论文标题
接近复杂系统的大偏差理论
Approaching a large deviation theory for complex systems
论文作者
论文摘要
标准的大偏差理论（LDT）在数学上用玻尔兹曼·基布斯因子说明，该因子描述了短距离相互作用的多体哈密顿系统的热平衡，其速度分布是麦克斯韦。它通常适用于满足中心极限定理（CLT）的系统。当我们专注于典型复杂系统的固定状态时（例如，经典的远程相互作用的多体汉密尔顿系统（例如自我磨损）），CLT以及可能的LDT需要概括。具体而言，当分布空间中的$ n \ to \ infty $吸引子（$ n $是自由度的数量）是一个$ q $ -Gaussian（一种基于标准高斯案例的基于$ q $ entrepropy的概括，以$ q $ q = 1 $恢复）与Q $ Q $ Q $ Quussian clt的概率分配的方法，该方法与clt clt a ippore a a a a aS a assm a ans a as a aS a aS a aS a asm a a aS a aS a aS a aS a aS a aS a aS a asm a aS a. 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 $和$ Q $预计将简单地连接，包括特定情况$ q = q = 1 $。这种$ Q $ - 指数的论点预计将与$ n $成比例，类似于多体汉密尔顿系统的热力学熵。我们在这里提供数值证据，通过分析标准图，生物灭绝和地震的相干噪声模型，Ehrenfest的狗 - 狗 - 丝 - 狗 - 丝状模型和随机游行雪崩，来支持渐近幂。
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.
