939 research outputs found
Optimized energy calculation in lattice systems with long-range interactions
We discuss an efficient approach to the calculation of the internal energy in
numerical simulations of spin systems with long-range interactions. Although,
since the introduction of the Luijten-Bl\"ote algorithm, Monte Carlo
simulations of these systems no longer pose a fundamental problem, the energy
calculation is still an O(N^2) problem for systems of size N. We show how this
can be reduced to an O(N logN) problem, with a break-even point that is already
reached for very small systems. This allows the study of a variety of, until
now hardly accessible, physical aspects of these systems. In particular, we
combine the optimized energy calculation with histogram interpolation methods
to investigate the specific heat of the Ising model and the first-order regime
of the three-state Potts model with long-range interactions.Comment: 10 pages, including 8 EPS figures. To appear in Phys. Rev. E. Also
available as PDF file at
http://www.cond-mat.physik.uni-mainz.de/~luijten/erikpubs.htm
Critical properties of the three-dimensional equivalent-neighbor model and crossover scaling in finite systems
Accurate numerical results are presented for the three-dimensional
equivalent-neighbor model on a cubic lattice, for twelve different interaction
ranges (coordination number between 18 and 250). These results allow the
determination of the range dependences of the critical temperature and various
critical amplitudes, which are compared to renormalization-group predictions.
In addition, the analysis yields an estimate for the interaction range at which
the leading corrections to scaling vanish for the spin-1/2 model and confirms
earlier conclusions that the leading Wegner correction must be negative for the
three-dimensional (nearest-neighbor) Ising model. By complementing these
results with Monte Carlo data for systems with coordination numbers as large as
52514, the full finite-size crossover curves between classical and Ising-like
behavior are obtained as a function of a generalized Ginzburg parameter. Also
the crossover function for the effective magnetic exponent is determined.Comment: Corrected shift of critical temperature and some typos. To appear in
Phys. Rev. E. 18 pages RevTeX, including 10 EPS figures. Also available as
PDF file at http://www.cond-mat.physik.uni-mainz.de/~luijten/erikpubs.htm
Polyelectrolyte Condensation Induced by Linear Cations
We examine the role of the condensing agent in the formation of
polyelectrolyte bundles, via grand-canonical Monte Carlo simulations. Following
recent experiments we use linear, rigid divalent cations of various lengths to
induce condensation. Our results clarify and explain the experimental results
for short cations. For longer cations we observe novel condensation behavior
owing to alignment of the cations. We also study the role of the
polyelectrolyte surface charge density, and find a nonmonotonic variation in
bundle stability. This nonmonotonicity captures two trends that have been
observed in separate experiments.Comment: To appear in Physical Review Letter
Test of renormalization predictions for universal finite-size scaling functions
We calculate universal finite-size scaling functions for systems with an
n-component order parameter and algebraically decaying interactions. Just as
previously has been found for short-range interactions, this leads to a
singular epsilon-expansion, where epsilon is the distance to the upper critical
dimension. Subsequently, we check the results by numerical simulations of spin
models in the same universality class. Our systems offer the essential
advantage that epsilon can be varied continuously, allowing an accurate
examination of the region where epsilon is small. The numerical calculations
turn out to be in striking disagreement with the predicted singularity.Comment: 6 pages, including 3 EPS figures. To appear in Phys. Rev. E. Also
available as PDF file at
http://www.cond-mat.physik.uni-mainz.de/~luijten/erikpubs.htm
Crossover scaling in two dimensions
We determine the scaling functions describing the crossover from Ising-like
critical behavior to classical critical behavior in two-dimensional systems
with a variable interaction range. Since this crossover spans several decades
in the reduced temperature as well as in the finite-size crossover variable, it
has up to now largely evaded a satisfactory numerical determination. Using a
new Monte Carlo method, we could obtain accurate results for sufficiently large
interactions ranges. Our data cover the full crossover region both above and
below the critical temperature and support the hypothesis that the crossover
functions are universal. Also the so-called effective exponents are discussed
and we show that these can vary nonmonotonically in the crossover region.Comment: 24 pages RevTeX 3.0/3.1, including 22 PostScript figures. Uses
epsf.st
Finite-size scaling above the upper critical dimension revisited: The case of the five-dimensional Ising model
Monte Carlo results for the moments of the magnetization distribution
of the nearest-neighbor Ising ferromagnet in a L^d geometry, where L (4 \leq L
\leq 22) is the linear dimension of a hypercubic lattice with periodic boundary
conditions in d=5 dimensions, are analyzed in the critical region and compared
to a recent theory of Chen and Dohm (CD) [X.S. Chen and V. Dohm, Int. J. Mod.
