Harmonic chain far from equilibrium: Single-file diffusion, long-range order, and hyperuniformity
Harukuni Ikeda
Abstract
In one dimension, particles can not bypass each other. As a consequence, the mean-squared displacement (MSD) in equilibrium shows sub-diffusion MSD(t)\sim t^{1/2} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>S</mml:mi> <mml:mi>D</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , instead of normal diffusion MSD(t)\sim t <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>S</mml:mi> <mml:mi>D</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>∼</mml:mo> <mml:mi>t</mml:mi> </mml:mrow> </mml:math> . This phenomenon is the so-called single-file diffusion. Here, we investigate how the above equilibrium behaviors are modified far from equilibrium. In particular, we want to uncover what kind of non-equilibrium driving force can suppress diffusion and achieve the long-range crystalline order in one dimension, which is prohibited by the Mermin-Wagner theorem in equilibrium. For that purpose, we investigate the harmonic chain driven by the following four types of driving forces that do not satisfy the detailed balance: (i) temporally correlated noise with the noise spectrum D(\omega)\sim \omega^{-2\theta} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>ω</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> <mml:mi>θ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , (ii) conserving noise, (iii) periodic driving force, and (iv) periodic deformations of particles. For the driving force (i) with \theta >-1/4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo>></mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , we observe MSD(t)\sim t^{1/2+2\theta} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>S</mml:mi> <mml:mi>D</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mi>θ</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> for large t <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>t</mml:mi> </mml:math> . On the other hand, for the driving forces (i) with \theta<-1/4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>θ</mml:mi> <mml:mo><</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mi>/</mml:mi> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> and (ii)-(iv), MSD remains finite. As a consequence, the harmonic chain exhibits the crystalline order even in one dimension. Furthermore, the density fluctuations of the model are highly suppressed in a large scale in the crystal phase. This phenomenon is known as hyperuniformity. We discuss that hyperuniformity of the noise fluctuations themselves is the relevant mechanism to stabilize the long-range crystalline order in one dimension and yield hyperuniformity of the density fluctuations.