论文标题
非线性pólyaurn的非平衡相变的通用功能
Universal function of the non-equilibrium phase transition of nonlinear Pólya urn
论文作者
论文摘要
我们研究了非线性pólyaurn的相变和临界特性，这是一个简单的二进制随机过程$ x（t）\ in \ {0,1 \}，t = 1，\ cdots $带有反馈机制。令$ f $是单位间隔到本身的连续功能，而$ z（t）$是第一个$ t $变量$ x（1），\ cdots，x（t）$的比例，该值将值1。$ x（t+1）$带带有概率$ f（z（t））$的值1。当$ f（z）$更改的稳定固定点的数量时，系统会经历非平衡相变，而顺序参数是自相关函数的限制值。当系统为$ z_ {2} $对称时，即$ f（z）= 1-f（1-z）$，会发生连续的相变，并且自相关函数在$ \ ln（t+1）^{ - 1/2} g（ - 1/2} g（\ ln（t+1）$和适当的定义$ corriation cortor corrial cortor corrial cortor corrial corrial cortor corrial cortor corrial cortor corrial cortor corrial cortor and cortor corrial cortor cortor cortor cortor cortor cortor cortor cortor cortor cortor cortor correl $ g（x）$。我们使用随机微分方程和关于随机噪声强度的扩展来分析$ g（x）$。 $ g（x）$确定自相关函数的渐近行为附近的临界点和相变的通用类别。
We study the phase transition and the critical properties of a nonlinear Pólya urn, which is a simple binary stochastic process $X(t)\in \{0,1\},t=1,\cdots$ with a feedback mechanism. Let $f$ be a continuous function from the unit interval to itself, and $z(t)$ be the proportion of the first $t$ variables $X(1),\cdots,X(t)$ that take the value 1. $X(t+1)$ takes the value 1 with probability $f(z(t))$. When the number of stable fixed points of $f(z)$ changes, the system undergoes a non-equilibrium phase transition and the order parameter is the limit value of the autocorrelation function. When the system is $Z_{2}$ symmetric, that is, $f(z)=1-f(1-z)$, a continuous phase transition occurs, and the autocorrelation function behaves asymptotically as $\ln(t+1)^{-1/2}g(\ln(t+1)/ξ)$, with a suitable definition of the correlation length $ξ$ and the universal function $g(x)$. We derive $g(x)$ analytically using stochastic differential equations and the expansion about the strength of stochastic noise. $g(x)$ determines the asymptotic behavior of the autocorrelation function near the critical point and the universality class of the phase transition.
