arXiv subjects

Jean-Bernard Bru

Publications and source records attributed to Jean-Bernard Bru.

13 recordsLinked to original sources

arXiv2026

Non-Closing Double-Commutator Flows and the Small-Coupling Limit of Spin-Boson Models

Spin-boson models are paradigmatic examples of open quantum systems and serve as the theoretical paradigm for current quantum computers based on single-ion trap technology. Despite their ubiquity, a complete spectral diagonalization of these models remains an open problem, except in highly singular regimes. This paper establishes a rigorous framework for the approximate diagonalization of generalized spin-boson systems. Our approach is inspired by the Brockett-Wegner double-commutator flow, which is a non-linear differential equation governing the evolution of (here unbounded) operators. Unlike relatively recent applications of this flow to quadratic Hamiltonians in quantum field theory, it does not close in the spin-boson context. We overcome this fundamental obstruction by performing a detailed analysis of the resulting non-closed algebraic structure, allowing us to explicitly bound the higher-order error term with respect to the spin-boson coupling strength. Consequently, this work provides the first mathematically rigorous justification for several heuristic diagonalization techniques widely employed in the theoretical physics literature for small-coupling regimes, in the simplest non-trivial cases. More broadly, our framework renders a flow-based algorithm feasible for systematic higher-order diagonalization and self-energy renormalization. This strategy is conceptually akin to multi-scale analysis, or, much more recently, to the iterative, local Lie-Schwinger block-diagonalization method by Fröhlich and Pizzo.

math-ph↗

arXiv2026

Abstract Theory of Bogoliubov Linearizations with Application to Nonlinear Thermodynamic Formalism

Bogoliubov's 1947 approximation, originally developed in the microscopic theory of superfluidity, laid the foundation for solving previously intractable quantum models and later became part of "quantum mathematics". Regarding mathematically rigorous results, one of its most advanced forms - the only one that handles quantum equilibrium states - was published in the Memoirs of the AMS in 2013. Building on key results from convex analysis, the present work significantly extends it to obtain a general mathematical theory that enables nonlinear variational problems on convex compact spaces to be fully studied via a linearization process, referred to here as the "Bogoliubov linearization". This problem is particularly timely, given the current development of quantum algorithms and computers, which are inherently linear machines. A deep connection with the optimal transport is also proven. As a paradigmatic example of application, the approach proposed here is applied to the nonlinear thermodynamic formalism - an emerging field that can have important impacts on various fields of mathematics, such as ergodic transport, the fractals and multifractal formalism, discrete-time linear dynamics, C*-algebras, etc. Notably, even in the case of finite alphabets the obtained results go beyond the scope of the existing literature in nonlinear thermodynamic formalism.

math.FA↗

arXiv2026

Quadratic Hamiltonians in Fermionic Fock Spaces

Quadratic Hamiltonians are important in quantum field theory and quantum statistical mechanics. Their general studies, which go back to the sixties, are relatively incomplete for the fermionic case studied here. Following Berezin, they are quadratic in the fermionic field and in this way well-defined self-adjoint operators acting on the fermionic Fock space. We analyze their diagonalization by applying a novel elliptic operator-valued differential equations studied in a companion paper. This allows for their ($\mathrm{N}$-) diagonalization under much weaker assumptions than before. Last but not least, in 1994 Bach, Lieb and Solovej defined them to be generators of strongly continuous unitary groups of Bogoliubov transformations. This is shown to be an equivalent definition, as soon as the vacuum state belongs to the domain of definition of these Hamiltonians. This second outcome is demonstrated to be reminiscent to the celebrated Shale-Stinespring condition on Bogoliubov transformations.

math-ph↗

arXiv2025

Nonlinear Thermodynamic Formalism: Mean-field Phase Transitions, Large Deviations and Bogoliubov's Variational Principle

