分形上选民模型精确关联中的老化现象
Ageing in the exact correlations of the voter model on a fractal
- Université de Lorraine（洛林大学）
- Universidade de Lisboa（里斯本大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究在分形基底上精确求解选民模型的老化现象，推导出动力学指数和自关联指数，并证实活性界面衰减指数，为分形系统中的非平衡动力学提供了理论依据。
AI中文摘要：
在具有最近邻相互作用的分形基底上的选民模型中，我们找到了单时间和双时间关联函数包络的精确行为。此处，分形基底的几何由其非整数几何分形维数$d_f$描述，其拓扑和扩散输运则由不同的谱维数$d_s$描述。在关联函数的运动方程层面上，这可以通过考虑空间依赖的扩散常数${\cal D}(r)\sim r^{-\theta}$来建模，该常数隐含谱指数$\theta$，而$\theta$本身是$d_f$和$d_s$的函数。通过标度假设，我们确认了老化的一般现象学，并推导出动力学指数${z}=2+\theta$和自关联指数$\lambda=d_f$。显式求得的动力学标度函数被证明仅依赖于谱维数$d_s$。对于$d_s<2$，活性界面密度包络随时间衰减由指数$\alpha=1-d_s/2$描述，这证实了先前数值模拟的结果。
英文摘要：
The exact behaviour of the enveloppes of the single-time and two-time correlators is found for the voter model on a fractal substrate, with nearest-neighbour interactions. Herein the geometry of the fractal substrate is described by its non-integer geometric fractal dimension $d_f$, and its topology and diffusive transport by the distinct spectral dimension $d_s$. On the level of the equations of motion of the correlators this can be modelled by considering a space-dependent diffusion constant ${\cal D}(r)\sim r^{-θ}$ which implies the spectral index $θ$, itself a function of $d_f$ and $d_s$. With a scaling ansatz, the generic phenomenology of ageing is confirmed and the dynamic exponent ${z}=2+θ$ and the autocorrelation exponent $λ=d_f$ are derived. The explicitly found dynamic scaling functions are shown to depend only on the spectral dimension $d_s$. The decay of the enveloppe of the density of active interfaces with time is described by the exponent $α=1-d_s/2$ for $d_s<2$, confirming the results of preexisting numerical simulations.
