复杂系统的大偏差理论探讨
Approaching a large deviation theory for complex systems
摘要：
标准的大偏差理论（LDT）在数学上可以用Boltzmann-Gibbs因子来描述，该因子描述了速度分布为麦克斯韦分布的短程相互作用多体哈密顿系统的热平衡。它一般适用于满足中心极限定理的系统。当我们转而关注典型复杂系统的定态时（例如，经典的远程相互作用多体哈密顿系统，如自引力系统），CLT，也可能是LDT，需要推广。具体来说，当分布空间中的$N\to\infty$吸引子（$N$是自由度的数量）是与$Q$广义CLT相关的$Q$ -高斯（标准高斯情况的非加性$q$ -基于熵的广义，对于$Q=1$恢复）时，我们期望LDT概率分布渐近地接近幂律。与概率模型的可用强数值指示一致，该行为可能与$q$ -指数相关（定义为$e_q^x\equiv\left[1+(1-q)x\right]^{1/(1-q)}$，这是标准指数形式的推广，直接恢复为$q=1$）; $q$和$Q$预计是简单连接的，包括特殊情况$q=Q=1$。这种$q$ -指数的辐角应该与$N$成比例，类似于多体哈密顿系统的熵。我们在这里提供的数值证据支持渐近幂律通过分析标准地图，相干噪声模型的生物防御和地震，Escherichfest狗蚤模型，和随机行走雪崩。
英文摘要：
The standard Large Deviation Theory (LDT) is mathematically illustrated by the Boltzmann-Gibbs factor which describes the thermal equilibrium of short-range-interacting many-body Hamiltonian systems, the velocity distribution of which is Maxwellian. It is generically applicable to systems satisfying the Central Limit Theorem (CLT). When we focus instead on stationary states of typical complex systems (e.g., classical long-range-interacting many-body Hamiltonian systems, such as self-gravitating ones), the CLT, and possibly also the LDT, need to be generalised. Specifically, when the $N\to\infty$ attractor ($N$ being the number of degrees of freedom) in the space of distributions is a $Q$-Gaussian (a nonadditive $q$-entropy-based generalisation of the standard Gaussian case, which is recovered for $Q=1$) related to a $Q$-generalised CLT, we expect the LDT probability distribution to asymptotically approach a power law. Consistently with available strong numerical indications for probabilistic models, this behaviour possibly is that associated to a $q$-exponential (defined as $e_q^x\equiv\left[1+(1-q)x\right]^{1/(1-q)}$, which is the generalisation of the standard exponential form, straightforwardly recovered for $q=1$); $q$ and $Q$ are expected to be simply connected, including the particular case $q=Q=1$. The argument of such $q$-exponential would be expected to be proportional to $N$, analogously to the thermodynamical entropy of many-body Hamiltonian systems. We provide here numerical evidence supporting the asymptotic power-law by analysing the standard map, the coherent noise model for biological extinctions and earthquakes, the Ehrenfest dog-flea model, and the random-walk avalanches.
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