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We give an introductory account of the recent hyperdensity functional theory for the equilibrium statistical mechanics of soft matter systems [F. Samm\"uller et al., Phys. Rev. Lett. 133, 098201 (2024); 10.1103/PhysRevLett.133.098201]. Hyperdensity functionals give access to the behaviour of arbitrary thermal observables in spatially inhomogeneous equilibrium many-body systems. The approach is based on classical density functional theory applied to an extended ensemble using standard functional techniques. The associated formally exact generalized Mermin-Evans functional relationships can be represented efficiently by neural functionals. These neural networks are trained via simulation-based supervised machine learning and they allow one to carry out efficient functional calculus using automatic differentiation and numerical functional line integration. Exact sum rules, including hard wall contact theorems and hyperfluctuation Ornstein-Zernike equations, interrelate the different correlation functions. We lay out close connections to hyperforce correlation sum rules [S. Robitschko et al., Commun. Phys. 7, 103 (2024); 10.1038/s42005-024-01568-y] that arise from statistical mechanical gauge invariance [J. M\"uller et al., Phys. Rev. Lett. (to appear); arXiv:2406.19235]. Further quantitative measures of collective self-organization are provided by hyperdirect correlation functionals and spatially resolved hyperfluctuation profiles. The theory facilitates to gain deep insight into the inherent structuring mechanisms that govern the behaviour of both simple and complex order parameters in coupled many-body systems., Comment: 23 pages, 5 figures, 75 equations

We give an introductory account of the recently identified gauge invariance of the equilibrium statistical mechanics of classical many-body systems [J. M\"uller et al., Phys. Rev. Lett. (to appear) arXiv:2406.19235]. The gauge transformation is a non-commutative shifting operation on phase space that keeps the differential phase space volume element and hence the Gibbs integration measure conserved. When thermally averaged any observable is an invariant, including thermodynamic and structural quantities. Shifting transformations are canonical in the sense of classical mechanics. They also form an infinite-dimensional group with generators of infinitesimal transformations that build a non-commutative Lie algebra. We lay out the connections with the underlying geometry of coordinate displacement and with Noether's theorem. Spatial localization of the shifting yields differential operators that satisfy commutator relationships, which we describe both in purely configurational and in full phase space setups. Standard operator calculus yields corresponding equilibrium hyperforce correlation sum rules for general observables and order parameters. Using Monte Carlos simulations we demonstrate explicitly the gauge invariance for finite shifting. We argue in favour of using the gauge invariance as a statistical mechanical construction principle for obtaining exact results and for formulating smart sampling algorithms., Comment: 18 pages, 7 figures, 90 equations

We use supervised machine learning together with the concepts of classical density functional theory to investigate the effects of interparticle attraction on the pair structure, thermodynamics, bulk liquid-gas coexistence, and associated interfacial phenomena in many-body systems. Local learning of the one-body direct correlation functional is based on Monte Carlo simulations of inhomogeneous systems with randomized thermodynamic conditions, randomized planar shapes of the external potential, and randomized box sizes. Focusing on the prototypical Lennard-Jones system, we test predictions of the resulting neural attractive density functional across a broad spectrum of physical behaviour associated with liquid-gas phase coexistence in bulk and at interfaces. We analyse the bulk radial distribution function $g(r)$ obtained from automatic differentiation and the Ornstein-Zernike route and determine i) the Fisher-Widom line, i.e.\ the crossover of the asymptotic (large distance) decay of $g(r)$ from monotonic to oscillatory, ii) the (Widom) line of maximal correlation length, iii) the line of maximal isothermal compressibility and iv) the spinodal by calculating the poles of the structure factor in the complex plane. The bulk binodal and the density profile of the free liquid-gas interface are obtained from density functional minimization and the corresponding surface tension from functional line integration. We also show that the neural functional describes accurately the phenomena of drying at a hard wall and of capillary evaporation for a liquid confined in a slit pore. Our neural framework yields results that improve significantly upon standard mean-field treatments of interparticle attraction. Comparison with independent simulation results demonstrates a consistent picture of phase separation even when restricting the training to supercritical states only., Comment: 21 pages, 9 figures

