# Mathematical Physics

## New submissions

[ total of 24 entries: 1-24 ]
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### New submissions for Thu, 22 Aug 19

[1]
Title: Entropy in Themodynamics: from Foliation to Categorization
Comments: 19 pages, 1 figure, a survey paper by no means an original research;
Subjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech)

We overview the notion of entropy in thermodynamics. We start from the smooth case using differential forms on the manifold, which is the natural language for thermodynamics. Then the axiomatic definition of entropy as ordering on set that is induced by adiabatic processes will be outlined. Finally, the viewpoint of category theory is provided, which reinterprets the ordering structure as a category of pre-ordered sets.

[2]
Title: Connection probabilities in the double-dimer model -- the case of two connectivity patterns
Subjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech)

We apply the Grassmannian representation of the dimer model, an equivalent approach to Kasteleyn's solution to the close-packed dimer problem, to calculate the connection probabilities for the double-dimer model with wired/free/wired/free boundary conditions, on a rectangular subdomain of the square lattice with four marked boundary points at the corners. Using some series identities related to Schwarz-Christoffel transformations, we show that the continuum of the result is consistent with the corresponding one in the upper half-plane (previously obtained by Kenyon-Wilson), which is in turn identical to the connection probabilities for 4SLE$_4$ emanating from the boundary, or equivalently, to a conditioned version of CLE$_4$ with wired/free/wired/free boundary conditions in the context of conformal loop ensembles.

[3]
Title: Liouvillian solutions for second order linear differential equations with polynomial coefficients
Subjects: Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA)

In this paper we present an algebraic study concerning the general second order linear differential equation with polynomial coefficients. By means of Kovacic's algorithm and asymptotic iteration method we find a degree independent algebraic description of the spectral set: the subset, in the parameter space, of Liouiville integrable differential equations. For each fixed degree, we prove that the spectral set is a countable union of non accumulating algebraic varieties. This algebraic description of the spectral set allow us to bound the number of eigenvalues for algebraically quasi-solvable potentials in the Schr\"odinger equation.

[4]
Title: Essential singularities of fractal zeta functions
Authors: Michel L. Lapidus (1), Goran Radunović (2), Darko Žubrinić (2) ((1) University of California, Riverside, (2) University of Zagreb)
Subjects: Mathematical Physics (math-ph)

We study the essential singularities of geometric zeta functions $\zeta_{\mathcal L}$, associated with bounded fractal strings $\mathcal L$. For any three prescribed real numbers $D_{\infty}$, $D_1$ and $D$ in $[0,1]$, such that $D_{\infty}<D_1\le D$, we construct a bounded fractal string $\mathcal L$ such that $D_{\rm par}(\zeta_{\mathcal L})=D_{\infty}$, $D_{\rm mer}(\zeta_{\mathcal L})=D_1$ and $D(\zeta_{\mathcal L})=D$. Here, $D(\zeta_{\mathcal L})$ is the abscissa of absolute convergence of $\zeta_{\mathcal L}$, $D_{\rm mer}(\zeta_{\mathcal L})$ is the abscissa of meromorphic continuation of $\zeta_{\mathcal L}$, while $D_{\rm par}(\zeta_{\mathcal L})$ is the infimum of all positive real numbers $\alpha$ such that $\zeta_{\mathcal L}$ is holomorphic in the open right half-plane $\{{\rm Re}\, s>\alpha\}$, except for possible isolated singularities in this half-plane. Defining $\mathcal L$ as the disjoint union of a sequence of suitable generalized Cantor strings, we show that the set of accumulation points of the set $S_{\infty}$ of essential singularities of $\zeta_{\mathcal L}$, contained in the open right half-plane $\{{\rm Re}\, s>D_{\infty}\}$, coincides with the vertical line $\{{\rm Re}\, s=D_{\infty}\}$. We extend this construction to the case of distance zeta functions $\zeta_A$ of compact sets $A$ in $\mathbb{R}^N$, for any positive integer $N$.

### Cross-lists for Thu, 22 Aug 19

[5]  arXiv:1908.07518 (cross-list from math.HO) [pdf, ps, other]
Title: Basel problem: a physicist's solution
Comments: 11 pages, to be published in The Mathematical Intelligencer
Subjects: History and Overview (math.HO); Mathematical Physics (math-ph)

Some time ago Wastlund reformulated the Basel problem in terms of a physical system using the proportionality of the apparent brightness of a star to the inverse square of its distance. Inspired by this approach, we give another physical interpretation which, in our opinion, is simpler, natural enough, and very Eulerian in its spirit.

