Mode-coupling theory for multiple-point and multiple-time correlation functions

Abstract

We present a theoretical framework for higher-order correlation functions involving multiple times and multiple points in a classical, many-body system. Such higher-order correlation functions have attracted much interest recently in the context of various forms of multi-dimensional spectroscopy, and have found an intriguing application as proposed measures of dynamical heterogeneities in structural glasses. The theoretical formalism is based upon projection operator techniques which are used to isolate the slow time evolution of dynamical variables by expanding the slowly-evolving component of arbitrary variables in an infinite, "multi-linear" basis composed of the products of slow variables of the system. Using the formalism, a formally exact mode coupling theory is derived for multi-point and multi-time correlation functions. The resulting expressions for higher-order correlation functions are made tractable by applying a rigorous perturbation scheme, called the N -ordering method, which is exact for systems with finite correlation lengths in the thermodynamic limit. The theory is contrasted with standard mode coupling theories in which the noise or fluctuating force appearing in the generalized Langevin equation is assumed to be Gaussian, and it is demonstrated that the non-Gaussian nature of the fluctuating forces leads to important contributions to higherorder correlation functions. Finally, the higher-order correlation functions are evaluated analytically for an ideal gas system for which it is shown that the mode coupling theory is exact.

Fifth-order two-dimensional Raman spectroscopy of liquid water, crystalline ice Ih and amorphous ices: Sensitivity to anharmonic dynamics and local hydrogen bond network structure

“…31 The effect of the nonlinear polarizabilities in real systems, i.e., liquid Xe, CS 2 , and water, was first analyzed by using normal mode description. [42][43][44][45][46][47][48][49] In particular, Denny and Reichman successfully reproduced the 2D-Raman signal of liquid Xe based on the mode coupling theory by using the factorizing 32 Ma and Stratt found that a clear echo signal of the 2D-Raman response exists in the instantaneous normal mode description of liquid Xe, but disappears when the full molecular dynamics ͑MD͒ calculation is employed.…”

Molecular hydrodynamic theory of nonresonant Raman spectra in liquids: Fifth-order spectra

“…Indeed, several sophisticated mode-coupling approaches have been formulated to compute corrections to a Gaussian theory. 36,37 One may view both the ''primitive'' INM method and the molecular hydrodynamic theory presented in this work as 0th-order theories from which corrections may be incorporated. The molecular hydrodynamic approach provides a vastly more successful 0th-order starting point than the primitive INM theory, and, in some respects, slightly better results than the anharmonic INM theory.…”

Multi-time density correlation functions in glass-forming liquids: Probing dynamical heterogeneity and its lifetime

“…[73][74][75][76][77][78] There exist promising theoretical treatments for the multi-time correlation function based on the mode-coupling theory. 79,80 These techniques have now become powerful and standard tools to study condensed phase dynamics. For example, there are often used to study the ultrafast dynamics of liquid water.…”