744 research outputs found
On a flow of transformations of a Wiener space
In this paper, we define, via Fourier transform, an ergodic flow of
transformations of a Wiener space which preserves the law of the
Ornstein-Uhlenbeck process and which interpolates the iterations of a
transformation previously defined by Jeulin and Yor. Then, we give a more
explicit expression for this flow, and we construct from it a continuous
gaussian process indexed by R^2, such that all its restriction obtained by
fixing the first coordinate are Ornstein-Uhlenbeck processes
A stochastic derivation of the geodesic rule
We argue that the geodesic rule, for global defects, is a consequence of the
randomness of the values of the Goldstone field in each causally
connected volume. As these volumes collide and coalescence, evolves by
performing a random walk on the vacuum manifold . We derive a
Fokker-Planck equation that describes the continuum limit of this process. Its
fundamental solution is the heat kernel on , whose leading
asymptotic behavior establishes the geodesic rule.Comment: 12 pages, No figures. To be published in Int. Jour. Mod. Phys.
Three-dimensional flows in slowly-varying planar geometries
We consider laminar flow in channels constrained geometrically to remain
between two parallel planes; this geometry is typical of microchannels obtained
with a single step by current microfabrication techniques. For pressure-driven
Stokes flow in this geometry and assuming that the channel dimensions change
slowly in the streamwise direction, we show that the velocity component
perpendicular to the constraint plane cannot be zero unless the channel has
both constant curvature and constant cross-sectional width. This result implies
that it is, in principle, possible to design "planar mixers", i.e. passive
mixers for channels that are constrained to lie in a flat layer using only
streamwise variations of their in-plane dimensions. Numerical results are
presented for the case of a channel with sinusoidally varying width
Derivation of quantum work equalities using quantum Feynman-Kac formula
On the basis of a quantum mechanical analogue of the famous Feynman-Kac
formula and the Kolmogorov picture, we present a novel method to derive
nonequilibrium work equalities for isolated quantum systems, which include the
Jarzynski equality and Bochkov-Kuzovlev equality. Compared with previous
methods in the literature, our method shows higher similarity in form to that
deriving the classical fluctuation relations, which would give important
insight when exploring new quantum fluctuation relations.Comment: 5 page
Large Deviations in Stochastic Heat-Conduction Processes Provide a Gradient-Flow Structure for Heat Conduction
We consider three one-dimensional continuous-time Markov processes on a
lattice, each of which models the conduction of heat: the family of Brownian
Energy Processes with parameter , a Generalized Brownian Energy Process, and
the Kipnis-Marchioro-Presutti process. The hydrodynamic limit of each of these
three processes is a parabolic equation, the linear heat equation in the case
of the BEP and the KMP, and a nonlinear heat equation for the GBEP().
We prove the hydrodynamic limit rigorously for the BEP, and give a formal
derivation for the GBEP().
We then formally derive the pathwise large-deviation rate functional for the
empirical measure of the three processes. These rate functionals imply
gradient-flow structures for the limiting linear and nonlinear heat equations.
We contrast these gradient-flow structures with those for processes describing
the diffusion of mass, most importantly the class of Wasserstein gradient-flow
systems. The linear and nonlinear heat-equation gradient-flow structures are
each driven by entropy terms of the form ; they involve dissipation
or mobility terms of order for the linear heat equation, and a
nonlinear function of for the nonlinear heat equation.Comment: 29 page
Quantum Diffusion and Delocalization for Band Matrices with General Distribution
We consider Hermitian and symmetric random band matrices in
dimensions. The matrix elements , indexed by , are independent and their variances satisfy \sigma_{xy}^2:=\E
\abs{H_{xy}}^2 = W^{-d} f((x - y)/W) for some probability density . We
assume that the law of each matrix element is symmetric and exhibits
subexponential decay. We prove that the time evolution of a quantum particle
subject to the Hamiltonian is diffusive on time scales . We
also show that the localization length of the eigenvectors of is larger
than a factor times the band width . All results are uniform in
the size \abs{\Lambda} of the matrix. This extends our recent result
\cite{erdosknowles} to general band matrices. As another consequence of our
proof we show that, for a larger class of random matrices satisfying
for all , the largest eigenvalue of is bounded
with high probability by for any ,
where M \deq 1 / (\max_{x,y} \sigma_{xy}^2).Comment: Corrected typos and some inaccuracies in appendix
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