arXiv subjects

C. R. Stephens

Publications and source records attributed to C. R. Stephens.

At least 19 recordsLinked to original sources

arXiv2010

Two-Loop Crossover Scaling Functions of the O(N) Model

Using Environmentally Friendly Renormalization, we present an analytic calculation of the series for the renormalization constants that describe the equation of state for the $O(N)$ model in the whole critical region. The solution of the beta-function equation, for the running coupling to order two loops, exhibits crossover between the strong coupling fixed point, associated with the Goldstone modes, and the Wilson-Fisher fixed point. The Wilson functions $γ_λ$, $γ_ϕ$ and $γ_{ϕ^2}$, and thus the effective critical exponents associated with renormalization of the transverse vertex functions, also exhibit non-trivial crossover between these fixed points.

cond-mat.stat-mech↗

arXiv2008

An ab initio Calculation of the Universal Equation of State for the O(N) Model

Using an Environmentally Friendly Renormalization Group we derive an ab initio universal scaling form for the equation of state for the O(N) model, y=f(x), that exhibits all required analyticity properties in the limits $x\to 0$, $x\to\infty$ and $x\to -1$. Unlike current methodologies based on a phenomenological scaling ansatz the scaling function is derived solely from the underlying Landau-Ginzburg-Wilson Hamiltonian and depends only on the three Wilson functions $γ_λ$, $γ_ϕ$ and $γ_{ϕ^2}$ which exhibit a non-trivial crossover between the Wilson-Fisher fixed point and the strong coupling fixed point associated with the Goldstone modes on the coexistence curve. We give explicit results for N=2, 3 and 4 to one-loop order and compare with known results.

cond-mat.stat-mech↗

arXiv2006

An Analytic Equation of State for Ising-like Models

Using an Environmentally Friendly Renormalization we derive, from an underlying field theory representation, a formal expression for the equation of state, $y=f(x)$, that exhibits all desired asymptotic and analyticity properties in the three limits $x\to 0$, $x\to \infty$ and $x\to -1$. The only necessary inputs are the Wilson functions $γ_λ$, $γ_ϕ$ and $γ_{ϕ^2}$, associated with a renormalization of the transverse vertex functions. These Wilson functions exhibit a crossover between the Wilson-Fisher fixed point and the fixed point that controls the coexistence curve. Restricting to the case N=1, we derive a one-loop equation of state for $2< d<4$ naturally parameterized by a ratio of non-linear scaling fields. For $d=3$ we show that a non-parameterized analytic form can be deduced. Various asymptotic amplitudes are calculated directly from the equation of state in all three asymptotic limits of interest and comparison made with known results. By positing a scaling form for the equation of state inspired by the one-loop result, but adjusted to fit the known values of the critical exponents, we obtain better agreement with known asymptotic amplitudes.

cond-mat.stat-mech↗

arXiv2004

Dimensional Reduction, Hard Thermal Loops and the Renormalization Group

We study the realization of dimensional reduction and the validity of the hard thermal loop expansion for lambda phi^4 theory at finite temperature, using an environmentally friendly finite-temperature renormalization group with a fiducial temperature as flow parameter. The one-loop renormalization group allows for a consistent description of the system at low and high temperatures, and in particular of the phase transition. The main results are that dimensional reduction applies, apart from a range of temperatures around the phase transition, at high temperatures (compared to the zero temperature mass) only for sufficiently small coupling constants, while the HTL expansion is valid below (and rather far from) the phase transition, and, again, at high temperatures only in the case of sufficiently small coupling constants. We emphasize that close to the critical temperature, physics is completely dominated by thermal fluctuations that are not resummed in the hard thermal loop approach and where universal quantities are independent of the parameters of the fundamental four-dimensional theory.

hep-th↗

arXiv2002

Dimensional Crossover in the Non-Linear Sigma Model

We consider dimensional crossover for an O(N) model on a d-dimensional layered geometry of thickness L, in the sigma-model limit, using ``environmentally friendly'' renormalization. We show how to derive critical temperature shifts, giving explicit results to one loop. We also obtain expressions for the effective critical exponents delta and beta effective that interpolate between their characteristic fixed point values assocliated with a d and (d-1)-dimensional system in the limits T -> T_c(L), with L(T-T_c(L))^ν->infty, and T->T_c(L) for L fixed respectively, where T_c(L) is the L-dependent critical temperature of the system.