Phys. C (1998)]. We show that this finite-size scaling theory (formulated in
terms of two scaling variables) can account for the longstanding discrepancies
between Monte Carlo results and the so-called ``lowest-mode'' theory, which
uses a single scaling variable tL^{d/2} where t=T/T_c-1 is the temperature
distance from the critical temperature, only to a very limited extent. While
the CD theory gives a somewhat improved description of corrections to the
``lowest-mode'' results (to which the CD theory can easily be reduced in the
limit t \to 0, L \to \infty, tL^{d/2} fixed) for the fourth-order cumulant,
discrepancies are found for the susceptibility (L^d ). Reasons for these
problems are briefly discussed.Comment: 9 pages, 13 Encapsulated PostScript figures. To appear in Eur. Phys.
J. B. Also available as PDF file at
http://www.cond-mat.physik.uni-mainz.de/~luijten/erikpubs.htm
Universal finite-size scaling analysis of Ising models with long-range interactions at the upper critical dimensionality: Isotropic case
We investigate a two-dimensional Ising model with long-range interactions
that emerge from a generalization of the magnetic dipolar interaction in spin
systems with in-plane spin orientation. This interaction is, in general,
anisotropic whereby in the present work we focus on the isotropic case for
which the model is found to be at its upper critical dimensionality. To
investigate the critical behavior the temperature and field dependence of
several quantities are studied by means of Monte Carlo simulations. On the
basis of the Privman-Fisher hypothesis and results of the renormalization group
the numerical data are analyzed in the framework of a finite-size scaling
analysis and compared to finite-size scaling functions derived from a
Ginzburg-Landau-Wilson model in zero mode (mean-field) approximation. The
obtained excellent agreement suggests that at least in the present case the
concept of universal finite-size scaling functions can be extended to the upper
critical dimensionality.Comment: revtex4, 10 pages, 5 figures, 1 tabl
Universality and the five-dimensional Ising model
We solve the long-standing discrepancy between Monte Carlo results and the
renormalization prediction for the Binder cumulant of the five-dimensional
Ising model. Our conclusions are based on accurate Monte Carlo data for systems
with linear sizes up to L=22. A detailed analysis of the corrections to scaling
allows the extrapolation of these results to L=\infinity. Our determination of
the critical point, K_c=0.1139150 (4), is more than an order of magnitude more
accurate than previous estimates.Comment: 6 pages LaTeX, 1 PostScript figure. Uses cite.sty (included) and
epsf.sty. Also available as PostScript and PDF file at
http://www.tn.tudelft.nl/tn/erikpubs.htm
Medium-range interactions and crossover to classical critical behavior
We study the crossover from Ising-like to classical critical behavior as a
function of the range R of interactions. The power-law dependence on R of
several critical amplitudes is calculated from renormalization theory. The
results confirm the predictions of Mon and Binder, which were obtained from
phenomenological scaling arguments. In addition, we calculate the range
dependence of several corrections to scaling. We have tested the results in
Monte Carlo simulations of two-dimensional systems with an extended range of
interaction. An efficient Monte Carlo algorithm enabled us to carry out
simulations for sufficiently large values of R, so that the theoretical
predictions could actually be observed.Comment: 16 pages RevTeX, 8 PostScript figures. Uses epsf.sty. Also available
as PostScript and PDF file at http://www.tn.tudelft.nl/tn/erikpubs.htm
Comment on ``Scaling Laws for a System with Long-Range Interactions within Tsallis Statistics''
In their recent Letter [Phys. Rev. Lett. 83, 4233 (1999)], Salazar and Toral
(ST) study numerically a finite Ising chain with non-integrable interactions
decaying like 1/r^(d+sigma) where -d <= sigma <= 0 (like ST, we discuss general
dimensionality d). In particular, they explore a presumed connection between
non-integrable interactions and Tsallis's non-extensive statistics. We point
out that (i) non-integrable interactions provide no more motivation for Tsallis
statistics than do integrable interactions, i.e., Gibbs statistics remain
meaningful for the non-integrable case, and in fact provide a {\em complete and
exact treatment}; and (ii) there are undesirable features of the method ST use
to regulate the non-integrable interactions.Comment: Accepted for publication in Phys. Rev. Let
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