arXiv subjects

Florian Sammüller

Publications and source records attributed to Florian Sammüller.

At least 19 recordsLinked to original sources

arXiv2026

Multivariate hyperdensity functional theory for inhomogeneous equilibrium fluids: From first principles to simulation-based machine learning

Hyperdensity functional theory facilitates the investigation of the equilibrium behavior of a general order parameter or statistical mechanical observable in spatially inhomogeneous classical many-body systems. The approach is based on applying the exact Mermin-Evans classical density functional mapping to an extended ensemble. Here we present the multivariate generalization for investigating simultaneously the properties and interrelations of several different hyperobservables of choice. The resulting framework gives rise to a systematic characterization and prediction scheme for general many-body phenomena. All pertinent equilibrium averages, variances, and covariances constitute universal density functionals, as we demonstrate explicitly. Associated one-body hyperfluctuation profiles quantify the degree of correlation of the local density with first- and second-order combinations of hyperobservables. These multivariate hyperfluctuation profiles are accessible in many-body simulations and they satisfy exact hyper-Ornstein-Zernike equations, which we derive from the minimization principle in the extended multivariate ensemble. The formal structure of the theory integrates naturally with supervised machine learning, which renders all hyperdensity functionals accessible in practice via training of neural networks on simulation data. We demonstrate all salient techniques using the illustrative case of clustering in confined hard rod fluids, thereby choosing the total number of particles and the largest cluster size as the representative hyperobservables of interest. Our numerical methodology enables the efficient and successful prediction of all statistical quantities induced by the chosen hyperobservables, which we verify via comparison to test data and which we attribute to the tight interplay of first-principles and machine-learning concepts that our general approach combines.

cond-mat.soft↗

arXiv2026

Spherical metadensity functional learning for inhomogeneous classical fluids

We develop classical density functional learning to address fluids with truncated pairwise interparticle interactions in three-dimensional spherical geometry. Simulation data for systems with randomized repulsive pair potentials provide the basis for supervised training of a neural metadensity functional, thereby making efficient use of results for radial distribution functions in the bulk fluid via the test particle route. Specifically, we develop spherical local learning in order to represent the one-body direct correlation functional in terms of a neural network, which captures spatial curvature effects as well as the metadensity functional dependence on the thermally scaled pair potential. The framework yields efficient access to inhomogeneous structuring and related physical phenomena that occur in fluids and general solvents when adsorbed against curved solutes and confined inside of spherical and planar cavities. Test particle setups facilitate accurate prediction of the bulk fluid pair structure and verification of thermodynamic test particle sum rules via functional line integration. Applying the metadensity functional for Henderson inversion allows one to infer accurately the pair potential from the bulk radial distribution function. We address implications of the geometrical setup for two-body quantities and obtain the two-body direct correlation functional from automatic differentiation. For the hard sphere fluid, we confirm metadensity functional predictions against results from a standard neural density functional with fixed pair potential as well as to an analytic functional as given by fundamental measure theory. Simulation results provide further reference and corroborate reliable results of the spherical neural metadensity functional across a broad range of applications.

cond-mat.soft↗

arXiv2026

Nonequilibrium scaling of drag forces in counterdriven fluid mixtures

We address the effective nonequilibrium drag force field that emerges from the microscopic interparticle interactions in steady states of counterdriven binary fluid mixtures. Using power functional scaling arguments for adaptive Brownian dynamics computer simulation results, we establish quantitatively the crossover between near-equilibrium linear response and far-nonequilibrium square root asymptotics. An algebraic expression captures both limiting cases and remains applicable in the crossover regime. Using simulation results as benchmarks, we verify that a local power functional approximation based on the scaling law reproduces the spatial nonequilibrium structure formation in inhomogenously driven systems. The crossover scenario transcends dynamical density functional theory and it sheds light on general nonequilibrium scaling of driven fluids.

cond-mat.soft↗

arXiv2026

Metadensity functional learning for classical fluids: Regularizing with pair correlations

We investigate and exploit consequences of the recent neural metadensity functional theory [Kampa et al., Phys. Rev. Lett. 134, 107301 (2025), 10.1103/PhysRevLett.134.107301] for describing the physics of inhomogeneous fluids. The metadensity dependence on the pair potential is relevant for soft matter design and Henderson inversion and it allows one to change the pair potential on the fly at prediction stage. Here we consider one-dimensional systems with short-ranged (truncated) interparticle forces and draw on the functional pair potential dependence to investigate 'metadirect' routes towards the bulk fluid pair correlation structure. Classical density functional theory provides the required functional relationships. Efficient variational calculus is implemented by neural functional line integration and automatic differentiation. We regularize local learning of neural functionals by comparing the pair structure from different routes. Thereby results from metadirect functional differentiation are matched against accurate test particle data from an initial locally trained metadensity functional. Accessing the pair structure via the metadensity functional dependence circumvents Ornstein-Zernike inversion and it is based on first principles.

cond-mat.soft↗

arXiv2025

Gauge invariance and hyperforce correlation theory for equilibrium fluid mixtures

