Power functional theory for Newtonian many-body dynamics

Why neural functionals suit statistical mechanics

We describe recent progress in the statistical mechanical description of many-body systems via machine learning combined with concepts from density functional theory and many-body simulations. We argue that the neural functional theory by Sammülleret al(2023Proc. Natl Acad. Sci.120e2312484120) gives a functional representation of direct correlations and of thermodynamics that allows for thorough quality control and consistency checking of the involved methods of artificial intelligence. Addressing a prototypical system we here present a pedagogical application to hard core particle in one spatial dimension, where Percus' exact solution for the free energy functional provides an unambiguous reference. A corresponding standalone numerical tutorial that demonstrates the neural functional concepts together with the underlying fundamentals of Monte Carlo simulations, classical density functional theory, machine learning, and differential programming is available online athttps://github.com/sfalmo/NeuralDFT-Tutorial.

Perspective: How to overcome dynamical density functional theory

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 (DDFT). 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 DDFT as well as the limited availability of its analytical functional approximations.

Perspective: New directions in dynamical density functional theory

Classical dynamical density functional theory (DDFT) has become one of the central modeling approaches in nonequilibrium soft matter physics. Recent years have seen the emergence of novel and interesting fields of application for DDFT. In particular, there has been a remarkable growth in the amount of work related to chemistry. Moreover, DDFT has stimulated research on other theories such as phase field crystal models and power functional theory. In this perspective, we summarize the latest developments in the field of DDFT and discuss a variety of possible directions for future research.

Shear and Bulk Acceleration Viscosities in Simple Fluids

Inhomogeneities in the velocity field of a moving fluid are dampened by the inherent viscous behavior of the system. Both bulk and shear effects, related to the divergence and the curl of the velocity field, are relevant. On molecular time scales, beyond the Navier-Stokes description, memory plays an important role. Using molecular and overdamped Brownian dynamics many-body simulations, we demonstrate that analogous viscous effects act on the acceleration field. This acceleration viscous behavior is associated with the divergence and the curl of the acceleration field, and it can be quantitatively described using simple exponentially decaying memory kernels. The simultaneous use of velocity and acceleration fields enables the description of fast dynamics on molecular scales.

Hydrodynamic density functional theory for mixtures from a variational principle and its application to droplet coalescence

Dynamic density functional theory (DDFT) allows the description of microscopic dynamical processes on the molecular scale extending classical DFT to non-equilibrium situations. Since DDFT and DFT use the same Helmholtz energy functionals, both predict the same density profiles in thermodynamic equilibrium. We propose a molecular DDFT model, in this work also referred to as hydrodynamic DFT, for mixtures based on a variational principle that accounts for viscous forces as well as diffusive molecular transport via the generalized Maxwell-Stefan diffusion. Our work identifies a suitable expression for driving forces for molecular diffusion of inhomogeneous systems. These driving forces contain a contribution due to the interfacial tension. The hydrodynamic DFT model simplifies to the isothermal multicomponent Navier-Stokes equation in continuum situations when Helmholtz energies can be used instead of Helmholtz energy functionals, closing the gap between micro- and macroscopic scales. We show that the hydrodynamic DFT model, although not formulated in conservative form, globally satisfies the first and second law of thermodynamics. Shear viscosities and Maxwell-Stefan diffusion coefficients are predicted using an entropy scaling approach. As an example, we apply the hydrodynamic DFT model with a Helmholtz energy density functional based on the perturbed-chain statistical associating fluid theory equation of state to droplet and bubble coalescence in one dimension and analyze the influence of additional components on coalescence phenomena.

Reconsidering power functional theory

The original derivation of power functional theory [M. Schmidt and J. M. Brader, J. Chem. Phys. 138, 214101 (2013)] is reworked in some detail with a view to clarifying and simplifying the logic and making explicit the various functional dependencies. We note various issues with the original development and suggest a modification that allows us to avoid them. In the process, we also suggest an alternative interpretation of our results, which bears surprising similarities to classical density functional theory.

Phase-field modelling of the effect of density change on solidification revisited: model development and analytical solutions for single component materials

In this paper the development of a physically consistent phase-field theory of solidification shrinkage is presented. The coarse-grained hydrodynamic equations are derived directly from the N-body Hamiltonian equations in the framework of statistical physics, while the constitutive relations are developed in the framework of the standard phase-field theory, by following the variational formalism and the principles of non-equilibrium thermodynamics. To enhance the numerical practicality of the model, quasi-incompressible hydrodynamic equations are derived, where sound waves are absent (but density change is still possible), and therefore the time scale of solidification is accessible in numerical simulations. The model development is followed by a comprehensive mathematical analysis of the equilibrium and propagating one-dimensional solid-liquid interfaces for different density-phase couplings. It is shown, that the fluid flow decelerates/accelerates the solidification front in case of shrinkage/expansion of the solid compared to the case when no density contrast is present between the phases. Furthermore, such a free energy construction is proposed, in which the shape of the equilibrium planar phase-field interface is independent from the density-phase coupling, and the equilibrium interface represents an exact propagating planar interface solution of the quasi-incompressible hydrodynamic equations. Our results are in agreement with previous theoretical predictions.

