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1.
The vicious random walker problem on a line is studied in the limit of a large number of walkers. The multidimensional integral representing the probability that the p walkers will survive a time t (denoted P
t
(p)
) is shown to be analogous to the partition function of a particular one-component Coulomb gas. By assuming the existence of the thermodynamic limit for the Coulomb gas, one can deduce asymptotic formulas for P
t
(p)
in the large- p, large- t limit. A straightforward analysis gives rigorous asymptotic formulas for the probability that after a time t the walkers are in their initial configuration (this event is termed a reunion). Consequently, asymptotic formulas for the conditional probability of a reunion, given that all walkers survive, are derived. Also, an asymptotic formula for the conditional probability density that any walker will arrive at a particular point in time t, given that all p walkers survive, is calculated in the limit tp. 相似文献
2.
In a position-space renormalization group (PSRG) approach to percolation one calculates the probability R(p,b) that a finite lattice of linear size b percolates, where p is the occupation probability of a site or bond. A sequence of percolation thresholds p
c
(b) is then estimated from R(p
c
, b)=p
c
(b) and extrapolated to the limit b to obtain p
c
= p
c
(). Recently, it was shown that for a certain spanning rule and boundary condition, R(p
c
, )=R
c
is universal, and since p
c
is not universal, the validity of PSRG approaches was questioned. We suggest that the equation R(p
c
, b)=, where is any number in (0,1), provides a sequence of p
c
(b)'s that always converges to p
c
as b. Thus, there is an envelope from any point inside of which one can converge to p
c
. However, the convergence is optimal if = R
c
. By calculating the fractal dimension of the sample-spanning cluster at p
c
, we show that the same is true about any critical exponent of percolation that is calculated by a PSRG method. Thus PSRG methods are still a useful tool for investigating percolation properties of disordered systems. 相似文献
3.
The fibre of the extension of the frame-bundle of a space-time over a b-boundary point p is a homogeneous space / G
p
. It is shown that G
p
can be found by a construction like that for a holonomy group, and that it contains a subgroup determined by the Riemann tensor. Near a curvature singularity one would expect G
p
= 相似文献
4.
Using the algebraic properties of Poisson brackets, we extend the three-dimensional brackets (for a single free particle) to conform to the demands of special relativity. This yields, in an essentially unique way, the manifestly covariant extension [ x
, p
v]= + p
p
v/ m
2
c
2. Position and time then become fully dynamical variables expressible in terms of the canonical conjugate q
i and p
i and the time parameter as x
i = q
i + p
i( q · p)/ m
2c 2 and t = + E( q · p)/ m
2c 4. In the quantized version, the length associated with a particle of mass m is shown to be an integral multiple of the Compton wavelength
C = /mc. 相似文献
5.
We discuss the spectral function of a single O hole generated in a two-dimensional CuO 2-lattice at half-filling. The latter constitutes the most important structural element of high- T
c
superconducting materials. The system is described by the so-called extended Hubbard or Emery model. Strong electronic correlations which are incorporated in the model prevent the usual evaluation of Green functions based on Wick's theorem and using diagram techniques. For that reason we apply a new cumulant approach to dynamical correlation functions introduced recently. As a result we find that the local one-O hole excitation spectrum has two structured absorption regions around the bare O energy
p
and around
p
+ due to charge fluctuations of Cu holes. Here is the bare charge transfer gap. The width of the absorption regime around
p
is of the order of several times t
pd
2
/, where t
pd
is the hopping integral between Cu and O holes. 相似文献
6.
In the present paper, we discuss spectral properties of a periodic Schrödinger operator which is perturbed by randomly distributed impurities; such operators occur as simple models for crystals (or semi-conductors) with impurities. While the spectrum itself is independent of the concentration p of impurities, for 0< p<1, we focus our attention on the limiting behavior of the integrated density of states
p
of the random Schrödinger operator, inside a spectral gap of the periodic operator, as p0. Denoting by U
0 the set of eigenvalues (in the gap) of the reference problem having precisely one impurity (located at the origin, say), we show that the integrated density of states concentrates around the points of U
0, in the sense that
p
( U
) is of order p, for any fixed -neighborhood U
of U
0, while
p
( K) C· p
2, for any compact subset K of the gap which does not intersect U
.Research partially supported by Deutsche Forschungsgemeinschaft 相似文献
7.
The behaviour of the numbers n
s of clusters with s sites each in the case of a chain of N sites is studied for free and cyclic boundary conditions. Explicit expressions for the n
s, which differ from (1– p) 2
p
s= q
2
p
s for the infinite lattice, are given. Also the total number G( p) of clusters and the mean cluster size S( p) are calculated. In the thermodynamic limit the correction terms are found to be of order 1/ N. An investigation of the scaling behaviour shows that the scaling of n
s is described by two independent variables in contrast to G( p) and S( p) which require only one variable. 相似文献
8.
