LPTMS Publications

Archives :

• Chaos and Interacting Electrons in Ballistic Quantum Dots

D. Ullmo 1, 2, H. U. Baranger 1, K. Richter 3, F. Von Oppen 4, R. A. Jalabert 5

Physical Review Letters 80 (1998) 895-899

We show that the classical dynamics of independent particles can determine the quantum properties of interacting electrons in the ballistic regime. This connection is established using diagrammatic perturbation theory and semiclassical finite-temperature Green functions. Specifically, the orbital magnetism is greatly enhanced over the Landau susceptibility by the combined effects of interactions and finite size. The presence of families of periodic orbits in regular systems makes their susceptibility parametrically larger than that of chaotic systems, a difference which emerges from correlation terms.

• 1. Bell Laboratories–Lucent Technologies, Bell Laboratories
• 2. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud
• 3. Max-Planck-Institut für Physik komplexer Systeme, Max-Planck-Institut
• 4. Department of Condensed Matter Physics, Weizmann Institute of Science
• 5. Institut de Physique et Chimie des Matériaux de Strasbourg (IPCMS), CNRS : UMR7504 – Université Louis Pasteur - Strasbourg I

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• Comparing Mean Field and Euclidean Matching Problems

J. Houdayer 1, J. H. Boutet de Monvel 1, 2, O. C. Martin 1

European Physical Journal B 6 (1998) 383-393

Combinatorial optimization is a fertile testing ground for statistical physics methods developed in the context of disordered systems, allowing one to confront theoretical mean field predictions with actual properties of finite dimensional systems. Our focus here is on minimum matching problems, because they are computationally tractable while both frustrated and disordered. We first study a mean field model taking the link lengths between points to be independent random variables. For this model we find perfect agreement with the results of a replica calculation. Then we study the case where the points to be matched are placed at random in a d-dimensional Euclidean space. Using the mean field model as an approximation to the Euclidean case, we show numerically that the mean field predictions are very accurate even at low dimension, and that the error due to the approximation is O(1/d^2). Furthermore, it is possible to improve upon this approximation by including the effects of Euclidean correlations among k link lengths. Using k=3 (3-link correlations such as the triangle inequality), the resulting errors in the energy density are already less than 0.5% at d>=2. However, we argue that the Euclidean model\'s 1/d series expansion is beyond all orders in k of the expansion in k-link correlations.

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud
• 2. Forschungszentrum BiBos, Fakultät für Physik, Universität Bielefeld

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• Droplet Phenomenology and Mean Field in a Frustrated and Disordered System

J. Houdayer 1, O. C. Martin 1

Physical Review Letters 81 (1998) 2554-2557

The low lying excited states of the three-dimensional minimum matching problem are studied numerically. The excitations\' energies grow with their size and confirm the droplet picture. However, some low energy, infinite size excitations create multiple valleys in the energy landscape. These states violate the droplet scaling ansatz, and are consistent with mean field predictions. A similar picture may apply to spin glasses whereby the droplet picture describes the physics at small length scales, while mean field describes that at large length scales.

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud

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• Exact Results on Sinai’s Diffusion

Alain Comtet 1, David S. Dean 1

Journal of Physics A 31 (1998) 8595

We study the continuum version of Sinai\'s problem of a random walker in a random force field in one dimension. A method of stochastic representations is used to represent various probability distributions in this problem (mean probability density function and first passage time distributions). This method reproduces already known rigorous results and also confirms directly some recent results derived using approximation schemes. We demonstrate clearly, in the Sinai scaling regime, that the disorder dominates the problem and that the thermal distributions tend to zero-one laws.

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud

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• Exponential functionals of Brownian motion and disordered systems

Alain Comtet 1, Cecile Monthus 1, Marc Yor 2

Journal of Applied Probability 35 (1998) 255-271

The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts such as continuous time finance models and one-dimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a heat bath in a Wiener potential. Explicit expressions for the distribution of the free energy are presented.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Laboratoire de Probabilités et Modèles Aléatoires (LPMA), CNRS : UMR7599 – Université Paris VI - Pierre et Marie Curie – Université Paris VII - Paris Diderot

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• Kinetics of Anchoring of Polymer Chains on Substrates with Chemically Active Sites

