10 research outputs found
Fluctuations and correlations in an individual-based model of biological coevolution
We extend our study of a simple model of biological coevolution to its
statistical properties. Staring with a complete description in terms of a
master equation, we provide its relation to the deterministic evolution
equations used in previous investigations. The stationary states of the
mutationless model are generally well approximated by Gaussian distributions,
so that the fluctuations and correlations of the populations can be computed
analytically. Several specific cases are studied by Monte Carlo simulations,
and there is excellent agreement between the data and the theoretical
predictions.Comment: 25 pages, 2 figure
Monte Carlo with Absorbing Markov Chains: Fast Local Algorithms for Slow Dynamics
A class of Monte Carlo algorithms which incorporate absorbing Markov chains
is presented. In a particular limit, the lowest-order of these algorithms
reduces to the -fold way algorithm. These algorithms are applied to study
the escape from the metastable state in the two-dimensional square-lattice
nearest-neighbor Ising ferromagnet in an unfavorable applied field, and the
agreement with theoretical predictions is very good. It is demonstrated that
the higher-order algorithms can be many orders of magnitude faster than either
the traditional Monte Carlo or -fold way algorithms.Comment: ReVTeX, Request 3 figures from [email protected]
A Method to Study Relaxation of Metastable Phases: Macroscopic Mean-Field Dynamics
We propose two different macroscopic dynamics to describe the decay of
metastable phases in many-particle systems with local interactions. These
dynamics depend on the macroscopic order parameter through the restricted
free energy and are designed to give the correct equilibrium
distribution for . The connection between macroscopic dynamics and the
underlying microscopic dynamic are considered in the context of a projection-
operator formalism. Application to the square-lattice nearest-neighbor Ising
ferromagnet gives good agreement with droplet theory and Monte Carlo
simulations of the underlying microscopic dynamic. This includes quantitative
agreement for the exponential dependence of the lifetime on the inverse of the
applied field , and the observation of distinct field regions in which the
derivative of the lifetime with respect to depends differently on . In
addition, at very low temperatures we observe oscillatory behavior of this
derivative with respect to , due to the discreteness of the lattice and in
agreement with rigorous results. Similarities and differences between this work
and earlier works on finite Ising models in the fixed-magnetization ensemble
are discussed.Comment: 44 pages RevTeX3, 11 uuencoded Postscript figs. in separate file
Modulation of the nucleation rate pre-exponential in a low-temperature Ising system
A metastable lattice gas with nearest-neighbor interactions and
continuous-time dynamics is studied using a generalized Becker-Doring approach
in the multidimensional space of cluster configurations. The pre-exponential of
the metastable state lifetime (inverse of nucleation rate) is found to exhibit
distinct peaks at integer values of the inverse supersaturation. Peaks are
unobservable (infinitely narrow) in the strict limit T->0, but become
detectable and eventually dominate at higher temperatures.Comment: 4 pages, 2 Postscript figures, LaTeX, submitted to Phys. Rev. Lett.
Changes: updated references, re-written section around eqs.(5),(6), typos,
minor wording changes in conclusion and other parts of text (mostly in
response to referees' comments). Paper resubmitted to PR
Punctuated equilibria and 1/f noise in a biological coevolution model with individual-based dynamics
We present a study by linear stability analysis and large-scale Monte Carlo
simulations of a simple model of biological coevolution. Selection is provided
through a reproduction probability that contains quenched, random interspecies
interactions, while genetic variation is provided through a low mutation rate.
Both selection and mutation act on individual organisms. Consistent with some
current theories of macroevolutionary dynamics, the model displays
intermittent, statistically self-similar behavior with punctuated equilibria.
The probability density for the lifetimes of ecological communities is well
approximated by a power law with exponent near -2, and the corresponding power
spectral densities show 1/f noise (flicker noise) over several decades. The
long-lived communities (quasi-steady states) consist of a relatively small
number of mutualistically interacting species, and they are surrounded by a
``protection zone'' of closely related genotypes that have a very low
probability of invading the resident community. The extent of the protection
zone affects the stability of the community in a way analogous to the height of
the free-energy barrier surrounding a metastable state in a physical system.