We formulate gauge invariance for the equilibrium statistical mechanics of classical multi-component systems. Species-resolved phase space shifting constitutes a gauge transformation which we analyze using Noether's theorem and shifting differential operators that encapsulate the gauge invariance. The approach yields exact equilibrium sum rules for general mixtures. Species-resolved gauge correlation functions for the force-force and force-gradient pair correlation structure emerge on the two-body level. Exact 3g-sum rules relate these correlation functions to the spatial Hessian of the partial pair distribution functions. General observables are associated with hyperforce densities that measure the covariance of the given observable with the interparticle, external, and diffusive partial force density observables. Exact hyperforce and Lie algebra sum rules interrelate these correlation functions with each other. The practical accessibility of the framework is demonstrated for binary Lennard-Jones mixtures using both adaptive Brownian dynamics and grand canonical Monte Carlo simulations. Specifically, we investigate the force-force pair correlation structure of the Kob-Andersen bulk liquid and we show results for representative hyperforce correlation functions in Wilding et al.'s symmetrical mixture confined between two asymmetric planar parallel walls.

cond-mat.stat-mech↗

arXiv2025

Determining the chemical potential via universal density functional learning

We demonstrate that the machine learning of density functionals allows one to determine simultaneously the equilibrium chemical potential across simulation datasets of inhomogeneous classical fluids. Minimization of a loss function based on an Euler-Lagrange equation yields both the universal one-body direct correlation functional, which is represented locally by a neural network, as well as the system-specific unknown chemical potential values. The method can serve as an efficient alternative to conventional computational techniques of measuring the chemical potential. It also facilitates using canonical data from Brownian dynamics, molecular dynamics, or Monte Carlo simulations as a basis for constructing neural density functionals, which are fit for accurate multiscale prediction of soft matter systems in equilibrium.

cond-mat.soft↗

arXiv2025

Learning the bulk and interfacial physics of liquid-liquid phase separation with neural density functionals

We use simulation-based supervised machine learning and classical density functional theory to investigate bulk and interfacial phenomena associated with phase coexistence in binary mixtures. For a prototypical symmetrical Lennard-Jones mixture our trained neural density functional yields accurate liquid-liquid and liquid-vapour binodals together with predictions for the variation of the associated interfacial tensions across the entire fluid phase diagram. From the latter we determine the contact angles at fluid-fluid interfaces along the line of triple-phase coexistence and confirm there can be no wetting transition in this symmetrical mixture.

cond-mat.soft↗

arXiv2025

Dynamical gauge invariance of statistical mechanics

We investigate gauge invariance against phase space shifting in nonequilibrium systems, as represented by time-dependent many-body Hamiltonians that drive an initial ensemble out of thermal equilibrium. The theory gives rise to gauge correlation functions that characterize spatial and temporal inhomogeneity with microscopic resolution on the one-body level. Analyzing the dynamical gauge invariance allows one to identify a specific localized shift gauge current as a fundamental nonequilibrium observable that characterizes particle-based dynamics. When averaged over the nonequilibrium ensemble, the shift current vanishes identically, which constitutes an exact nonequilibrium conservation law that generalizes the Yvon-Born-Green equilibrium balance of the vanishing sum of ideal, interparticle, and external forces. Any given observable is associated with a corresponding dynamical hyperforce density and hypercurrent correlation function. An exact nonequilibrium sum rule interrelates these one-body functions, in generalization of the recent hyperforce balance for equilibrium systems. We demonstrate the physical consequences of the dynamical gauge invariance using both harmonically confined ideal gas setups, for which we present analytical solutions, and molecular dynamics simulations of interacting systems, for which we demonstrate the shift current and hypercurrent correlation functions to be accessible both via finite-difference methods and via trajectory-based automatic differentiation. We show that the theory constitutes a starting point for developing nonequilibrium reduced-variance sampling algorithms and for investigating thermally-activated barrier crossing.

cond-mat.stat-mech↗

arXiv2025

Why gauge invariance applies to statistical mechanics

We give an introductory account of the recently identified gauge invariance of the equilibrium statistical mechanics of classical many-body systems [J. Müller et al., Phys. Rev. Lett. Phys. Rev. Lett. 133, 217101 (2024)]. 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.

cond-mat.stat-mech↗

arXiv2025

Metadensity functional theory for classical fluids: Extracting the pair potential

The excess free energy functional of classical density functional theory depends upon the type of fluid model, specifically on the choice of (pair) potential, is unknown in general, and is approximated reliably only in special cases. We present a machine learning scheme for training a neural network that acts as a generic metadensity functional for truncated but otherwise arbitrary pair potentials. Automatic differentiation and neural functional calculus then yield, for one-dimensional fluids, accurate predictions for inhomogeneous states and immediate access to the pair distribution function. The approach provides a means of addressing a fundamental problem in the physics of liquids, and for soft matter design: How best to invert structural data to obtain the pair potential?

cond-mat.soft↗

arXiv2024

Why hyperdensity functionals describe any equilibrium observable

We give an introductory account of the recent hyperdensity functional theory for the equilibrium statistical mechanics of soft matter systems [F. Sammüller 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 accurately 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. Müller et al., Phys. Rev. Lett. 133, 217101 (2024); 10.1103/PhysRevLett.133.217101]. 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.

cond-mat.soft↗

arXiv2024

Neural density functional theory of liquid-gas phase coexistence

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 behavior 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.

cond-mat.soft↗

arXiv2024

Gauge Invariance of Equilibrium Statistical Mechanics

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.

cond-mat.stat-mech↗

arXiv2024

Neural density functionals: Local learning and pair-correlation matching

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 [Sammüller 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.

cond-mat.soft↗

arXiv2024

Hyper-density functional theory of soft matter

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.

cond-mat.soft↗

arXiv2024

Neural force functional for non-equilibrium many-body colloidal systems

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.

cond-mat.soft↗

arXiv2024

Noether invariance theory for the equilibrium force structure of soft matter

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. Sammüller $\textit{et al.}$, 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.

cond-mat.soft↗

arXiv2024

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 Sammüller 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.

cond-mat.soft↗