arXiv subjects

Johannes Zimmer

Publications and source records attributed to Johannes Zimmer.

At least 19 recordsLinked to original sources

arXiv2026

Layered Dissipative Neural Fields: Bounding the Dimension of the Population Activity Manifold with Infinitely Many Neurons

We consider a class of neural field models describing the macroscopic activity of an infinite population of neurons organized into finitely many stacked layers, written as a system of semi-linear higher-order parabolic equations. For this class of biologically inspired equations, we show that the higher-order dissipation, modeling gap junction activity regulation, induces a spectral gap ensuring that the associated semigroup possesses an inertial manifold, that is, a finite-dimensional manifold attracting all orbits of the infinite-dimensional system of neural activity. Our main result is an upper bound on the dimension of this manifold, giving its explicit scaling in the dissipation order and strength, the leaking rate, and the norm of the connection operator. We interpret this as a step toward the mathematical modeling of the empirically motivated concept of neural manifold, namely the idea that the collective activity of a large neural population lies on a low-dimensional manifold. For this class of equations, we further prove universal approximation properties, namely stationary pattern expressivity in terms of the input and input-dependent dynamical expressivity over finite time horizons. Finally, we provide sufficient conditions for the existence of traveling waves and nontrivial stationary solutions.

math.AP↗

arXiv2026

Kinetic Theory with Fluctuations: Well-Posedness of The Vlasov--Fokker--Planck--Dean--Kawasaki Equations

We study Vlasov--Fokker--Planck--Dean--Kawasaki equations driven by correlated conservative noise. For regular noise coefficients and bounded nonlocal interactions, we establish probabilistically strong existence and uniqueness in a renormalized kinetic framework. For the square-root coefficient, we treat the non-interacting case and construct a probabilistically weak solution. Key challenges stem from the complexity of the kinetic operator and the irregularity introduced by the conservative noise with square-root-type coefficients. The proof relies on a novel combination of kinetic semigroup estimates and the framework of renormalized kinetic solutions.

math.PR↗

arXiv2026

Entropy Production and Reversibility Criteria for Stochastic Evolution Equations

This paper develops a path-space theory of entropy production for a class of stochastic evolution equations on infinite-dimensional Hilbert spaces. Since such spaces have no canonical Lebesgue reference measure, the usual finite-dimensional density formulas do not extend directly. We instead work relative to the invariant Gaussian measure of a reversible Ornstein--Uhlenbeck reference process. Combining an infinite-dimensional Girsanov transform, time reversal of the reference process, and the stationary Fokker--Planck equation relative to the Gaussian measure, we derive an explicit entropy-production formula in terms of an irreversibility field. On the natural test class, this field represents the difference between the forward and reversed nonlinear drifts. Under the standing assumptions, vanishing entropy production is equivalent to vanishing stationary probability current, self-adjointness of the generator in the invariant Hilbert space, detailed balance, and invariance of the stationary path law under time reversal. The reversible case is therefore characterized by a Gaussian-reference gradient structure for the nonlinear drift.

math.AP↗

arXiv2026

On the generalized Langevin equation and the Mori projection operator technique

In statistical physics, the Nakajima-Mori-Zwanzig projection operator formalism is used to derive an integro-differential equation for observables in a Hilbert space, the generalized Langevin equation (GLE). This technique relies on the splitting of the dynamics into a projected and an orthogonal part. However, the well-posedness of the abstract Cauchy problem for the orthogonal dynamics remains an open problem. Moreover, it is rarely discussed under which assumptions the Dyson identity, which is used to derive the GLE, holds. In this article, we address this issue for rank-one projections (Mori's projection). For the Mori projection, the orthogonal dynamics is obtained from the bounded perturbation theorem. The variation of constants formula for strongly continuous semigroups then yields the GLE and the second fluctuation dissipation theorem (2FDT). We show that the variation of constants can be replaced by a limiting process in order to give a general proof of the GLE and 2FDT that does not require the differentiability of the fluctuating forces. In addition, we offer an alternative approach that does not require the bounded perturbation theorem. Our starting point is the observation that the GLE and 2FDT uniquely determine the fluctuating forces as well as the memory kernel. Furthermore, the orbit maps for the orthogonal dynamics can be directly defined via solutions of linear Volterra equations. All desired properties of the orthogonal dynamics are then proven directly from this definition. In particular, the orthogonal dynamics is a strongly continuous semigroup generated by $\overline{\mathcal{QL}}\mathcal{Q}=\mathcal{QLQ}$, where $\mathcal{L}$ is the generator of the time evolution operator, and $\mathcal{P}=1-\mathcal{Q}$ is the Mori projection operator. Our results apply to general autonomous dynamical systems whose time evolution is given by a strongly continuous semigroup.

