Hyperuniformity in phase ordering: the roles of activity, noise, and non-constant mobility

Abstract

Abstract Hyperuniformity emerges generically in the coarsening regime of phase-separating fluids. Numerical studies of active and passive systems have shown that the structure factor S ( q ) behaves as q ς for q → 0, with hyperuniformity exponent ς = 4. For passive systems, this result was explained in 1991 by a qualitative scaling analysis of Tomita, exploiting isotropy at scales much larger than the coarsening length. Here we reconsider and extend Tomita’s argument to address cases of active phase separation and of non-constant mobility, again finding ς = 4. We further show that dynamical noise of variance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>D</mml:mi> </mml:mrow> </mml:math> creates a transient ς = 2 regime for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mover> <mml:mi>q</mml:mi> <mml:mo stretchy="true">^</mml:mo> </mml:mover> </mml:mrow> <mml:mo>≪</mml:mo> <mml:msub> <mml:mrow> <mml:mover> <mml:mi>q</mml:mi> <mml:mo stretchy="true">^</mml:mo> </mml:mover> </mml:mrow> <mml:mo>∗</mml:mo> </mml:msub> <mml:mo>∼</mml:mo> <mml:msqrt> <mml:mi>D</mml:mi> </mml:msqrt> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mi>ν</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , crossing over to ς = 4 at larger <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mover> <mml:mi>q</mml:mi> <mml:mo stretchy="true">^</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> </mml:math> . Here, ν is the coarsening exponent for the domain size <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:math> , such that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ℓ</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mi>ν</mml:mi> </mml:msup> </mml:mrow> </mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mover> <mml:mi>q</mml:mi> <mml:mo stretchy="true">^</mml:mo> </mml:mover> </mml:mrow> <mml:mo>∝</mml:mo> <mml:mi>q</mml:mi> <mml:mi>ℓ</mml:mi> </mml:mrow> </mml:math> is the rescaled wavenumber. In diffusive coarsening <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , so the rescaled crossover wavevector <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mover> <mml:mi>q</mml:mi> <mml:mo stretchy="true">^</mml:mo> </mml:mover> </mml:mrow> <mml:mo>∗</mml:mo> </mml:msub> </mml:mrow> </mml:math> vanishes at large times when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . The slowness of this decay suggests a natural explanation for experiments that observe a long-lived ς = 2 scaling in phase-separating active fluids (where noise is typically large). Conversely, in d = 1, we demonstrate that with noise the ς = 2 regime survives as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> </mml:math> , with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mover> <mml:mi>q</mml:mi> <mml:mo stretchy="true">^</mml:mo> </mml:mover> </mml:mrow> <mml:mo>∗</mml:mo> </mml:msub> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>D</mml:mi> <mml:mrow> <mml:mn>5</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>6</mml:mn> </mml:mrow>

Topics & Concepts