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Author: Jansen, Sabine × End date: 2019 × Start date: 2019 × End date: 2019 × Start date: 2010 × DDC: 510 × Type: Text ×

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Now showing 1 - 5 of 5
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    Fermionic and bosonic Laughlin state on thick cylinders
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Jansen, Sabine
    We investigate a many-body wave function for particles on a cylinder known as Laughlin's function. It is the power of a Vandermonde determinant times a Gaussian. Our main result is: in a many-particle limit, at fixed radius, all correlation functions have a unique limit, and the limit state has a non-trivial period in the axial direction. The result holds regardless how large the radius is, for fermions as well as bosons. In addition, we explain how the algebraic structure used in proofs relates to a ground state perturbation series and to quasi-state decompositions, and we show that the monomer-dimer function introduced in an earlier work is an exact, zero energy, ground state of a suitable finite range Hamiltonian; this is interesting because of formal analogies with some quantum spin chains.
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    Symmetry breaking in quasi-1D Coulomb systems
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2010) Aizenman, Michael; Jansen, Sabine; Jung, Paul
    Let a high-dimensional random vector vecX can be represented as a sum of two components - a signa vecS, which belongs to some low-dimensional subspace mathcalS, and a noise component vecN. This paper presents a new approach for estimating the subspace mathcalS based on the ideas of the Non-Gaussian Component Analysis. Our approach avoids the technical difficulties that usually exist in similar methods - it doesn't require neither the estimation of the inverse covariance matrix of vecX nor the estimation of the covariance matrix of vecN
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    Mayer and virial series at low temperature
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Jansen, Sabine
    We analyze the Mayer pressure-activity and virial pressure-density series for a classical system of particles in continuous configuration space at low temperature. Particles interact via a finite range potential with an attractive tail. We propose physical interpretations of the Mayer and virial series' radius of convergence, valid independently of the question of phase transition: the Mayer radius corresponds to a fast increase from very small to finite density, and the virial radius corresponds to a cross-over from monatomic to polyatomic gas. Our results have consequences for the search of a low density, low temperature solid-gas phase transition, consistent with the Lee-Yang theorem for lattice gases and with the continuum Widom-Rowlinson mode.
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    Large deviations for cluster size distributions in a continuous classical many-body system
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2011) Jansen, Sabine; König, Wolfgang; Metzger, Bernd
    An interesting problem in statistical physics is the condensation of classical particles in droplets or clusters when the pair-interaction is given by a stable Lennard-Jones-type potential. We study two aspects of this problem. We start by deriving a large deviations principle for the cluster size distribution for any inverse temperature $betain(0,infty)$ and particle density $rhoin(0,rho_rmcp)$ in the thermodynamic limit. Here $rho_rmcp >0$ is the close packing density. While in general the rate function is an abstract object, our second main result is the $Gamma$-convergence of the rate function towards an explicit limiting rate function in the low-temperature dilute limit $betatoinfty$, $rho downarrow 0$ such that $-beta^-1logrhoto nu$ for some $nuin(0,infty)$. The limiting rate function and its minimisers appeared in recent work, where the temperature and the particle density were coupled with the particle number. In the de-coupled limit considered here, we prove that just one cluster size is dominant, depending on the parameter $nu$. Under additional assumptions on the potential, the $Gamma$-convergence along curves can be strengthened to uniform bounds, valid in a low-temperature, low-density rectangle.
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    Ideal mixture approximation of cluster size distributions at low density
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2012) Jansen, Sabine; König, Wolfgang
    We consider an interacting particle system in continuous configuration space. The pair interaction has an attractive part. We show that, at low density, the system behaves approximately like an ideal mixture of clusters (droplets): we prove rigorous bounds (a) for the constrained free energy associated with a given cluster size distribution, considered as an order parameter, (b) for the free energy, obtained by minimising over the order parameter, and (c) for the minimising cluster size distributions. It is known that, under suitable assumptions, the ideal mixture has a transition from a gas phase to a condensed phase as the density is varied; our bounds hold both in the gas phase and in the coexistence region of the ideal mixture.