math-ph↗

arXiv2026

Quantitative Error Estimates for Learning Macroscopic Mobilities from Microscopic Fluctuations

We develop quantitative error estimates connecting microscopic fluctuation of interacting particle systems with the mobilities of their hydrodynamic limits. Focusing on the Symmetric Simple Exclusion Process and systems of independent Brownian particles, we provide explicit bounds for the discrepancy between the quadratic variation of fluctuation fields and the corresponding mobilities, in terms of time and spatial discretization parameters. In addition, we establish analogous error estimates for a class of fluctuating hydrodynamic stochastic PDEs with regularized coefficients. For stochastic PDEs with irregular square-root type coefficients, including Dean-Kawasaki type equations, we further identify the asymptotic behavior of the associated fluctuation structures within the framework of renormalized kinetic solutions. Our results provide quantitative insights into the relationship between microscopic fluctuation mechanisms and macroscopic mobilities, and contribute to a structured comparison between discrete particle systems and continuum fluctuating hydrodynamic descriptions.

math.PR↗

arXiv2025

Reversibility, covariance and coarse-graining for Langevin dynamics: On the choice of multiplicative noise

We study the interplay between reversibility, geometry, and the choice of multiplicative noise (in particular Itô, Stratonovich, Klimontovich) in stochastic differential equations (SDEs). Building on a unified geometric framework, we derive algebraic conditions under which a diffusion process is reversible with respect to a Gibbs measure on a Riemannian manifold. The condition depends continuously on a parameter $λ\in [0,1]$ which interpolates between the conventions of Itô ($λ= 0$), Stratonovich ($λ= \frac 1 2$) and Klimontovich ($λ= 1$). For reversible slow-fast systems of SDEs with a block-diagonal diffusion structure, we show, using the theory of Dirichlet forms, that both reversibility and the Klimontovich noise interpretation are preserved under coarse-graining. In particular, we prove that the effective dynamics for the slow variables, obtained via projection onto a lower-dimensional manifold, retain the Klimontovich interpretation and remain reversible with respect to the marginal Gibbs measure/free energy. Our results provide a flexible variational framework for modeling coarse-grained reversible dynamics with nontrivial geometric and noise structures.

math.PR↗

arXiv2025

Well-Posedness for Dean-Kawasaki Models of Vlasov-Fokker-Planck Type

We consider systems of interacting particles which are described by a second order Langevin equation. The class of equations considered includes the situation where the particle evolution is governed by Hamiltonian dynamics with additional damping and noise satisfying a fluctuation-dissipation relation. Also covered are systems of two equations describing an evolution of interacting agents, as arising in several descriptions of active matter, including models for flocking and swarming. We first show that such particle systems can be represented exactly by so-called equations of fluctuating hydrodynamics, which in this case are stochastic versions of a Vlasov-Fokker-Planck type equation. While the derivation given here is simple, it is a blueprint for the rigorous derivation of equations of fluctuating hydrodynamics. We then show a dichotomy previously known for purely diffusive (first order) systems carries over to the second order setting considered here: Solutions exist for suitable atomic initial data, in which case the solution is, properly scaled, the empirical density describing the particle system. For smooth initial data, however, we prove that no solution exists.

math.AP↗

arXiv2025

Deriving a GENERIC system from a Hamiltonian system

We reconsider the fundamental problem of coarse-graining infinite-dimensional Hamiltonian dynamics to obtain a macroscopic system which includes dissipative mechanisms. In particular, we study the thermodynamical implications concerning Hamiltonians, energy, and entropy and the induced geometric structures such as Poisson and Onsager brackets (symplectic and dissipative brackets). We start from a general finite-dimensional Hamiltonian system that is coupled linearly to an infinite-dimensional heat bath with linear dynamics. The latter is assumed to admit a compression to a finite-dimensional dissipative semigroup (i.e., the heat bath is a dilation of the semigroup) describing the dissipative evolution of new macroscopic variables. Already in the finite-energy case (zero-temperature heat bath) we obtain the so-called GENERIC structure (General Equations for Non-Equilibrium Reversible Irreversibe Coupling), with conserved energy, nondecreasing entropy, a new Poisson structure, and an Onsager operator describing the dissipation. However, their origin is not obvious at this stage. After extending the system in a natural way to the case of positive temperature, giving a heat bath with infinite energy, the compression property leads to an exact multivariate Ornstein-Uhlenbeck process that drives the rest of the system. Thus, we are able to identify a conserved energy, an entropy, and an Onsager operator (involving the Green-Kubo formalism) which indeed provide a GENERIC structure for the macroscopic system.

