Abstract
This Lecture treats one of the standard systematic methods for analyzing the time dependence of dynamic correlation functions. The Mori-Zwanzig formalism has as an overall goal the systematization of nonequilibrium calculations in a way that reduces the likelihood that any category of physical effect will be overlooked during the calculation. The formalism is supposed to keep track of possible classes of effect, leaving the computor only with the (very difficult) task of evaluating a specific set of correlation functions specified by the formalism. The ability of the formalism to track some aspects of the calculation in a systematic way saves a substantial amount of effort. The price one pays, if one lets the formalism do some of one’s thinking, is that the formalism then constrains precisely which correlation functions you must calculate, perhaps demanding expressions that are not very easy to obtain [1].
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For an interesting example of such constraints, consider several published treatments of the light scattering spectrum of a two-component fluid, namely the results of R. D. Mountain and J. M. Deutch, J. Chem. Phys 50, 1103 (1976), P. Madden and D. Kivelson, J. Stat Phys. 12, 167 (1975), and G. D. J. Phillies and D. Kivelson Molecular Physics 38, 1393 (1979). The calculations differ in their choice of independent variables. Madden and Kivelson, and Phillies and Kivelson each include in their work a chemical reaction linking the two species. Mountain and Deutch chose as their independent variables a set which has the important feature that the equal-time fluctuations in these variables are not cross correlated. Consequently, in their final results all equal-time cross-correlation functions involving two of these variables vanish, simplifying greatly the evaluation and inversion of the matrices of equal-time correlation functions found in the vector form of the Mori-Zwanzig identity. The price paid by Mountain and Deutch is that, in their subsequent evaluation of the light-scattering spectrum, they needed three thermodynamic derivatives of the index of refraction, namely. These derivatives have the disadvantages that they are evaluated along unconventional paths, and that it is not clear a priori whether any of these derivatives are significant or are vanishingly small. In contrast, the two papers with Kivelson chose as their variables the number density N, the concentration difference ΔN, and an energy and a momentum density, namely (N, ΔN, E, i k.U M). Fluctuations in N, ΔN, and E are highly correlated, so construction and inversion of the matrices of equal-time correlation functions becomes quite tedious. However, with this choice of variables calculation of the light-scattering spectrum requires only two thermodynamic derivatives of the index of refraction, namely (ie364-3) and (ie364-4). Furthermore, in at least some systems the latter of these derivatives vanishes. For example, if the two components are linked by a dimerization reaction (Phillies and Kivelson were originally interested in the acetic acid monomer-dimer reaction), the derivative with respect to ΔN vanishes unless the reaction is effective at changing the index of refraction increment, which is rarely the case. (The third derivative, referring to the variation of ∈ with E at constant N and ΔN, is very certainly vanishingly small.) The reduction in the number of required derivatives of ∈ between the Mountain and Deutch paper and the Kivelson et al. papers greatly simplifies the estimation of the light-scattering spectrum. One pays a price: the intervening Mori matrices are more tedious to evaluate.
H. Mori, Progress Theoret. Phys. (Kyoto) 33, 423; ibid. 34, 399 (1965).
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I. Daubechies, Ten Lectures on Wavelets, SIAM Press, Philadelphia (1992); G. D. J. Phillies, Computers in Physics 10, 247 (1996).
B. Berne and R. Pecora, Dynamic Light Scattering, Wiley, New York (1976).
For example, the vector cross product is anticommuting, so. In general, Grassman algebras give noncommuting multiplication. Noncommuting variables are common in quantum mechanics. D. F. Nelson and B. Chen [Phys. Rev. B 50, 1023 (1994)] show that introducing noncommuting variables into classical mechanics permits the inclusion of electron spin density into the classical Lagrangian for the long-wavelength excitations of dielectric crystals.
T. Keyes, Phys. Rev. A 30, 2590 (1984); Principles of Mode-Mode Coupling Theory, in Modern Physical Chemistry, Volume 6, Part B, Time Dependent Processes, B. J. Berne, Editor (Plenum, New York, 1977).
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Phillies, G.D. (2000). Projection Operators and the Mori-Zwanzig Formalism. In: Elementary Lectures in Statistical Mechanics. Graduate Texts in Contemporary Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1264-5_32
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