arXiv subjects

Johanna Müller

Publications and source records attributed to Johanna Müller.

6 recordsLinked to original sources

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

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

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↗

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↗