math-ph↗

arXiv2024

Hydrodynamic limits and non-equilibrium fluctuations for the Symmetric Inclusion Process with long jumps

We consider a d-dimensional symmetric inclusion process (SIP), where particles are allowed to jump arbitrarily far apart. We establish both the hydrodynamic limit and non-equilibrium fluctuations for the empirical measure of particles. With the help of self-duality and Mosco convergence of Dirichlet forms, we extend structural parallels between exclusion and inclusion dynamics from the short-range scenario to the long-range setting. The hydrodynamic equation for the symmetric inclusion process turns out to be of non-local type. At the level of fluctuations from the hydrodynamic limit, we demonstrate that the density fluctuation field converges to a time-dependent generalized Ornstein-Uhlenbeck process whose characteristics are again non-local.

math.PR↗

arXiv2023

Statistical-Physics-Informed Neural Networks (Stat-PINNs): A Machine Learning Strategy for Coarse-graining Dissipative Dynamics

Machine learning, with its remarkable ability for retrieving information and identifying patterns from data, has emerged as a powerful tool for discovering governing equations. It has been increasingly informed by physics, and more recently by thermodynamics, to further uncover the thermodynamic structure underlying the evolution equations, i.e., the thermodynamic potentials driving the system and the operators governing the kinetics. However, despite its great success, the inverse problem of thermodynamic model discovery from macroscopic data is in many cases non-unique, meaning that multiple pairs of potentials and operators can give rise to the same macroscopic dynamics, which significantly hinders the physical interpretability of the learned models. In this work, we propose a machine learning framework, named as Statistical-Physics-Informed Neural Networks (Stat-PINNs), which further encodes knowledge from statistical mechanics and resolves this non-uniqueness issue for the first time. The framework is here developed for purely dissipative isothermal systems. It only uses data from short-time particle simulations to learn the thermodynamic structure, which can be used to predict long-time macroscopic evolutions. We demonstrate the approach for particle systems with Arrhenius-type interactions, common to a wide range of phenomena, such as defect diffusion in solids, surface absorption and chemical reactions. Stat-PINNs can successfully recover the known analytic solution for the case with long-range interaction and discover the hitherto unknown potential and operator governing the short-range interaction cases. We compare our results with an analogous approach that solely excludes statistical mechanics, and observe that, in addition to recovering the unique thermodynamic structure, statistical mechanics relations can increase the robustness and predictability of the learning strategy.

cond-mat.stat-mech↗

arXiv2022

Second-order asymptotic expansion and thermodynamic interpretation of a fast-slow Hamiltonian system

This article includes a short survey of selected averaging and dimension reduction techniques for deterministic fast-slow systems. This survey includes, among others, classical techniques, such as the WKB approximation or the averaging method, as well as modern techniques, such as the GENERIC formalism. The main part of this article combines ideas of some of these techniques and addresses the problem of deriving a reduced system for the slow degrees of freedom (DOF) of a fast-slow Hamiltonian system. In the first part, we derive an asymptotic expansion of the averaged evolution of the fast-slow system up to second-order, using weak convergence techniques and two-scale convergence. In the second part, we determine quantities which can be interpreted as temperature and entropy of the system and expand these quantities up to second-order, using results from the first part. The results give new insights into the thermodynamic interpretation of the fast-slow system at different scales.

math-ph↗

arXiv2021

Second-order fast-slow dynamics of non-ergodic Hamiltonian systems: Thermodynamic interpretation and simulation

A class of fast-slow Hamiltonian systems with potential $U_\varepsilon$ describing the interaction of non-ergodic fast and slow degrees of freedom is studied. The parameter $\varepsilon$ indicates the typical timescale ratio of the fast and slow degrees of freedom. It is known that the Hamiltonian system converges for $\varepsilon\to0$ to a homogenised Hamiltonian system. We study the situation where $\varepsilon$ is small but positive. First, we rigorously derive the second-order corrections to the homogenised (slow) degrees of freedom. They can be decomposed into explicitly given terms that oscillate rapidly around zero and terms that trace the average motion of the corrections, which are given as the solution to an inhomogeneous linear system of differential equations. Then, we analyse the energy of the fast degrees of freedom expanded to second-order from a thermodynamic point of view. In particular, we define and expand to second-order a temperature, an entropy and external forces and show that they satisfy to leading-order, as well as on average to second-order, thermodynamic energy relations akin to the first and second law of thermodynamics. Finally, we analyse for a specific fast-slow Hamiltonian system the second-order asymptotic expansion of the slow degrees of freedom from a numerical point of view. Their approximation quality for short and long time frames and their total computation time are compared with those of the solution to the original fast-slow Hamiltonian system of similar accuracy.

