排斥性排序扩散的非平衡动力学：膨胀晶体
Out of equilibrium dynamics of repulsive ranked diffusions: the expanding crystal
AI总结：
本文研究一维库仑排斥排序扩散的非平衡动力学，借助 Lieb-Liniger 映射与含噪 Burgers 方程，精确刻画膨胀晶体的位置分布、高斯涨落及高阶长程关联。
摘要：
我们研究了一维中具有库仑排斥线性相互作用的 $N$ 个粒子的非平衡 Langevin 动力学。这是所谓 jellium 模型（无约束）的动力学版本，也称为排序扩散。利用到量子玻色子 Lieb-Liniger 模型的映射，我们得到了所有粒子均从原点出发时，$N$ 个粒子在时刻 $t$ 的位置联合分布的精确公式。鞍点分析表明，系统在长时间收敛为线性膨胀的晶体。经过适当重标度后，这一动力学状态类似于含时有效二次势中的平衡晶体。这一类比使我们能够研究围绕完美晶体的涨落，其在领头阶为高斯型。然而，仍存在偏离该高斯行为的部分，这些偏差体现了纯粹源于动力学的长程关联，并由例如粒子间间距的高阶累积量所表征，我们对其进行了精确计算。我们利用本文作者之一近期提出的基于含噪 Burgers 方程的方法对这些结果加以补充。在大 $N$ 极限下，气体的平均密度可在任意时刻由确定性黏性 Burgers 方程的解得到。该方法定量描述了较短时间下的稠密区域。我们的预测与有限 $N$ 和大 $N$ 情形的数值模拟吻合良好。
英文摘要：
We study the non-equilibrium Langevin dynamics of $N$ particles in one dimension with Coulomb repulsive linear interactions. This is a dynamical version of the so-called jellium model (without confinement) also known as ranked diffusion. Using a mapping to the Lieb-Liniger model of quantum bosons, we obtain an exact formula for the joint distribution of the positions of the $N$ particles at time $t$, all starting from the origin. A saddle point analysis shows that the system converges at large time to a linearly expanding crystal. Properly rescaled, this dynamical state resembles the equilibrium crystal in a time dependent effective quadratic potential. This analogy allows to study the fluctuations around the perfect crystal, which, to leading order, are Gaussian. There are however deviations from this Gaussian behavior, which embody long-range correlations of purely dynamical origin, characterized by the higher order cumulants of, e.g., the gaps between the particles, that we calculate exactly. We complement these results using a recent approach by one of us in terms of a noisy Burgers equation. In the large $N$ limit, the mean density of the gas can be obtained at any time from the solution of a deterministic viscous Burgers equation. This approach provides a quantitative description of the dense regime at shorter times. Our predictions are in good agreement with numerical simulations for finite and large $N$.
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