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1.
We consider the narrow escape problem in two-dimensional Riemannian manifolds (with a metric g) with corners and cusps, in an annulus, and on a sphere. Specifically, we calculate the mean time it takes a Brownian particle
diffusing in a domain Ω to reach an absorbing window when the ratio
between the absorbing window and the otherwise reflecting boundary is small. If the boundary is smooth, as in the cases of
the annulus and the sphere, the leading term in the expansion is the same as that given in part I of the present series of
papers, however, when it is not smooth, the leading order term is different. If the absorbing window is located at a corner
of angle α, then
if near a cusp, then
grows algebraically, rather than logarithmically. Thus, in the domain bounded between two tangent circles, the expected lifetime
is
, where
is the ratio of the radii. For the smooth boundary case, we calculate the next term of the expansion for the annulus and
the sphere. It can also be evaluated for domains that can be mapped conformally onto an annulus. This term is needed in real
life applications, such as trafficking of receptors on neuronal spines, because
is not necessarily large, even when
is small. In these two problems there are additional parameters that can be small, such as the ratio δ of the radii of the
annulus. The contributions of these parameters to the expansion of the mean escape time are also logarithmic. In the case
of the annulus the mean escape time is
. 相似文献
2.
We consider Brownian motion in a circular disk Ω, whose boundary
is reflecting, except for a small arc,
, which is absorbing. As
decreases to zero the mean time to absorption in
, denoted
, becomes infinite. The narrow escape problem is to find an asymptotic expansion of
for
. We find the first two terms in the expansion and an estimate of the error. The results are extended in a straightforward
manner to planar domains and two-dimensional Riemannian manifolds that can be mapped conformally onto the disk. Our results
improve the previously derived expansion for a general domain,
(
is the diffusion coefficient) in the case of a circular disk. We find that the mean first passage time from the center of
the disk is
. The second term in the expansion is needed in real life applications, such as trafficking of receptors on neuronal spines,
because
is not necessarily large, even when ε is small. We also find the singular behavior of the probability flux profile into
at the endpoints of
, and find the value of the flux near the center of the window. 相似文献
3.
Yakov Sinai 《Journal of statistical physics》2005,121(5-6):779-803
In this paper we study the Fourier transform of the
-Navier-Stokes System without external forcing on the whole space R
3. The properties of solutions depend very much on the space in which the system is considered. In this paper we deal with
the space
of functions
where
and c (k) is bounded,
. We construct the power series which converges for small t and gives solutions of the system for bounded intervals of time. These solutions can be estimated at infinity (in k-space) by
. 相似文献
4.
Sergei B. Kuksin 《Journal of statistical physics》2006,122(1):101-114
We use the balance relations for the stationary in time solutions of the randomly forced 2D Navier-Stokes equations, found
in [10], to study these solutions further. We show that the vorticity ξ(t,x) of a stationary solution has a finite exponential moment, and that for any
the expectation of the integral of
over the level-set
, up to a constant factor equals the expectation of the integral of
over the same set. 相似文献
5.
Xuguang Lu 《Journal of statistical physics》2005,119(5-6):1027-1067
Long time behavior of solutions of the spatially homogeneous Boltzmann equation for Bose–Einstein particles is studied for hard potentials with certain cutoffs and for the hard sphere model. It is proved that in the cutoff case solutions as time
converge to the Bose–Einstein distribution in L1 topology with the weighted measure
, where
for temperature
and
for T<Tc. In particular this implies that if T<Tc then the solutions in the velocity regions
(with
) converge to a unique Dirac delta function (velocity concentration). All these convergence are uniform with respect to the cutoff constants. For the hard sphere model, these results hold also for weak or distributional solutions. Our methods are based on entropy inequalities and an observation that the convergence to Bose–Einstein distributions can be reduced to the convergence to Maxwell distributions. 相似文献
6.