math-ph↗

arXiv2021

A Hamilton-Jacobi PDE associated with hydrodynamic fluctuations from a nonlinear diffusion equation

We study a class of Hamilton-Jacobi partial differential equations in the space of probability measures. In the first part of this paper, we prove comparison principles (implying uniqueness) for this class. In the second part, we establish the existence of a solution and give a representation using a family of partial differential equations with control. A large part of our analysis exploits special structures of the Hamiltonian, which might look mysterious at first sight. However, we show that this Hamiltonian structure arises naturally as limit of Hamiltonians of microscopical models. Indeed, in the third part of this paper, we informally derive the Hamiltonian studied before, in a context of fluctuation theory on the hydrodynamic scale. The analysis is carried out for a specific model of stochastic interacting particles in gas kinetics, namely a version of the Carleman model. We use a two-scale averaging method on Hamiltonians defined in the space of probability measures to derive the limiting Hamiltonian.

math.AP↗

arXiv2021

Well-posedness for a regularised inertial Dean-Kawasaki model for slender particles in several space dimensions

A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension $d\in\mathbb{N}$. It is a regularised and inertial version of the Dean-Kawasaki model. A high-probability well-posedness theory for this model is developed. This theory improves significantly on the spatial scaling restrictions imposed in an earlier work of the same authors, which applied only to significantly larger particles in one dimension. The well-posedness theory now applies in $d$-dimensions when the particle-width $ε$ is proportional to $N^{-1/θ}$ for $θ>2d$ and $N$ is the number of particles. This scaling is optimal in a certain Sobolev norm. Key tools of the analysis are fractional Sobolev spaces, sharp bounds on Bessel functions, separability of the regularisation in the $d$-spatial dimensions, and use of the Faà di Bruno's formula.

math.AP↗

arXiv2019

Orthogonality of Fluxes in General Nonlinear Reaction Networks

We consider the chemical reaction networks and study currents in these systems. Reviewing recent decomposition of rate functionals from large deviation theory for Markov processes, we adapt these results for reaction networks. In particular, we state a suitable generalisation of orthogonality of forces in these systems, and derive an inequality that bounds the free energy loss and Fisher information by the rate functional.

math-ph↗

arXiv2019

From weakly interacting particles to a regularised Dean--Kawasaki model

The evolution of finitely many particles obeying Langevin dynamics is described by Dean-Kawasaki equations, a class of stochastic equations featuring a non-Lipschitz multiplicative noise in divergence form. We derive a regularised Dean-Kawasaki model based on second order Langevin dynamics by analysing a system of particles interacting via a pairwise potential. Key tools of our analysis are the propagation of chaos and Simon's compactness criterion. The model we obtain is a small-noise stochastic perturbation of the undamped McKean-Vlasov equation. We also provide a high-probability result for existence and uniqueness for our model.

math.PR↗

arXiv2019

A regularised Dean-Kawasaki model: derivation and analysis

The Dean-Kawasaki model consists of a nonlinear stochastic partial differential equation featuring a conservative, multiplicative, stochastic term with non-Lipschitz coefficient, and driven by space-time white noise; this equation describes the evolution of the density function for a system of finitely many particles governed by Langevin dynamics. Well-posedness for the Dean-Kawasaki model is open except for specific diffusive cases, corresponding to overdamped Langevin dynamics. There, it was recently shown by Lehmann, Konarovskyi, and von Renesse that no regular (non-atomic) solutions exist. We derive and analyse a suitably regularised Dean-Kawasaki model of wave equation type driven by coloured noise, corresponding to second order Langevin dynamics, in one space dimension. The regularisation can be interpreted as considering particles of finite size rather than describing them by atomic measures. We establish existence and uniqueness of a solution. Specifically, we prove a high-probability result for the existence and uniqueness of mild solutions to this regularised Dean-Kawasaki model.

math.PR↗

arXiv2018

A Variational Structure for Interacting Particle Systems and their Hydrodynamic Scaling Limits

We consider hydrodynamic scaling limits for a class of reversible interacting particle systems, which includes the symmetric simple exclusion process and certain zero-range processes. We study a (non-quadratic) microscopic action functional for these systems. We analyse the behaviour of this functional in the hydrodynamic limit and we establish conditions under which it converges to the (quadratic) action functional of Macroscopic Fluctuation Theory. We discuss the implications of these results for rigorous analysis of hydrodynamic limits.

math-ph↗