arXiv subjects

Matthias Schmidt

Publications and source records attributed to Matthias Schmidt.

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

Entropy power functional theory for Brownian many-body dynamics

We present a formally exact variational scheme for the overdamped Brownian dynamics of pairwise interacting many-body systems in general spatiotemporal nonequilibrium. A joint free power minimization principle determines instantaneously the one-body current and the global interparticle distance flux. The intrinsic free power functional splits into entropic and energetic rates, where the latter are treated explicitly. The adiabatic contribution to the entropy rate is the time derivative of the equilibrium entropy metadensity functional. Genuine nonequilibrium effects originate from a universal entropy superpower functional. Two continuity equations close the dynamical description.

cond-mat.soft↗

arXiv2026

Entropy density functional universality: Correlation, response, and entropic Ornstein-Zernike structure

We give a comprehensive account of the recent entropy density functional theory for the equilibrium statistical mechanics of classical many-body systems (arXiv:2606.28240). The approach is formally exact and based on a joint grand potential minimization principle for the one-body density and the global pair distance distribution. These variational fields depend respectively on position and on scalar distance, which retains the low computational complexity of standard density functional theory. Correlations effects are contained in a unique excess entropy functional, which is universal across all systems with pairwise interparticle potentials. Functional differentiation yields entropic direct correlation functionals that generate entropic response and fluctuation correlation functions via coupled Ornstein-Zernike equations. Two alternative proofs are given for the existence and uniqueness of the underlying metadensity functional map, based on generalizations of either Levy's constrained search method or Mermin-Evans proof by contradiction. Simple excess entropy approximations yield the standard mean-field and second-virial excess free energy density functionals. We describe exact entropic functional line integrals, make connections to the recent one-body fluctuation profiles, and generalize the entropy approach beyond pairwise interparticle potentials.

cond-mat.soft↗

arXiv2026

Entropy density functional theory for inhomogeneous fluids

We present an exact variational scheme for the physics of inhomogeneous classical fluids in thermal equilibrium. A joint metadensity minimization principle is proven for the one-body density and the global interparticle distance distribution. The theory bypasses the inhomogeneous two-body density and thus remains computationally simple. A universal excess entropy functional accounts for all many-body correlations in arbitrary pairwise interacting systems. The framework is relevant for neural functional machine learning, for soft matter design, and for predicting structural correlation functions via entropic test-particle and meta-Ornstein-Zernike routes.

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

Quantum statistical mechanics: Gauge invariance, operator shifting, hyperdensity functionals, and nonequilibrium sum rules

We provide an extended acount of the recent statistical mechanical theory of gauge invariance against operator shifting in quantum many-body systems (arXiv:2509.20494). The gauge transformation is enacted by a shifting superoperator that displaces the fundamental position and momentum degrees of freedom. The shifting superoperator constitutes a map between Hilbert space operators and it features Lie algebra commutator structure. Averages of general observables remain invariant under the shifting both in and out of thermal equilibrium, as well as in groundstates. The gauge invariance induces exact sum rules that interconnect global observables and associated locally resolved correlation functions. In particular we describe the resulting one-body force, hyperforce, product, and two-body sum rules. We relate the shifting superoperator to a previously formulated quantum canonical transformation and present the generalization of quantum shifting to multi-component systems. The gauge theory respects fundamental fermionic and bosonic particle properties, as we demonstrate by proving the compatibility of operator shifting and exchange symmetry. We formulate the quantum version of hyperdensity functional theory to provide formal access to hyperforces as well as to general averaged quantum observables via universal density functionals. For time-dependent situations, we describe quantum dynamical gauge invariance and prove exact dynamical sum rules for nonequilibrium situations, as generated by Hamiltonian time dependence. We argue for the fundamental status of statistical mechanical gauge invariance based on the compliance of the underlying geometry with canonical quantization according to Dirac's correspondence principle. Analogies and differences of the quantum mechanical sum rules with their classical counterparts remain indicative of the respective levels of description.

cond-mat.stat-mech↗

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

Quantum statistical mechanical gauge invariance

We address gauge invariance in the statistical mechanics of quantum many-body systems. The gauge transformation acts on the position and momentum degrees of freedom and it is represented by a quantum shifting superoperator that maps quantum observables onto each other. The shifting superoperator is anti-self-adjoint and it has noncommutative Lie algebra structure. These properties induce exact equilibrium sum rules that connect locally-resolved force and hyperforce densities for any given observable. We argue that the framework is amenable to tight integration into quantum hyperdensity functional theory and that it generalizes naturally to nonequilibrium.

quant-ph↗

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

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

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

All-optical radio-frequency phase detection for Rydberg atom sensors using oscillatory dynamics

Rydberg atom radio frequency sensors are a unique platform for precision electromagnetic field measurement, e.g. they have extraordinary carrier bandwidth spanning MHz-THz and can be self-calibrated. These photonic sensors use lasers to prepare and read out the atomic response to a radio frequency electromagnetic field. Most work on Rydberg atom sensors centers on radio frequency electric field strength because the sensor functions as a square law detector, unless an external radio frequency heterodyning field is used. A heterodyning field acts as a local oscillator and enables phase read out at the expense of the radio frequency equipment necessary to generate it. In order to overcome the disadvantages of a radio frequency local oscillator, we investigate all-optical phase-sensitive detection using a five-level closed-loop excitation scheme. We show that under finite detuning of the loop fields, the atomic response oscillates at the frequency of the detuning. The oscillation is transferred to a probe laser absorption signal. The phase, frequency and amplitude of the radio frequency signal are imprinted on the oscillatory dynamics and can be determined using demodulation and matched filter techniques applied to the probe laser transmission signal.

physics.atom-ph↗

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↗

arXiv2024

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. 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 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. M\"uller 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

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. M\"uller 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↗

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↗