Mathematical Physics

New submissions

[ total of 22 entries: 1-22 ]
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New submissions for Fri, 27 Mar 20

[1]  arXiv:2003.11605 [pdf, ps, other]
Title: On the theory of kinetic equations for interacting particle systems with long range interactions
Comments: 89 pages
Subjects: Mathematical Physics (math-ph)

In this paper we review the formal derivation of different classes of kinetic equations for long range potentials. We consider suitable scaling limits for Lorentz and Rayleigh gases as well as interacting particle systems whose dynamics can be approximated by means of kinetic equations. The resulting kinetic equations are the Landau and the Balescu-Lenard equations. In the derivation of the kinetic equations particular emphasis is made in the fact that all the kinetic regimes can be obtained approximating the dynamics of the interacting particle systems by the evolution of a single particle in a random force field with a friction term which is due to the interaction with the surrounding particles. The case of particles interacting by means of Coulombian potentials as well as the cutoffs which yield the so-called Coulombian logarithm are discussed in detail.

[2]  arXiv:2003.11704 [pdf, ps, other]
Title: Gaussian Fluctuations and Free Energy Expansion for 2D and 3D Coulomb Gases at Any Temperature
Authors: Sylvia Serfaty
Comments: 71 pages
Subjects: Mathematical Physics (math-ph); Probability (math.PR)

We prove a Central Limit Theorem for the fluctuations of linear statistics of Coulomb gases in dimensions 2 and 3, which is valid down to microscales and for a broad temperature regime. This is the first such result in dimension 3. We show that the result can also be obtained in any dimension as soon as one can obtain a precise enough error rate for the expansion of the free energy -- such an expansion is obtained in any dimension, but the rate is good enough only in dimensions 2 and 3. In dimension 3 or larger, to obtain the CLT we need to make a "no phase transition" assumption. The CLT holds as soon as the test-function lives on a scale larger than the temperature-dependent minimal scale $\rho_\beta$ introduced in our previous work \cite{as}. It can be interpreted as a convergence to the Gaussian Free Field.

[3]  arXiv:2003.11892 [pdf, ps, other]
Title: On the geometry of discrete contact mechanics
Subjects: Mathematical Physics (math-ph); Numerical Analysis (math.NA)

In this paper, we continue the construction of variational integrators adapted to contact geometry started in \cite{VBS}, in particular, we introduce a discrete Herglotz Principle and the corresponding discrete Herglotz Equations for a discrete Lagrangian in the contact setting. This allows us to develop convenient numerical integrators for contact Lagrangian systems that are conformally contact by construction. The existence of an exact Lagrangian function is also discussed.

[4]  arXiv:2003.12038 [pdf, ps, other]
Title: Hydrogen atom bound states whose spectral measures have positive upper fractal dimensions
Subjects: Mathematical Physics (math-ph); Spectral Theory (math.SP)

It is shown that, Baire generically, the bound states of the Hamiltonian of the Hydrogen atom have spectral measures with exact $0$-lower and $1/3$-upper generalized fractal dimensions; the relation to (a weak form of) dynamical delocalization along orthonormal bases is also discussed. Such result is a consequence of the distribution of the Hamiltonian eigenvalues.

Cross-lists for Fri, 27 Mar 20

[5]  arXiv:2003.11616 (cross-list from math.CA) [pdf, ps, other]
Title: Reflective prolate-spheroidal operators and the adelic Grassmannian
Comments: 33 pages
Subjects: Classical Analysis and ODEs (math.CA); Mathematical Physics (math-ph); Algebraic Geometry (math.AG); Spectral Theory (math.SP)

Beginning with the work of Landau, Pollak and Slepian in the 1960s on time-band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory and integrable systems. Previously, such pairs were constructed by ad hoc methods, which worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every point $W$ of Wilson's infinite dimensional adelic Grassmannian $\mathrm Gr^ad$ gives rise to an integral operator $T_W$, acting on $L^2(\Gamma)$ for a contour $\Gamma\subset\mathbb C$, which reflects a differential operator $R(z,\partial_z)$ in the sense that $R(-z,-\partial_z)\circ T_W=T_W\circ R(w,\partial_w)$ on a dense subset of $L^2(\Gamma)$. By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function $\psi_W(x,z)$. The size of this algebra with respect to a bifiltration is in turn determined using algebro-geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property in place of plain commutativity. Furthermore, we prove that the time-band limited operators of the generalized Laplace transforms with kernels given by all rank one bispectral functions $\psi_W(x,-z)$ reflect a differential operator. A $90^\circ$ rotation argument is used to prove that the time-band limited operators of the generalized Fourier transforms with kernels $\psi_W(x,iz)$ admit a commuting differential operator. These methods produce vast collections of integral operators with prolate-spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s.

