Harmonic chain far from equilibrium: single-file diffusion, long-range order, and hyperuniformity

Abstract

In one dimension, particles can not bypass each other. As a consequence, the mean-squared displacement (MSD) in equilibrium shows sub-diffusion $\mathrm{MSD}(t)\sim t^{1/2}$, instead of normal diffusion $\mathrm{MSD}(t)\sim t$. 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 $D(\omega )\sim {\omega }^{-2\theta }$, (ii) conserving noise, (iii) periodic driving force, and (iv) periodic deformations of particles. For the driving force (i) with $\theta >-1/4$, we observe $\mathrm{MSD}(t)\sim t^{1/2+2\theta }$ for large $t$. On the other hand, for the driving forces (i) with $\theta <-1/4$ 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.

arXiv comment23 pages, 7figures

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• Singular density correlations in chiral active fluids in three dimensions
• Emergent universal long-range structure in random-organizing systems
• Interaction-free ergodicity-breaking driven by temporally hyperuniform noise
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