AI 中文总结
该研究分析经典分形子哈密顿量的尺度与形状动力学，发现特殊分形子模型可呈现类宇宙学演化，为无微调的宇宙学动力学提供了玩具模型。
AI 中文摘要
经典分形子是哈密顿系统，在投影到构型或形状变量后可形成吸引子，尽管其完整相空间不存在吸引子。我们研究一类尺度不变、偶极守恒的双参数分形子哈密顿量$H_{\alpha,\beta}$。通过将坐标分离为尺度和形状，我们得到自治的形状动力学，其允许不动点，这些不动点给出纯尺度演化形式$R(t)\propto |t|^{\alpha/(\alpha-\beta)}$。决定膨胀粒子分布的形状不动点是幂律里斯势的中心构型。特殊模型$(\alpha,\beta)=(-2,1)$具有唯一性：其尺度演化呈现爱因斯坦-德西特形式$R(t)\propto |t|^{2/3}$，其不动点方程是等质量牛顿中心构型，其大$N$分布为均匀球，且其同宿轨迹存在零能牛顿引力对偶。这些不动点是局部稳定的，中等$N$的模拟从随机初始数据趋近于它们。大$N$模拟揭示了更丰富的不动点类：近似固定物理尺寸的束缚团簇保留内部运动，而其中心趋近于不等质量牛顿中心构型并保持大尺度均匀性。尺度分离猜想为这些中心产生有效的不等质量分形子动力学及对应的零能牛顿引力对偶。轨迹通常呈现双向时间箭头：尺度和形状复杂度从雅努斯点增长，而玻尔兹曼熵对数增长。这些特征共同再现了平坦物质主导宇宙学的显著结构。在特殊分形子模型中，所有这些宇宙学类似物均作为吸引子性质出现，使其成为无需微调的宇宙学动力学玩具模型。
英文摘要
Classical fractons are Hamiltonian systems that can develop attractors after projection onto configuration or shape variables, although the full phase space admits none. We study a scale-invariant, dipole-conserving two-parameter family of fracton Hamiltonians $H_{α,β}$. By separating coordinates into scale and shape, we obtain autonomous shape dynamics that admit fixed points which leave a purely scale evolution of the form $R(t)\propto |t|^{α/(α-β)}$. The shape fixed points, which determine the distribution of the expanding particles, are central configurations of power-law Riesz potentials. The distinguished model $(α,β)=(-2,1)$ is unique: its scale evolution takes the Einstein-de Sitter form $R(t)\propto |t|^{2/3}$, its fixed-point equation is the equal-mass Newtonian central-configuration, its large-$N$ distribution is a homogeneous ball, and its homothetic trajectories admit a zero-energy Newtonian gravitational dual. The fixed points are locally stable, and simulations at moderate $N$ approach them from random initial data. Large $N$ simulations reveal a richer class of fixed-points: bound clusters of approximately fixed physical size retain internal motion, while their centers approach unequal-mass Newtonian central configurations and preserve large-scale homogeneity. A scale-separation conjecture yields an effective unequal-mass fracton dynamics for the centers and a corresponding zero-energy Newtonian gravitational dual. Trajectories generically exhibit a bidirectional arrow of time: scale and shape complexity grow away from a Janus point, while Boltzmann entropy grows logarithmically. Together, these features reproduce the salient structure of a flat matter-dominated cosmology. In the distinguished fracton model, all these cosmological analogues emerge as attractor properties, making it a toy model for cosmological dynamics without fine-tuning.
Comments64 pages, 10 figures
