24,962 research outputs found
An extension of the Kac ring model
We introduce a unitary dynamics for quantum spins which is an extension of a
model introduced by Mark Kac to clarify the phenomenon of relaxation to
equilibrium. When the number of spins gets very large, the magnetization
satisfies an autonomous equation as function of time with exponentially fast
relaxation to the equilibrium magnetization as determined by the microcanonical
ensemble. This is proven as a law of large numbers with respect to a class of
initial data. The corresponding Gibbs-von Neumann entropy is also computed and
its monotonicity in time discussed.Comment: 15 pages, v2 -> v3: minor typographic correctio
Fibrational induction rules for initial algebras
This paper provides an induction rule that can be used to prove properties of data structures whose types are inductive, i.e., are carriers of initial algebras of functors. Our results are semantic in nature and are inspired by Hermida and Jacobs’ elegant algebraic formulation of induction for polynomial data types. Our contribution is to derive, under slightly different assumptions, an induction rule that is generic over all inductive types, polynomial or not. Our induction rule is generic over the kinds of properties to be proved as well: like Hermida and Jacobs, we work in a general fibrational setting and so can accommodate very general notions of properties on inductive types rather than just those of particular syntactic forms. We establish the correctness of our generic induction rule by reducing induction to iteration. We show how our rule can be instantiated to give induction rules for the data types of rose trees, finite hereditary sets, and hyperfunctions. The former lies outside the scope of Hermida and Jacobs’ work because it is not polynomial; as far as we are aware, no induction rules have been known to exist for the latter two in a general fibrational framework. Our instantiation for hyperfunctions underscores the value of working in the general fibrational setting since this data type cannot be interpreted as a set
Combinatorial models of rigidity and renormalization
We first introduce the percolation problems associated with the graph
theoretical concepts of -sparsity, and make contact with the physical
concepts of ordinary and rigidity percolation. We then devise a renormalization
transformation for -percolation problems, and investigate its domain of
validity. In particular, we show that it allows an exact solution of
-percolation problems on hierarchical graphs, for . We
introduce and solve by renormalization such a model, which has the interesting
feature of showing both ordinary percolation and rigidity percolation phase
transitions, depending on the values of the parameters.Comment: 22 pages, 6 figure
Experimental Demonstration of a Quantum Circuit using Linear Optics Gates
One of the main advantages of an optical approach to quantum computing is the
fact that optical fibers can be used to connect the logic and memory devices to
form useful circuits, in analogy with the wires of a conventional computer.
Here we describe an experimental demonstration of a simple quantum circuit of
that kind in which two probabilistic exclusive-OR (XOR) logic gates were
combined to calculate the parity of three input qubits.Comment: v2 is final PRA versio
Generic Fibrational Induction
This paper provides an induction rule that can be used to prove properties of
data structures whose types are inductive, i.e., are carriers of initial
algebras of functors. Our results are semantic in nature and are inspired by
Hermida and Jacobs' elegant algebraic formulation of induction for polynomial
data types. Our contribution is to derive, under slightly different
assumptions, a sound induction rule that is generic over all inductive types,
polynomial or not. Our induction rule is generic over the kinds of properties
to be proved as well: like Hermida and Jacobs, we work in a general fibrational
setting and so can accommodate very general notions of properties on inductive
types rather than just those of a particular syntactic form. We establish the
soundness of our generic induction rule by reducing induction to iteration. We
then show how our generic induction rule can be instantiated to give induction
rules for the data types of rose trees, finite hereditary sets, and
hyperfunctions. The first of these lies outside the scope of Hermida and
Jacobs' work because it is not polynomial, and as far as we are aware, no
induction rules have been known to exist for the second and third in a general
fibrational framework. Our instantiation for hyperfunctions underscores the
value of working in the general fibrational setting since this data type cannot
be interpreted as a set.Comment: For Special Issue from CSL 201
Conditions for the Quantum to Classical Transition: Trajectories vs. Phase Space Distributions
We contrast two sets of conditions that govern the transition in which
classical dynamics emerges from the evolution of a quantum system. The first
was derived by considering the trajectories seen by an observer (dubbed the
``strong'' transition) [Bhattacharya, et al., Phys. Rev. Lett. 85: 4852
(2000)], and the second by considering phase-space densities (the ``weak''
transition) [Greenbaum, et al., Chaos 15, 033302 (2005)]. On the face of it
these conditions appear rather different. We show, however, that in the
semiclassical regime, in which the action of the system is large compared to
, and the measurement noise is small, they both offer an essentially
equivalent local picture. Within this regime, the weak conditions dominate
while in the opposite regime where the action is not much larger than Planck's
constant, the strong conditions dominate.Comment: 8 pages, 2 eps figure
Spin transport in a unitary Fermi gas close to the BCS transition
We consider spin transport in a two-component ultracold Fermi gas with
attractive interspecies interactions close to the BCS pairing transition. In
particular, we consider the spin-transport relaxation rate and the
spin-diffusion constant. Upon approaching the transition, the scattering
amplitude is enhanced by pairing fluctuations. However, as the system
approaches the transition, the spectral weight for excitations close to the
Fermi level is decreased by the formation of a pseudogap. To study the
consequence of these two competing effects, we determine the spin-transport
relaxation rate and the spin-diffusion constant using both a Boltzmann approach
and a diagrammatic approach. The former ignores pseudogap physics and finite
lifetime effects. In the latter, we incorporate the full pseudogap physics and
lifetime effects, but we ignore vertex corrections, so that we effectively
calculate single-particle relaxation rates instead of transport relaxation
rates. We find that there is qualitative agreement between these two approaches
although the results for the transport coefficients differ quantitatively.Comment: 9 pages, 10 figure
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