Networks of coadjoint orbits: From geometric to statistical mechanics
Networks of coadjoint orbits: From geometric to statistical mechanics
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共交轨道网络：从几何力学到统计力学
DOI：
10.3934/jgm.2019023
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发表时间：
2018
影响因子：
0.8
通讯作者：
So Takao
中科院分区：
文献类型：
--
作者：
Alexis Arnaudon;So Takao
在几何力学的框架下，推广了统计力学中经典的Heisenberg网络模型，得到了一类具有对称群G$的网络模型，该模型演化为Lie-Poisson系统.我们考虑了两种自旋耦合方式：一种是通过动量耦合，另一种是通过位置耦合，并利用能量Casimir方法详细研究了平衡解及其相应的非线性稳定性。然后我们以G=SO（3）$为例，看到动量耦合系统简化为具有大质量自旋的经典海森堡模型，位置耦合情况简化为具有类似于重顶的破缺对称群SO（3）/SO（2）$的新系统。在后一个系统中，我们在数值上观察到了一个有趣的同步现象，为某一类初始条件。添加一种类型的噪声和耗散，保持网络模型的余伴随轨道，我们发现不变测度由吉布斯测度给出，温度的概念由此定义。然后，我们观察到一个令人惊讶的“三峰”相变在重顶状晶格模型，自旋从一个平衡位置切换到另一个之前失去磁化，因为我们增加了温度。这项工作只是连接几何力学与统计力学的第一步，还有几个有趣的问题有待进一步研究。
A class of network models with symmetry group $G$ that evolve as a Lie-Poisson system is derived from the framework of geometric mechanics, which generalises the classical Heisenberg model studied in statistical mechanics. We considered two ways of coupling the spins: one via the momentum and the other via the position and studied in details the equilibrium solutions and their corresponding nonlinear stability properties using the energy-Casimir method. We then took the example $G=SO(3)$ and saw that the momentum-coupled system reduces to the classical Heisenberg model with massive spins and the position-coupled case reduces to a new system that has a broken symmetry group $SO(3)/SO(2)$ similar to the heavy top. In the latter system, we numerically observed an interesting synchronisation-like phenomenon for a certain class of initial conditions. Adding a type of noise and dissipation that preserves the coadjoint orbit of the network model, we found that the invariant measure is given by the Gibbs measure, from which the notion of temperature is defined. We then observed a surprising `triple-humped' phase transition in the heavy top-like lattice model, where the spins switched from one equilibrium position to another before losing magnetisation as we increased the temperature. This work is only a first step towards connecting geometric mechanics with statistical mechanics and several interesting problems are open for further investigation.
随机能量-卡西米尔法
DOI：
10.1016/j.crme.2018.01.003
发表时间：
2018
期刊：
Comptes Rendus Mécanique
影响因子：
--
作者：
Arnaudon A
通讯作者：
Arnaudon A
影响因子：
3
作者：
Arnaudon A
通讯作者：
Arnaudon A