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In one dimension, particles can not bypass each other. As a consequence, themean-squared displacement (MSD) in equilibrium shows sub-diffusion ${\rmMSD}(t)\sim t^{1/2}$, instead of normal diffusion ${\rm MSD}(t)\sim t$. Thisphenomenon is the so-called single-file diffusion. Here, we investigate how theabove equilibrium behaviors are modified far from equilibrium. In particular,we want to uncover what kind of non-equilibrium driving force can suppressdiffusion and achieve the long-range crystalline order in one dimension, whichis prohibited by the Mermin-Wagner theorem in equilibrium. For that purpose, weinvestigate the harmonic chain driven by the following four types of drivingforces that do not satisfy the detailed balance: (i) temporally correlatednoise with the noise spectrum $D(\omega)\sim \omega^{-2\theta}$, (ii)center-of-mass conserving noise, (iii) periodic driving force, and (iv)periodic deformations of particles. For the driving force (i) with$\theta>-1/4$, we observe ${\rm MSD}(t)\sim t^{1/2+2\theta}$ for large $t$. Onthe other hand, for the driving forces (i) with $\theta<-1/4$ and (ii)-(iv),MSD remains finite. As a consequence, the harmonic chain exhibits thecrystalline order even in one dimension. Furthermore, the density fluctuationsof the model are highly suppressed in a large scale for the driving forces (i)with $\theta<0$ and (ii)-(iv). This phenomenon is known as hyperuniformity. Wediscuss that hyperuniformity of the noise fluctuations themselves is therelevant mechanism to stabilize the long-range crystalline order in onedimension and yield hyperuniformity of the density fluctuations.

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