254,488 research outputs found
Convergence of the Poincare Constant
The Poincare constant R(Y) of a random variable Y relates the L2 norm of a
function g and its derivative g'. Since R(Y) - Var(Y) is positive, with
equality if and only if Y is normal, it can be seen as a distance from the
normal distribution. In this paper we establish a best possible rate of
convergence of this distance in the Central Limit Theorem. Furthermore, we show
that R(Y) is finite for discrete mixtures of normals, allowing us to add rates
to the proof of the Central Limit Theorem in the sense of relative entropy.Comment: 11 page
Molecular Dynamics Study of Orientational Cooperativity in Water
Recent experiments on liquid water show collective dipole orientation
fluctuations dramatically slower then expected (with relaxation time 50 ns)
[D. P. Shelton, Phys. Rev. B {\bf 72}, 020201(R) (2005)]. Molecular dynamics
simulations of SPC/E water show large vortex-like structure of dipole field at
ambient conditions surviving over 300 ps [J. Higo at al. PNAS, {\bf 98} 5961
(2001)]. Both results disagree with previous results on water dipoles in
similar conditions, for which autocorrelation times are a few ps. Motivated by
these recent results, we study the water dipole reorientation using molecular
dynamics simulations in bulk SPC/E water for temperatures ranging from ambient
300 K down to the deep supercooled region of the phase diagram at 210 K. First,
we calculate the dipole autocorrelation function and find that our simulations
are well-described by a stretched exponential decay, from which we calculate
the {\it orientational autocorrelation time} . Second, we define a
second characteristic time, namely the time required for the randomization of
molecular dipole orientation, the {\it self-dipole randomization time}
, which is an upper limit on ; we find that
. Third, to check if there are correlated domains
of dipoles in water which have large relaxation times compared to the
individual dipoles, we calculate the randomization time of the
site-dipole field, the net dipole moment formed by a set of molecules belonging
to a box of edge . We find that the {\it site-dipole randomization
time} for \AA, i.e.
it is shorter than the same quantity calculated for the self-dipole. Finally,
we find that the orientational correlation length is short even at low .Comment: 25 Pages, 10 figure
The generalized KP hierarchy
We propose one possible generalization of the KP hierarchy, which possesses
multi bi--hamiltonian structures, and can be viewed as several KP hierarchies
coupled together.Comment: 12
The short-time critical behaviour of the Ginzburg-Landau model with long-range interaction
The renormalisation group approach is applied to the study of the short-time
critical behaviour of the -dimensional Ginzburg-Landau model with long-range
interaction of the form in momentum space. Firstly the
system is quenched from a high temperature to the critical temperature and then
relaxes to equilibrium within the model A dynamics. The asymptotic scaling laws
and the initial slip exponents and of the order
parameter and the response function respectively, are calculated to the second
order in .Comment: 18 pages, 4 figures, 1 tabl
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