论文标题
耦合振荡器的同步 - 相变和熵
Synchronization of Coupled Oscillators -- Phase Transitions and Entropies
论文作者
论文摘要
在过去的半个世纪中，液态气相过渡和磁化相变已得到充分理解。在定义订单参数后，$ r $，可以得出$ r = 0 $的$ r = 0 $，$ t> t_c $以及$ r \ propto（t_c -t）^γ$在$ t <t_c $的最低订单下如何。 $γ$的值似乎不取决于系统的物理细节，而是在很大程度上取决于维度。一维系统不存在相变。对于四个或多个维度的系统，每个单元都与许多邻居相互作用，以保证均值场上的方法。平均场近似导致$γ= 1/2 $。在本文中，我们制定了耦合振荡器的现实系统。每个振荡器都通过$ n $状态的环状1D阵列向前移动，振荡器从州$ i $ $ $ $ $ $ i+1 $进行的速度取决于州$ i+1 $ $ 1 $和$ i-1 $的人群。我们研究如何从状态上的均匀分布到簇分布的相变。聚类的分布意味着振荡器已同步。我们定义一个订单参数，发现关键指数的平均字段值为任何$ n $的1/2。但是，随着状态数量的增加，相转换的发生量为$ t_c $的较小值。我们提出了严格的数学和简单的近似，以发展对该系统中相变的理解。我们解释了为什么1/2的临界指数值预计将是强大的，我们讨论了湿的实验设置以证实我们的发现。
Over the last half century the liquid-gas phase transition and the magnetization phase transition have come to be well understood. After an order parameter, $r$, is defined, it can be derived how $r=0$ for $T>T_c$ and how $r \propto (T_c - T)^γ$ at lowest order for $T < T_c$. The value of $γ$ appears to not depend on physical details of the system, but very much on dimensionality. No phase transitions exist for one-dimensional systems. For systems of four or more dimensions, each unit is interacting with sufficiently many neighbors to warrant a mean-field approach. The mean-field approximation leads to $γ= 1/2$. In this article we formulate a realistic system of coupled oscillators. Each oscillator moves forward through a cyclic 1D array of $n$ states and the rate at which an oscillator proceeds from state $i$ to state $i+1$ depends on the populations in states $i+1$ and $i-1$. We study how the phase transitions occur from a homogeneous distribution over the states to a clustered distribution. A clustered distribution means that oscillators have synchronized. We define an order parameter and we find that the critical exponent takes on the mean-field value of 1/2 for any $n$. However, as the number of states increases, the phase transition occurs for ever smaller values of $T_c$. We present rigorous mathematics and simple approximations to develop an understanding of the phase transitions in this system. We explain why and how the critical exponent value of 1/2 is expected to be robust and we discuss a wet-lab experimental setup to substantiate our findings.