The central production of 0, f
2 and
3
0
mesons is observed for the first time in processes which are originated by +
p reactions proceeding via baryon exchange mechanism. The data come from the CERN WA56 experiment designed to separate the baryon exchange reactions in +
p-collisions at 20 GeV/c. We report on the measured integral and differential cross sections and also give the density matrix elements of the meson resonances observed. 相似文献
9.
In classical logic (Boolean algebras) probability systems involving correlations are fully characterized by the system of generalized Bell inequalities. On the other hand, probability systems with pairwise correlations on orthomodular lattices (OML) representing quantum logics are so general that the only inequalities that hold universally are the trivial inequalities 0 p
i1, 0 p
ijmin { p
i, p
j}. In this paper it is shown that every correlation sequence p=(p 1,..., p
n,..., P
ij,...) satisfying the above inequalities can be represented by a probability measure on an orthomodular lattice L admitting a full set of {0,1}-valued probability measures with the additional property that is L ortho-Arguesian. 相似文献
10.
A disordered n-vector model with p spin interactions previously introduced is studied for the quenched case by means of the replica method and a generalized Parisi theory. We present formal solutions for general n and p and then study the case p . The high-temperature solution is stable at all temperatures and there is only one phase transition at a temperature T
g. Only longitudinal lowtemperature solutions are possible. There is one spin-glass solution, and it is stable for all T
g. The phase transition atT
g is of first order and displays a jump discontinuity in the order parametersq
j
(L)
andd. The spin-glass free energy is temperature dependent forn > 1 while it is constant whenn = 1. 相似文献
11.
We study the two-point correlation function of a freely decaying scalar in Kraichnan's model of advection by a Gaussian random velocity field that is stationary and white noise in time, but fractional Brownian in space with roughness exponent 0< <2, appropriate to the inertial-convective range of the scalar. We find all self-similar solutions by transforming the scaling equation to Kummer's equation. It is shown that only those scaling solutions with scalar energy decay exponent a( d/ )+1 are statistically realizable, where d is space dimension and =2– . An infinite sequence of invariants J
p, p=0, 1, 2,..., is pointed out, where J
0 is Corrsin's integral invariant but the higher invariants appear to be new. We show that at least one of the invariants J
0 or J
1 must be nonzero (possibly infinite) for realizable initial data. Initial datum with a finite, nonzero invariant—the first being J
p—converges at long times to a scaling solution
p with a=( d/ )+ p, p=0, 1. The latter belongs to an exceptional series of self-similar solutions with stretched-exponential decay in space. However, the domain of attraction includes many initial data with power-law decay. When the initial datum has all invariants zero or infinite and also it exhibits power-law decay, then the solution converges at long times to a nonexceptional scaling solution with the same power-law decay. These results support a picture of a two-scale decay with breakdown of self-similarity for a range of exponents ( d+ )/ < a<( d+2)/ , analogous to what has recently been found in the decay of Burgers turbulence. 相似文献
12.
An excess term exists when using hermitian form of Cartesian momentum p
i
(i = 1, 2, 3) in usual kinetic energy 1/(2) p
i
2 for a particle moving on the 2D surface, and the correct kinetic energy turns to be 1/(2) 1/f ip i f i p i where the f
i are dummy factors in classical mechanics and nontrivial in quantum mechanics. In this paper, the explicit form of the dummy functions f
i is given for some surfaces of nonspherical topology, such as toroidal surface, paraboloid of revolution, the hyperboloid of revolution of two sheets, and the hyperboloid of revolution of one sheets. 相似文献
13.
We consider percolation on the sites of a graph G, e.g., a regular d-dimensional lattice. All sites of G are occupied (vacant) with probability p (respectively, q=1–p), independently of each other. W denotes the cluster of occupied sites containing a fixed site (which will usually be taken to be the origin) and W the cardinality of W. The percolation probability is the probability that # W=, i.e., (p)=P
p{# W=}. Some critical values of p,p
H and p
T, are defined, respectively, as the smallest value of p for which (p)> 0, and for which the expectation of # W is infinite. Formally, p
H=inf { p(p)>0} and p
T=inf{ p E
p{# W}=}. We show for fairly general graphs Gthat if p
T, thenP
P{#W n} decreases exponentially inn. For the special casesG =G
0= the simple quadratic lattice andG
1= the graph which corresponds to bond-percolation on 2, we obtain upper and lower bounds for(p) of the formC¦p¦-P
H¦, and bounds forEp{#W} of the formC¦p–p
H¦–. We also investigate smoothness properties of (p)=E
p{number of clusters per site} =E
p {(#W)–1; (#W) 1}. This function was introduced by Sykes and Essam, who assumed that (·) has exactly one singularity, namely, atp=p
H. For the graphsG
0 andG
1, (i.e., site or bond percolation on 2) we show that (p) is analytic atp p
H and has two continuous derivatives atp=p
H. The emphasis is on rigorous proofs.Research supported by the NSF through a grant to Cornell University. 相似文献
14.
A study is made of the spreading of damage in the random but deterministic Kauffman model on the square lattice with the spreading from one edge of the lattice. The critical value of the parameter p
c above which the system becomes chaotic is found to be p
c0.298. The possibility of suppression of the chaotic phase by noise is also studied. It is found that for pp
c, an extremely large noise level g>0.99 is required, if possible at all.On leave from Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, China. 相似文献
15.