Let $Ω=\{1,2,\ldots ,d\}^{\mathbb{N}}$, $T$ be the shift acting on $Ω$, $\mathcal{P}(T)$ the set of $T$-invariant probabilities. Given a Hölder potential $A$ and a continuous function $F$, we investigate the probabilities $ρ_{F,A}$ that are maximizers of the nonlinear pressure $\mathfrak{P}_{F,A}:=\sup_{ρ\in \mathcal{P}(T)}\{ F(\int A(x)ρ(\mathrm{d}x))+h(ρ)\} .$ $ρ_{F,A}$} is called a nonlinear equilibrium; a nonlinear phase transition occurs when there is more than one. In the case $F$\ is convex or concave, we combine Varadhan's lemma and Bogoliubov's variational principle to characterize them via the linear pressure problem and self-consistency conditions. Let $μ\in \mathcal{P}(T)$ be the maximal entropy measure, $φ_{n}(x)=n^{-1}(φ(x)+φ(T(x))+\cdots +φ(T^{n-1}(x)))$ and $β>0$.}\newline (I) We also consider the limit measure $\mathfrak{m}$ on $ Ω$, so that $\forall ψ\in C(Ω)$, $\int ψ(x)\,\mathfrak{m}\,( \mathrm{d}x)\,\,=\lim_{n\rightarrow \infty }\frac{\,\int \,ψ(x)\,\,\,e^{ \frac{βn}{2}\,\,A_{n}((x)^{2}}\,\,μ\,(\mathrm{d}x)\,}{\int e^{\frac{ βn}{2}\,\,A_{n}((x)^{2}}μ\,(\mathrm{d}x)\,\,}.$ We call $\mathfrak{m}$ a \textit{quadratic mean-field Gibbs probability (II) Via subsequences $n_{k}$, $k\in \mathbb{N}$, we study the limit measure $\mathfrak{M}$ on $Ω$, so that $\forall ψ\in C(Ω)$, $\int ψ(x)\mathfrak{M}(\mathrm{d} x)=\lim_{k\rightarrow \infty }\frac{\,\int ψ_{n_{k}}(x)e^{\frac{βn_{k}}{2}A_{n_{k}}(x)^{2}}μ(\mathrm{d}x)}{\int e^{\frac{βn_{k}}{2} A_{n_{k}}(x)^{2}}μ(\mathrm{d}x)}.$ We call $\mathfrak{M}$ a quadratic mean-field equilibrium probability; it is shift-invariant. Explicit examples are given.

math.DS↗

arXiv2025

Charge Transport at Atomic Scales in 1D-Semiconductors: A Quantum Statistical Model Allowing Rigorous Numerical Studies

There has been a recent surge of interest in understanding charge transport at atomic scales. The motivations are myriad, including understanding the conductance properties of peptides measured experimentally. In this study, we propose a model of quantum statistical mechanics which aims to investigate the transport properties of 1D-semiconductor at nanoscales. The model is a two-band Hamiltonian in which electrons are assumed to be quasi-free. It allows us to investigate the behaviour of current and quantum fluctuations under the influence of numerous parameters, showing the response with respect to varying voltage, temperature and length. We compute the current observable at each site and demonstrate the local behaviour generating the current.

physics.bio-ph↗

arXiv2024

Non-Linear Operator-valued Elliptic Flows with Application to Quantum Field Theory

Differential equations on spaces of operators are very little developed in Mathematics, being in general very challenging. Here, we study a novel system of such (non-linear) differential equations. We show it has a unique solution for all times, for instance in the operator or Hilbert-Schmidt norm topologies. This system presents remarkable ellipticity properties that turn out to be crucial for the study of the infinite-time limit of its solution, which is proven under relatively weak, albeit probably not necessary, hypotheses on the initial data. This system of differential equations is the elliptic counterpart of an hyperbolic flow applied to quantum field theory to diagonalize Hamiltonians that are quadratic in the bosonic field. In a similar way, this elliptic flow, in particular its asymptotics, has application in quantum field theory: it can be used to diagonalize Hamiltonians that are quadratic in the fermionic field while giving new explicit expressions and properties of these pivotal Hamiltonians of quantum field theory and quantum statistical mechanics.

math-ph↗

arXiv2018

The discreteness-driven relaxation of collisionless gravitating systems: entropy evolution in external potentials, N-dependence and the role of chaos

