Mathematical Physics
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New submissions for Thu, 22 Aug 19
 [1] arXiv:1908.07583 [pdf, other]

Title: Entropy in Themodynamics: from Foliation to CategorizationAuthors: Radosław A. KyciaComments: 19 pages, 1 figure, a survey paper by no means an original research;Subjects: Mathematical Physics (mathph); Statistical Mechanics (condmat.statmech)
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 preordered sets.
 [2] arXiv:1908.07595 [pdf, other]

Title: Connection probabilities in the doubledimer model  the case of two connectivity patternsAuthors: Nahid GhodratipourSubjects: Mathematical Physics (mathph); Statistical Mechanics (condmat.statmech)
We apply the Grassmannian representation of the dimer model, an equivalent approach to Kasteleyn's solution to the closepacked dimer problem, to calculate the connection probabilities for the doubledimer 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 SchwarzChristoffel transformations, we show that the continuum of the result is consistent with the corresponding one in the upper halfplane (previously obtained by KenyonWilson), 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] arXiv:1908.07666 [pdf, ps, other]

Title: Liouvillian solutions for second order linear differential equations with polynomial coefficientsComments: 17 pagesSubjects: Mathematical Physics (mathph); 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 quasisolvable potentials in the Schr\"odinger equation.
 [4] arXiv:1908.07845 [pdf, ps, other]

Title: Essential singularities of fractal zeta functionsAuthors: Michel L. Lapidus (1), Goran Radunović (2), Darko Žubrinić (2) ((1) University of California, Riverside, (2) University of Zagreb)Comments: 20 pages, 1 figureSubjects: Mathematical Physics (mathph)
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 halfplane $\{{\rm Re}\, s>\alpha\}$, except for possible isolated singularities in this halfplane. 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 halfplane $\{{\rm Re}\, s>D_{\infty}\}$, 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 $N$.
Crosslists for Thu, 22 Aug 19
 [5] arXiv:1908.07518 (crosslist from math.HO) [pdf, ps, other]

Title: Basel problem: a physicist's solutionAuthors: Z.K. SilagadzeComments: 11 pages, to be published in The Mathematical IntelligencerSubjects: History and Overview (math.HO); Mathematical Physics (mathph)
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 (crosslist from quantph) [pdf, other]

Title: The Complementary Information Principle of Quantum MechanicsComments: 20 pages, 7 figures, comments are most welcome!Subjects: Quantum Physics (quantph); Mathematical Physics (mathph)
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 posttesting 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 outerapproximating all uncertainty regions and also establish fundamental limits for all qualified joint uncertainties.
 [7] arXiv:1908.07863 (crosslist from math.PR) [pdf, other]

Title: Derivation of coupled KPZBurgers equation from multispecies zerorange processesSubjects: Probability (math.PR); Statistical Mechanics (condmat.statmech); Mathematical Physics (mathph)
We consider the fluctuation fields of multispecies weaklyasymmetric zerorange 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 multispecies 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 multispecies 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 (crosslist from math.AP) [pdf, ps, other]

Title: On an optimal potential of Schrödinger operator with prescribed $m$ eigenvalueComments: 8 pagesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (mathph); 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 onedimensional 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 (crosslist from condmat.disnn) [pdf, other]

Title: On the multifractal dimensions and statistical properties of critical ensembles characterized by the three classical WignerDyson symmetry classesComments: 9 pages, 4 figuresSubjects: Disordered Systems and Neural Networks (condmat.disnn); Mathematical Physics (mathph)
We introduce a powerlaw banded random matrix model for the third of the three classical WignerDyson 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 timereversal symmetric systems with strong spinorbit interaction. For the sake of completeness, we also review the statistical properties of eigenvectors and eigenvalues of the powerlaw 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 powerlaw random banded matrix model corresponding to the three classical WignerDyson ensembles.
 [10] arXiv:1908.08002 (crosslist from condmat.disnn) [pdf, other]

Title: Virtual clusters model on branching random graphs for confined fluid thermodynamics in heterogeneous solid geometryComments: 24 pages, 7 figuresSubjects: Disordered Systems and Neural Networks (condmat.disnn); Mesoscale and Nanoscale Physics (condmat.meshall); Statistical Mechanics (condmat.statmech); Mathematical Physics (mathph)
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 fluidsolid interactions that have not been discussed in the literature. Our results demonstrate that at low temperature a significant impact to fluidsolid 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 (crosslist from hepth) [pdf, ps, other]

