http://arxiv.org/ Mathematical Physics (math-ph) updates on the arXiv.org e-print archive en-us 2010-02-08T20:30:00-05:00 www-admin@arxiv.org Mathematical Physics 1901-01-01T00:00+00:00 1 daily http://arxiv.org/icons/sfx.gif http://arxiv.org/ http://arxiv.org/abs/1002.1453 <p>We address the following inverse problem in quantum statistical physics: does the quantum free energy (von Neumann entropy + kinetic energy) admit a unique minimizer among the density operators having a given local density $n(x)$? We give a positive answer to that question, in dimension one. This enables to define rigourously the notion of local quantum equilibrium, or quantum Maxwellian, which is at the basis of recently derived quantum hydrodynamic models and quantum drift-diffusion models. We also characterize this unique minimizer, which takes the form of a global thermodynamic equilibrium (canonical ensemble) with a quantum chemical potential. </p> Florian M&#xe9;hats (IRMAR), Olivier Pinaud (ICJ) http://arxiv.org/abs/1002.1454 <p>We present, for both minkowskian and euclidean signatures, short derivations of the diagonal Einstein metrics for Bianchi type II, III and V. For the first two cases we show the integrability of the geodesic flow while for the third case a somewhat unusual bifurcation phenomenon takes place: for minkowskian signature elliptic functions are essential in the metric while for euclidean signature only elementary functions appear. </p> Galliano Valent (LPTHE) http://arxiv.org/abs/1002.1463 <p>The two-dimensional, periodic Lorentz gas, is the dynamical system corresponding with the free motion of a point particle in a planar system of fixed circular obstacles centered at the vertices of a square lattice in the Euclidian plane. Assuming elastic collisions between the particle and the obstacles, we consider this dynamical system in the Boltzmann-Grad limit, where the obstacle radius r and the reciprocal mean free path are asymptotically equivalent small quantities, and that the particle's distribution function is slowly varying in the space variable. While it is known that the particle's distribution function in that limit cannot be governed by a linear Boltzmann equation [F.Golse, Ann. Fac. Sci. Toulouse 17 (2008), 735--749], we propose a limiting theory involving an extended phase space larger than the usual phase space of a classical point particle in which the limiting particle's distribution function evolves under an integro-differential equation somewhat analogous to a linear Boltzmann equation. The particle's distribution function in the classical phase space is obtained by averaging out the additional variables involved in the extended phase space. We derive explicit formulas for the coefficients appearing in that integro-differential equation, and study its basic dynamical properties -- especially, we establish an analogue of Boltzmann's H theorem, provide a complete description of the equilibrium distribution functions and investigate the convergence to these equilibrium distribution functions in the long time limit. This article provides complete proofs of the results announced in [E. Caglioti, F. Golse, C.R. Acad. Sci. Ser. I Math. 346 (2008) 477--482]. </p> Emanuele Caglioti, Fran&#xe7;ois Golse http://arxiv.org/abs/1002.1487 <p>We review the basic ideas lying at the foundation of the recently developed theory of twisted symmetries of differential equations, and some of its developments. </p> Giuseppe Gaeta http://arxiv.org/abs/1002.1489 <p>Symmetry properties are at the basis of integrability. In recent years, it appeared that so called "twisted symmetries" are as effective as standard symmetries in many respects (integrating ODEs, finding special solutions to PDEs). Here we discuss how twisted symmetries can be used to detect integrability of Lagrangian systems which are not integrable via standard symmetries. </p> G. Cicogna, G. Gaeta http://arxiv.org/abs/1002.1490 <p>We derive the BBGKY hierarchy for the Fermi and Bose many-particle systems, using the von Neumann hierarchy for the correlation operators. The solution of the Cauchy problem of the formulated hierarchy for the case of a n-body interaction potential is constructed in the space of sequences of trace-class operators. </p> D.O. Polishchuk http://arxiv.org/abs/1002.1494 <p>In this paper Fokker-Planck-Kolmogorov type equations associated with stochastic differential equations driven by a time-changed fractional Brownian motion are derived. Two equivalent forms are suggested. The time-change process considered is either the first