[6]  arXiv:1908.07694 (cross-list from quant-ph) [pdf, other]
Title: The Complementary Information Principle of Quantum Mechanics
Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)

The uncertainty principle bounds the uncertainties about incompatible measurements, clearly setting quantum theory apart from the classical world. Its mathematical formulation via uncertainty relations, plays an irreplaceable role in quantum technologies. However, neither the uncertainty principle nor uncertainty relations can fully describe the complementarity between quantum measurements. As an attempt to advance the efforts of complementarity in quantum theories, we formally propose a complementary information principle, significantly extending the one introduced by Heisenberg. First, we build a framework of black box testing consisting of pre- and post-testing with two incompatible measurements, introducing a rigorous mathematical expression of complementarity with definite information causality. Second, we provide majorization lower and upper bounds for the complementary information by utilizing the tool of semidefinite programming. In particular, we prove that our bounds are optimal under majorization due to the completeness of the majorization lattice. Finally, as applications to our framework, we present a general method to outer-approximating all uncertainty regions and also establish fundamental limits for all qualified joint uncertainties.

[7]  arXiv:1908.07863 (cross-list from math.PR) [pdf, other]
Title: Derivation of coupled KPZ-Burgers equation from multi-species zero-range processes
Subjects: Probability (math.PR); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)

We consider the fluctuation fields of multi-species weakly-asymmetric zero-range interacting particle systems in one dimension, where the mass density of each species is conserved. Although such fields have been studied in systems with a single species, the multi-species setting is much less understood. Among other results, we show that, when the system starts from stationary states, with a particular property, the scaling limits of the multi-species fluctuation fields, seen in a characteristic traveling frame, solve a coupled Burgers SPDE, which is a formal spatial gradient of a coupled KPZ equation.

[8]  arXiv:1908.07876 (cross-list from math.AP) [pdf, ps, other]
Title: On an optimal potential of Schrödinger operator with prescribed $m$ eigenvalue
Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Spectral Theory (math.SP)

The purpose of this paper is twofold: firstly, we present a new type of relationship between inverse problems and nonlinear differential equations. Secondly, we introduce a new type of inverse spectral problem, posed as follows: for a priori given potential $V_0$ find the closest function $\hat{V}$ such that $m$ eigenvalues of one-dimensional space Schrodinger operator with potential $\hat{V}$ would coincide with the given values $E_1$, $\ldots$, $E_m \in \mathbb {R}$. In our main result, we prove the existence of a solution to this problem, and more importantly, we show that such a solution can be directly found by solving a system of nonlinear differential equations.

[9]  arXiv:1908.07950 (cross-list from cond-mat.dis-nn) [pdf, other]
Title: On the multifractal dimensions and statistical properties of critical ensembles characterized by the three classical Wigner-Dyson symmetry classes
Subjects: Disordered Systems and Neural Networks (cond-mat.dis-nn); Mathematical Physics (math-ph)

We introduce a power-law banded random matrix model for the third of the three classical Wigner-Dyson ensembles, i.e., the symplectic ensemble. A detailed analysis of the statistical properties of its eigenvectors and eigenvalues, at criticality, is presented. This ensemble is relevant for time-reversal symmetric systems with strong spin-orbit interaction. For the sake of completeness, we also review the statistical properties of eigenvectors and eigenvalues of the power-law random banded matrix model for the corresponding systems in the presence and absence of time reversal invariance, previously considered in the literature. Our results show a good agreement with heuristic relations for the eigenstate and eigenenergy statistics at criticality, proposed in previous studies. With this, we provide a full picture of the power-law random banded matrix model corresponding to the three classical Wigner-Dyson ensembles.

[10]  arXiv:1908.08002 (cross-list from cond-mat.dis-nn) [pdf, other]
Title: Virtual clusters model on branching random graphs for confined fluid thermodynamics in heterogeneous solid geometry
Subjects: Disordered Systems and Neural Networks (cond-mat.dis-nn); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)

Fluid properties near rough surfaces are crucial in both a description of fundamental surface phenomena and modern industrial material design implementations. One of the most powerful approach to model real rough materials is based on the surface representation in terms of random geometry. Understanding the influence of random solid geometry on the low temperature fluid thermodynamics is a cutting edge problem. Therefor this work extends recent studies bypassing high temperature expansion and small heterogeneity scale. We introduce random branching trees whose topology reflects the hierarchical properties of random solid geometry. This mathematical representation allows to obtain averaged free energy using novel statistical model of virtual clusters interacting through random ultrametric pairwise potentials. Excellent agreement with direct Monte Carlo calculations is obtained. Moreover, we find that this model leads to interesting features of fluid-solid interactions that have not been discussed in the literature. Our results demonstrate that at low temperature a significant impact to fluid-solid interface energy is induced by hierarchical structure of random geometry. Due to the interdisciplinary nature of the study, our approach can be generalized and applied to a wide range of quenched disorder systems on random graphs. Cooperative phenomena in biological populations and social networks seem most attractive.

