Perspective: How to overcome dynamical density functional theory

Abstract

We argue in favour of developing a comprehensive dynamical theory for rationalizing, predicting, designing, and machine learning nonequilibrium phenomena that occur in soft matter. To give guidance for navigating the theoretical and practical challenges that lie ahead, we discuss and exemplify the limitations of dynamical density functional theory. Instead of the implied adiabatic sequence of equilibrium states that this approach provides as a makeshift for the true time evolution, we posit that the pending theoretical tasks lie in developing a systematic understanding of the dynamical functional relationships that govern the genuine nonequilibrium physics. While static density functional theory gives a comprehensive account of the equilibrium properties of many-body systems, we argue that power functional theory is the only present contender to shed similar insights into nonequilibrium dynamics, including the recognition and implementation of exact sum rules that result from the Noether theorem. As~a~demonstration of the power functional point of view, we consider an idealized steady sedimentation flow of the three-dimensional Lennard-Jones fluid and machine-learn the kinematic map from the mean motion to the internal force field. The trained model is capable of both predicting and designing the steady state dynamics universally for various target density modulations. This demonstrates the significant potential of using such techniques in nonequilibrium many-body physics and overcomes both the conceptual constraints of dynamical density functional theory as well as the limited availability of its analytical functional approximations.

Competition between drift and topological transport of colloidal particles in twisted magnetic patterns

“…It would be interesting to study the effects that the many-body interparticle interactions have on the transport. Many-body non-equilibrium superadiabatic forces [30,31] might alter the dynamical phase diagram and new states such as the occurrence of solitons [32,33] might appear. Interparticle repulsion might scatter particles away from the flat channels, while interparticle attraction could drag particles together through flat channels resulting in an increased mobility.…”

Local measures of fluctuations in inhomogeneous liquids: statistical mechanics and illustrative applications

“…If appropriate closure relations could be found in future work, one might get a feasible scheme for obtaining the inhomogeneous fluctuation profiles-akin to liquid integral equation theory [13]-from these Ornstein-Zernike equations. A more immediate application arises by considering the fluctuation OZ equations as easily accessible sum rules, which could prove to be useful in the development and verification of novel numerical methods and machine learning techniques [48].…”

Why neural functionals suit statistical mechanics

“…The neural nonequilibrium force fields were successfully compared against analytical power functional approximations, where simple and physically motivated semi-local dependence on both the local density and local velocity gradients was shown to capture correctly the essence of the forces that occur in the nonequilibrium situation. Together with the exact force balance equation, this allows to predict and to design nonequilibrium steady states [40]. The approach offers a systematic way to go beyond dynamical density functional theory and to address genuine nonequilibrium beyond a free energy description.…”

“…Density functional theory lends itself towards machine learning due the necessity of finding an approximation for the central functional. Corresponding research was carried out in the classical [31][32][33][34][35][36][37][38][39][40][41][42] and quantum realms [43][44][45][46][47][48][49][50][51]. The classical work addressed liquid crystals in complex confinement [31], the functional construction of a convolutional network [32] and of an equation-learning network [33], the improvement of the standard mean-field approximation for the threedimensional Lennard-Jones system [34] with the aim of addressing gas solubility in nanopores [35], the use physicsinformed Bayesian inference [36,37], active learning with error control [38], and the physics of patchy particles [39].…”

“…In recent work, de las Heras et al [40] and Sammüller et al [41] have put forward machine learning strategies that operate on the one-body level of correlation functions. Here we address in detail the neural functional theory [41] for inhomogeneous fluids in equilibrium.…”