oriented reference systems. It can nevertheless be used to find the scalar product (A7.7), that is the orientational correlation function. 

Boltzmann distribution 13 change of average z projection 17 change per collision 18-19 correlation functions 12, 25-7, 28 calculation 14-15 correlation times quasi-free rotation 218 various molecules 69 and energy relaxation 164-6 impact theory 92 torque 18-19, 27... 

---