We identify a recently proposed shifting operation on classical phase space as a gauge transformation for statistical mechanical microstates. The infinitesimal generators of the continuous gauge group form a non-commutative Lie algebra, which induces exact sum rules when thermally averaged. Gauge invariance with respect to finite shifting is demonstrated via Monte Carlo simulation in the transformed phase space which generates identical equilibrium averages. Our results point towards a deeper basis of statistical mechanics than previously known and they offer avenues for systematic construction of exact identities and of sampling algorithms., Comment: 7 pages, 3 figures

Recently Dijkman et al. (arXiv:2403.15007) proposed training classical neural density functionals via bulk pair-correlation matching. We show their method to be an efficient regularizer for neural functionals based on local learning of inhomogeneous one-body direct correlations [Samm\"uller et al., Proc. Natl. Acad. Sci. 120, e2312484120 (2023), 10.1073/pnas.2312484120]. While Dijkman et al. demonstrated pair-correlation matching of a global neural free energy functional, we argue in favor of local one-body learning for flexible neural modelling of the full Mermin-Evans density functional map. Using spatial localization gives access to accurate neural free energy functionals, including convolutional neural networks, that transcend the training box., Comment: 6 pages, 2 figures + supplementary material (7 pages, 6 figures)

We combine power functional theory and machine learning to study non-equilibrium overdamped many-body systems of colloidal particles at the level of one-body fields. We first sample in steady state the one-body fields relevant for the dynamics from computer simulations of Brownian particles under the influence of randomly generated external fields. A neural network is then trained with this data to represent locally in space the formally exact functional mapping from the one-body density and velocity profiles to the one-body internal force field. The trained network is used to analyse the non-equilibrium superadiabatic force field and the transport coefficients such as shear and bulk viscosities. Due to the local learning approach, the network can be applied to systems much larger than the original simulation box in which the one-body fields are sampled. Complemented with the exact non-equilibrium one-body force balance equation and a continuity equation, the network yields viable predictions of the dynamics in time-dependent situations. Even though training is based on steady states only, the predicted dynamics is in good agreement with simulation results. A neural dynamical density functional theory can be straightforwardly implemented as a limiting case in which the internal force field is that of an equilibrium system. The framework is general and directly applicable to other many-body systems of interacting particles following Brownian dynamics.

We present a scheme for investigating arbitrary thermal observables in spatially inhomogeneous equilibrium many-body systems. Extending the grand canonical ensemble yields any given observable as an explicit hyper-density functional. Associated local fluctuation profiles follow from an exact hyper-Ornstein-Zernike equation. While the local compressibility and simple observables permit analytic treatment, complex order parameters are accessible via simulation-based supervised machine learning of neural hyper-direct correlation functionals. We exemplify efficient and accurate neural predictions for the cluster statistics of hard rods, square well rods, and hard spheres. The theory allows to treat complex observables, as is impossible in standard density functional theory., Comment: 7 pages, 2 figures + supplementary material (4 pages, 1 figure)

