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1.
Dyson’s model is a one-dimensional system of Brownian motions with long-range repulsive forces acting between any pair of
particles with strength proportional to the inverse of distances with proportionality constant β/2. We give sufficient conditions for initial configurations so that Dyson’s model with β = 2 and an infinite number of particles is well defined in the sense that any multitime correlation function is given by
a determinant with a continuous kernel. The class of infinite-dimensional configurations satisfying our conditions is large
enough to study non-equilibrium dynamics. For example, we obtain the relaxation process starting from a configuration, in
which every point of
\mathbbZ{\mathbb{Z}} is occupied by one particle, to the stationary state, which is the determinantal point process with the sine kernel. 相似文献
2.
Makoto Katori 《Journal of statistical physics》2012,146(2):249-263
When the number of particles N is finite, the noncolliding Brownian motion (BM) and the noncolliding squared Bessel process with index ν>−1 (BESQ(ν)) are determinantal processes for arbitrary fixed initial configurations. In the present paper we prove that, if initial configurations
are distributed with orthogonal symmetry, they are Pfaffian processes in the sense that any multitime correlation functions
are expressed by Pfaffians. The 2×2 skew-symmetric matrix-valued correlation kernels of the Pfaffians processes are explicitly
obtained by the equivalence between the noncolliding BM and an appropriate dilatation of a time reversal of the temporally
inhomogeneous version of noncolliding BM with finite duration in which all particles start from the origin, Nδ
0, and by the equivalence between the noncolliding BESQ(ν) and that of the noncolliding squared generalized meander starting from Nδ
0. 相似文献
3.
We introduce and study a 2-parameter family of unitarily invariant probability measures on the space of infinite Hermitian
matrices. We show that the decomposition of a measure from this family on ergodic components is described by a determinantal
point process on the real line. The correlation kernel for this process is explicitly computed.
At certain values of parameters the kernel turns into the well-known sine kernel which describes the local correlation in
Circular and Gaussian Unitary Ensembles. Thus, the random point configuration of the sine process is interpreted as the random
set of “eigenvalues” of infinite Hermitian matrices distributed according to the corresponding measure.
Received: 22 January 2001 / Accepted: 30 May 2001 相似文献
4.
One-dimensional system of Brownian motions called Dyson’s model is the particle system with long-range repulsive forces acting
between any pair of particles, where the strength of force is β/2 times the inverse of particle distance. When β=2, it is realized as the Brownian motions in one dimension conditioned never to collide with each other. For any initial
configuration, it is proved that Dyson’s model with β=2 and N particles,
$\mbox {\boldmath $\mbox {\boldmath
, is determinantal in the sense that any multitime correlation function is given by a determinant with a continuous kernel.
The Airy function
(z){\rm Ai}(z)
is an entire function with zeros all located on the negative part of the real axis ℝ. We consider Dyson’s model with β=2 starting from the first N zeros of
Ai(z){\rm Ai}(z)
, 0>a
1>⋅⋅⋅>a
N
, N≥2. In order to properly control the effect of such initial confinement of particles in the negative region of ℝ, we put the
drift term to each Brownian motion, which increases in time as a parabolic function: Y
j
(t)=X
j
(t)+t
2/4+{d
1+∑
ℓ=1
N
(1/a
ℓ
)}t,1≤j≤N, where
d1=Ai¢(0)/Ai(0)d_{1}={\rm Ai}'(0)/{\rm Ai}(0)
. We show that, as the N→∞ limit of
$\mbox {\boldmath $\mbox {\boldmath
, we obtain an infinite particle system, which is the relaxation process from the configuration, in which every zero of
(z){\rm Ai}(z)
on the negative ℝ is occupied by one particle, to the stationary state
mAi\mu_{{\rm Ai}}
. The stationary state
mAi\mu_{{\rm Ai}}
is the determinantal point process with the Airy kernel, which is spatially inhomogeneous on ℝ and in which the Tracy-Widom
distribution describes the rightmost particle position. 相似文献
5.
We study a 3-parametric family of stochastic point processes on the one-dimensional lattice originated from a remarkable family
of representations of the infinite symmetric group. We prove that the correlation functions of the processes are given by
determinantal formulas with a certain kernel. The kernel can be expressed through the Gauss hypergeometric function; we call
it the hypergeometric kernel.
In a scaling limit our processes approximate the processes describing the decomposition of representations mentioned above
into irreducibles. As we showed in previous works, the correlation functions of these limit processes also have determinantal
form with so-called Whittaker kernel. We show that the scaling limit of the hypergeometric kernel is the Whittaker kernel.
integrable operator as defined by Its, Izergin, Korepin, and Slavnov. We argue that the hypergeometric kernel can be considered
as a kernel defining a ‘discrete integrable operator’.
