Risler Thomas, Prost Jacques, Jülicher Frank
Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzerstrasse 38, 01187 Dresden, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2005 Jul;72(1 Pt 2):016130. doi: 10.1103/PhysRevE.72.016130. Epub 2005 Jul 29.
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-epsilon)-dimensional space. We develop an RG scheme that eliminates the phase and frequency of the oscillations using a scale-dependent oscillating reference frame. Within Callan-Symanzik's 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 O2 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.
我们表明，大量有噪声的耦合振子的同步转变是远离热力学平衡的动态临界点的一个例子。对于这类临界振子，它们排列在d维空间的晶格中并通过最近邻相互作用耦合，其普适行为可以用场论方法来研究。与均匀振荡不稳定性（或耦合振子的霍普夫分岔）临界点相关的场论是带有加性噪声的复金兹堡 - 朗道方程。我们在（4 - ε）维空间中进行微扰重整化群（RG）研究。我们开发了一种RG方案，该方案使用与尺度相关的振荡参考系来消除振荡的相位和频率。在微扰理论中到两圈阶的卡伦 - 西曼齐克RG方案内，我们发现RG不动点在形式上与具有序参量O(2)对称性的实金兹堡 - 朗道理论的模型A动力学的不动点相关。因此，耦合振子的主导临界指数与这个平衡场论的相同。与平衡临界点的这种形式联系在临界区域中对耦合振子的关联函数和响应函数施加了一种关系。由于系统远离热力学平衡运行，波动耗散关系会被强烈违反，其特征是有效温度的普适发散。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位序。