Shear-induced deconfinement of hard disks

Using Brownian dynamics simulations, we investigate the response to shear of a two-dimensional system of quasi-hard disks that are confined in the velocity gradient direction by a smooth external potential. Shearing the confined system leads to a homogenization of the one-body density profile. In order to rationalize this deconfinement effect, we split the internal one-body force field into adiabatic and superadiabatic contributions. We demonstrate that the superadiabatic force field consists of viscous and of structural contributions. We give an empirical scaling law that yields results for the superadiabatic force profiles both in the flow and in the gradient direction, in excellent agreement with the simulation data.

Superadiabatic Forces via the Acceleration Gradient in Quantum Many-Body Dynamics

We apply the formally exact quantum power functional framework (J. Chem. Phys.2015, 143, 174108) to a one-dimensional Hooke’s helium model atom. The physical dynamics are described on the one-body level beyond the density-based adiabatic approximation. We show that gradients of both the microscopic velocity and acceleration field are required to correctly describe the effects due to interparticle interactions. We validate the proposed analytical forms of the superadiabatic force and transport contributions by comparison to one-body data from exact numerical solution of the Schrödinger equation. Superadiabatic contributions beyond the adiabatic approximation are important in the dynamics and they include effective dissipation.

Deriving phase field crystal theory from dynamical density functional theory: Consequences of the approximations

Phase field crystal (PFC) theory is extensively used for modeling the phase behavior, structure, thermodynamics, and other related properties of solids. PFC theory can be derived from dynamical density functional theory (DDFT) via a sequence of approximations. Here, we carefully identify all of these approximations and explain the consequences of each. One approximation that is made in standard derivations is to neglect a term of form ∇·[n∇Ln], where n is the scaled density profile and L is a linear operator. We show that this term makes a significant contribution to the stability of the crystal, and that dropping this term from the theory forces another approximation, that of replacing the logarithmic term from the ideal gas contribution to the free energy with its truncated Taylor expansion, to yield a polynomial in n. However, the consequences of doing this are (i) the presence of an additional spinodal in the phase diagram, so the liquid is predicted first to freeze and then to melt again as the density is increased; and (ii) other periodic structures, such as stripes, are erroneously predicted to be thermodynamic equilibrium structures. In general, L consists of a nonlocal convolution involving the pair direct correlation function. A second approximation sometimes made in deriving PFC theory is to replace L with a gradient expansion involving derivatives. We show that this leads to the possibility of the density going to zero, with its logarithm going to -∞ while being balanced by the fourth derivative of the density going to +∞. This subtle singularity leads to solutions failing to exist above a certain value of the average density. We illustrate all of these conclusions with results for a particularly simple model two-dimensional fluid, the generalized exponential model of index 4 (GEM-4), chosen because a DDFT is known to be accurate for this model. The consequences of the subsequent PFC approximations can then be examined. These include the phase diagram being both qualitatively incorrect, in that it has a stripe phase, and quantitatively incorrect (by orders of magnitude) regarding the properties of the crystal phase. Thus, although PFC models are very successful as phenomenological models of crystallization, we find it impossible to derive the PFC as a theory for the (scaled) density distribution when starting from an accurate DDFT, without introducing spurious artifacts. However, we find that making a simple one-mode approximation for the logarithm of the density distribution lnρ(x) rather than for ρ(x) is surprisingly accurate. This approach gives a tantalizing hint that accurate PFC-type theories may instead be derived as theories for the field lnρ(x), rather than for the density profile itself.

Structural Nonequilibrium Forces in Driven Colloidal Systems

We identify a structural one-body force field that sustains spatial inhomogeneities in nonequilibrium overdamped Brownian many-body systems. The structural force is perpendicular to the local flow direction, it is free of viscous dissipation, it is microscopically resolved in both space and time, and it can stabilize density gradients. From the time evolution in the exact (Smoluchowski) low-density limit, Brownian dynamics simulations, and a novel power functional approximation, we obtain a quantitative understanding of viscous and structural forces, including memory and shear migration.