We evaluate the virial coefficients Bk for
for hard spheres in dimensions
Virial coefficients with k even are found to be negative when
This provides strong evidence that the leading singularity for the virial series lies away from the positive real axis when
. Further analysis provides evidence that negative virial coefficients will be seen for some k > 10 for D = 4, and there is a distinct possibility that negative virial coefficients will also eventually occur for D = 3. 相似文献
7.
8.
For a Lie algebra with Lie bracket got by taking commutators in a nonunital associative algebra
, let
be the vector space of tensors over
equipped with the Itô Hopf algebra structure derived from the associative multiplication in
. It is shown that a necessary and sufficient condition that the double product integral
satisfy the quantum Yang–Baxter equation over
is that
satisfy the same equation over the unital associative algebra
got by adjoining a unit element to
. In particular, the first-order coefficient r1 of r[h] satisfies the classical Yang–Baxter equation. Using the fact that the multiplicative inverse of
is
where
is the inverse of
in
we construct a quantisation of an arbitrary quasitriangular Lie bialgebra structure on
in the unital associative subalgebra of
consisting of formal power series whose zero order coefficient lies in the space
of symmetric tensors. The deformation coproduct acts on
by conjugating the undeformed coproduct by
and the coboundary structure r of
is given by
where
is the flip.Mathematical Subject Classification (2000). 53D55, 17B62 相似文献
9.
It is shown that solvable mixed spin ladder models can be constructed from su(N) permutators. Heisenberg rung interactions appear as chemical potential terms in the Bethe Ansatz solution. Explicit examples given are a mixed spin-
spin-1 ladder, a mixed spin-
spin-
ladder and a spin-1 ladder with biquadratic interactions. 相似文献
10.
Ruqian Lu 《International Journal of Theoretical Physics》2005,44(9):1495-1530
We propose the concept of finite stop quantum automata (ftqa) based on Hilbert space and compare it with the finite state
quantum automata (fsqa) proposed by Moore and Crutchfield (Theoretical Computer Science 237(1–2), 2000, 275–306). The languages accepted by fsqa form a proper subset of the languages accepted by ftqa. In addition,
the fsqa form an infinite hierarchy of language inclusion with respect to the dimensionality of unitary matrices. We introduce
complex-valued acceptance degrees and two types of finite stop quantum automata based on them: the invariant ftqa (icftq)
and the variant ftqa (vcftq). The languages accepted by icftq form a proper subset of the languages accepted by vcftq. In
addition, the icftq form an infinite hierarchy of language inclusion with respect to the dimensionality of unitary matrices.
In this way, we establish two proper inclusion relations
(fsqa) ⊂
(ftqa) and
(icftq) ⊂
(vcftq), where the symbol
means languages, and two infinite language hierarchies
(fsqa) ⊂
(fsqa),
(icftq)
(icftq). 相似文献
11.
Let
be a Hilbert space with an inner product
. In Jajte, R., and Paszkiewicz, A. (1978, Vector measure on the closed subspaces of a Hilbert space, Studia Mathematica
63, 229–251), the
-measure on the logic of all orthogonal projections on
H was studied. We examine the
-measure on the hyperbolic logic of all J-projections on a Krein space.
PACS: 03.65.Ta, 03.65.Db, 03.65.Ca. 相似文献
12.
In their 1936 founding paper on quantum logic, Birkhoff and von Neumann postulated that the lattice describing the experimental
propositions concerning a quantum system is orthocomplemented. We prove that this postulate fails for the lattice
sep describing a compound system consisting of so called separated quantum systems. By separated we mean two systems prepared in different “rooms” of the lab, and before any interaction takes
place. In that case, the state of the compound system is necessarily a product state. As a consequence, Dirac’s superposition
principle fails, and therefore
sep cannot satisfy all Piron’s axioms. In previous works, assuming that
sep is orthocomplemented, it was argued that
sep is not orthomodular and fails to have the covering property. Here we prove that
sep cannot admit an orthocomplementation. Moreover, we propose a natural model for
sep which has the covering property.
PACS: 03.65.Ta, 03.65.Ca 相似文献
13.
