3,640 research outputs found
The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory
Let X be a noetherian scheme of finite Krull dimension, having 2 invertible
in its ring of regular functions, an ample family of line bundles, and a global
bound on the virtual mod-2 cohomological dimensions of its residue fields. We
prove that the comparison map from the hermitian K-theory of X to the homotopy
fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence
in general, and an integral equivalence when X has no formally real residue
field. We also show that the comparison map between the higher
Grothendieck-Witt (hermitian K-) theory of X and its \'etale version is an
isomorphism on homotopy groups in the same range as for the Quillen-Lichtenbaum
conjecture in K-theory. Applications compute higher Grothendieck-Witt groups of
complex algebraic varieties and rings of 2-integers in number fields, and hence
values of Dedekind zeta-functions.Comment: 17 pages, to appear in Adv. Mat
Quantification of coarse-graining error in Langevin and overdamped Langevin dynamics
In molecular dynamics and sampling of high dimensional Gibbs measures
coarse-graining is an important technique to reduce the dimensionality of the
problem. We will study and quantify the coarse-graining error between the
coarse-grained dynamics and an effective dynamics. The effective dynamics is a
Markov process on the coarse-grained state space obtained by a closure
procedure from the coarse-grained coefficients. We obtain error estimates both
in relative entropy and Wasserstein distance, for both Langevin and overdamped
Langevin dynamics. The approach allows for vectorial coarse-graining maps.
Hereby, the quality of the chosen coarse-graining is measured by certain
functional inequalities encoding the scale separation of the Gibbs measure. The
method is based on error estimates between solutions of (kinetic) Fokker-Planck
equations in terms of large-deviation rate functionals
Parametric forcing approach to rough-wall turbulent channel flow
The effects of rough surfaces on turbulent channel flow are modelled by an extra force term in the Navier–Stokes equations. This force term contains two parameters, related to the density and the height of the roughness elements, and a shape function, which regulates the influence of the force term with respect to the distance from the channel wall. This permits a more flexible specification of a rough surface than a single parameter such as the equivalent sand grain roughness. The effects of the roughness force term on turbulent channel flow have been investigated for a large number of parameter combinations and several shape functions by direct numerical simulations. It is possible to cover the full spectrum of rough flows ranging from hydraulically smooth through transitionally rough to fully rough cases. By using different parameter combinations and shape functions, it is possible to match the effects of different types of rough surfaces. Mean flow and standard turbulence statistics have been used to compare the results to recent experimental and numerical studies and a good qualitative agreement has been found. Outer scaling is preserved for the streamwise velocity for both the mean profile as well as its mean square fluctuations in all but extremely rough cases. The structure of the turbulent flow shows a trend towards more isotropic turbulent states within the roughness layer. In extremely rough cases, spanwise structures emerge near the wall and the turbulent state resembles a mixing layer. A direct comparison with the study of Ashrafian, Andersson & Manhart (Intl J. Heat Fluid Flow, vol. 25, 2004, pp. 373–383) shows a good quantitative agreement of the mean flow and Reynolds stresses everywhere except in the immediate vicinity of the rough wall. The proposed roughness force term may be of benefit as a wall model for direct and large-eddy numerical simulations in cases where the exact details of the flow over a rough wall can be neglecte
Integral analysis of laminar indirect free convection boundary layers with weak blowing for Schmidt no. ~ 1
Laminar natural convection at unity Schmidt number over a horizontal surface
with a weak normal velocity at the wall is studied using an integral analysis.
