9,430 research outputs found
Gibbs measures on permutations over one-dimensional discrete point sets
We consider Gibbs distributions on permutations of a locally finite infinite
set , where a permutation of is assigned
(formal) energy . This is motivated by Feynman's
path representation of the quantum Bose gas; the choice and
is of principal interest. Under suitable regularity
conditions on the set and the potential , 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
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
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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