We also show that the hypergeometric kernel degenerates for certain values of parameters to the Christoffel–Darboux kernel
for Meixner orthogonal polynomials. This fact is parallel to the degeneration of the Whittaker kernel to the Christoffel–Darboux
kernel for Laguerre polynomials.
Received: 22 September 1999 / Accepted: 23 November 1999 相似文献
6.
Makoto Katori 《Journal of statistical physics》2012,149(3):411-431
The O??Connell process is a softened version (a geometric lifting with a parameter a>0) of the noncolliding Brownian motion such that neighboring particles can change the order of positions in one dimension within the characteristic length a. This process is not determinantal. Under a special entrance law, however, Borodin and Corwin gave a Fredholm determinant expression for the expectation of an observable, which is a softening of an indicator of a particle position. We rewrite their integral kernel to a form similar to the correlation kernels of determinantal processes and show, if the number of particles is?N, the rank of the matrix of the Fredholm determinant is N. Then we give a representation for the quantity by using an N-particle system of complex Brownian motions (CBMs). The complex function, which gives the determinantal expression to the weight of CBM paths, is not entire, but in the combinatorial limit a??0 it becomes an entire function providing conformal martingales and the CBM representation for the noncolliding Brownian motion is recovered. 相似文献
7.
We study a generic class of inelastic soft sphere models with a binary collision rate g^ν that depends on the relative velocity g. This includes previously studied inelastic hard spheres (ν = 1) and inelastic Maxwell molecules (ν = 0). We develop a new asymptotic method for analyzing large deviations from Gaussian behavior for the velocity distribution function f(c). The framework is that of the spatially uniform nonlinear Boltzmann equation and special emphasis is put on the situation where the system is driven by white noise. Depending on the value of exponent ν, three different situations are reported. For ν < −2, the non-equilibrium steady state is a repelling fixed point of the dynamics. For ν > −2, it becomes an attractive fixed point, with velocity distributions f(c) having stretched exponential behavior at large c. The corresponding dominant behavior of f(c) is computed together with sub-leading corrections. In the marginally stable case ν = −2, the high energy tail of f(c) is of power law type and the associated exponents are calculated. Our analytical predictions are confronted with Monte Carlo simulations, with a remarkably good agreement. 相似文献
8.
A system of one-dimensional Brownian motions (BMs) conditioned never to collide with each other is realized as (i) Dyson’s BM model, which is a process of eigenvalues of hermitian matrix-valued diffusion process in the Gaussian unitary ensemble (GUE), and as (ii) the h-transform of absorbing BM in a Weyl chamber, where the harmonic function h is the product of differences of variables (the Vandermonde determinant). The Karlin–McGregor formula gives determinantal expression to the transition probability density of absorbing BM. We show from the Karlin–McGregor formula, if the initial state is in the eigenvalue distribution of GUE, the noncolliding BM is a determinantal process, in the sense that any multitime correlation function is given by a determinant specified by a matrix-kernel. By taking appropriate scaling limits, spatially homogeneous and inhomogeneous infinite determinantal processes are derived. We note that the determinantal processes related with noncolliding particle systems have a feature in common such that the matrix-kernels are expressed using spectral projections of appropriate effective Hamiltonians. On the common structure of matrix-kernels, continuity of processes in time is proved and general property of the determinantal processes is discussed. 相似文献
9.
Alexei Borodin Patrik L. Ferrari Michael Prähofer Tomohiro Sasamoto 《Journal of statistical physics》2007,129(5-6):1055-1080
We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process
(TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then
consider certain periodic initial conditions and determine the kernel in the scaling limit. This result has been announced
first in a letter by one of us (Sasamoto in J. Phys. A 38:L549–L556, 2005) and here we provide a self-contained derivation. Connections to last passage directed percolation and random matrices are
also briefly discussed. 相似文献
10.
Motivated by the problem of domino tilings of the Aztec diamond, a weighted particle system is defined on N lines, with line j containing j particles. The particles are restricted to lattice points from 0 to N, and particles on successive lines are subject to an interlacing constraint. It is shown that this particle system is exactly
solvable, to the extent that not only can the partition function be computed exactly, but so too can the marginal distributions.
