Cumulant expansions

3 Cumulant Expansion The inverse-Laplace transform is a convenient analysis method when the distribution is broad, especially bimodal or trimodal. When the distribution is narrow and gi(T) is close to a single exponential decay, a simpler analysis method, called a cumulant expansion, is more useful. In this 

Problem 3.8 proves this expansion, hi the absence of distribution, i.e., AF = 0, the second- and higher-order terms disappear, and In gi(T) is a straight line. Cnrve-fitting of the measured hi gi(T) by a polynomial gives an estimate of (F) and (AF ). As found in Problem 3.10, the dififnsion coefficient estimated from (F) for a solntion of a polydisperse polymer is a z-average diffusion coefficient. Often, we nse a simple symbol of F for (F). 

Despite such a fault, the attempt to utilize the experimental data undertaken in [117] proves that the inverse problem is solvable. It is even simpler to do starting from the differential kinetic equation (2.25) whose integral is [118, 119] 

The long-time behaviour of K (t) is obviously exponential, and the rate is proportional to the diffusion coefficient defined in Eq. (2.42). Therefore the expression 

Owing to this relation the true shape of Kj(t) may be recovered from Kt(t) found from the optical spectra or MD simulations. 

In this section we consider how to express the response of a system to noise employing a method of cumulant expansions [38], The averaging of the dynamical equation (2.19) performed by this technique is a rigorous continuation of the iteration procedure (2.20)-(2.22). It enables one to get the higher order corrections to what was found with the simplest perturbation theory. Following Zatsepin [108], let us expound the above technique for a density of the conditional probability which is the average 

Here g and go are a set of angular variables, which define a molecular orientation at instants of time 0 and t, respectively and ft is the orientation at instant t which was g0 at t = 0. By difference of arguments we mean the difference of turns. In the molecular frame (MS), where the axes are oriented along the main axes of the inertia tensor, Wj = Ji/Ii. Thus the analogue of Eq. (2.19) is the equation 

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The same result can also be obtained directly from the virial equation of state given above and the low-density fonn of g(r). B2(T) is called the second virial coefficient and the expansion of P in powers of is known as the virial expansion, of which the leading non-ideal temi is deduced above. The higher-order temis in the virial expansion for P and in the density expansion of g(r) can be obtained using the methods of cluster expansion and cumulant expansion. 

Kubo R. Generalized cumulant expansion method. J. Phys. Soc. Japan. 

Q-branch rotational structure 179-82 spectra of nitrogen in argon 180 spectral collapse theory 150 spectral width 107 strong collision model 188 cumulant expansions 85-91... 

The remaining terms in Eq. (4-24) are the nth-order corrections to approximate the real system, in which the expectation value ( c is called cumulant, which can be written in terms of the standard expectation value ( by cumulant expansion in terms of Gaussian smearing convolution integrals ... 

In the last equation Hi(x) is the th Hermite polynomial. The reader may readily recognize that the functions look familiar. Indeed, these functions are identical to the wave functions for the different excitation levels of the quantum harmonic oscillator. Using the expansion (2.56), it is possible to express AA as a series, as has been done before for the cumulant expansion. To do so, one takes advantage of the linearization theorem for Hermite polynomials [42] and the fact that exp(-t2 + 2tx) is the generating function for these polynomials. In practice, however, it is easier to carry out the integration in (2.12) numerically, using the representation of Po(AU) given by expressions (2.56) and (2.57). 

For computer simulations, (5.35) leads to accurate estimates of free energies. It is also the basis for higher-order cumulant expansions [20] and applications of Bennett s optimal estimator [21-23], We note that Jarzynski s identity (5.8) follows from (5.35) simply by integration over w because the probability densities are normalized to 1 ... 

Instead of estimating -/ 1 ln(exp(-/W(f)) directly using (5.44), one can use cumulant expansion approaches, as in regular free energy perturbation theory (see e.g., [20, 39] for combining cumulant expansions about the initial and final states). Unbiased estimators for cumulants can be used. Probably the most useful relations involve averages and variances of the work ... 

These are the first two terms in a cumulant expansion [50]. We note here that the convergence of cumulant expansions is a subtle issue. Generally, if the statistics are nearly Gaussian, the cumulant expansion yields a good approximation. If the statistical distribution is not Gaussian, however, the cumulant expansion diverges with the inclusion of higher-order terms. See [29] and references therein for more discussion of this point. 

At this point it might be helpful to summarize what has been done so far in terms of effective potentials. To obtain the QFH correction, we started with an exact path integral expression and obtained the effective potential by making a first-order cumulant expansion of the Boltzmann factor and analytically performing all of the Gaussian kinetic energy integrals. Once the first-order cumulant approximation is made, the rest of the derivation is exact up to (11.26). A second-order expansion of the potential then leads to the QFH approximation. 

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

We now turn to the quantum version of these results. In this case, the analogous cumulant expansion gives exactly the same equations for the centroids as above, while the equations for the higher cumulants are different. We can again investigate whether a trajectory limit exists. Localization holds in the weakly nonlinear case if the classical condition above is satisfied. In the case of strong nonlinearity, the inequality becomes... 

New optimism was brought into the field of RDMs by Hiroshi Nakatsuji, Carmela Valdemoro, and David Mazziotti with their cumulant expansion, the hierarchy of equations connecting the 2-RDM with 4-RDMs, and the contracted Schrd-dinger equation. John Coleman continues to be the motor for further progress. 

W. Kutzelnigg and D. Mukherjee, Cumulant expansion of the reduced density matrices. J. Chem. Phys. 110, 2800 (1999). 

There are several other studies of cumulant expansions of the ROMs. Thus Kutzelnigg and Mukheijee also published in 1999 [64] an RDM expansion that is similar to Mazziotti s. An extended study of this cumulant approach was given by Ziesche [65]. Also, a particularly interesting analysis of the cumulant expansions was given by Harris [66], who proposed a systematic way for obtaining the different terms of the expansion. 

P. Ziesche, Cumulant expansions of reduced density matrices, reduced density matrices and Green functions, in Many-Electron Densities and Reduced Density Matrices (J. Cioslowski, ed.), Kluwer, Norwell, MA, 2000. 

M. D. Benayoun and A. Y. Lu, Invariance of the cumulant expansion under l-particle unitary transformations in reduced density matrix theory. Chem. Phys. Lett. 387, 485 (2004). 

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