经典惯性α-XY铁磁体中q统计与玻尔兹曼-吉布斯统计之间的扩散交叉
Diffusion crossover from/to $q$-statistics to/from Boltzmann-Gibbs statistics in the classical inertial $α$-XY ferromagnet
AI总结：
本研究针对经典d维惯性XY模型的角扩散，揭示超扩散到正常扩散的转变规律，首次建立反常扩散指数与q统计熵指数的关联，明确系统尺寸和时间对扩散交叉方向的影响。
摘要：
我们研究了经典d维惯性XY模型中的角扩散，该模型中自旋间相互作用随距离以r^{-α}形式衰减，其中α≥0。在极短时间的弹道区（σ_θ²~t²）之后，观测到一个超扩散区，其σ_θ²~t^{α_D}，α_D约为1.45，该区域的持续时间覆盖初始准稳态及其向以玻尔兹曼-吉布斯温度T_BG为特征的第二个平台的转变。在达到T_BG很久之后，观测到向正常扩散的交叉，即σ_θ²~t。我们首次通过表达式α_D=2/(3-q)，将反常扩散指数α_D与表征时间平均角度和动量概率分布函数（pdfs）的熵指数q关联起来，这些分布由所谓的q-高斯分布f_q(x)∝e_q(-βx²)给出，其中e_q(u)≡[1+(1-q)u]^{1/(1-q)}（e₁(u)=exp(u)）。对于固定尺寸N和足够长的时间，表征角度概率分布函数的指数q_θ趋近于1，表明最终弛豫到玻尔兹曼-吉布斯平衡态。对于固定时间和足够大的N，交叉以相反的方向发生。
英文摘要：
We study the angular diffusion in a classical $d-$dimensional inertial XY model with interactions decaying with the distance between spins as $r^{-α}$, wiht $α\geqslant 0$. After a very short-time ballistic regime, with $σ_θ^2\sim t^2$, a super-diffusive regime, for which $σ_θ^2\sim t^{α_D}$, with $α_D \simeq 1\text{.}45$ is observed, whose duration covers an initial quasistationary state and its transition to a second plateau characterized by the Boltzmann-Gibbs temperature $T_\text{BG}$. Long after $T_\text{BG}$ is reached, a crossover to normal diffusion, $σ_θ^2\sim t$, is observed. We relate, for the first time, via the expression $α_D = 2/(3 - q)$, the anomalous diffusion exponent $α_D$ with the entropic index $q$ characterizing the time-averaged angles and momenta probability distribution functions (pdfs), which are given by the so called $q-$Gaussian distributions, $f_q(x)\propto e_q(-βx^2)$, where $e_q (u) \equiv [1 + (1 - q)u]^{\frac{1}{1 - q}}$ ($e_1(u) = \exp(u)$). For fixed size $N$ and large enough times, the index $q_θ$ characterizing the angles pdf approaches unity, thus indicating a final relaxation to Boltzmann-Gibbs equilibrium. For fixed time and large enough $N$, the crossover occurs in the opposite sense.
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