Ensemble-averaged correlation function

In the limit of long times and if the ergodic assumption holds, then we have 7 = (/), where (I) is the ensemble average. As usual, we may generate many intensity trajectories one at a time, to obtain the ensemble-averaged correlation function... 

The time average correlation function is random. The ensemble average correlation function exhibits aging. Hence data analysis should be made with care. 

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... 

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. 

The operators Fk(t) defined in Eq.(49) are taken as fluctuations based on the idea that at t=0 the initial values of the bath operators are uncertain. Ensemble averages over initial conditions allow for a definite specification of statistical properties. The statistical average of the stochastic forces Fk(t) is calculated over the solvent effective ensemble by taking the trace of the operator product pmFk (this is equivalent to sum over the diagonal matrix elements of this product), so that = Trace(pmFk) is identically zero (Fjk(t)=Fk(t) in this particular case). The non-zero correlation functions of the fluctuations are solvent statistical averages over products of operator forces,... 

The theory of statistical mechanics provides the formalism to obtain observables as ensemble averages from the microscopic configurations generated by such a simulation. From both the MC and MD trajectories, ensemble averages can be formed as simple averages of the properties over the set of configurations. From the time-ordered properties of the MD trajectory, additional dynamic information can be calculated via the time correlation function formalism. An autocorrelation function Caa( = (a(r) a(t + r)) is the ensemble average of the product of some function a at time r and at a later time t + r. 

Consider a general system described by the Hamiltonian of Eq. (5), where = Huif) describes the interaction between the spin system (7) and its environment (the lattice, L). The interaction is characterized by a strength parameter co/i- When deriving the WBR (or the Redfield relaxation theory), the time-dependence of the density operator is expressed as a kind of power expansion in Huif) or (17-20). The first (linear) term in the expansion vanishes if the ensemble average of HiL(t) is zero. If oo/z,Tc <5c 1, where the correlation time, t, describes the decay rate of the time correlation functions of Huif), the expansion is convergent and it is sufficient to retain the first non-zero term corresponding to oo/l. This leads to the Redfield equation of motion as stated in Eq. (18) or (19). In the other limit, 1> the expan-... 

Ensemble average fluctuation with time, CORRELATION FUNCTION ENTATIC STATE ENTERING GROUP ENTEROPEPTIDASE ENTEROTOXIN ENTHALPY... 

Note that the energy is minimized with respect to all choices of the orbital basis and subject to the (1, conditions on p, = F, ,- this ensures that there exists an ensemble of Slater determinants with the desired electron density. Because an ensemble average of Slater determinants does not describe electron correlation, these variational energy expressions include a correlation functional, Ec p, which corrects the energy for the effects of electron correlation. Reasonable approximations for Ec[p] exist, though they tend to work only in conjunction with approximate exchange-energy functionals, Ex p. ... 

The parameter ( r(f) - r(O)p) is the ensemble average of the square of the distances between the initial and the later positions for each particle in the system, and the factor of 3 takes care of the three-dimensional nature of the random walk. Given an ensemble of M particles, the correlation function of properties r t) and r(0) is given by... 

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

The double sum in Eq. (IS) can now be expressed by a cylindrical density correlation function n(r) which counts the number of molecular sites lying in a circular shell of thickness dr with distance r to a given point of reference on the symmetry axis of the monodomam. This RDCF contains the lateral distance statistics of segments within a domain - procedure (3), as well as the average of the domain ensemble - procedure (4). Due to the cylindrical symmetry we finally obtain... 

First we discuss and construct monodisperse two-dimensional arrangements of impenetrable cylinders in terms of radial distance correlation functions, the lateral packing fraction and number density. In the second step, these hard cylinders are covered by the mean electronic density functions of the RISA chain segment ensemble. Last of all, the Fourier transformation and final averaging is... 

Since these characteristics are time-dependent, let us assume particle birth-death and migration to be the Markov stochastic processes. Note that making use of the stochastic models, we discuss below in detail, does not contradict the deterministic equations employed for these processes. Say, the equations for nv t), Xu(r,t), Y(r,t) given in Section 2.3.1 are deterministic since both the concentrations and joint correlation functions are defined by equations (2.3.2), (2.3.4) just as ensemble average quantities. Note that the... 

The Wiener-Khinchin theorem is a special case of Bocliner theorem applicable to time averages of stationary stochastic variables. Bochner s theorem enables the Wiener-Khinchin theorem to be applied to ensemble averaged time-correlation functions in quantum mechanics where it is difficult to think of properties as stochastic processes. 

The important point to note here is that the 2nd moment of Ky(t) depends on the 2nd and 4th moments of y(t). The 2nd moments of each of the three previously mentioned autocorrelation functions may be calculated from ensemble averages of appropriate functions of the positions, velocities, and accelerations created in the dynamics calculations. Likewise, the 4th moment of the dipolar autocorrelation function may also be calculated in this manner. However the 4th moments of the velocity and angular momentum correlation functions depend on the derivative with respect to time of the force and torque acting on a molecule and, hence, cannot be evaluated directly from the primary dynamics information. Therefore, these moments must be calculated in another manner before Eq. (B.3) may be used. 

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