Correlation delta-function

The Boltzmann constant is ks and T the absolute temperature. — is the Dirac delta function. Below we assume for convenience (equation (5)) that the delta function is narrow, but not infinitely narrow. The random force has a zero mean and no correlation in time. For simplicity we further set the friction to be a scalar which is independent of time or coordinates. 

We assume that the sequential errors are not correlated in time, we can write the probability of sampling a sequence of errors as the product of the individual probabilities. We further use the finite time approximation for the delta function and have ... 

Vgiec and Vxc represent the electron-nuclei, electron-electron and exchange-correlation dionals, respectively. The delta function is zero unless G = G, in which case it has lue of 1. There are two potential problems with the practical use of this equation for a croscopic lattice. First, the summation over G (a Fourier series) is in theory over an rite number of reciprocal lattice vectors. In addition, for a macroscropic lattice there effectively an infinite number of k points within the first Brillouin zone. Fortunately, e are practical solutions to both of these problems. 

In order to improve upon the mean-field approximation given in equation 7.112, we must somehow account for possible site-site correlations. Let us go back to the deterministic version of the basic Life rule (equation 7.110). We could take a formal expectation of this equation but we first need a way to compute expectation values of Kronecker delta functions. Schulman and Seiden [schul78] provide a simple means to do precisely that. We state their result without proof... 

D(co)VV (cn)/ii m equal to the expression (l/rr) y y co2). Integrating from frequency zero up to infinite, one gets the empirical formula K(t-x)= (X/ft) y exp(-y t-x ). Here, 1/y represents the memory time of the dissipation and is essentially the inverse of the phonon bandwidth of the heat bath excitations that can be coupled to the oscillator. It reduces to a delta function when y->infinite. The correlation function (t-t), in this model is [133]... 

In order to perform calculations of the correlation function of particles 1 and 2 (which is the same quantity as the density of the 1st pseudoparticle, g ( )) using basis (49), we need the matrix elements of the delta function, 5(ri — ). They can be evaluated by replacing Ay Ay + uJn in expression (64) and then differentiating p times with respect to u. 

A point that has not been investigated is the possibility of considering u(k) a coloured noise instead of white noise, and therefore a non diagonal E. For example, the choice of a tridiagonal Ey would imply the assumption of u(k) a random walk process. On the one hand, by imposing a correlation among successive values of u(k), the flexibility of the output is reduced, and for example a delta function could not be recuperated. On the other hand, smoother outputs and better solutions could be obtained if good "a priori" estimations of the real autocorrelations of u(k) could be provided. 

If the random force has a delta function correlation function then K(t) is a delta function and the classical Langevin theory results. The next obvious approximation to make is that F is a Gaussian-Markov process. Then is exponential by Doob s theorem and K t) is an exponential. The velocity autocorrelation function can then be found. This approximation will be discussed at length in a subsequent section. The main thing to note here is that the second fluctuation dissipation theorem provides an intuitive understanding of the memory function. ... 

The phenomenological Langevin Eqs. (227) and (228) are only applicable to a very restricted class of physical processes. In particular, they are only valid when the stochastic forces and torques have infinitely short correlation times, i.e., their autocorrelation functions are proportional to Dirac delta functions. As was shown in the previous section, these restrictions can be removed by a suitable generalization of these Langevin equations. As we saw in the particular case of the velocity, the modified Langevin equation is... 

The one-electron crystal field Hamiltonian does not take into account electron correlation effects. For some systems, it has been useful to augment the crystal field Hamiltonian with additional terms representing the two-electron, correlated crystal field. The additional terms most commonly used (see, for example, Peijzel et al., 2005b Wegh et al., 2003) are from the simplified delta-function correlation crystal field model first proposed by Judd (1978) that assumes electron interaction takes place only when two electrons are located at the same position (hence the name delta-function ). This simplified model, developed by Lo and Reid (1993), adds additional terms, given as,... 

The matrix elements of the total correlation function, h, are related to all pairs of atoms. The intramolecular correlation function, to, introduced here represents the shape of the molecule. 8(r) in the diagonal element is the Dirac delta function and represents the position of an atom. The function appearing in the off-diagonal element is given by,... 

A pair correlation function g(7 ) was defined in Section 2.7 as the probability of finding a monomer in a unit volume at distance 7 away from a given monomer (labeled by y — 1). Note that y = 1 is not necessarily the end monomer of any chain. The pair correlation function g r ) can be written in terms of the delta function summed over all monomers except for the one at 7 i... 

The correlation function corresponds to the memory function, which indicates to which degree values of one function at time t are comparable to values of another function at time t — a before. For statistical signals, the similarity usually decreases rapidly with increasing shift a. For white noise, all values are independent of the others, and the auto-correlation function is proportional to a delta function. The proportionality factor is the second moment (12 of the noise signal. 

Through the delta function in the first step of (5.4.22) the relation (5.4.20) is taken care of. In the second step, the displacement and position variables R and r, respectively, are introduced. In the final result, the inner integral is identified as the auto-correlation... 

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