cond-mat.stat-mech↗

arXiv2001

Some Exact Results from a Coarse Grained Formulation of Genetic Dynamics

We extend a recently developed exact schema based, or coarse grained, formulation of genetic dynamics \cite{stewael,stewael1,stewael2} and its associated exact Schema theorem to an arbitrary selection scheme and a general crossover operator. We show that the intuitive``building block'' interpretation of the former is preserved leading to hierarchical formal solutions of the equations that upon iteration lead to new results for the limiting distribution of a population in the case of 1-point crossover and ``weak'' selection, where we define quantitatively ``weak''. We also derive an exact, analytic form for the population distribution as a function of time for a flat landscape and 1-point crossover.

nlin.AO↗

arXiv2000

Bound States from Regge Trajectories in a Scalar Model

The calculation of bound state properties using renormalization group techniques to compute the corresponding Regge trajectories is presented. In particular, we investigate the bound states in different charge sectors of a scalar theory with interaction (phi^dagger phi chi). The resulting bound state spectrum is surprisingly rich. Where possible we compare and contrast with known results of the Bethe-Salpeter equation in the ladder approximation and, in the non-relativistic limit, with the corresponding Schr"odinger equation.

hep-th↗

arXiv2000

Schemata Evolution and Building Blocks

In the light of a recently derived evolution equation for genetic algorithms we consider the schema theorem and the building block hypothesis. We derive a schema theorem based on the concept of effective fitness showing that schemata of higher than average effective fitness receive an exponentially increasing number of trials over time. The equation makes manifest the content of the building block hypothesis showing how fit schemata are constructed from fit sub-schemata. However, we show that generically there is no preference for short, low-order schemata. In the case where schema reconstruction is favored over schema destruction large schemata tend to be favored. As a corollary of the evolution equation we prove Geiringer's theorem.

nlin.AO↗

arXiv2000

Schemata as Building Blocks: Does Size Matter?

We analyze the schema theorem and the building block hypothesis using a recently derived, exact schemata evolution equation. We derive a new schema theorem based on the concept of effective fitness showing that schemata of higher than average effective fitness receive an exponentially increasing number of trials over time. The building block hypothesis is a natural consequence in that the equation shows how fit schemata are constructed from fit sub-schemata. However, we show that generically there is no preference for short, low-order schemata. In the case where schema reconstruction is favoured over schema destruction large schemata tend to be favoured. As a corollary of the evolution equation we prove Geiringer's theorem. We give supporting numerical evidence for our claims in both non-epsitatic and epistatic landscapes.

nlin.AO↗

arXiv2000

Effect of Mutation and Recombination on the Genotype-Phenotype Map

The effect of genetic operators other than selection, such as mutation and recombination, on the genotype-phenotype map is considered. In particular, when the genotypic fitness landscape exhibits a ``symmetry'', i.e. many genotypes corresponding to the same phenotype have equal fitness values, it is shown that such operators can break this symmetry. The consequences of this ``induced symmetry breaking'' are investigated. Specifically, it is shown that it generically leads to an increase in order or self-organization in the system and to the phenomenon of orthogenesis. Additionally, it is shown that it potentially leads to a more robust evolution circumventing some of the problems of brittleness. The above points are supported by explicit, analytic results associated with some simple one and two-locus models and also by some much more complicated numerical simulations.

nlin.AO↗

arXiv2000

Effective Fitness Landscapes for Evolutionary Systems

In evolution theory the concept of a fitness landscape has played an important role, evolution itself being portrayed as a hill-climbing process on a rugged landscape. In this article it is shown that in general, in the presence of other genetic operators such as mutation and recombination, hill-climbing is the exception rather than the rule. This descrepency can be traced to the different ways that the concept of fitness appears --- as a measure of the number of fit offspring, or as a measure of the probability to reach reproductive age. Effective fitness models the former not the latter and gives an intuitive way to understand population dynamics as flows on an effective fitness landscape when genetic operators other than selection play an important role. The efficacy of the concept is shown using several simple analytic examples and also some more complicated cases illustrated by simulations.

nlin.AO↗

arXiv1999

Glueball Spectrum from an Effective Hamiltonian

Using simple and general arguments we propose an effective Hamiltonian for the description of low-energy pure QCD. The Hamiltonian is a function of spatially constant collective modes. Its eigenstates can be organized into bands classified by the irreducible representations of an O(8) group. The latter also determine parity and charge conjugation of the states. The energy spectrum agrees well with the glueball spectrum as measured on the lattice, and in particular the level ordering with respect to spin is naturally explained.

hep-ph↗

arXiv1998

Quantum Field Theory in the Limit x << 1

The asymptotic high momentum behaviour of quantum field theories with cubic interactions is investigated using renormalization group techniques in the asymmetric limit x << 1. Particular emphasis is paid to theories with interactions involving more than one field where it is found that a matrix renormalization is necessary. Asymptotic scaling forms, in agreement with Regge theory, are derived for the elastic two-particle scattering amplitude and verified in one-loop renormalization group improved perturbation theory, corresponding to the summation of leading logs to all orders. We give explicit forms for the Regge trajectories of different scalar theories in this approximation and determine the signatures.

