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Related Concept Videos

Radical Chain-Growth Polymerization: Chain Branching01:17

Radical Chain-Growth Polymerization: Chain Branching

The skeletal structure of polymers synthesized via radical polymerization is always branched. For example, the polymerization of ethylene by radical polymerization results in a low-density grade of polyethylene with a heavily branched skeletal structure. Here, the radical site abstracts hydrogen from the growing chain, and the radical site shifts from the end (a primary carbon center) to anywhere within the growing chain (a secondary carbon center). Consequently, the part of the chain from the...
Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
Random Variables01:09

Random Variables

A random variable is a single numerical value that indicates the outcome of a procedure. The concept of random variables is fundamental to the probability theory and was introduced by a Russian mathematician, Pafnuty Chebyshev, in the mid-nineteenth century.
Uppercase letters such as X or Y denote a random variable. Lowercase letters like x or y denote the value of a random variable. If X is a random variable, then X is written in words, and x is given as a number.
For example, let X = the...
Mutation, Gene Flow, and Genetic Drift01:09

Mutation, Gene Flow, and Genetic Drift

In a population that is not at Hardy-Weinberg equilibrium, the frequency of alleles changes over time. Therefore, any deviations from the five conditions of Hardy-Weinberg equilibrium can alter the genetic variation of a given population. Conditions that change the genetic variability of a population include mutations, natural selection, non-random mating, gene flow, and genetic drift (small population size).
Radical Chain-Growth Polymerization: Overview01:10

Radical Chain-Growth Polymerization: Overview

Chain-growth or addition polymerization is successive addition reactions of monomers with a polymer chain. In radical chain-growth polymerization, the reaction proceeds via a free-radical intermediate. The free radical is formed from radical initiators, which spontaneously generate free radicals by homolytic fission. Organic peroxides (such as dibenzoyl peroxide, as shown in Figure 1) or azo compounds are popular radical initiators. A low concentration ratio of radical initiator to monomer is...
Radical Chain-Growth Polymerization: Mechanism01:09

Radical Chain-Growth Polymerization: Mechanism

The radical chain-growth polymerization mechanism consists of three steps: initiation, propagation, and termination of polymerization. The polymerization initiates when a free radical generated from the radical initiator adds to the unsaturated bond in the monomer. The unpaired electron of the free radical and one π electron in the unsaturated bond creates a σ bond between the free radical and the monomer. As a result, the other π electron in the unsaturated bond converts this species into the...

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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
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Branching-rate expansion around annihilating random walks.

Federico Benitez1, Nicolás Wschebor

  • 1LPTMC, CNRS-UMR 7600, Université Pierre et Marie Curie, 75252 Paris, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
PubMed
Summary

We provide exact results for branching and annihilating random walks, determining critical annihilation rates for phase transitions in reaction-diffusion systems. Our findings refine understanding of universality classes and require improvements to existing models.

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Area of Science:

  • Statistical Physics
  • Complex Systems

Background:

  • Branching and annihilating random walks are fundamental models in statistical physics.
  • Understanding phase transitions in reaction-diffusion systems is crucial for various scientific fields.

Purpose of the Study:

  • To derive exact results for branching and annihilating random walks.
  • To compute the nonuniversal threshold for phase transitions in directed percolation and parity conserving universality classes.

Main Methods:

  • Exact solution of pure annihilation in any dimension.
  • Expansion in the branching rate around pure annihilation.

Main Results:

  • Computed the nonuniversal threshold value of the annihilation rate for a phase transition in the directed percolation universality class.
  • Showed that the accepted scenario for phase transitions in the parity conserving universality class needs improvement.

Conclusions:

  • The study provides exact analytical results for complex reaction-diffusion systems.
  • Findings necessitate a revision of theoretical models for phase transitions in certain universality classes.