[6]  arXiv:2003.11792 (cross-list from math.FA) [pdf, ps, other]
Title: The Essential Spectrum of the Discrete Laplacian on Klaus-sparse Graphs
Authors: Sylvain Golenia (IMB), Françoise Truc (IF)
Subjects: Functional Analysis (math.FA); Mathematical Physics (math-ph); Spectral Theory (math.SP)

In 1983, Klaus studied a class of potentials with bumps and computed the essential spectrum of the associated Schr{\"o}dinger operator with the help of some localisations at infinity. A key hypothesis is that the distance between two consecutive bumps tends to infinity at infinity. In this article, we introduce a new class of graphs (with patterns) that mimics this situation, in the sense that the distance between two patterns tends to infinity at infinity. These patterns tend, in some way, to asymptotic graphs. They are the localisations at infinity. Our result is that the essential spectrum of the Laplacian acting on our graph is given by the union of the spectra of the Laplacian acting on the asymptotic graphs. We also discuss the question of the stability of the essential spectrum in the appendix.

[7]  arXiv:2003.11810 (cross-list from quant-ph) [pdf, other]
Title: Searching for Coherent States, From Origins to Quantum Gravity
Subjects: Quantum Physics (quant-ph); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph); History and Philosophy of Physics (physics.hist-ph)

We discuss the notion of coherent states from three different perspectives: the seminal approach of Schroedinger, the experimental take of quantum optics, and the theoretical developments in quantum gravity. This comparative study tries to emphasise the connections between the approaches, and to offer a coherent short story of the field, so to speak. It may be useful for pedagogical purposes, as well as for specialists of quantum optics and quantum gravity willing to embed their perspective within a wider landscape.

[8]  arXiv:2003.11868 (cross-list from math.PR) [pdf, ps, other]
Title: Mean Field Limits of Particle-Based Stochastic Reaction-Diffusion Models
Subjects: Probability (math.PR); Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Dynamical Systems (math.DS)

Particle-based stochastic reaction-diffusion (PBSRD) models are a popular approach for studying biological systems involving both noise in the reaction process and diffusive transport. In this work we derive coarse-grained deterministic partial integro-differential equation (PIDE) models that provide a mean field approximation to the volume reactivity PBSRD model, a model commonly used for studying cellular processes. We formulate a weak measure-valued stochastic process (MVSP) representation for the volume reactivity PBSRD model, demonstrating for a simplified but representative system that it is consistent with the commonly used Doi Fock Space representation of the corresponding forward equation. We then prove the convergence of the general volume reactivity model MVSP to the mean field PIDEs in the large-population (i.e. thermodynamic) limit.

[9]  arXiv:2003.11926 (cross-list from math.QA) [pdf, ps, other]
Title: Twisting on pre-Lie algebras and quasi-pre-Lie bialgebras
Authors: Jiefeng Liu
Comments: 26 pages. arXiv admin note: text overlap with arXiv:1902.03033 by other authors
Subjects: Quantum Algebra (math.QA); Mathematical Physics (math-ph)

We study (quasi-)twilled pre-Lie algebras and the associated $L_\infty$-algebras and differential graded Lie algebras. Then we show that certain twisting transformations on (quasi-)twilled pre-Lie algbras can be characterized by the solutions of Maurer-Cartan equations of the associated differential graded Lie algebras ($L_\infty$-algebras). Furthermore, we show that $\huaO$-operators and twisted $\huaO$-operators are solutions of the Maurer-Cartan equations. As applications, we study (quasi-)pre-Lie bialgebras using the associated differential graded Lie algebras ($L_\infty$-algebras) and the twisting theory of (quasi-)twilled pre-Lie algebras. In particular, we give a construction of quasi-pre-Lie bialgebras using symplectic Lie algebras, which is parallel to that a Cartan $3$-form on a semi-simple Lie algebra gives a quasi-Lie bialgebra.

