arXiv subjects

Walter de Siqueira Pedra

Publications and source records attributed to Walter de Siqueira Pedra.

12 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↗

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

Minimum-entropy constraints on galactic potentials

A tracer sample in a gravitational potential, starting from a generic initial condition, phase-mixes towards a stationary state. This evolution is accompanied by an entropy increase, and the final state is characterized by a distribution function (DF) that depends only on integrals of motion (Jeans' theorem). We present a method to constrain a gravitational potential assuming a stationary (phase mixed) sample by minimizing the entropy the sample would have if it were allowed to phase-mix in trial potentials. This method avoids modeling the DF, and is applicable to any sets of integrals. We provide expressions for the entropy of DFs depending on energy, $f(E)$, energy and angular momentum, $f(E,L)$, or three actions, $f(\vec{J})$, and investigate the bias and statistical uncertainties in their estimates. We show that the method correctly recovers the parameters for spherical and axisymmetric potentials. We also present a methodology to characterize the posterior probability distribution of the parameters with an Approximate Bayesian Computation, indicating a pathway for application to observational data. Using $10^4$ tracers with $10\% (20\%)$-uncertainties in the 6D coordinates, we recover the flattening parameter $q$ of an axisymmetric potential with $σ_q/q\sim 5\% (10\%)$. The python module for the entropy estimators, \texttt{tropygal}, is made publicly available.

astro-ph.GA↗

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↗

arXiv2018

The discreteness-driven relaxation of collisionless gravitating systems: entropy evolution and the Nyquist-Shannon theorem

The time irreversibility and fast relaxation of collapsing $N$-body gravitating systems (as opposed to the time reversibility of the equations of motion for individual stars or particles) are traditionally attributed to information loss due to coarse-graining in the observation. We show that this subjective element is not necessary once one takes into consideration the fundamental fact that these systems are discrete, i.e. composed of a finite number $N$ of stars or particles. We show that a connection can be made between entropy estimates for discrete systems and the Nyquist-Shannon sampling criterion. Specifically, given a sample with $N$ points in a space of $d$ dimensions, the Nyquist-Shannon criterion constrains the size of the smallest structures defined by a function in the continuum that can be uniquely associated with the discrete sample. When applied to an $N$-body system, this theorem sets a lower limit to the size of phase-space structures (in the continuum) that can be resolved in the discrete data. As a consequence, the finite $N$ system tends to a uniform distribution after a relaxation time that typically scales as $N^{1/d}$. This provides an explanation for the fast achievement of a stationary state in collapsing $N$-body gravitating systems such as galaxies and star clusters, without the need to advocate for the subjective effect of coarse-graining.

astro-ph.GA↗

arXiv2017

Bounds on the Pure Point Spectrum of Lattice Schrödinger Operators

In dimension $d\geq 3$, a variational principle for the size of the pure point spectrum of (discrete) Schrödinger operators $H(\mathfrak{e},V)$ on the hypercubic lattice $\mathbb{Z}^{d}$, with dispersion relation $\mathfrak{e}$ and potential $V$, is established. The dispersion relation $\mathfrak{e}$ is assumed to be a Morse function and the potential $V(x)$ to decay faster than $|x|^{-2(d+3)}$, but not necessarily to be of definite sign. Our estimate on the size of the pure-point spectrum yields the absence of embedded and threshold eigenvalues of $H(\mathfrak{e},V)$ for a class ot potentials of this kind. The proof of the variational principle is based on a limiting absorption principle combined with a positive commutator (Mourre) estimate, and a Virial theorem. A further observation of crucial importance for our argument is that, for any selfadjoint operator $B$ and positive number $λ>0$, the number of negative eigenvalues of $λB$ is independent of $λ$.

math-ph↗

arXiv2017

Bounds on the Discrete Spectrum of Lattice Schrödinger Operators

We discuss the validity of the Weyl asymptotics -- in the sense of two-sided bounds -- for the size of the discrete spectrum of (discrete) Schrödinger operators on the $d$--dimensional, $d\geq 1$, cubic lattice $\mathbb{Z}^{d}$ at large couplings. We show that the Weyl asymptotics can be violated in any spatial dimension $d\geq 1$ -- even if the semi-classical number of bound states is finite. Furthermore, we prove for all dimensions $d\geq 1$ that, for potentials well-behaved at infinity and fulfilling suitable decay conditions, the Weyl asymptotics always hold. These decay conditions are mild in the case $d\geq 3$, while stronger for $d=1,2$. It is well-known that the semi-classical number of bound states is -- up to a constant -- always an upper bound on the size of the discrete spectrum of Schrödinger operators if $d\geq 3$. We show here how to construct general upper bounds on the number of bound states of Schrödinger operators on $\mathbb{Z}^{d}$ from semi-classical quantities in all space dimensions $d\geq 1$ and independently of the positivity-improving property of the free Hamiltonian.

math-ph↗

arXiv2017

The Arrow of Time in the collapse of collisionless self-gravitating systems: non-validity of the Vlasov-Poisson equation during violent relaxation

The collapse of a collisionless self-gravitating system, with the fast achievement of a quasi-stationary state, is driven by violent relaxation, with a typical particle interacting with the time-changing collective potential. It is traditionally assumed that this evolution is governed by the Vlasov-Poisson equation, in which case entropy must be conserved. We run N-body simulations of isolated self-gravitating systems, using three simulation codes: NBODY-6 (direct summation without softening), NBODY-2 (direct summation with softening) and GADGET-2 (tree code with softening), for different numbers of particles and initial conditions. At each snapshot, we estimate the Shannon entropy of the distribution function with three different techniques: Kernel, Nearest Neighbor and EnBiD. For all simulation codes and estimators, the entropy evolution converges to the same limit as N increases. During violent relaxation, the entropy has a fast increase followed by damping oscillations, indicating that violent relaxation must be described by a kinetic equation other than the Vlasov-Poisson, even for N as large as that of astronomical structures. This indicates that violent relaxation cannot be described by a time-reversible equation, shedding some light on the so-called "fundamental paradox of stellar dynamics". The long-term evolution is well described by the orbit-averaged Fokker-Planck model, with Coulomb logarithm values in the expected range 10-12. By means of NBODY-2, we also study the dependence of the 2-body relaxation time-scale on the softening length. The approach presented in the current work can potentially provide a general method for testing any kinetic equation intended to describe the macroscopic evolution of 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↗

arXiv2014

Suppression of Decoherence by Periodic Forcing

We consider a finite-dimensional quantum system coupled to a thermal reservoir and subject to a time-periodic, energy conserving forcing. We show that, if a certain dynamical decoupling condition is fulfilled, then the periodic forcing counteracts the decoherence induced by the reservoir: for small system-reservoir coupling $λ$ and small forcing period $T$, the system dynamics is approximated by an energy conserving and non-dissipative dynamics, which preserves coherences. For times up to order $(λT)^{-1}$, the difference between the true and approximated dynamics is of size $λ+T$. Our approach is rigorous and combines Floquet and spectral deformation theory. We illustrate our results on the spin-fermion model and recover previously known, heuristically obtained results.

quant-ph↗