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有噪声耦合振子的普适临界行为：重整化群研究

Universal critical behavior of noisy coupled oscillators: A renormalization group study

Thomas Risler, Jacques Prost, Frank Julicher

备注：
24 pages, published in 2005
发表信息：
Phys. Rev. E 72, 016130 (2005)
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摘要：

我们表明，大量的噪声耦合振子的同步过渡是一个例子，远离热力学平衡的动态临界点。在$d$维空间中排列在晶格上并通过最近邻相互作用耦合的这种临界振子的普遍行为可以使用场论方法来研究。与齐次振荡不稳定性（或耦合振子的Hopf分叉）的临界点相关的场论是具有加性噪声的复Ginzburg-Landau方程。我们在$4-ε$维空间中进行微扰重整化群（RG）研究。我们开发了一个RG计划，消除了相位和频率的振荡使用的尺度依赖振荡参考系。在微扰理论中，在一个双圈阶的Callan-Symanzik RG格式中，我们发现RG不动点形式上与实Ginzburg-Landau理论的模型$A$动力学的不动点有关，其序参量具有O（2）对称性.因此，耦合振子的主要临界指数与平衡场论的相同。与平衡临界点的这种形式上的联系强加了临界状态下耦合振子的相关性和响应函数之间的关系。由于系统远离热力学平衡运行，波动耗散关系发生了强烈的破坏，其特征在于有效温度的普遍发散。临界振子与平衡临界点之间的形式关系表明，在二维以上的临界振子中存在长程相序。

英文摘要：

We show that the synchronization transition of a large number of noisy coupled oscillators is an example for a dynamic critical point far from thermodynamic equilibrium. The universal behaviors of such critical oscillators, arranged on a lattice in a $d$-dimensional space and coupled by nearest neighbors interactions, can be studied using field theoretical methods. The field theory associated with the critical point of a homogeneous oscillatory instability (or Hopf bifurcation of coupled oscillators) is the complex Ginzburg-Landau equation with additive noise. We perform a perturbative renormalization group (RG) study in a $4-ε$ dimensional space. We develop an RG scheme that eliminates the phase and frequency of the oscillations using a scale-dependent oscillating reference frame. Within a Callan-Symanzik RG scheme to two-loop order in perturbation theory, we find that the RG fixed point is formally related to the one of the model $A$ dynamics of the real Ginzburg-Landau theory with an O(2) symmetry of the order parameter. Therefore, the dominant critical exponents for coupled oscillators are the same as for this equilibrium field theory. This formal connection with an equilibrium critical point imposes a relation between the correlation and response functions of coupled oscillators in the critical regime. Since the system operates far from thermodynamic equilibrium, a strong violation of the fluctuation-dissipation relation occurs and is characterized by a universal divergence of an effective temperature. The formal relation between critical oscillators and equilibrium critical points suggests that long-range phase order exists in critical oscillators above two dimensions.

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