Marcus Pivato 《Journal of statistical physics》2003,110(1-2):247-267
If
, and
is a finite (nonabelian) group, then
is a compact group; a multiplicative cellular automaton (MCA) is a continuous transformation
which commutes with all shift maps, and where nearby coordinates are combined using the multiplication operation of
. We characterize when MCA are group endomorphisms of
, and show that MCA on
inherit a natural structure theory from the structure of
. We apply this structure theory to compute the measurable entropy of MCA, and to study convergence of initial measures to Haar measure. 相似文献
14.
15.
Let
be a finite dimensional complex Lie algebra and
a Lie subalgebra equipped with the structure of a factorizable quasitriangular Lie bialgebra. Consider the Lie group Exp
with the Semenov-Tjan-Shansky Poisson bracket as a Poisson Lie manifold for the double Lie bialgebra
. Let
be an open domain parameterizing a neighborhood of the identity in Exp
by the exponential map. We present dynamical r-matrices with values in
over the Poisson Lie base manifold
.*This research is partially supported by the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany, the Excellency Center Group Theoretic Methods in the study of Algebraic Varieties of the Israel Science foundation, and by the RFBR grant no. 03-01-00593. 相似文献
16.
We study the (strong-) Gibbsian character on
of the law at time t of an infinite- imensional gradient Brownian diffusion, when the initial distribution is Gibbsian 相似文献
17.
It has been maintained that the physical content of a model of a system is completely contained in the C∗-algebra of quasi-local observables
that is associated with the system. The reason given for this is that the unitarily inequivalent representations of
are physically equivalent. But, this view is dubious for at least two reasons. First, it is not clear why the physical content
does not extend to the elements of the von Neumann algebras that are generated by representations of
. It is shown here that although the unitarily inequivalent representations of
are physically equivalent, the extended representations are not. Second, this view detracts from special global features
of physical systems such as temperature and chemical potential by effectively relegating them to the status of fixed parameters.
It is desirable to characterize such observables theoretically as elements of the algebra that is associated with a system
rather than as parameters, and thereby give a uniform treatment to all observables. This can be accomplished by going to larger
algebras. One such algebra is the universal enveloping von Neumann algebra, which is generated by the universal representation
of
; another is the direct integral of factor representations that are associated with the set of values of the global features.
Placing interpretive significance on the von Neumann algebras mentioned earlier sheds light on the significance of unitarily
inequivalent representations of
, and it serves to show the limitations of the notion of physical equivalence. 相似文献
18.
In this paper, assuming a linear change of the gravitational potential V in the universe, i.e.
, some consequences are obtained. 1. The Hubble red shift is explained by the potential difference between the considered galaxy long time ago and the observer at this epoch. 2. The anomalous acceleration a
P
from the spacecraft Pioneer 10 and 11 [1] is explained. 3. The deformations of the trajectories of planets are studied. It is shown that the planetary orbits are not axially symmetric and the angle from the perihelion to the aphelion is
, while the angle from the aphelion to the perihelion is
, where is the orbital period. There is no perihelion precession caused by the time dependent gravitational potential V. The quotient of two consecutive orbital periods 1 and 2 is equal to
This formula is tested for the pulsars B1885+09 and B1534+12, and the results are good.This revised version was published online in April 2005. The publishing date was inserted. 相似文献
19.
We study a family of holomorphic functions defined by infinite products of the form
(a, r real, ar > 0) which generalize Eulers definition since
. We obtain analogues of classical formulas (e.g. Gauss multiplication and complement formulas) for these functions a,r(s) 相似文献
20.
We consider a fixed quantum measurement performed over n identical copies of quantum states. Using a rigorous notion of distinguishability based on Shannon’s 12th theorem, we show that in the case of a single qubit, the number of distinguishable states is
, where (α1,α2) is the angle interval from which the states are chosen. In the general case of an N-dimensional Hilbert space and an area Ω of the domain on the unit sphere from which the states are chosen, the number of distinguishable states is
. The optimal distribution is uniform over the domain in Cartesian coordinates. 相似文献