hitting time process for a stable subordinator or a mixture of stable subordinators. A family of operators arising in the representation of the Fokker-Plank-Kolmogorov equations is shown to have the semigroup property. </p> Marjorie Hahn, Kei Kobayashi, Sabir Umarov http://arxiv.org/abs/1002.1514 <p>We establish a series representation of the Hill discriminant based on the spectral parameter power series recently introduced by V. Kravchenko. We also show the invariance of the Hill discriminant under a Darboux transformation and the feasibility of the new series form for numerical calculations of the eigenspectrum </p> K.V. Khmelnytskaya, H.C. Rosu http://arxiv.org/abs/1002.1565 <p>A self-consistent description of degenerate Lagrangian theories is made by introducing a Clairaut-like version of the Hamiltonian formalism. A generalization of the Legendre transform to the case when the Hessian is zero is done using the mixed (envelope/general) solutions of the multidimensional Clairaut equation. The corresponding system of equations of motion is equivalent to the Lagrange equations and has a subsytem for "unresolved" velocities. Then it is presented in the Hamiltonian-like form by introducing a new (non-Lie) bracket. This is a "shortened" formalism since finally it does not contain "nondynamical" (degenerate) momenta at all, and therefore there is no notion of constraint: nothing to constrain. It is shown that any classical degenerate Lagrangian theory in its Clairaut-like Hamiltonian form is equivalent to the many-time classical dynamics. </p> Steven Duplij http://arxiv.org/abs/1002.1590 <p>We study focussing discrete nonlinear Schr\"{o}dinger equations and present a new variational existence proof for homoclinic standing waves (bright solitons). Our approach relies on the constrained maximization of an energy functional and provides the existence of two one-parameter families of waves with unimodal and even profile function for a wide class of nonlinearities. Finally, we illustrate our results by numerical simulations. </p> Michael Herrmann http://arxiv.org/abs/1002.1591 <p>We study heteroclinic standing waves (dark solitons) in discrete nonlinear Schr\"{o}dinger equations with defocussing nonlinearity. Our main result is a quite elementary existence proof for waves with monotone and odd profile, and relies on minimizing an appropriately defined energy functional. We also study the continuum limit and the numerical approximation of standing waves. </p> Michael Herrmann http://arxiv.org/abs/1002.1620 <p>We find a general solution to the unique 7th order ODE admitting ten dimensional group of contact symmetries. The integral curves of this ODE are rational contact curves in $\PP^3$ which give rise to rational plane curves of degree six. The moduli space of these curves is a real form of the homogeneous space $Sp(4)/SL(2)$. </p> Maciej Dunajski, Vladimir Sokolov http://arxiv.org/abs/1002.1623 <p>In this work we demonstrate that the Yang-Baxter algebra can also be employed in order to derive a functional relation for the partition function of the six vertex model with domain wall boundary conditions. The homogeneous limit is studied for small lattices and the properties determining the partition function are also discussed. </p> W. Galleas http://arxiv.org/abs/1002.1667 <p>We have constructed a set of non-Hermitian operators that generate rotations in the configuration space and not in the momentum space but in a modified non-Hermitian momentum space. We show that by means of this new momentum operators we can construct supersymmetric Hamiltonians as well as Hamiltonians that naturally arise in Quantum Finance. It is shown that the rotations generators represent conserved quantities for a non-Hermitian Hamiltonian. This generators are related with a new type of spherical harmonics that result to be PT-orthonormal. Finally, a non-Hermitian hydrogen atom Hamiltonian is constructed and its states are obtained. </p> Juan M. Romero, R. Bernal-Jaquez, O. Gonzalez-Gaxiola http://arxiv.org/abs/1002.1676 <p>We consider the moduli space of bordered Riemann surfaces with boundary and marked points. Such spaces appear in open-closed string theory, particularly with respect to holomorphic curves with Lagrangian submanifolds. We consider a combinatorial framework to view the compactification of this space based on the pair-of-pants decomposition of the surface, relating it to the well-known phenomenon of bubbling. Our main result classifies all such spaces that can be realized as convex polytopes. A new polytope is