[11]  arXiv:1908.08030 (cross-list from hep-th) [pdf, ps, other]
Title: Integrability and cycles of deformed ${\cal N}=2$ gauge theory
Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)

To analyse pure ${\cal N}=2$ $SU(2)$ gauge theory in the Nekrasov-Shatashvili (NS) limit (or deformed Seiberg-Witten (SW)), we use the Ordinary Differential Equation/Integrable Model (ODE/IM) correspondence, and in particular its (broken) discrete symmetry in its extended version with {\it two} singular irregular points. Actually, this symmetry appears to be 'manifestation' of the spontaneously broken $\mathbb{Z}_2$ R-symmetry of the original gauge problem and the two deformed SW cycles are simply connected to the Baxter's $T$ and $Q$ functions, respectively, of the Liouville conformal field theory at the self-dual point. The liaison is realised via a second order differential operator which is essentially the 'quantum' version of the square of the SW differential. Moreover, the constraints imposed by the broken $\mathbb{Z}_2$ R-symmetry acting on the moduli space (Bilal-Ferrari equations) seem to have their quantum counterpart in the $TQ$ and the $T$ periodicity relations, and integrability yields also a useful Thermodynamic Bethe Ansatz (TBA) for the cycles ($Y(\theta,\pm u)$ or their square roots, $Q(\theta,\pm u)$). A latere, two efficient asymptotic expansion techniques are presented. Clearly, the whole construction is extendable to gauge theories with matter and/or higher rank groups.

### Replacements for Thu, 22 Aug 19

[12]  arXiv:1408.2903 (replaced) [pdf, other]
Title: Poincaré--Birkhoff--Witt isomorphisms and Kapranov dg-manifolds
Subjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Algebraic Geometry (math.AG); Algebraic Topology (math.AT); Quantum Algebra (math.QA)
[13]  arXiv:1703.04811 (replaced) [pdf, other]
Title: Equilibrium configurations for generalized Frenkel-Kontorova models on quasicrystals
Authors: Rodrigo Treviño
Subjects: Mathematical Physics (math-ph); Dynamical Systems (math.DS)
[14]  arXiv:1703.07021 (replaced) [pdf, ps, other]
Title: Regularity of Schroedinger's functional equation and mean field PDEs for h-path processes
Authors: Toshio Mikami
Subjects: Probability (math.PR); Mathematical Physics (math-ph); Optimization and Control (math.OC)
[15]  arXiv:1809.08806 (replaced) [pdf, ps, other]
Title: The theory of N-Mixed-Spin-P fields
Subjects: Algebraic Geometry (math.AG); Mathematical Physics (math-ph)
[16]  arXiv:1903.05070 (replaced) [pdf, other]
Title: A generalized Noether theorem for scaling symmetry
Comments: Extended version. 18 pages, 3 figures. Details and relations to the Bargmann framework clarified and illustrated with further examples. v4: earlier results summarized
Subjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th)
[17]  arXiv:1903.11211 (replaced) [pdf, ps, other]
Authors: John R. Klauder
Comments: 14 pages, a novel approach to quantizing general relativity, minor corrections; NEW RESULTS, SHARPER EXAMPLES; Important correction
Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
[18]  arXiv:1905.00659 (replaced) [pdf, ps, other]
Title: Lie Algebroid Gauging of Non-linear Sigma Models
Authors: Kyle Wright
Comments: 32 pages. V2: fixed small typos to match published version
Journal-ref: J. Geom. Phys. 146 (2019) 103490
Subjects: Differential Geometry (math.DG); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
[19]  arXiv:1905.00864 (replaced) [pdf, other]
Title: Elliptic Blowup Equations for 6d SCFTs. II: Exceptional Cases
Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
[20]  arXiv:1906.01229 (replaced) [pdf, ps, other]
Title: An optimization problem for finite point interaction families
Authors: Pavel Exner
Comments: typos corrected, to appear in J. Phys. A: Math. Theor
Subjects: Spectral Theory (math.SP); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
[21]  arXiv:1906.02619 (replaced) [pdf, other]
Title: Hyperbolic spin Ruijsenaars-Schneider model from Poisson reduction
Comments: 16 pages. The statement about coincidence of the Poisson structure of spin variables at generic $N$ and $\ell$ with that of 1811.08727 was corrected
Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)
[22]  arXiv:1906.07238 (replaced) [pdf, ps, other]
Title: Heterotic/$F$-theory Duality and Narasimhan-Seshadri Equivalence
Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
[23]  arXiv:1907.08204 (replaced) [pdf, other]
Title: Topological theory of Lieb-Schultz-Mattis theorems in quantum spin systems
Comments: 27 pages + 12 pages of appendices. v2 updated references
Subjects: Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph); Quantum Physics (quant-ph)
[24]  arXiv:1908.07065 (replaced) [pdf, other]
Title: Non-perturbative approaches to the quantum Seiberg-Witten curve
Comments: 46 pages, 4 figures, 12 tables; new references added, typos corrected
Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
[ total of 24 entries: 1-24 ]
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