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Exact Volterra series for mean field dynamics

Ion Santra, Matthias Krüger

Abstract

We derive an exact Volterra series expansion for a mean field of an interacting particle system subject to a potential perturbation, expressing the Volterra expansion kernels in terms of the field's response functions, to any order. Applying this formalism to the mean particle density of a simple fluid, we identify a form reminiscent of dynamical density functional theory, with, however, fundamental differences: A nonlocal mobility kernel appears, and forces derive from a functional of the {\it history} of mean density. The equilibrium density functional is shown to be recovered in the limit of slowly varying perturbation. We identify a freedom in deriving this expansion, which allows different forms of mobility kernels. These developments allow for a systematic improvement of established mean field formalisms.

Exact Volterra series for mean field dynamics

Abstract

We derive an exact Volterra series expansion for a mean field of an interacting particle system subject to a potential perturbation, expressing the Volterra expansion kernels in terms of the field's response functions, to any order. Applying this formalism to the mean particle density of a simple fluid, we identify a form reminiscent of dynamical density functional theory, with, however, fundamental differences: A nonlocal mobility kernel appears, and forces derive from a functional of the {\it history} of mean density. The equilibrium density functional is shown to be recovered in the limit of slowly varying perturbation. We identify a freedom in deriving this expansion, which allows different forms of mobility kernels. These developments allow for a systematic improvement of established mean field formalisms.
Paper Structure (4 sections, 50 equations, 1 figure)

This paper contains 4 sections, 50 equations, 1 figure.

Table of Contents

  1. General expressions
  2. Equivalence of third-order TDDCF and equilibrium DCF
  3. Ornstein-Zernicke relation
  4. Integrated response for static perturbation

Figures (1)

  • Figure 1: Mean field dynamics for systems subject to the perturbation of Eq. \ref{['eq:forcing']}, as obtained from the two different gauge choices and standard DDFT, compared to the exact solution. Panel (a) shows a system with single relaxation time, in which case all the lines coincide. Panel (b) shows a system with two distinct relaxation times $t_k^{(2)}/t^{(1)}_k=40$, and $g_k^{(1)}=0.2$ and $g_k^{(2)}=0.8$ (see Eq. \ref{['glass:sk2']}). While local and non-local gauge choices agree with the exact solution, DDFT deviates. Inset shows the agreement of DDFT for short times.