Title: Integrability and cycles of deformed ${\cal N}=2$ gauge theoryComments: 15 pages, LatexSubjects: High Energy Physics  Theory (hepth); Mathematical Physics (mathph); Exactly Solvable and Integrable Systems (nlin.SI)
To analyse pure ${\cal N}=2$ $SU(2)$ gauge theory in the NekrasovShatashvili (NS) limit (or deformed SeibergWitten (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 $\mathbb{Z}_2$ Rsymmetry 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 selfdual 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 $\mathbb{Z}_2$ Rsymmetry acting on the moduli space (BilalFerrari 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: PoincaréBirkhoffWitt isomorphisms and Kapranov dgmanifoldsComments: 40 pages, major revisionSubjects: Differential Geometry (math.DG); Mathematical Physics (mathph); Algebraic Geometry (math.AG); Algebraic Topology (math.AT); Quantum Algebra (math.QA)
 [13] arXiv:1703.04811 (replaced) [pdf, other]

Title: Equilibrium configurations for generalized FrenkelKontorova models on quasicrystalsAuthors: Rodrigo TreviñoComments: 14 pages, comments welcomeSubjects: Mathematical Physics (mathph); Dynamical Systems (math.DS)
 [14] arXiv:1703.07021 (replaced) [pdf, ps, other]

Title: Regularity of Schroedinger's functional equation and mean field PDEs for hpath processesAuthors: Toshio MikamiSubjects: Probability (math.PR); Mathematical Physics (mathph); Optimization and Control (math.OC)
 [15] arXiv:1809.08806 (replaced) [pdf, ps, other]

Title: The theory of NMixedSpinP fieldsSubjects: Algebraic Geometry (math.AG); Mathematical Physics (mathph)
 [16] arXiv:1903.05070 (replaced) [pdf, other]

Title: A generalized Noether theorem for scaling symmetryComments: Extended version. 18 pages, 3 figures. Details and relations to the Bargmann framework clarified and illustrated with further examples. v4: earlier results summarizedSubjects: Mathematical Physics (mathph); High Energy Physics  Theory (hepth)
 [17] arXiv:1903.11211 (replaced) [pdf, ps, other]

Title: Quantum Gravity Made EasyAuthors: John R. KlauderComments: 14 pages, a novel approach to quantizing general relativity, minor corrections; NEW RESULTS, SHARPER EXAMPLES; Important correctionSubjects: General Relativity and Quantum Cosmology (grqc); High Energy Physics  Theory (hepth); Mathematical Physics (mathph); Quantum Physics (quantph)
 [18] arXiv:1905.00659 (replaced) [pdf, ps, other]

Title: Lie Algebroid Gauging of Nonlinear Sigma ModelsAuthors: Kyle WrightComments: 32 pages. V2: fixed small typos to match published versionJournalref: J. Geom. Phys. 146 (2019) 103490Subjects: Differential Geometry (math.DG); High Energy Physics  Theory (hepth); Mathematical Physics (mathph)
 [19] arXiv:1905.00864 (replaced) [pdf, other]

Title: Elliptic Blowup Equations for 6d SCFTs. II: Exceptional CasesComments: 117 pages, 5 figures; typos corrected, new references addedSubjects: High Energy Physics  Theory (hepth); Mathematical Physics (mathph)
 [20] arXiv:1906.01229 (replaced) [pdf, ps, other]

Title: An optimization problem for finite point interaction familiesAuthors: Pavel ExnerComments: typos corrected, to appear in J. Phys. A: Math. TheorSubjects: Spectral Theory (math.SP); Mathematical Physics (mathph); Quantum Physics (quantph)
 [21] arXiv:1906.02619 (replaced) [pdf, other]

Title: Hyperbolic spin RuijsenaarsSchneider model from Poisson reductionComments: 16 pages. The statement about coincidence of the Poisson structure of spin variables at generic $N$ and $\ell$ with that of 1811.08727 was correctedSubjects: High Energy Physics  Theory (hepth); Mathematical Physics (mathph); Exactly Solvable and Integrable Systems (nlin.SI)
 [22] arXiv:1906.07238 (replaced) [pdf, ps, other]

Title: Heterotic/$F$theory Duality and NarasimhanSeshadri EquivalenceComments: 23 pagesSubjects: High Energy Physics  Theory (hepth); Mathematical Physics (mathph)
 [23] arXiv:1907.08204 (replaced) [pdf, other]

Title: Topological theory of LiebSchultzMattis theorems in quantum spin systemsComments: 27 pages + 12 pages of appendices. v2 updated referencesSubjects: Strongly Correlated Electrons (condmat.strel); Mathematical Physics (mathph); Quantum Physics (quantph)
 [24] arXiv:1908.07065 (replaced) [pdf, other]

Title: Nonperturbative approaches to the quantum SeibergWitten curveComments: 46 pages, 4 figures, 12 tables; new references added, typos correctedSubjects: High Energy Physics  Theory (hepth); Mathematical Physics (mathph)
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