355 research outputs found
Extrapolation of power series by self-similar factor and root approximants
The problem of extrapolating the series in powers of small variables to the
region of large variables is addressed. Such a problem is typical of quantum
theory and statistical physics. A method of extrapolation is developed based on
self-similar factor and root approximants, suggested earlier by the authors. It
is shown that these approximants and their combinations can effectively
extrapolate power series to the region of large variables, even up to infinity.
Several examples from quantum and statistical mechanics are analysed,
illustrating the approach.Comment: 21 pages, Latex fil
Quenched Averages for self-avoiding walks and polygons on deterministic fractals
We study rooted self avoiding polygons and self avoiding walks on
deterministic fractal lattices of finite ramification index. Different sites on
such lattices are not equivalent, and the number of rooted open walks W_n(S),
and rooted self-avoiding polygons P_n(S) of n steps depend on the root S. We
use exact recursion equations on the fractal to determine the generating
functions for P_n(S), and W_n(S) for an arbitrary point S on the lattice. These
are used to compute the averages and over different positions of S. We find that the connectivity constant
, and the radius of gyration exponent are the same for the annealed
and quenched averages. However, , and , where the exponents
and take values different from the annealed case. These
are expressed as the Lyapunov exponents of random product of finite-dimensional
matrices. For the 3-simplex lattice, our numerical estimation gives ; and , to be
compared with the annealed values and .Comment: 17 pages, 10 figures, submitted to Journal of Statistical Physic
Non-perturbative calculations for the effective potential of the symmetric and non-Hermitian field theoretic model
We investigate the effective potential of the symmetric
field theory, perturbatively as well as non-perturbatively. For the
perturbative calculations, we first use normal ordering to obtain the first
order effective potential from which the predicted vacuum condensate vanishes
exponentially as in agreement with previous calculations. For the
higher orders, we employed the invariance of the bare parameters under the
change of the mass scale to fix the transformed form totally equivalent to
the original theory. The form so obtained up to is new and shows that all
the 1PI amplitudes are perurbative for both and regions. For
the intermediate region, we modified the fractal self-similar resummation
method to have a unique resummation formula for all values. This unique
formula is necessary because the effective potential is the generating
functional for all the 1PI amplitudes which can be obtained via and thus we can obtain an analytic calculation for the 1PI
amplitudes. Again, the resummed from of the effective potential is new and
interpolates the effective potential between the perturbative regions.
Moreover, the resummed effective potential agrees in spirit of previous
calculation concerning bound states.Comment: 20 page
Critical Indices as Limits of Control Functions
A variant of self-similar approximation theory is suggested, permitting an
easy and accurate summation of divergent series consisting of only a few terms.
The method is based on a power-law algebraic transformation, whose powers play
the role of control functions governing the fastest convergence of the
renormalized series. A striking relation between the theory of critical
phenomena and optimal control theory is discovered: The critical indices are
found to be directly related to limits of control functions at critical points.