We investigate the old problem of the fast relaxation of collisionless $N$-body systems which are collapsing or perturbed, emphasizing the importance of (non-collisional) discreteness effects. We integrate orbit ensembles in fixed external potentials, estimating the entropy of the ensemble to analyze the time evolution of the distribution function. We show that these estimates capture the correct physical behavior expected from the 2nd Law of Thermodynamics, without any spurious entropy production. For self-consistent (i.e. stationary) samples, the entropy is conserved, while for non-self-consistent samples, it increases within a few dynamical times up to a maximum where it stabilizes (even in integrable potentials). Our results shed light on the main ingredients for this fast collisionless relaxation. The fundamental ingredient is the discreteness (finite $N$) of gravitational systems in any potential. Additionally, in non-integrable potentials, the presence of chaotic orbits accelerates the entropy production. Contrary to the traditional violent relaxation scenario, our results indicate that a time-dependent potential is not necessary for this fast relaxation. For the first time, in connection with the Nyquist-Shannon theorem we determine the $N$-dependence of this discreteness-driven relaxation, deriving a typical relaxation time $T/τ_{cr}\approx 0.1 N^{1/6}$, with slightly weaker $N$-dependencies for non-integrable potentials with substantial fractions of chaotic orbits. This timescale is much smaller than the collisional timescale even for small-$N$ systems such as open clusters and represents an upper limit for real collisionless $N$-body systems. Additionally, our results reinforce the conclusion draw in Beraldo e Silva et al. (2017) that the Vlasov equation does not provide an adequate kinetic description of the fast relaxation of collapsing collisionless $N$-body systems.

astro-ph.GA↗

arXiv2017

Microscopic Conductivity of Lattice Fermions at Equilibrium - Part II: Interacting Particles

We apply Lieb-Robinson bounds for multi-commutators we recently derived to study the (possibly non-linear) response of interacting fermions at thermal equilibrium to perturbations of the external electromagnetic field. This analysis leads to an extension of the results for quasi-free fermions of \cite{OhmI,OhmII} to fermion systems on the lattice with short-range interactions. More precisely, we investigate entropy production and charge transport properties of non-autonomous $C^{\ast }$-dynamical systems associated with interacting lattice fermions within bounded static potentials and in presence of an electric field that is time- and space-dependent. We verify the 1st law of thermodynamics for the heat production of the system under consideration. In linear response theory, the latter is related with Ohm and Joule's laws. These laws are proven here to hold at the microscopic scale, uniformly with respect to the size of the (microscopic) region where the electric field is applied. An important outcome is the extension of the notion of conductivity measures to interacting fermions.

math-ph↗

arXiv2016

Characterization of the Quasi-Stationary State of an Impurity Driven by Monochromatic Light I - The Effective Theory

We consider an impurity ($N$--level atom) driven by monochromatic light in a host environment which is a fermionic thermal reservoir. The external light source is a time--periodic perturbation of the atomic Hamiltonian stimulating transitions between two atomic energy levels $E_{1}$ and $E_{N}$ and thus acts as an optical pump. The purpose of the present work is the analysis of the effective atomic dynamics resulting from the full microscopic time--evolution of the compound system. We prove, in particular, that the atomic dynamics of population relaxes for large times to a quasi-stationary state, that is, a stationary state up to small oscillations driven by the external light source. This state turns out to be uniquely determined by a balance condition. The latter is related to \textquotedblleft generalized Einstein relations\textquotedblright relations of spontaneous/stimulated emission/absorption rates, which are conceptually similar to the phenomenological relations derived by Einstein in 1916. As an application we show from quantum mechanical first principles how an inversion of population of energy levels of an impurity in a crystal can appear. Our results are based on the spectral analysis of the generator of the evolution semigroup related to a non--autonomous Cauchy problem effectively describing the atomic dynamics.

math-ph↗

arXiv2016

Diagonalizing Quadratic Bosonic Operators by Non-Autonomous Flow Equation

We study a non-autonomous, non-linear evolution equation on the space of operators on a complex Hilbert space. We specify assumptions that ensure the global existence of its solutions and allow us to derive its asymptotics at temporal infinity. We demonstrate that these assumptions are optimal in a suitable sense and more general than those used before. The evolution equation derives from the Brocket-Wegner flow that was proposed to diagonalize matrices and operators by a strongly continuous unitary flow. In fact, the solution of the non-linear flow equation leads to a diagonalization of Hamiltonian operators in boson quantum field theory which are quadratic in the field.