We give details and derivations for the Noether invariance theory that characterizes the spatial equilibrium structure of inhomogeneous classical many-body systems, as recently proposed and investigated for bulk systems [F. Samm\"uller $\textit{et al.}$, Phys. Rev. Lett. $\textbf{130}$, 268203 (2023)]. Thereby an intrinsic thermal symmetry against a local shifting transformation on phase space is exploited on the basis of the Noether theorem for invariant variations. We consider the consequences of the shifting that emerge at second order in the displacement field that parameterizes the transformation. In a natural way the standard two-body density distribution is generated. Its second spatial derivative (Hessian) is thereby balanced by two further and different two-body correlation functions, which respectively introduce thermally averaged force correlations and force gradients in a systematic and microscopically sharp way into the framework. Separate exact self and distinct sum rules hold expressing this balance. We exemplify the validity of the theory on the basis of computer simulations for the Lennard-Jones gas, liquid, and crystal, the Weeks-Chandler-Andersen fluid, monatomic Molinero-Moore water at ambient conditions, a three-body-interacting colloidal gel former, the Yukawa and soft-sphere dipolar fluids, and for isotropic and nematic phases of Gay-Berne particles. We describe explicitly the derivation of the sum rules based on Noether's theorem and also give more elementary proofs based on partial phase space integration following Yvon's theorem., Comment: 20 pages, 3 figures

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 Samm\"uller et al. [Proc. Nat. Acad. Sci. 120, e2312484120 (2023)] 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 at https://github.com/sfalmo/NeuralDFT-Tutorial., Comment: 25 pages, 7 figures, 170 references; for associated online tutorial, see: https://github.com/sfalmo/NeuralDFT-Tutorial

Noether invariance in statistical mechanics provides fundamental connections between the symmetries of a physical system and its conservation laws and sum rules. The latter are exact identities that involve statistically averaged forces and force correlations and they are derived from statistical mechanical functionals. However, the implications for more general observables and order parameters are unclear. Here, we demonstrate that thermally averaged classical phase space functions are associated with exact hyperforce sum rules that follow from translational Noether invariance. Both global and locally resolved identities hold and they relate the mean gradient of a phase-space function to its negative mean product with the total force. Similar to Hirschfelder's hypervirial theorem, the hyperforce sum rules apply to arbitrary observables in equilibrium. Exact hierarchies of higher-order sum rules follow iteratively. As applications we investigate via computer simulations the emerging one-body force fluctuation profiles in confined liquids. These local correlators quantify spatially inhomogeneous self-organization and their measurement allows for the development of stringent convergence tests and enhanced sampling schemes in complex systems., Comment: 15 pages, 6 figures

We present a hybrid scheme based on classical density functional theory and machine learning for determining the equilibrium structure and thermodynamics of inhomogeneous fluids. The exact functional map from the density profile to the one-body direct correlation function is represented locally by a deep neural network. We substantiate the general framework for the hard sphere fluid and use grand canonical Monte Carlo simulation data of systems in randomized external environments during training and as reference. Functional calculus is implemented on the basis of the neural network to access higher-order correlation functions via automatic differentiation and the free energy via functional line integration. Thermal Noether sum rules are validated explicitly. We demonstrate the use of the neural functional in the self-consistent calculation of density profiles. The results outperform those from state-of-the-art fundamental measure density functional theory. The low cost of solving an associated Euler-Lagrange equation allows to bridge the gap from the system size of the original training data to macroscopic predictions upon maintaining near-simulation microscopic precision. These results establish the machine learning of functionals as an effective tool in the multiscale description of soft matter., Comment: 20 pages, 11 figures

We show in detail how three one-body fluctuation profiles, namely the local compressibility, the local thermal susceptibility, and the reduced density, can be obtained from a statistical mechanical many-body description of classical particle-based systems. We present several different and equivalent routes to the definition of each fluctuation profile, facilitating their explicit numerical calculation in inhomogeneous equilibrium systems. This underlying framework is used for the derivation of further properties such as hard wall contact theorems and novel types of inhomogeneous one-body Ornstein-Zernike equations. The practical accessibility of all three fluctuation profiles is exemplified by grand canonical Monte Carlo simulations that we present for hard sphere, Gaussian core and Lennard-Jones fluids in confinement., Comment: 17 pages, 8 figures

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., Comment: 21 pages, 5 figures. J. Phys.: Condens. Matter (invited Perspective)

Liquid structure carries deep imprints of an inherent thermal invariance against a spatial transformation of the underlying classical many-body Hamiltonian. At first order in the transformation field the Noether theorem yields the local force balance. Three distinct two-body correlation functions emerge at second order, namely the standard two-body density, the localized force-force correlation function, and the localized force gradient. An exact Noether sum rule interrelates these correlators. Simulations of Lennard-Jones, Yukawa, soft-sphere dipolar, Stockmayer, Gay-Berne and Weeks-Chandler-Andersen liquids, of monatomic water and of a colloidal gel former demonstrate their fundamental role in the characterization of spatial structure., Comment: Previous title was: "What is liquid, from Noether's perspective?"

