9,430 research outputs found

    Gibbs measures on permutations over one-dimensional discrete point sets

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    We consider Gibbs distributions on permutations of a locally finite infinite set X⊂RX\subset\mathbb{R}, where a permutation σ\sigma of XX is assigned (formal) energy ∑x∈XV(σ(x)−x)\sum_{x\in X}V(\sigma(x)-x). This is motivated by Feynman's path representation of the quantum Bose gas; the choice X:=ZX:=\mathbb{Z} and V(x):=αx2V(x):=\alpha x^2 is of principal interest. Under suitable regularity conditions on the set XX and the potential VV, we establish existence and a full classification of the infinite-volume Gibbs measures for this problem, including a result on the number of infinite cycles of typical permutations. Unlike earlier results, our conclusions are not limited to small densities and/or high temperatures.Comment: Published in at http://dx.doi.org/10.1214/14-AAP1013 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Information-theoretic equilibration: the appearance of irreversibility under complex quantum dynamics

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    The question of how irreversibility can emerge as a generic phenomena when the underlying mechanical theory is reversible has been a long-standing fundamental problem for both classical and quantum mechanics. We describe a mechanism for the appearance of irreversibility that applies to coherent, isolated systems in a pure quantum state. This equilibration mechanism requires only an assumption of sufficiently complex internal dynamics and natural information-theoretic constraints arising from the infeasibility of collecting an astronomical amount of measurement data. Remarkably, we are able to prove that irreversibility can be understood as typical without assuming decoherence or restricting to coarse-grained observables, and hence occurs under distinct conditions and time-scales than those implied by the usual decoherence point of view. We illustrate the effect numerically in several model systems and prove that the effect is typical under the standard random-matrix conjecture for complex quantum systems.Comment: 15 pages, 7 figures. Discussion has been clarified and additional numerical evidence for information theoretic equilibration is provided for a variant of the Heisenberg model as well as one and two-dimensional random local Hamiltonian

    Universal behaviour of 3D loop soup models

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    These notes describe several loop soup models and their {\it universal behaviour} in dimensions greater or equal to 3. These loop models represent certain classical or quantum statistical mechanical systems. These systems undergo phase transitions that are characterised by changes in the structures of the loops. Namely, long-range order is equivalent to the occurrence of macroscopic loops. There are many such loops, and the joint distribution of their lengths is always given by a {\it Poisson-Dirichlet distribution}. This distribution concerns random partitions and it is not widely known in statistical physics. We introduce it explicitly, and we explain that it is the invariant measure of a mean-field split-merge process. It is relevant to spatial models because the macroscopic loops are so intertwined that they behave effectively in mean-field fashion. This heuristics can be made exact and it allows to calculate the parameter of the Poisson-Dirichlet distribution. We discuss consequences about symmetry breaking in certain quantum spin systems.Comment: 31 pages, 11 figures. Notes prepared for the 6th Warsaw School of Statistical Physics, held from 25 June to 2 July 2016 in Sandomierz, Polan
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