论文标题
在存在较强相关性的情况下，熵扩张和较大的偏差
Entropic extensivity and large deviations in the presence of strong correlations
论文作者
论文摘要
标准大偏差理论（LDT）反映了描述短距离汉密尔顿系统的热平衡的玻尔兹曼 - 吉布斯（BG）因子，其速度分布是麦克斯韦。它通常适用于满足中心限制定理（CLT）等的系统。当我们专注于典型复杂系统（例如经典的远程哈密顿系统）的固定状态时，CLT和LDT都需要概括。我们将重点放在涉及强相关的可交换随机变量的比例不变随机过程上，该变量通过Laplace-de Finetti定理，众所周知，该变量可产生长尾$ Q $ -Gaussian $ n \ n \ to \ f \ hoftty $ in \ \ in \ infty $吸引者在分布空间中的吸引力（$ 1 <q <q <3）$。我们提出了强烈的数值指示：相应的LDT概率分布由$ p（n，z）= p_0 \，e_q^{ - r_q（z）n} = p_0 [1-（1- q）r_q（z）n] The rate function $r_q(z)$ seemingly equals the relative nonadditive $q_r$-entropy per particle, with $q_r \simeq \frac{7}{10} + \frac{6}{10}\frac{1}{Q-1}$, thus exhibiting a singularity at $Q=1$ and recovering the BG value $q_r=1$ in the $ q \至3 $限制。让我们强调，从热力学的传说结构中，$ r_q（z）n $的扩展似乎与预期的结构保持一致。目前对一个相对简单模型的分析在某种程度上反映了Spin-1/2远程相互作用的铁磁体（例如，具有强烈的各向异性XY耦合）可能有助于更深入地了解具有全球相关性和其他复杂系统的非平衡系统。
The standard Large Deviation Theory (LDT) mirrors the Boltzmann-Gibbs (BG) factor which describes the thermal equilibrium of short-range Hamiltonian systems, the velocity distribution of which is Maxwellian. It is generically applicable to systems satisfying the Central Limit Theorem (CLT), among others. When we focus instead on stationary states of typical complex systems (e.g., classical long-range Hamiltonian systems), both the CLT and LDT need to be generalized. We focus here on a scale-invariant stochastic process involving strongly-correlated exchangeable random variables which, through the Laplace-de Finetti theorem, is known to yield a long-tailed $Q$-Gaussian $N\to\infty$ attractor in the space of distributions ($1 < Q<3)$. We present strong numerical indications that the corresponding LDT probability distribution is given by $P(N,z)=P_0\,e_q^{-r_q(z)N}=P_0[1-(1-q)r_q(z)N]^{1/(1-q)}$ with $q=2-1/Q \in (1,5/3)$. The rate function $r_q(z)$ seemingly equals the relative nonadditive $q_r$-entropy per particle, with $q_r \simeq \frac{7}{10} + \frac{6}{10}\frac{1}{Q-1}$, thus exhibiting a singularity at $Q=1$ and recovering the BG value $q_r=1$ in the $Q \to 3$ limit. Let us emphasize that the extensivity of $r_q(z)N$ appears to be verified, consistently with what is expected, from the Legendre structure of thermodynamics, for a total entropy. The present analysis of a relatively simple model somewhat mirroring spin-1/2 long-range-interacting ferromagnets (e.g., with strongly anisotropic XY coupling) might be helpful for a deeper understanding of nonequilibrium systems with global correlations and other complex systems.