introduced based on truncations of cubes, and its combinatorial and algebraic structures are related to generalizations of associahedra and multiplihedra. </p> Satyan L. Devadoss, Timothy Heath, Cid Vipismakul http://arxiv.org/abs/1002.1695 <p>We consider Hermitian random band matrices $H$ in $d \geq 1$ dimensions. The matrix elements $H_{xy}$, indexed by $x,y\in \Z^d$, are independent, uniformly distributed random variables if $\abs{x-y}$ is less than the band width $W$, and zero otherwise. We prove that the time evolution of a quantum particle subject to the Hamiltonian $H$ is diffusive on time scales $t \ll W^{d/3}$. We also show that the localization length of the overwhelming majority of the eigenfunctions is larger than a factor $W^{d/6}$ times the band width. All results are uniform in the size of the matrix. </p> Laszlo Erdos, Antti Knowles http://arxiv.org/abs/1002.1396 <p>The paper contains a stability analysis of the plane-wave Riemann problem for the two-dimensional hyperbolic conservation laws for an ideal compressible gas. It is proved that the contact discontinuity in the plane-wave Riemann problem is unstable under perturbations. The implications for Godunovs method are discussed and it is shown that numerical post shock noise can set of a contact instability. A relation to carbuncle instabilities is established. </p> B. Einfeldt http://arxiv.org/abs/1002.1371 <p>Wigner and Husimi transforms have long been used for the phase-space reformulation of Schr\"odinger-type equations, and the study of the corresponding semiclassical limits. Most of the existing results provide approximations in appropriate weak topologies. In this work we are concerned with semiclassical limits in the strong topology, i.e. approximation of Wigner functions by solutions of the Liouville equation in $L^2$ and Sobolev norms. The results obtained improve the state of the art, and highlight the role of potential regularity, especially through the regularity of the Wigner equation. It must be mentioned that the strong convergence can be shown up to $O(log \frac{1}\epsilon)$ time-scales, which is well known to be, in general, the limit of validity of semiclassical asymptotics. </p> Agissilaos Athanassoulis, Thierry Paul http://arxiv.org/abs/1002.1344 <p>We propose an alternative factorization for the simple harmonic oscillator hamiltonian which includes Mielnik's isospectral factorization as a particular case. This factorization is realized in two non-mutually adjoint operators whose inverse product, in the simplest case, lead to a new Sturm-Liouville eigenvalue equation which includes Schrodinger's equation for the oscillator and Hermite's equation as particular cases for limiting values of the factorization's parameter, and whose eigenfunctions allow us to define new generalized Hermite functions. </p> Marco A. Reyes, M. Ranferi Gutierrez http://arxiv.org/abs/1002.1356 <p>This paper is on tilings of polygons by rectangles. A celebrated physical interpretation of such tilings due to R.L. Brooks, C.A.B. Smith, A.H. Stone and W.T. Tutte uses direct-current circuits. The new approach of the paper is an application of alternating-current circuits. </p> <p>The following results are obtained: </p> <p>- a necessary condition for a rectangle to be tilable by rectangles of given shapes; </p> <p>- a criterion for a rectangle to be tilable by rectangles similar to it but not all homothetic to it; </p> <p>- a criterion for a generic polygon to be tilable by squares. </p> <p>These results generalize the ones of C. Freiling, R. Kenyon, M. Laczkovich, D. Rinne and G. Szekeres. </p> M. Prasolov, M. Skopenkov http://arxiv.org/abs/1002.1698 <p>In this thesis we discuss some applications of the theory of Anosov systems to nonequilibrium statistical mechanics. In particular, we perform a perturbative check of the Gallavotti-Cohen fluctuation relation for a simple Anosov system; we find that the lack of differentiability of the time reversal operator implies a violation of the Gallavotti-Cohen fluctuation relation. </p> Marcello Porta http://arxiv.org/abs/0810.0994 <p>We prove an important partial case of the pseudo-Riemannian version of the projective Lichnerowicz conjecture stating that a complete manifold admitting an essential group of projective transformations is the round sphere (up to a finite cover). </p> Volodymyr Kiosak, Vladimir S. Matveev http://arxiv.org/abs/0901.1454 <p>The space, on which quantum field operators are given, is constructed in any theory, in which the usual product between test functions is substituted by the $\star$-product (the Moyal-type product). The important example of such a theory is noncommutative quantum field theory (NC QFT). This construction is the key point in the derivation of the Wightman reconstruction theorem. </p> M.N. Mnatsakanova, Yu.S. Vernov http://arxiv.org/abs/0903.1185 <p>First we review some of the attempts made to find exact spherically symmetric solutions of Einstein field equations in the presence of scalar fields .Wyman solution in both static and non static scalar field is discussed briefly and it is show that why in the case of non static homogenous matter field, static metric can not be represented in terms of elementary functions. We mention here that if our spacetime be static, according to EFE there is two option for choose scalar field matter: static (time independent) and non static (time dependent). All these solutions are limited to the minimally coupled massless scalar fields and also in the absence of the cosmological constant . Then we show that if we are interesting to have a homogenous isotropic scalar field matter one can construct a series solution in terms of scalar field's mass and cosmological constant. This metric is static and posses a locally flat case as a special chooses of mass of scalar field and can be interpreted as an effective vacuum. Therefore mass of scalar field eliminates any locally gravitational effect as tidal forces. Finally we describe why this system is unstable in the language of dynamical systems. </p> Mohammad Mehrpooya, D. Momeni http://arxiv.org/abs/0905.1551 <p>We study the asymptotic behavior and the asymptotic stability of the two-dimensional Euler equations and of the two-dimensional linearized Euler equations close to parallel flows. We focus on spectrally stable jet profiles $U(y)$ with stationary streamlines $y_{0}$ such that $U'(y_{0})=0$, a case that has not been studied previously. We describe a new dynamical phenomenon: the depletion of the vorticity at the stationary streamlines. An unexpected consequence, is that the velocity decays for large times with power laws, similarly to what happens in the case of the Orr mechanism for base flows without stationary streamlines. The asymptotic behaviors of velocity and the asymptotic profiles of vorticity are theoretically predicted and compared with direct numerical simulations. We argue on the asymptotic stability of these flow velocities even in the absence of any dissipative mechanisms. </p> Freddy Bouchet (Phys-ENS, INLN), Hidetoshi Morita (INLN) http://arxiv.org/abs/0909.2974 <p>We establish large deviation formulas for linear statistics on the $N$ transmission eigenvalues $\{T_i\}$ of a chaotic cavity, in the framework of Random Matrix Theory. Given any linear statistics of interest $A=\sum_{i=1}^N a(T_i)$, the probability distribution $\mathcal{P}_A(A,N)$ of $A$ generically satisfies the large deviation formula $\lim_{N\to\infty}[-2\log\mathcal{P}_A(Nx,N)/\beta N^2]=\Psi_A(x)$, where $\Psi_A(x)$ is a rate function that we compute explicitly in many cases (conductance, shot noise, moments) and $\beta$ corresponds to different symmetry classes. Using these large deviation expressions, it is possible to recover easily known results and to produce new formulas, such as a closed form expression for $v(n)=\lim_{N\to\infty}\mathrm{var}(\mathcal{T}_n)$ (where $\mathcal{T}_n=\sum_{i}T_i^n$) for arbitrary integer $n$. The universal limit $v^\star=\lim_{n\to\infty} v(n)=1/2\pi\beta$ is also computed exactly. The distributions display a central Gaussian region flanked on both sides by non-Gaussian tails. At the junction of the two regimes, weakly non-analytical points appear, a direct consequence of phase transitions in an associated Coulomb gas problem. Numerical checks are also provided, which are in full agreement with our asymptotic results in both real and Laplace space even for moderately small $N$. Part of the results have been announced in [P. Vivo, S.N. Majumdar and O. Bohigas, {\it Phys. Rev. Lett.