A study is made of the gap exponents for percolation processes with the triangle condition in the subcritical region. It is show that the gaps are given by
t
=2 for t=2, 3,. Scaling theory predicts that P
p
(¦ C
0¦ S(p))–( p
c
– p) and E
p
(1/¦ C
0¦; ¦ C
0¦ S(p))–( p
c
– p) 3, where S(p) is the typical cluster size. It is found that ( p
c
– p) P
p
(| C
0S(p)
1–)( p
c
– p) 1–2 and ( p
c
– p) 3E
p
(1/| C
0|;| C
0| S(p)
1–))( p
c
– p) 3–4. 相似文献
16.
The exchange part of the third cluster integral can be divided into two parts: b
3(exch-1), which arises from the exchange of two particles, and b
3(exch-2), which arises from the cyclic exchange of all three particles. The first few terms of b
3(exch-1) are calculated by arguing that b
3(exch-1) =-[9a 3/(4 3)]b 2(exch)[1 + O(/a)], where b
2(exch) is the exchange second cluster integral, is the thermal de Broglie wavelength, and a is the hardsphere diameter. The first three terms of b
3(exch-2) are calculated by writing it in path integral form and expanding about the shortest path. 相似文献
17.
The dynamics of a classical system, specified by a Hamiltonian H
0( q,p) and a number of constraint functions
k
( q,p) in Euclidean phase space ( q,p), is described by means of a new Hamiltonian H( q,p), which is an invariant of the (closed) Poisson-bracket Lie algebra ( H
0,
k
). Fixed values of
k
(not necessarily zero) are given by initial conditions, and they are conserved along the trajectories determined by the Hamilton equations. The quantization is performed by the standard Heisenberg commutation relations in the embedding phase space, while the constraint functions are put in correspondence with constraint operators
which generate the Lie algebra of quantum commutators. A subset of commuting constraint operators may be chosen to have certain values in the initial state; and as soon as the Hamiltonian operator is an invariant of the Lie algebra, these conditions are maintained permanently. Simple examples are presented. Systems with both Bose and Fermi degrees of freedom (and constraints) can be treated universally. 相似文献
18.
The field-driven Kawasaki model with a fraction p admixture of Glauber dynamics is studied by computer simulation: p=0 corresponds to the order-parameter-onserving driven diffusive system, while p=1 is the equilibrium Ising model. For p=0.1 our best estimates of critical exponents based on a system of size 4096×128 are 0.22, RS0.45, and v
v
1. These exponents differ from both the values predicted by a field-theoretic method for p=0 and those of the equilibrium Ising model. Anisotropic finite-size scaling analyses are carried out, both for subsystems of the large system and for fully periodic systems. The results of the latter, however, are inconsistent, probably due to the complexity of the size effects. This leaves open the possibility that we are in a crossover regime from p=0 to p0 and that our critical exponents are effective ones. For p=0 our results are consistent with the predictions v
> v
. 相似文献
19.
Recently there are several evidences of the increase of the total cross section σtot to be log 2s consistent with the Froissart unitarity bound, and the COMPETE collaborations in the PDG have further assumed σtotBlog 2( s/ s0) to extend its universal rise with a common value of B for all the hadronic scatterings. However, there is no rigorous proof yet based only on QCD. Therefore, it is worthwhile to prove this universal rise of σtot even empirically. In this Letter we attempt to obtain the value of B for πp scattering, Bπp, with reasonable accuracy by taking into account the rich πp data in all the energy regions. We use the finite-energy sum rule (FESR) expressed in terms of the πp scattering data in the low and intermediate energies as a constraint between high-energy parameters. We then have searched for the simultaneous best fit to the σtot and ρ ratios, the ratios of the real to imaginary parts of the forward scattering amplitudes. The lower energy data are included in the integral of FESR, the more precisely determined is the non-leading term such as log s, and then helps to determine the leading terms like log 2s. We have derived the value of Bπp as Bπp=0.311±0.044 mb. This value is to be compared with the value of B for scattering, Bpp, in our previous analysis [M. Ishida, K. Igi, Eur. Phys. J. C 52 (2007) 357], Bpp=0.289±0.023 mb. Thus, our result appears to support the universality hypothesis. 相似文献
20.
The average number n
s
( p) of percolation clusters with s sites is calculated for the triangular lattice using real-space renormalization. For s up to 2,000 the whole range of concentrations p was analyzed; n
s
varied over sixty decades. We found log n
s
– s for p below p
c
=0.5, and n
s
s
–, =2.35 at p=p
c
. For small p one has n
s
( p · )
s
. Near p
c
we found the scaling from n
s
s
–
f(( p
c
–p) · s
) with =0.53. Presumably for the first time renormalization methods were used to calculate percolation properties not only near p
c
but also far away from the critical point.Sonderforschungsbereich 125 Aachen-Jülich-Köln 相似文献
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