G. Oshanin 1, S. Nechaev 2, A. M. Cazabat 3, M. Moreau 1

Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 58 (1998) 6134-6144

We consider dynamics of an isolated polymer chain with a chemically active end-bead on a 2D solid substrate containing immobile, randomly placed chemically active sites (traps). For a particular situation when the end-bead can be irreversibly trapped by any of these sites, which results in a complete anchoring of the whole chain, we calculate the time evolution of the probability $P_{ch}(t)$ that the initially non-anchored chain remains mobile until time $t$. We find that for relatively short chains $P_{ch}(t)$ follows at intermediate times a standard-form 2D Smoluchowski-type decay law $ln P_{ch}(t) \\sim - t/ln(t)$, which crosses over at very large times to the fluctuation-induced dependence $ln P_{ch}(t) \\sim - t^{1/2}$, associated with fluctuations in the spatial distribution of traps. We show next that for long chains the kinetic behavior is quite different; here the intermediate-time decay is of the form $ln P_{ch}(t) \\sim - t^{1/2}$, which is the Smoluchowski-type law associated with subdiffusive motion of the end-bead, while the long-time fluctuation-induced decay is described by the dependence $ln P_{ch}(t) \\sim - t^{1/4}$, stemming out of the interplay between fluctuations in traps distribution and internal relaxations of the chain.

• 1. Laboratoire de Physique Théorique des Liquides (LPTL), CNRS : UMR7600 – Université Paris VI - Pierre et Marie Curie
• 2. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud
• 3. Laboratoire de Physique de la Matière Condensée (LPMC), CNRS : UMR7125 – Collège de France

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• Normal and Anomalous Diffusion in a Deterministic Area-preserving Map

P. Leboeuf 1

Physica D: Nonlinear Phenomena 116 (1998) 8-20

Chaotic deterministic dynamics of a particle can give rise to diffusive Brownian motion. In this paper, we compute analytically the diffusion coefficient for a particular two-dimensional stochastic layer induced by the kicked Harper map. The variations of the transport coefficient as a parameter is varied are analyzed in terms of the underlying classical trajectories with particular emphasis in the appearance and bifurcations of periodic orbits. When accelerator modes are present, anomalous diffusion of the L\\évy type can occur. The exponent characterizing the anomalous diffusion is computed numerically and analyzed as a function of the parameter.

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud

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• Persistent Currents and Magnetization in two-dimensional Magnetic Quantum Systems

Jean Desbois 1, Stephane Ouvry 1, Christophe Texier 1

Nuclear Physics B 528 (1998) 727-745

Persistent currents and magnetization are considered for a two-dimensional electron (or gas of electrons) coupled to various magnetic fields. Thermodynamic formulae for the magnetization and the persistent current are established and the classical\'\' relationship between current and magnetization is shown to hold for systems invariant both by translation and rotation. Applications are given, including the point vortex superposed to an homogeneous magnetic field, the quantum Hall geometry (an electric field and an homogeneous magnetic field) and the random magnetic impurity problem (a random distribution of point vortices).

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud

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• Phases of random antiferromagnetic spin-1 chains

C. Monthus 1, O. Golinelli 2, Th. Jolicoeur 2

Physical Review B 58 (1998) 805-815

We formulate a real-space renormalization scheme that allows the study of the effects of bond randomness in the Heisenberg antiferromagnetic spin-1 chain. There are four types of bonds that appear during the renormalization flow. We implement numerically the decimation procedure. We give a detailed study of the probability distributions of all these bonds in the phases that occur when the strength of the disorder is varied. Approximate flow equations are obtained in the weak-disorder regime as well as in the strong disorder case where the physics is that of the random singlet phase.

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud
• 2. Service de Physique Théorique (SPhT), CNRS : URA2306 – CEA : DSM/SPHT

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• Quantum liquids of particles with generalized statistics

Serguei B. Isakov 1, 2

Physics Letters A 242 (1998) 130-138

We propose a phenomenological approach to quantum liquids of particles obeying generalized statistics of a fermionic type, in the spirit of the Landau Fermi liquid theory. The approach is developed for fractional exclusion statistics. We discuss both equilibrium (specific heat, compressibility, and Pauli spin susceptibility) and nonequilibrium (current and thermal conductivities, thermopower) properties. Low temperature quantities have the same temperature dependences as for the Fermi liquid, with the coefficients depending on the statistics parameter. The novel quantum liquids provide explicit realization of systems with a non-Fermi liquid Lorentz ratio in two and more dimensions. Consistency of the theory is verified by deriving the compressibility and $f$-sum rules.

