162 research outputs found
Ramsey Properties of Countably Infinite Partial Orderings
A partial ordering β is chain-Ramsey if, for every natural number n and every coloring of the n-element chains from β in finitely many colors, there is a monochromatic subordering β isomorphic to β. Chain-Ramsey partial orderings stratify naturally into levels. We show that a countably infinite partial ordering with finite levels is chain-Ramsey if and only if it is biembeddable with one of a canonical collection of examples constructed from certain edge-Ramsey families of finite bipartite graphs. A similar analysis applies to a large class of countably infinite partial orderings with infinite levels
Motion of vortices in inhomogeneous Bose-Einstein condensates
We derive a general and exact equation of motion for a quantised vortex in an
inhomogeneous two-dimensional Bose-Einstein condensate. This equation expresses
the velocity of a vortex as a sum of local ambient density and phase gradients
in the vicinity of the vortex. We perform Gross-Pitaevskii simulations of
single vortex dynamics in both harmonic and hard-walled disk-shaped traps, and
find excellent agreement in both cases with our analytical prediction. The
simulations reveal that, in a harmonic trap, the main contribution to the
vortex velocity is an induced ambient phase gradient, a finding that
contradicts the commonly quoted result that the local density gradient is the
only relevant effect in this scenario. We use our analytical vortex velocity
formula to derive a point-vortex model that accounts for both density and phase
contributions to the vortex velocity, suitable for use in inhomogeneous
condensates. Although good agreement is obtained between Gross-Pitaevskii and
point-vortex simulations for specific few-vortex configurations, the effects of
nonuniform condensate density are in general highly nontrivial, and are thus
difficult to efficiently and accurately model using a simplified point-vortex
description.Comment: 13 pages, 8 figure
A Basis Theorem for Perfect Sets
We show that if there is a nonconstructible real, then every perfect set has a nonconstructible element, answering a question of K. Prikry. This is a specific instance of a more general theorem giving a sufficient condition on a pair M β N of models of set theory implying that every perfect set in N has an element in N which is not in M
Decaying quantum turbulence in a two-dimensional Bose-Einstein condensate at finite temperature
We numerically model decaying quantum turbulence in two-dimensional
disk-shaped Bose-Einstein condensates, and investigate the effects of finite
temperature on the turbulent dynamics. We prepare initial states with a range
of condensate temperatures, and imprint equal numbers of vortices and
antivortices at randomly chosen positions throughout the fluid. The initial
states are then subjected to unitary time-evolution within the c-field
methodology. For the lowest condensate temperatures, the results of the zero
temperature Gross-Pitaevskii theory are reproduced, whereby vortex evaporative
heating leads to the formation of Onsager vortex clusters characterised by a
negative absolute vortex temperature. At higher condensate temperatures the
dissipative effects due to vortex-phonon interactions tend to drive the vortex
gas towards positive vortex temperatures dominated by the presence of vortex
dipoles. We associate these two behaviours with the system evolving toward an
anomalous non-thermal fixed point, or a Gaussian thermal fixed point,
respectively.Comment: 20 pages, 6 figures, SciPost format. Updated version includes a new
section with further analysis and an additional figur
Vortex thermometry for turbulent two-dimensional fluids
We introduce a new method of statistical analysis to characterize the dynamics of turbulent fluids in two dimensions. We establish that, in equilibrium, the vortex distributions can be uniquely connected to the temperature of the vortex gas, and we apply this vortex thermometry to characterize simulations of decaying superfluid turbulence. We confirm the hypothesis of vortex evaporative heating leading to Onsager vortices proposed in Phys. Rev. Lett. 113, 165302 (2014)PRLTAO0031-900710.1103/PhysRevLett.113.165302, and we find previously unidentified vortex power-law distributions that emerge from the dynamics
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