Kooperative Phänomene in raum-zeitlichen chaotischen Modellen und ihre Beziehung zu Nichtgleichgewichtsphasenübergängen

该项目旨在为高维混沌系统领域做出贡献，高维混沌系统仍然对我们的理解构成许多挑战。在这里可能会出现新的合作现象，如相变，这在几个自由度的系统中是不存在的。为了模拟这样的特征，一类特殊的时空混沌系统，即耦合映射格子，已经被证明是非常有效的。最近，人们发展了一类映射格子，它可以在微扰区通过解析的粗粒化过程映射到随机自旋动力学。在我的博士论文中，我们对一维对称耦合链进行了应用，得到了一个平衡的动力学伊辛模型。在空间二维系统中，特别是对于非对称耦合，当相应的随机自旋动力学不服从详细的平衡时，期望有新的特征，从而引起真正的非平衡现象，如通过系统的能量通量。为了揭示非平衡相变，我们将从混沌动力学系统中推导出相应的主方程，并用动力学平均场和重整化群方法进行研究。通过这种方式，与施密特曼教授和齐亚教授的研究产生了许多联系，他们用类似的方法研究了不同的非平衡自旋模型。

The project intends to contribute to the field of high dimensional chaotic systems which still pose many challenges to our understanding. Here new cooperative phenomena like phase transitions can emerge, which are not present in systems with few degrees of freedom. To model such features a particular class of spatio-temporal chaotic systems, namely coupled map lattices, have turned out to be very fruitful. Recently, a class of map lattices has been developed which can be mapped to a stochastic spin dynamics by an analytical coarse graining procedure in a perturbative regime. An application to a one dimensional symmetrically coupled chain was worked out in my Ph.D. thesis, resulting in an equilibrium kinetic Ising model. Novel features are expected in spatially two dimensional systems and in particular for asymmetric couplings, when the corresponding stochastic spin dynamics does not obey detailed balance giving rise to genuine non-equilibrium phenomena like energy fluxes through the system. In order to shed some light on non-equilibrium phase transitions the corresponding master equation will be derived from the chaotic dynamical system and will be studied with dynamical mean field and renormalization group approaches. In this way many connections to the research of Prof. Schmittmann and Prof. Zia arise who have studied diverse nonequilibrium spin models with similar methods.