We reexamine results obtained with the recently proposed density functional theory framework based on forces (force-DFT) [Tschopp et al., Phys. Rev. E 106, 014115 (2022)]. We compare inhomogeneous density profiles for hard sphere fluids to results from both standard density functional theory and from computer simulations. Test situations include the equilibrium hard sphere fluid adsorbed against a planar hard wall and the dynamical relaxation of hard spheres in a switched harmonic potential. The comparison to grand canonical Monte Carlo simulation profiles shows that equilibrium force-DFT alone does not improve upon results obtained with the standard Rosenfeld functional. Similar behavior holds for the relaxation dynamics, where we use our event-driven Brownian dynamics data as benchmark. Based on an appropriate linear combination of standard and force-DFT results, we investigate a simple hybrid scheme which rectifies these deficiencies in both the equilibrium and the dynamical case. We explicitly demonstrate that although the hybrid method is based on the original Rosenfeld fundamental measure functional, its performance is comparable to that of the more advanced White Bear theory.

We investigate the stationary flow of a colloidal gel under an inhomogeneous external shear force using adaptive Brownian dynamics simulations. The interparticle forces are derived from the Stillinger-Weber potential, where the three-body term is tuned to enable network formation and gelation in equilibrium. When subjected to the shear force field, the system develops remarkable modulations in the one-body density profile. Depending on the shear magnitude, particles accumulate either in quiescent regions or in the vicinity of maximum net flow, and we deduce this strong non-equilibrium response to be characteristic of the gel state. Studying the components of the internal force parallel and perpendicular to the flow direction reveals that the emerging flow and structure of the stationary state are driven by significant viscous and structural superadiabatic forces. Thereby, the magnitude and nature of the observed non-equilibrium phenomena differs from the corresponding behavior of simple fluids. We demonstrate that a simple power functional theory reproduces accurately the viscous force profile, giving a rationale of the complex dynamical behavior of the system.

A framework for performant Brownian Dynamics (BD) many-body simulations with adaptive timestepping is presented. Contrary to the Euler-Maruyama scheme in common non-adaptive BD, we employ an embedded Heun-Euler integrator for the propagation of the overdamped coupled Langevin equations of motion. This enables the derivation of a local error estimate and the formulation of criteria for the acceptance or rejection of trial steps and for the control of optimal stepsize. Introducing erroneous bias in the random forces is avoided by Rejection Sampling with Memory (RSwM) due to Rackauckas and Nie, which makes use of the Brownian bridge theorem and guarantees the correct generation of a specified random process even when rejecting trial steps. For test cases of Lennard-Jones fluids in bulk and in confinement, it is shown that adaptive BD solves performance and stability issues of conventional BD, already outperforming the latter even in standard situations. We expect this novel computational approach to BD to be especially helpful in long-time simulations of complex systems, e.g. in non-equilibrium, where concurrent slow and fast processes occur.

Three one-body profiles that correspond to local fluctuations in energy, in entropy, and in particle number are used to describe the equilibrium properties of inhomogeneous classical many-body systems. Local fluctuations are obtained from thermodynamic differentiation of the density profile or equivalently from average microscopic covariances. The fluctuation profiles follow from functional generators and they satisfy Ornstein-Zernike relations. Computer simulations reveal markedly different fluctuations in confined fluids with Lennard-Jones, hard sphere, and Gaussian core interactions.