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

In one dimension, particles can not bypass each other. As a consequence, the mean-squared displacement (MSD) in equilibrium shows sub-diffusion MSD(t)~ t1/2 MSD(t)∼t1/2 , instead of normal diffusion MSD(t)~ t MSD(t)∼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(ω)~ ω-2θ D(ω)∼ω−2θ , (ii) conserving noise, (iii) periodic driving force, and (iv) periodic deformations of particles. For the driving force (i) with θ >-1/4 θ>−1/4 , we observe MSD(t)~ t1/2+2θ MSD(t)∼t1/2+2θ for large t t . On the other hand, for the driving forces (i) with θ<-1/4 θ<−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.