• 1. Laboratoire de Physique Théorique et Modèles Statistiques (LPTMS), CNRS : UMR8626 – Université Paris XI - Paris Sud
• 2. Department of Physics, University of Oslo

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• Random Operator Approach for Word Enumeration in Braid Groups

Alain Comtet 1, Sergei K. Nechaev 1, 2

Journal of Physics A 31 (1998) 5609-5630

We investigate analytically the problem of enumeration of nonequivalent primitive words in the braid group B_n for n >> 1 by analysing the random word statistics and the target space on the basis of the locally free group approximation. We develop a \'symbolic dynamics\' method for exact word enumeration in locally free groups and bring arguments in support of the conjecture that the number of very long primitive words in the braid group is not sensitive to the precise local commutation relations. We consider the connection of these problems with the conventional random operator theory, localization phenomena and statistics of systems with quenched disorder. Also we discuss the relation of the particular problems of random operator theory to the theory of modular functions

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud
• 2. L.D. Landau Institute for Theoretical Physics, Landau Institute for Theoretical Physics

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• Random Walks, Reaction-Diffusion, and Nonequilibrium Dynamics of Spin Chains in One-dimensional Random Environments

Daniel Fisher 1, Pierre Le Doussal 2, Cecile Monthus 3

Physical Review Letters 80 (1998) 3539-3542

Sinai\'s model of diffusion in one-dimension with random local bias is studied by a real space renormalization group which yields asymptotically exact long time results. The distribution of the position of a particle and the probability of it not returning to the origin are obtained, as well as the two-time distribution which exhibits äging\' with $\\frac{\\ln t}{\\ln t\'}$ scaling and a singularity at $\\ln t =\\ln t\'$. The effects of a small uniform force are also studied. Extension to motion of many domain walls yields non-equilibrium time dependent correlations for the 1D random field Ising model with Glauber dynamics and \'persistence\' exponents of 1D reaction-diffusion models with random forces.

• 1. Lyman Laboratory of Physics, University of Harvard
• 2. Laboratoire de Physique Théorique de l'ENS (LPTENS), CNRS : UMR8549 – Université Paris VI - Pierre et Marie Curie – Ecole Normale Supérieure de Paris - ENS Paris
• 3. Service de Physique Théorique (SPhT), CNRS : URA2306 – CEA : DSM/SPHT

Details Citations to the Article (88)
• Rough droplet model for spherical metal clusters

N. Pavloff 1, C. Schmit 1

Physical Review B 58 (1998) 4942-4951

We study the thermally activated oscillations, or capillary waves, of a neutral metal cluster within the liquid drop model. These deformations correspond to a surface roughness which we characterize by a single parameter $\\Delta$. We derive a simple analytic approximate expression determining $\\Delta$ as a function of temperature and cluster size. We then estimate the induced effects on shell structure by means of a periodic orbit analysis and compare with recent data for shell energy of sodium clusters in the size range $50 < N < 250$. A small surface roughness $\\Delta\\simeq 0.6$ \Å~ is seen to give a reasonable account of the decrease of amplitude of the shell structure observed in experiment. Moreover -- contrary to usual Jahn-Teller type of deformations -- roughness correctly reproduces the shape of the shell energy in the domain of sizes considered in experiment.

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud

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• Scaling universalities of kth-nearest neighbor distances on closed manifolds

A. G. Percus 1, O. C. Martin 2

Advances in Applied Mathematics 21 (1998) 424-436

Take N sites distributed randomly and uniformly on a smooth closed surface. We express the expected distance from an arbitrary point on the surface to its kth-nearest neighboring site, in terms of the function A(l) giving the area of a disc of radius l about that point. We then find two universalities. First, for a flat surface, where A(l)=\\pi l^2, the k-dependence and the N-dependence separate in . All kth-nearest neighbor distances thus have the same scaling law in N. Second, for a curved surface, the average \\int d\\mu over the surface is a topological invariant at leading and subleading order in a large N expansion. The 1/N scaling series then depends, up through O(1/N), only on the surface\'s topology and not on its precise shape. We discuss the case of higher dimensions (d>2), and also interpret our results using Regge calculus.

• 1. CIC-3 and Center for Nonlinear Studies, Los Alamos National Laboratory
• 2. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud

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• Statistics of reduced words in locally free and braid groups: Abstract studies and application to ballistic growth model

Jean Desbois 1, Sergei K. Nechaev 1, 2

Journal of Physics A 31 (1998) 2767-2784

We study numerically and analytically the average length of reduced (primitive) words in so-called locally free and braid groups. We consider the situations when the letters in the initial words are drawn either without or with correlations. In the latter case we show that the average length of the reduced word can be increased or lowered depending on the type of correlation. The ideas developed are used for analytical computation of the average number of peaks of the surface appearing in some specific ballistic growth model

• 1. Division de Physique Théorique, IPN, Université Paris XI - Paris Sud
• 2. L D Landau Institute for Theoretical Physics, Landau Institute for Theoretical Physics