Aggregation driven by a localized source. (19th June 2015)

• Record Type: Journal Article
• Authors: Krapivsky, P L
• Is Part Of: Journal of physics. Volume 48:Number 24(2015)
• Page Start: 806
• Publication Date: 2015-06-19
• Subjects: Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8113/48/24/245002
• Languages: English
• ISSNs: 1751-8113

Abstract

<abstract> <title>Abstract</title> <p>We study aggregation driven by a localized source of monomers. The densities become stationary and have algebraic tails far away from the source. We show that in a model with mass-independent reaction rates and diffusion coefficients, the density of monomers decays as <inline-formula><tex-math><?CDATA ${{r}^{-\beta (d)}}$?></tex-math><?MML <mml:math> <mml:msup> <mml:mrow> <mml:mi>r</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>β</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> </mml:mrow> </mml:msup> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj14kmzx25" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> in <italic>d</italic> dimensions. The decay exponent has irrational values in physically relevant dimensions: <inline-formula><tex-math><?CDATA $\beta (3)=(\sqrt{17}+1)/2$?></tex-math><?MML <mml:math> <mml:mi>β</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow> <!--false buildup--> <mml:mo stretchy="true">(</mml:mo> <mml:msqrt> <mml:mrow> <mml:mn>17</mml:mn> </mml:mrow> </mml:msqrt> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj14kmzrkv" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula> and <inline-formula><tex-math><?CDATA $\beta (2)=\sqrt{8}$?></tex-math><?MML <mml:math> <mml:mi>β</mml:mi> <mml:mrow> <mml:mo stretchy="true">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="true">)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:msqrt> <mml:mrow> <mml:mn>8</mml:mn> </mml:mrow> </mml:msqrt> </mml:math>?><inline-graphic xlink:href="ark:/27927/pgj14kmzxt9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /></inline-formula>. We also study Brownian coagulation with a localized source and establish the behavior of the total cluster density and the total number of of clusters in the system. The latter quantity exhibits a logarithmic growth with time.</p> </abstract>