Measures of biological diversity are on average stationary with no discernible
trends, even over our very long simulation runs of approximately 3.4x10^7
generations.Comment: 20 pages RevTex. Minor revisions consistent with published versio
Test of the Kolmogorov-Johnson-Mehl-Avrami picture of metastable decay in a model with microscopic dynamics
The Kolmogorov-Johnson-Mehl-Avrami (KJMA) theory for the time evolution of
the order parameter in systems undergoing first-order phase transformations has
been extended by Sekimoto to the level of two-point correlation functions.
Here, this extended KJMA theory is applied to a kinetic Ising lattice-gas
model, in which the elementary kinetic processes act on microscopic length and
time scales. The theoretical framework is used to analyze data from extensive
Monte Carlo simulations. The theory is inherently a mesoscopic continuum
picture, and in principle it requires a large separation between the
microscopic scales and the mesoscopic scales characteristic of the evolving
two-phase structure. Nevertheless, we find excellent quantitative agreement
with the simulations in a large parameter regime, extending remarkably far
towards strong fields (large supersaturations) and correspondingly small
nucleation barriers. The original KJMA theory permits direct measurement of the
order parameter in the metastable phase, and using the extension to correlation
functions one can also perform separate measurements of the nucleation rate and
the average velocity of the convoluted interface between the metastable and
stable phase regions. The values obtained for all three quantities are verified
by other theoretical and computational methods. As these quantities are often
difficult to measure directly during a process of phase transformation, data
analysis using the extended KJMA theory may provide a useful experimental
alternative.Comment: RevTex, 21 pages including 14 ps figures. Submitted to Phys. Rev. B.
One misprint corrected in Eq.(C1
Analytical and computational study of magnetization switching in kinetic Ising systems with demagnetizing fields
An important aspect of real ferromagnetic particles is the demagnetizing
field resulting from magnetostatic dipole-dipole interaction, which causes
large particles to break up into domains. Sufficiently small particles,
however, remain single-domain in equilibrium. This makes such small particles
of particular interest as materials for high-density magnetic recording media.
In this paper we use analytic arguments and Monte Carlo simulations to study
the effect of the demagnetizing field on the dynamics of magnetization
switching in two-dimensional, single-domain, kinetic Ising systems. For systems
in the ``Stochastic Region,'' where magnetization switching is on average
effected by the nucleation and growth of fewer than two well-defined critical
droplets, the simulation results can be explained by the dynamics of a simple
model in which the free energy is a function only of magnetization. In the
``Multi-Droplet Region,'' a generalization of Avrami's Law involving a
magnetization-dependent effective magnetic field gives good agreement with our
simulations.Comment: 29 pages, REVTeX 3.0, 10 figures, 2 more figures by request.
Submitted Phys. Rev.
Metastable lifetimes in a kinetic Ising model: Dependence on field and system size
The lifetimes of metastable states in kinetic Ising ferromagnets are studied
by droplet theory and Monte Carlo simulation, in order to determine their
dependences on applied field and system size. For a wide range of fields, the
dominant field dependence is universal for local dynamics and has the form of
an exponential in the inverse field, modified by universal and nonuniversal
power-law prefactors. Quantitative droplet-theory predictions are numerically
verified, and small deviations are shown to depend nonuniversally on the
details of the dynamics. We identify four distinct field intervals in which the
field dependence and statistical properties of the lifetimes are different. The
field marking the crossover between the weak-field regime, in which the decay
is dominated by a single droplet, and the intermediate-field regime, in which
it is dominated by a finite droplet density, vanishes logarithmically with
system size. As a consequence the slow decay characteristic of the former
regime may be observable in systems that are macroscopic as far as their
equilibrium properties are concerned.Comment: 18 pages single spaced. RevTex Version 3. FSU-SCRI-94-1