These results in turn are used to give new derivations within the particle picture of a number of known fundamental properties
of the tiling problem, for example that the number of distinct configurations is 2
N(N+1)/2, and that there is a limit to the GUE minor process, which we show at the level of the joint PDFs. It is shown too that the
study of tilings of the half Aztec diamond—not known from earlier literature—also leads to an interlaced particle system,
now with successive lines 2n−1 and 2n (n=1,…,N/2−1) having n particles. Its exact solution allows for an analysis of the half Aztec diamond tilings analogous to that given for the Aztec
diamond tilings. 相似文献
11.
12.
YeBing Xing XiaoHong Zhou YuHu Zhang YingXiang Guo Long Ma XiangGuo Lei WenTao Guo M. Oshima Y. Toh M. Koizumi A. Osa Y. Hatsukawa FuRong Xu M. Sugawara 《中国科学G辑(英文版)》2008,51(8):1053-1071
High-spin states in 187Pt were studied via the 173Yb(18O, 4n) reaction. Rotational bands based on the νi13/2, ν7/2−[503], νi2
13/2νj, ν3/2−[512] and ν1/2−[521] configurations were observed, and interpreted within the framework of the cranked shell model. The TRS calculations
show that the νi13/2 band has an appreciable negative γ deformation, and the negative-parity bands tend to have a near prolate shape with small positive γ values. Experimental values of B(M1)/B(E2) ratios have been extracted and compared with theoretical values from the semi-classical D?nau and Frauendof approach,
strongly suggesting a low frequency πh9/2 alignment in the ν7/2−[503] band.
Supported by the National Natural Science Foundation of China (Grant Nos. 10475097 and 10505025) and the Chinese Academy of
Sciences 相似文献
13.
Otto?H?nninen Irene?Brüske-Hohlfeld Miranda?Loh Tobias?Stoeger Wolfgang?Kreyling Otmar?Schmid Annette?Peters 《Journal of nanoparticle research》2010,12(1):91-99
Several studies have reported laser printers as significant sources of nanosized particles (<0.1 μm). Laser printers are used
occupationally in office environments and by consumers in their homes. The current work combines existing epidemiological
and toxicological evidence on particle-related health effects, measuring doses as mass, particle number and surface area,
to estimate and compare the potential risks in occupational and consumer exposure scenarios related to the use of laser printers.
The daily uptake of laser printer particles was estimated based on measured particle size distributions and lung deposition
modelling. The obtained daily uptakes (particle mass 0.15–0.44 μg d−1; particle number 1.1–3.1 × 109 d−1) were estimated to correspond to 4–13 (mass) or 12–34 (number) deaths per million persons exposed on the basis of epidemiological
risk estimates for ambient particles. These risks are higher than the generally used definition of acceptable risk of 1 × 10−6, but substantially lower than the estimated risks due to ambient particles. Toxicological studies on ambient particles revealed
consistent values for lowest observed effect levels (LOELs) which were converted into equivalent daily uptakes using allometric
scaling. These LOEL uptakes were by a factor of about 330–1,000 (mass) and 1,000–2,500 (particle surface area) higher than
estimated uptakes from printers. This toxicological assessment would indicate no significant health risks due to printer particles.
Finally, our study suggests that particle number (not mass) and mass (not surface area) are the most conservative risk metrics
for the epidemiological and toxicological risks presented here, respectively. 相似文献
14.
We formalize a classification of pair interactions based on the convergence properties of the forces acting on particles as a function of system size. We do so by considering the behavior of the probability distribution function
(PDF) P(F) of the force field F in a particle distribution in the limit that the size of the system is taken to infinity at constant particle density, i.e.,
in the “usual” thermodynamic limit. For a pair interaction potential V(r) with V(r→∞)∼1/r
γ
defining a bounded pair force, we show that P(F) converges continuously to a well-defined and rapidly decreasing PDF if and only if the pair force is absolutely integrable, i.e., for γ>d−1, where d is the spatial dimension. We refer to this case as dynamically short-range, because the dominant contribution to the force on a typical particle in this limit arises from particles in a finite neighborhood
around it. For the dynamically long-range case, i.e., γ≤d−1, on the other hand, the dominant contribution to the force comes from the mean field due to the bulk, which becomes undefined
in this limit. We discuss also how, for γ≤d−1 (and notably, for the case of gravity, γ=d−2) P(F) may, in some cases, be defined in a weaker sense. This involves a regularization of the force summation which is generalization
of the procedure employed to define gravitational forces in an infinite static homogeneous universe. We explain that the relevant
classification in this context is, however, that which divides pair forces with γ>d−2 (or γ<d−2), for which the PDF of the difference in forces is defined (or not defined) in the infinite system limit, without any regularization. In the former case dynamics can, as
for the (marginal) case of gravity, be defined consistently in an infinite uniform system. 相似文献
15.