[10]  arXiv:2003.12031 (cross-list from math.SP) [pdf, other]
Title: Schrödinger and polyharmonic operators on infinite graphs: Parabolic well-posedness and p-independence of spectra
Comments: comments welcome
Subjects: Spectral Theory (math.SP); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)

We analyze properties of semigroups generated by Schr\"odinger operators $-\Delta+V$ or polyharmonic operators $-(-\Delta)^m$, on metric graphs both on $L^p$-spaces and spaces of continuous functions. In the case of spatially constant potentials, we provide a semi-explicit formula for their kernel. Under an additional sub-exponential growth condition on the graph, we prove analyticity, ultracontractivity, and pointwise kernel estimates for these semigroups; we also show that their generators' spectra coincide on all relevant function spaces and present a Kre\u{\i}n-type dimension reduction, showing that their spectral values are determined by the spectra of generalized discrete Laplacians acting on various spaces of functions supported on combinatorial graphs.

Replacements for Fri, 27 Mar 20

[11]  arXiv:1807.05152 (replaced) [pdf, ps, other]
Title: Information theory with finite vector spaces
Comments: Presented in part at the Latin American Week on Coding and Information 2018 (Campinas, Brazil)
Journal-ref: IEEE Transactions on Information Theory, vol. 65, no. 9, pp. 5674-5687, Sept. 2019
Subjects: Mathematical Physics (math-ph); Information Theory (cs.IT); Probability (math.PR)
[12]  arXiv:1808.07890 (replaced) [pdf, ps, other]
Title: TAP free energy, spin glasses, and variational inference
Subjects: Probability (math.PR); Mathematical Physics (math-ph); Statistics Theory (math.ST)
[13]  arXiv:1904.08578 (replaced) [pdf, ps, other]
Title: Classification of simple weight modules for the $N=2$ superconformal algebra
Comments: 18 pages, Latex, in this version we delete the Section 7 for application to the $N=1$ superconformal algebra
Subjects: Representation Theory (math.RT); Mathematical Physics (math-ph); Quantum Algebra (math.QA); Rings and Algebras (math.RA)
[14]  arXiv:1905.02173 (replaced) [pdf, other]
Title: Assisted concentration of Gaussian resources
Comments: 21 pages, 3 figures. In v2 we changed the title and added the new Figures 1 and 2, illustrating the one-way and two-way Gaussian collaboration protocols, respectively
Subjects: Quantum Physics (quant-ph); Other Condensed Matter (cond-mat.other); Mathematical Physics (math-ph)
[15]  arXiv:1905.04250 (replaced) [pdf, ps, other]
Title: From path integrals to dynamical algebras: a macroscopic view of quantum physics
Comments: 9 pages, no figures; v2: exposition improved, reference added
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); History and Philosophy of Physics (physics.hist-ph)
[16]  arXiv:1908.05631 (replaced) [pdf, ps, other]
Title: Sharp polynomial decay rates for the damped wave equation with Hölder-like damping
Comments: 8 pages, minor revision, final changes before publication
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Spectral Theory (math.SP)
[17]  arXiv:1911.08069 (replaced) [pdf, other]
Title: Scale Invariance of the Homentropic Inviscid Euler Equations with Application to the Noh Problem
Comments: 21 pages, 9 .png figures, For submission to Physical Review E, major revisions and format changes
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Fluid Dynamics (physics.flu-dyn)
[18]  arXiv:2003.00560 (replaced) [pdf, ps, other]
Title: Solid-On-Solid interfaces with disordered pinning
Authors: Hubert Lacoin
Comments: 43 pages 3 figures (minor changes)
Subjects: Probability (math.PR); Mathematical Physics (math-ph)
[19]  arXiv:2003.04660 (replaced) [pdf, other]
Title: Impossible measurements require impossible apparatus
Comments: 5pp. v2: font issue fixed, minor textual changes, reference added
Subjects: Quantum Physics (quant-ph); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
[20]  arXiv:2003.07001 (replaced) [pdf, ps, other]
Title: Resonances and viscosity limit for the Wigner-von Neumann type Hamiltonian
Subjects: Mathematical Physics (math-ph); Analysis of PDEs (math.AP)
[21]  arXiv:2003.10755 (replaced) [pdf, ps, other]
Title: Contact interactions in Q.M. Gamma convergence and Bose-Einstein condensation
Subjects: Mathematical Physics (math-ph)
[22]  arXiv:2003.11049 (replaced) [pdf, ps, other]
Title: A Note on the Disentanglement of Gaussian Quantum States by Symplectic Rotations
Comments: Submitted to CRAS, Paris, in French
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Operator Algebras (math.OA); Symplectic Geometry (math.SG)
[ total of 22 entries: 1-22 ]
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