Order-preserving dynamics in one dimension – single-file diffusion and caging from the perspective of dynamical density functional theory

Abstract

Dynamical density functional theory (DDFT) is a powerful variational framework to study the nonequilibrium properties of colloids by only considering a time-dependent one-body number density. Despite the large number of recent successes, properly modelling the long-time dynamics in interacting systems within DDFT remains a notoriously difficult problem, since structural information, accounting for temporary or permanent particle cages, gets lost. Here we address such a caging scenario by reducing it to a clean one-dimensional problem, where the particles are naturally ordered (arranged on a line) by perfect cages created by their two next neighbours. In particular, we construct a DDFT approximation based on an equilibrium system with an asymmetric pair potential, such that the corresponding one-body densities still carry the footprint of particle order. Applied to a system of confined hard rods, this order-preserving dynamics (OPD) yields exact results at the system boundaries, in addition to the imprinted correct long-time behaviour of density profiles representing individual particles. In an open system, our approach correctly reproduces the reduced long-time diffusion coefficient and subdiffusion, characteristic for a single-file setup. These observations cannot be made using current forms of DDFT without particle order.

GRAPHICAL ABSTRACT

Keywords:

• Single-file diffusion
• dynamical density functional theory
• Brownian dynamics
• ergodicity
• caging

Acknowledgments

The authors would like to thank Thomas Schindler, Abhinav Sharma, Michael te Vrugt, Raphael Wittkowski, Suvendu Mandal and Daniel de las Heras for stimulating discussions. We also acknowledge Thomas Schindler for providing the BD code used in this study and careful proofreading of the manuscript. This paper is dedicated to the late Gerhard Findenegg.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Supplemental material for this article, containing extended versions of Figures 4–7, 9–11 and A1, is available online.

2 Describing particle order as a type of interaction is particularly convenient in DFT, since most explicit density functionals are derived from a given pair-interaction potential between two particles. In contrast, it is not possible to incorporate particle order through boundary conditions on the level of the one-body density profiles ${\rho }_N^{(\nu )}(x)$ of individual particles, since working with ${\rho }_N^{(\nu )}(x)$ implies that all configurational integrals have already been carried out. The conditional densities ${\varrho }_N^{(\nu )}(x|x_0)$, however, are not fully ensemble-averaged, such that a boundary condition involving the field variable x and the remaining configurational variable $x_0$ can be properly included for the three-component mixture considered here.

3 Note that the grand-canonical intrinsic free energy functional $\mathcal{F}[\{{\rho }^{(\nu )}(x)\}]$ expressed in a form analogous to Equation (30(30) $\begin{array}{r}\beta {\mathcal{F}}_N[\{{\rho }_N^{(\nu )}\}]=-\ln Z_N-\sum _{\nu }\int \mathrm{d}x{\rho }_N^{(\nu )}(x)\beta V^{(\nu )}(x)\end{array}$(30) ) contains additional terms including the chemical potentials ${\mu }_{\nu }$, which are linear in the densities and thus vanish after taking the spatial derivative of the functional derivative.

Additional information

Funding

Financial support is provided by the Deutsche Forschungsgemeinschaft (DFG) through the SPP 2265, under grant numbers WI 5527/1-1 (R.W.) and LO 418/25-1 (H.L.).