To characterise the strength of the blowing, we define a non-dimensional
parameter called the blowing parameter. After benchmarking with the no blowing
case, the effect of the blowing parameter on boundary layer thickness, velocity
and concentration profiles is obtained. Weak blowing is seen to increase the
wall shear stress. For blowing parameters greater than unity, the diffusional
flux at the wall becomes negligible and the flux is almost entirely due to the
blowing.Comment: 10 pages, published in International Communications in heat and mass
transfer,Vol31,No8, 2004, pp 1199 -120
Formation of Kuiper Belt Binaries
The discovery that a substantial fraction of Kuiper Belt objects (KBOs)
exists in binaries with wide separations and roughly equal masses, has
motivated a variety of new theories explaining their formation. Goldreich et
al. (2002) proposed two formation scenarios: In the first, a transient binary
is formed, which becomes bound with the aid of dynamical friction from the sea
of small bodies (L^2s mechanism); in the second, a binary is formed by three
body gravitational deflection (L^3 mechanism). Here, we accurately calculate
the L^2s and L^3 formation rates for sub-Hill velocities. While the L^2s
formation rate is close to previous order of magnitude estimates, the L^3
formation rate is about a factor of 4 smaller. For sub-Hill KBO velocities (v
<< v_H) the ratio of the L^3 to the L^2s formation rate is 0.05 (v/v_H)
independent of the small bodies' velocity dispersion, their surface density or
their mutual collisions. For Super-Hill velocities (v >> v_H) the L^3 mechanism
dominates over the L^2s mechanism. Binary formation via the L^3 mechanism
competes with binary destruction by passing bodies. Given sufficient time, a
statistical equilibrium abundance of binaries forms. We show that the frequency
of long-lived transient binaries drops exponentially with the system's lifetime
and that such transient binaries are not important for binary formation via the
L^3 mechanism, contrary to Lee et al. (2007). For the L^2s mechanism we find
that the typical time, transient binaries must last, to form Kuiper Belt
binaries (KBBs) for a given strength of dynamical friction, D, increases only
logarithmically with D. Longevity of transient binaries only becomes important
for very weak dynamical friction (i.e. D \lesssim 0.002) and is most likely not
crucial for KBB formation.Comment: 20 pages, 3 figures, Accepted for publication in ApJ, correction of
minor typo
Barriers of the McKean–Vlasov energy via a mountain pass theorem in the space of probability measures
We show that the empirical process associated with a system of weakly interacting diffusion processes exhibits a form of noise-induced metastability. The result is based on an analysis of the associated McKean–Vlasov free energy, which, for suitable attractive interaction potentials, has at least two distinct global minimisers at the critical parameter value . On the torus, one of these states is the spatially homogeneous constant state, and the other is a clustered state. We show that a third critical point exists at this value. As a result, we obtain that the probability of transition of the empirical process from the constant state scales like , with Δ the energy gap at . The proof is based on a version of the mountain pass theorem for lower semicontinuous and λ-geodesically convex functionals on the space of probability measures equipped with the 2-Wasserstein metric, where M is a complete, connected, and smooth Riemannian manifold
Analytic study of the urn model for separation of sand
We present an analytic study of the urn model for separation of sand recently
introduced by Lipowski and Droz (Phys. Rev. E 65, 031307 (2002)). We solve
analytically the master equation and the first-passage problem. The analytic
results confirm the numerical results obtained by Lipowski and Droz. We find
that the stationary probability distribution and the shortest one among the
characteristic times are governed by the same free energy. We also analytically
derive the form of the critical probability distribution on the critical line,
which supports their results obtained by numerically calculating Binder
cumulants (cond-mat/0201472).Comment: 6 pages including 3 figures, RevTe
On a Class of Nonlocal Continuity Equations on Graphs
Motivated by applications in data science, we study partial differential
equations on graphs. By a classical fixed-point argument, we show existence and
uniqueness of solutions to a class of nonlocal continuity equations on graphs.
We consider general interpolation functions, which give rise to a variety of
different dynamics, e.g., the nonlocal interaction dynamics coming from a
solution-dependent velocity field. Our analysis reveals structural differences
with the more standard Euclidean space, as some analogous properties rely on
the interpolation chosen
Noise activated granular dynamics
We study the behavior of two particles moving in a bistable potential,
colliding inelastically with each other and driven by a stochastic heat bath.
The system has the tendency to clusterize, placing the particles in the same
well at low drivings, and to fill all of the available space at high
temperatures. We show that the hopping over the potential barrier occurs
following the Arrhenius rate, where the heat bath temperature is replaced by
the granular temperature. Moreover, within the clusterized ``phase'' one
encounters two different scenarios: for moderate inelasticity, the jumps from
one well to the other involve one particle at a time, whereas for strong
inelasticity the two particles hop simultaneously.Comment: RevTex4, 4 pages, 4 eps figures, Minor revisio
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