math-ph↗

arXiv2006

Large deviations for trapped interacting Brownian particles and paths

We introduce two probabilistic models for $N$ interacting Brownian motions moving in a trap in $\mathbb {R}^d$ under mutually repellent forces. The two models are defined in terms of transformed path measures on finite time intervals under a trap Hamiltonian and two respective pair-interaction Hamiltonians. The first pair interaction exhibits a particle repellency, while the second one imposes a path repellency. We analyze both models in the limit of diverging time with fixed number $N$ of Brownian motions. In particular, we prove large deviations principles for the normalized occupation measures. The minimizers of the rate functions are related to a certain associated operator, the Hamilton operator for a system of $N$ interacting trapped particles. More precisely, in the particle-repellency model, the minimizer is its ground state, and in the path-repellency model, the minimizers are its ground product-states. In the case of path-repellency, we also discuss the case of a Dirac-type interaction, which is rigorously defined in terms of Brownian intersection local times. We prove a large-deviation result for a discrete variant of the model. This study is a contribution to the search for a mathematical formulation of the quantum system of $N$ trapped interacting bosons as a model for Bose--Einstein condensation, motivated by the success of the famous 1995 experiments. Recently, Lieb et al. described the large-N behavior of the ground state in terms of the well-known Gross--Pitaevskii formula, involving the scattering length of the pair potential. We prove that the large-N behavior of the ground product-states is also described by the Gross--Pitaevskii formula, however, with the scattering length of the pair potential replaced by its integral.

math.PR↗

arXiv2005

Large systems of path-repellent Brownian motions in a trap at positive temperature

We study a model of $ N $ mutually repellent Brownian motions under confinement to stay in some bounded region of space. Our model is defined in terms of a transformed path measure under a trap Hamiltonian, which prevents the motions from escaping to infinity, and a pair-interaction Hamiltonian, which imposes a repellency of the $N$ paths. In fact, this interaction is an $N$-dependent regularisation of the Brownian intersection local times, an object which is of independent interest in the theory of stochastic processes. The time horizon (interpreted as the inverse temperature) is kept fixed. We analyse the model for diverging number of Brownian motions in terms of a large deviation principle. The resulting variational formula is the positive-temperature analogue of the well-known Gross-Pitaevskii formula, which approximates the ground state of a certain dilute large quantum system; the kinetic energy term of that formula is replaced by a probabilistic energy functional. This study is a continuation of the analysis in \cite{ABK04} where we considered the limit of diverging time (i.e., the zero-temperature limit) with fixed number of Brownian motions, followed by the limit for diverging number of motions. \bibitem[ABK04]{ABK04} {\sc S.~Adams, J.-B.~Bru} and {\sc W.~König}, \newblock Large deviations for trapped interacting Brownian particles and paths, \newblock {\it Ann. Probab.}, to appear (2004).

math.PR↗

arXiv2002

The equilibrium states for a model with two kinds of Bose condensation

We study the equilibrium Gibbs states for a Boson gas model, defined by Bru and Zagrebnov, which has two phase transitions of the Bose condensation type. The two phase transitions correspond to two distinct mechanisms by which these condensations can occur. The first (non-conventional) Bose condensation is mediated by a zero-mode interaction term in the Hamiltonian. The second is a transition due to saturation quite similar to the conventional Bose-Einstein (BE) condensation in the ideal Bose gas. Due to repulsive interaction in non-zero modes the model manifests a generalized type III, i.e., non-extensive BE condensation. Our main result is that, as in the ideal Bose gas, the conventional condensation is accompanied by a loss of strong equivalence of the canonical and grand canonical ensembles whereas the non-conventional one, due to the interaction, does not break the equivalence of ensembles. It is also interesting to note that the type of (generalized) condensate, I, II, or III (in the terminology of van den Berg, Lewis and Pule), has no effect on the equivalence of ensembles. These results are proved by computing the generating functional of the cyclic representation of the Canonical Commutation Relation (CCR) for the corresponding equilibrium Gibbs states.

math-ph↗