} {\bf 101}, 216809 (2008)]. </p> Pierpaolo Vivo, Satya N. Majumdar, Oriol Bohigas http://arxiv.org/abs/0910.1549 <p>We investigate the classical limit of non-Hermitian quantum dynamics arising from a coherent state approximation, and show that the resulting classical phase space dynamics can be described by generalised "canonical" equations of motion, for both conservative and dissipative motion. The dynamical equations combine a symplectic flow associated with the Hermitian part of the Hamiltonian with a metric gradient flow associated with the anti-Hermitian part of the Hamiltonian. We derive this structure of the classical limit of quantum systems in the case of a Euclidean phase space geometry. As an example we show that the classical dynamics of a damped and driven oscillator can be linked to a non-Hermitian quantum system, and investigate the quantum classical correspondence. Furthermore, we present an example of an angular momentum system whose classical phase space is spherical and show that the generalised canonical structure persists for this nontrivial phase space geometry. </p> Eva-Maria Graefe, Michael Hoening, Hans Juergen Korsch http://arxiv.org/abs/0911.5466 <p>We give an example of infinite order rational transformation that leaves a linear differential equation covariant. This example can be seen as a non-trivial but still simple illustration of an exact representation of the renormalization group. </p> A. Bostan, S. Boukraa, S. Hassani, J.-M. Maillard, J-A. Weil, N. Zenine, N. Abarenkova http://arxiv.org/abs/1001.3702 <p>We give a rigorous computer-assisted proof that the triangular bi-pyramid is the unique configuration of 5 points on the 2-sphere that globally minimizes the Coulomb (1/r) potential. We also prove the same result for the (1/r^2) potential. The main mathematical contribution of the paper is a fairly efficient energy estimate that works for any number of points and any power-law potential. </p> Richard Evan Schwartz http://arxiv.org/abs/1002.0104 <p>Holographic functional methods are introduced as probes of discrete time-stepped maps that lead to chaotic behavior. The methods provide continuous time interpolation between the time steps, thereby revealing the maps to be splintered Hamiltonian systems underlain by novel potentials. A sequence of successively deepening switchback potentials for a particle driven by Hamiltonian dynamics explains, in very physical terms, the frequency doubling and trajectory folding that occur on the particular route to chaos revealed by the logistic map, x --&gt; 4x(1-x). </p> Thomas L. Curtright, Cosmas K. Zachos http://arxiv.org/abs/1002.0742 <p>We examine the completeness of bi-orthogonal sets of eigenfunctions for non-Hermitian Hamiltonians possessing a spectral singularity. The correct resolutions of identity are constructed for delta like and smooth potentials. Their form and the contribution of a spectral singularity depend on the class of functions employed for physical states. With this specification there is no obstruction to completeness originating from a spectral singularity. </p> A.A. Andrianov, F. Cannata, A.V. Sokolov http://arxiv.org/abs/1002.1278 <p>Careful exploration of the idea that equation for radial wave function must be compatible with the full Schrodinger equation shows appearance of the delta-function while reduction of full Schrodinger equation in spherical coordinates. Elimination of this extra term produces a boundary condition for the radial wave function, which is the same both for regular and singular potentials. </p> Anzor A.Khelashvili, Teimuraz P. Nadareishvili http://arxiv.org/abs/math/0612765 <p>We describe a new method to bound certain higher-dimensional exponential sums which are associated with tori in symplectic groups over finite fields. Our method is based on the self-reducibility property of the Weil representation. As a result, we obtain a sharp form of the Hecke quantum unique ergodicity theorem for generic linear symplectomorphisms of the 2N-dimensional torus. </p> Shamgar Gurevich (IAS), Ronny Hadani (Austin) http://arxiv.org/abs/1002.0851 <p>A restriction of the baby Skyrme model consisting of the quartic and potential terms only is investigated in detail for a wide range of potentials. Further, its properties are compared with those of the corresponding full baby Skyrme models. We find that topological (charge) as well as geometrical (nucleus/shell shape) features of baby skyrmions are captured already by the soliton solutions of the restricted model. Further, we find a coincidence between the compact or non-compact nature of solitons in the restricted model, on the one hand, and the existence or non-existence of multi-skyrmions in the full baby Skyrme model, on the other hand. </p> C. Adam, T. Romanczukiewicz, J. Sanchez-Guillen, A. Wereszczynski http://arxiv.org/abs/1002.1123 <p>We discuss the Poisson structures, Lax matrices, $r$-matrices, bi-hamiltonian structures, the variables of separation and other attributes of the modern theory of dynamical systems in application to the integrable Euler top and to the nonholonomic Chaplygin ball. </p> A V Tsiganov