The method is applied to calculating the critical indices for several difficult
problems. The results are in very good agreement with accurate numerical data.Comment: 1 file, 5 pages, RevTe
Log-periodic route to fractal functions
Log-periodic oscillations have been found to decorate the usual power law
behavior found to describe the approach to a critical point, when the
continuous scale-invariance symmetry is partially broken into a discrete-scale
invariance (DSI) symmetry. We classify the `Weierstrass-type'' solutions of the
renormalization group equation F(x)= g(x)+(1/m)F(g x) into two classes
characterized by the amplitudes A(n) of the power law series expansion. These
two classes are separated by a novel ``critical'' point. Growth processes
(DLA), rupture, earthquake and financial crashes seem to be characterized by
oscillatory or bounded regular microscopic functions g(x) that lead to a slow
power law decay of A(n), giving strong log-periodic amplitudes. In contrast,
the regular function g(x) of statistical physics models with
``ferromagnetic''-type interactions at equibrium involves unbound logarithms of
polynomials of the control variable that lead to a fast exponential decay of
A(n) giving weak log-periodic amplitudes and smoothed observables. These two
classes of behavior can be traced back to the existence or abscence of
``antiferromagnetic'' or ``dipolar''-type interactions which, when present,
make the Green functions non-monotonous oscillatory and favor spatial modulated
patterns.Comment: Latex document of 29 pages + 20 ps figures, addition of a new
demonstration of the source of strong log-periodicity and of a justification
of the general offered classification, update of reference lis
Multifractality in Time Series
We apply the concepts of multifractal physics to financial time series in
order to characterize the onset of crash for the Standard & Poor's 500 stock
index x(t). It is found that within the framework of multifractality, the
"analogous" specific heat of the S&P500 discrete price index displays a
shoulder to the right of the main peak for low values of time lags. On
decreasing T, the presence of the shoulder is a consequence of the peaked,
temporal x(t+T)-x(t) fluctuations in this regime. For large time lags (T>80),
we have found that C_{q} displays typical features of a classical phase
transition at a critical point. An example of such dynamic phase transition in
a simple economic model system, based on a mapping with multifractality
phenomena in random multiplicative processes, is also presented by applying
former results obtained with a continuous probability theory for describing
scaling measures.Comment: 22 pages, Revtex, 4 ps figures - To appear J. Phys. A (2000
Self-Similar Interpolation in Quantum Mechanics
An approach is developed for constructing simple analytical formulae
accurately approximating solutions to eigenvalue problems of quantum mechanics.
This approach is based on self-similar approximation theory. In order to derive
interpolation formulae valid in the whole range of parameters of considered
physical quantities, the self-similar renormalization procedure is complimented
here by boundary conditions which define control functions guaranteeing correct
asymptotic behaviour in the vicinity of boundary points. To emphasize the
generality of the approach, it is illustrated by different problems that are
typical for quantum mechanics, such as anharmonic oscillators, double-well
potentials, and quasiresonance models with quasistationary states. In addition,
the nonlinear Schr\"odinger equation is considered, for which both eigenvalues
and wave functions are constructed.Comment: 1 file, 30 pages, RevTex, no figure
Chern-Simons Theory for Magnetization Plateaus of Frustrated - Heisenberg model
The magnetization curve of the two-dimensional spin-1/2 -
Heisenberg model is investigated by using the Chern-Simons theory under a
uniform mean-field approximation. We find that the magnetization curve is
monotonically increasing for , where the system under zero
external field is in the antiferromagnetic N\'eel phase. For larger ratios of
, various plateaus will appear in the magnetization curve. In
particular, in the disordered phase, our result supports the existence of the
plateau and predicts a new plateau at .
By identifying the onset ratio for the appearance of the 1/2-plateau
with the boundary between the N\'eel and the spin-disordered phases in zero
field, we can determine this phase boundary accurately by this mean-field
calculation. Verification of these interesting results would indicate a strong
connection between the frustrated antiferromagnetic system and the quantum Hall
system.Comment: RevTeX 4, 4 pages, 3 EPS figure
Critical properties of 1-D spin 1/2 antiferromagnetic Heisenberg model
We discuss numerical results for the 1-D spin 1/2 antiferromagnetic
Heisenberg model with next-to-nearest neighbour coupling and in the presence of
an uniform magnetic field. The model develops zero frequency excitations at
field dependent soft mode momenta. We compute critical quantities from finite
size dependence of static structure factors.Comment: talk given by H. Kr{\"o}ger at Heraeus Seminar Theory of Spin
Lattices and Lattice Gauge Models, Bad Honnef (1996), 20 pages, LaTeX + 18
figures, P
Self-Similar Bootstrap of Divergent Series
A method is developed for calculating effective sums of divergent series.
This approach is a variant of the self-similar approximation theory. The
novelty here is in using an algebraic transformation with a power providing the
maximal stability of the self-similar renormalization procedure. The latter is
to be repeated as many times as it is necessary in order to convert into closed
self-similar expressions all sums from the series considered. This multiple and
complete renormalization is called self-similar bootstrap. The method is
illustrated by several examples from statistical physics.Comment: 1 file, 22 pages, RevTe
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