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Title:  Superadiabatic Forces in the Dynamics of the Hard Sphere Fluid
Author:  Lucas L. Treffenstädt
As Contributor:   Lucas L. Treffenstädt
Type: Ph.D.
Field: Physics
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational
Degree granting institution:  University of Bayreuth
Supervisor(s): Matthias Schmidt
Defense date:  2022-07-08


In this thesis we investigate the Brownian dynamics of the monodisperse hard sphere fluid. The hard sphere fluid is a very simple model fluid, in which the interparticle interactions prevent particle overlap. The pair potential of the spheres is therefore characterised entirely by the diameter of the spheres. Despite its simplicity, the hard sphere fluid has many interesting properties and provides opportunities to study many fundamental properties of the fluid state. Brownian dynamics is a model for the time evolution of colloidal particles suspended in a solvent. The microscopic interactions between particles and solvent are modelled via stochastic external forces. The solvent itself is not modelled, and inertia of the particles is neglected. This results in a stochastic equation of motion for the many-body system, which can be integrated in time to obtain a powerful simulation method. For hard sphere interactions, special care must be given to particle collisions. One possible method is event-driven Brownian dynamics. We use event-driven Brownian dynamics simulations as a reference for a study of hard sphere dynamics. We are interested in a statistical description of a thermodynamic system in the one-body picture, where we consider one-body correlation functions such as the particle density profile and particle current profile. These quantities can be obtained via averages of many-body observables like particle positions and velocities. A one-body description is desirable, as the computational cost of many-body simulations can be prohibitive for large and/or dense systems. Additionally, a one-body description enables the understanding of collective phenomena which may be hidden in the many-body picture. One such description is given by power functional theory, which is a formally exact theoretical framework for one-body dynamics of classical (and quantum) nonequilibrium systems. This framework allows for a splitting of the internal force density, which acts on the current profile into an adiabatic and a superadiabatic contribution. Adiabatic forces depend on the instantaneous density profile of the system and can be derived from the free energy functional known from classical density functional theory. Superadiabatic forces are true nonequilibrium forces and depend on density and current as a function of time. This memory of previous density and current can be spatially nonlocal. One central aim of this thesis is to establish a universal functional for the calculation of superadiabatic forces in the dynamics of the hard sphere fluid. The development of this functional is based on simulations of an inhomogeneously sheared hard sphere fluid[1], and of the time evolution of the van Hove function[2,3]. The van Hove correlation function G(r,t) gives the correlation of a particle at the origin at time 0 with particles at distance r from the origin at time t. This function is related via the dynamical test particle limit to the time evolution of a binary fluid with a special initial condition. In this initial condition, the density of one fluid component, the self component, corresponds to a single particle fixed at the origin. The other component, the distinct component, is distributed according to the equilibrium radial distribution function. We find that superadiabatic forces play a major role both in sheared fluids and in the test particle limit and we propose an approximation which accurately describes these forces. The bottom-up construction of this approximation allows for a splitting of the superadiabatic force field into several distinct forces. These are the viscoelastic force, the drag force, and the structural force. In our study of the sheared fluid, we show that an accurate description of the viscoelastic force requires nonlocal memory. We demonstrate the quantitative accuracy of our corresponding approximation of this force by comparing with simulation data. In our studies of the van Hove function we arrive at the insight that our power functional approximation, which was first developed for the description of nonequilibrium systems, is equally applicable for equilibrium dynamics. We conclude that the described forces represent universal effects in fluids. [1] Treffenstädt, L. L. & Schmidt, M. „Memory-induced motion reversal in Brownian liquids“. Soft Matter 16, 1518 (2020). [2] Treffenstädt, L. L. & Schmidt, M. „Universality in Driven and Equilibrium Hard Sphere Liquid Dynamics“. Phys. Rev. Lett. 126, 058002 (2021). [3] Treffenstädt, L. L., Schindler, T. & Schmidt, M. „Dynamic Decay and Superadiabatic Forces in the van Hove Dynamics of Bulk Hard Sphere Fluids“. SciPost Phys. 12, 133 (2022).

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