Fedotov Sergei, Bashkirtseva Irina, Ryashko Lev
School of Mathematics, The University of Manchester, M60 1QD, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2006 Jun;73(6 Pt 2):066307. doi: 10.1103/PhysRevE.73.066307. Epub 2006 Jun 22.
The effects of stochastic perturbations in a nonlinear alpha Omega-dynamo model are investigated. By using transformation of variables we identify a "slow" variable that determines the global evolution of the non-normal alpha Omega-dynamo system in the subcritical case. We apply an adiabatic elimination procedure to derive a closed stochastic differential equation for the slow variable for which the dynamics is determined along one of the eigenvectors of the full system. We derive the corresponding Fokker-Planck equation and show that the generation of a large scale magnetic field can be regarded as a first-order phase transition. We show that the an advantage of the reduced system is that we have explicit expressions for both the stochastic and deterministic potentials. We also obtain the stationary solution of the Fokker-Planck equation and show that an increase in the intensity of the multiplicative noise leads to qualitative changes in the stationary probability density function. The latter can be interpreted as a noise-induced phase transition. By a numerical simulation of the stochastic galactic dynamo model, we show that the qualitative behavior of the "empirical" stationary pdf of the slow variable is accurately predicted by the stationary pdf of the reduced system.
研究了非线性αΩ发电机模型中随机扰动的影响。通过变量变换，我们确定了一个“慢”变量，它决定了亚临界情况下非正规αΩ发电机系统的全局演化。我们应用绝热消除程序来推导慢变量的一个封闭随机微分方程，其动力学沿着完整系统的一个特征向量确定。我们推导了相应的福克 - 普朗克方程，并表明大规模磁场的产生可被视为一阶相变。我们表明简化系统的一个优点是我们对随机势和确定性势都有明确的表达式。我们还得到了福克 - 普朗克方程的平稳解，并表明乘性噪声强度的增加会导致平稳概率密度函数的定性变化。后者可被解释为噪声诱导的相变。通过对随机星系发电机模型的数值模拟，我们表明简化系统的平稳概率密度函数准确地预测了慢变量“经验”平稳概率密度函数的定性行为。
