池田 晴國

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}, instead of normal diffusion 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 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.

Abstract In this work, we investigate a symmetric deformed random matrix, which is obtained by perturbing the diagonal elements of the Wigner matrix. The eigenvector $\boldsymbol{x}_\mathrm{min}$ of the minimal eigenvalue $\lambda_\mathrm{min}$ of the deformed random matrix tends to condensate at a single site. In certain types of perturbations and in the limit of the large components, this condensation becomes a sharp phase transition, the mechanism of which can be identified with the Bose–Einstein condensation in a mathematical level. We study this Bose–Einstein like condensation phenomenon by means of the replica method. We first derive a formula to calculate the minimal eigenvalue and the statistical properties of $\boldsymbol{x}_\mathrm{min}$ . Then, we apply the formula for two solvable cases: when the distribution of the perturbation has the double peak, and when it has a continuous distribution. For the double peak, we find that at the transition point, the participation ratio changes discontinuously from a finite value to zero. On the contrary, in the case of a continuous distribution, the participation ratio goes to zero either continuously or discontinuously, depending on the distribution.

Significance The jamming transition is a key property of granular materials, including sand and dense suspensions. In the generic situation of nonspherical particles, its scaling properties are not completely understood. Previous empirical and theoretical work in ellipsoids and spherocylinders indicates that both structural and vibrational properties are fundamentally affected by shape. Here we explain these observations using a combination of marginal stability arguments and the replica method. We unravel a universality class for particles with internal degrees of freedom and derive how the structure of packings and their vibrations scale as the particles evolve toward spheres.

The swap Monte Carlo algorithm combines the translational motion with the exchange of particle species and is unprecedentedly efficient for some models of glass former. In order to clarify the physics underlying this acceleration, we study the problem within the mean field replica liquid theory. We extend the Gaussian Ansatz so as to take into account the exchange of particles of different species, and we calculate analytically the dynamical glass transition points corresponding to the swap and standard Monte Carlo algorithms. We show that the system evolved with the standard Monte Carlo algorithm exhibits the dynamical transition before that of the swap Monte Carlo algorithm. We also test the result by performing computer simulations of a binary mixture of the Mari-Kurchan model, both with standard and swap Monte Carlo. This scenario provides a possible explanation for the efficiency of the swap Monte Carlo algorithm. Finally, we discuss how the thermodynamic theory of the glass transition should be modified based on our results.