We numerically study the relaxation dynamics and associated criticality of non-Brownian frictionless soft spheres below jamming in spatial dimensions \(d=2\), 3, 4, and 8, and in the mean-field Mari–Kurchan model. We discover non-trivial finite-size and volume fraction dependences of the relaxation time associated to the relaxation of unjammed packings. In particular, the relaxation time is shown to diverge logarithmically with system size at any density below jamming, and no critical exponent can characterise its behaviour approaching jamming. In mean-field, the relaxation time is instead well-defined: it diverges at jamming with a critical exponent that we determine numerically and differs from an earlier mean-field prediction. We rationalise the finite d logarithmic divergence using an extreme-value statistics argument in which the relaxation time is dominated by the most connected region of the system. The same argument shows that the earlier proposition that relaxation dynamics and shear viscosity are directly related breaks down in large systems. The shear viscosity of non-Brownian packings is well-defined in all d in the thermodynamic limit, but large finite-size effects plague its measurement close to jamming.

我们在空间尺寸\（d = 2 \），3、4和8以及平均场Mari-Kurchan模型中，数值模拟了干扰以下非布朗无摩擦软球的松弛动力学和相关的临界度。我们发现与非阻塞填料的松弛相关的松弛时间的非平凡的有限尺寸和体积分数依赖性。特别是，松弛时间显示为在干扰以下的任何密度下与系统大小成对数关系，并且没有关键指数可以表征其接近干扰的行为。相反，在平均场中，弛豫时间定义明确：它在干扰时与我们通过数值确定的临界指数不同，并且不同于先前的平均场预测。我们理顺有限d使用极值统计参数的对数散度，其中弛豫时间由系统中最相关的区域控制。相同的论点表明，在大型系统中，松弛动力学和剪切粘度直接相关的早期命题被打破了。非布朗填料的剪切粘度在热力学极限的所有d内都有很好的定义，但是较大的有限尺寸效应困扰着它的测量，接近堵塞。