Starting with Berry's hypothesis for fixed energy waves
in a classically chaotic system, and casting it in a Green
function form, we derive wavefunction correlations and density
matrices for few or many particles. Universal features of fixed
energy (microcanonical) random wavefunction correlation functions
appear which reflect the emergence of the canonical ensemble as
N↦∞. This arises through a little known asymptotic limit
of Bessel functions. The Berry random wave hypothesis in many
dimensions may be viewed as an alternative approach to quantum
statistical mechanics, when extended to include constraints and
potentials. 相似文献
16.
We study the transition probabilities for the totally asymmetric simple exclusion process (TASEP) on the infinite integer
lattice with a finite, but arbitrary number of first and second class particles. Using the Bethe ansatz we present an explicit
expression of these quantities in terms of the Bethe wave function. In a next step it is proved rigorously that this expression
can be written in a compact determinantal form for the case where the order of the first and second class particles does not
change in time. An independent geometrical approach provides insight into these results and enables us to generalize the determinantal
solution to the multi-class TASEP. 相似文献
17.
R. F.S. Andrade S. T.R. Pinho 《The European Physical Journal B - Condensed Matter and Complex Systems》2006,50(1-2):33-37
The critical behavior of the Ising chain with
long-range ferromagnetic interactions decaying with distance
rα, 1<α<2, is investigated using a numerically
efficient transfer matrix (TM) method. Finite size approximations
to the infinite chain are considered, in which both the number of
spins and the number of interaction constants can be independently
increased. Systems with interactions between spins up to 18
sites apart and up to 2500 spins in the chain are considered. We
obtain data for the critical exponents ν associated with the
correlation length based on the Finite Range Scaling (FRS)
hypothesis. FRS expressions require the evaluation of derivatives
of the thermodynamical properties, which are calculated with the
help of analytical recurrence expressions obtained within the TM
framework. The Van den Broeck extrapolation procedure is applied
in order to estimate the convergence of the exponents. The TM
procedure reduces the dimension of the matrices and circumvents
several numerical matrix operations. 相似文献
18.
Eunghyun Lee 《Journal of statistical physics》2010,140(4):635-647
In this paper we give the distribution of the position of a particle in the asymmetric simple exclusion process (ASEP) with
the alternating initial condition. That is, we find ℙ(X
m
(t)≤x) where X
m
(t) is the position of the particle at time t which was at m=2k−1, k∈ℤ at t=0. As in the ASEP with step initial condition, there arises a new combinatorial identity for the alternating initial condition,
and this identity relates the integrand of the integral formula for ℙ(X
m
(t)≤x) to a determinantal form together with an extra product. 相似文献
19.
S. N. Migunov A. A. Volkov G. A. Komandin A. N. Lobanov B. P. Gorshunov Yu. I. Golovko V. M. Mukhortov Yu. I. Yuzyuk 《Technical Physics》2008,53(11):1485-1489
Using submillimeter and infrared spectroscopies, the reflectance R(ν) and transmittance T(ν) spectra of heteroepitaxial barium-strontium titanate films of different thicknesses on MgO substrates are taken for the
first time in the frequency range 10 < ν < 3000 cm−1. By modeling the experimental spectra by the Fresnel formulas for layered media, the spectra of complex permittivity ɛ*(ν)
= ɛ′(ν) + iɛ″(ν) of the films are determined. It is shown that when the film thicknesses decrease down to 10 nm, there appear tensile
stresses in the direction parallel to the substrate surface. As a result, the dielectric contribution of a low-frequency soft
mode becomes several times larger than before. 相似文献
20.
K. J. Falconer 《Communications in Mathematical Physics》1999,206(1):235-245
A Laplacian may be defined on self-similar fractal domains in terms of a suitable self-similar Dirichlet form, enabling discussion
of elliptic PDEs on such domains. In this context it is shown that that semilinear equations such as Δu+u
p
= 0, with zero Dirichlet boundary conditions, have non-trivial non-negative solutions if 0<ν≤ 2 and p>1, or if ν >2 and 1<p< (ν+ 2)/(ν− 2), where ν is the “intrinsic dimension” or “spectral dimension” of the system. Thus the intrinsic dimension
takes the r\^{o}le of the Euclidean dimension in the classical case in determining critical exponents of semilinear problems.
Received: 11 December 1998 / Accepted: 22 March 1999 相似文献