hep-th↗

arXiv1997

Universal Scaling Properties of Superconductors in Magnetic Fields

Based on renormalization group arguments we establish that for a superconductor in the presence of a weak external magnetic field, $B$, the dependence on $B$ and the deviation from the critical temperature, $τ$, of a thermodynamic quantity, $P$, takes the scaling form $P=t^θX({B\overΦ_0}τ^{-2ν},qτ^{-νω_e})$, where $θ$ and $ν$ are XY exponents, $q$ is the scaled electromagnetic coupling and $νω_e$ is the associated crossover exponent. For $q/τ^{νω_e}\ll1$, the experimentally accessible region in high-$T_c$ superconductors, there is a reduction to one-variable scaling plus small corrections. In this region we find the shift in the specific heat maximum is given by $Δ=x_0{(B/Φ_0)}^{1/2ν}$ and that the singular part of the free energy at the critical temperature takes the form $F_{sing}=c(d){(B/Φ_0)}^{d/2}$ where $c(d)$ is a universal amplitude. A one loop approximation in three dimensions gives $c(3)\sim0.22$. The results presented here should have equal applicability to the nematic to smectic-A transition.

cond-mat.supr-con↗

arXiv1997

Regge Behaviour from an Environmentally Friendly Renormalization Group

The asymptotic behaviour of cubic field theories is investigated in the Regge limit using the techniques of environmentally friendly renormalization, environmentally friendly in the present context meaning asymmetric in its momentum dependence. In particular we consider the crossover between large and small energies at fixed momentum transfer for a model scalar theory of the type phi^2 psi. The asymptotic forms of the crossover scaling functions are exhibited for all two particle scattering processes in this channel to one loop in a renormalization group improved perturbation theory.

hep-th↗

arXiv1996

Analysis of the Effective Degrees of Freedom in Genetic Algorithms

An evolution equation for a population of strings evolving under the genetic operators: selection, mutation and crossover is derived. The corresponding equation describing the evolution of schematas is found by performing an exact coarse graining of this equation. In particular exact expressions for schemata reconstruction are derived which allows for a critical appraisal of the ``building-block hypothesis'' of genetic algorithms. A further coarse-graining is made by considering the contribution of all length-l schematas to the evolution of population observables such as fitness growth. As a test function for investigating the emergence of structure in the evolution the increase per generation of the in-schemata fitness averaged over all schematas of length l, $Δ_l$, is introduced. In finding solutions of the evolution equations we concentrate more on the effects of crossover, in particular we consider crossover in the context of Kauffman Nk models with k=0,2. For k=0, with a random initial population, in the first step of evolution the contribution from schemata reconstruction is equal to that of schemata destruction leading to a scale invariant situation where the contribution to fitness of schematas of size l is independent of l. This balance is broken in the next step of evolution leading to a situation where schematas that are either much larger or much smaller than half the string size dominate over those with $l \approx N/2$. The balance between block destruction and reconstruction is also broken in a k>0 landscape. It is conjectured that the effective degrees of freedom for such landscapes are landscape connective trees that break down into effectively fit smaller blocks, and not the blocks themselves. Numerical simulations confirm this ``connective tree hypothesis'' by showing that correlations drop off with connective distance and not with intrachromosomal distance.

adap-org↗

arXiv1996

Boson Mapping in Field Theory

A boson mapping of pair field operators is presented. The mapping preserves all hermiticity properties and the Poisson bracket relations between fields and momenta. The most practical application of the boson mapping is to field theories which exhibit bound states of pairs of fields. As a concrete application we consider, in the low energy limit, the Wick-Cutkosky model with equal mass for the charged fields.

hep-th↗

arXiv1996

Why Two Renormalization Groups are Better than One

The advantages of using more than one renormalization group (RG) in problems with more than one important length scale are discussed. It is shown that: i) using different RG's can lead to complementary information, i.e. what is very difficult to calculate with an RG based on one flow parameter may be much more accessible using another; ii) using more than one RG requires less physical input in order to describe via RG methods the theory as a function of its parameters; iii) using more than one RG allows one to solve problems with more than one diverging length scale. The above points are illustrated concretely in the context of both particle physics and statistical physics using the techniques of environmentally friendly renormalization. Specifically, finite temperature $λϕ^4$ theory, an Ising-type system in a film geometry, an Ising-type system in a transverse magnetic field, the QCD coupling constant at finite temperature and the crossover between bulk and surface critical behaviour in a semi-infinite geometry are considered.

hep-th↗