http://arxiv.org/abs/1909.04836
We study the dynamics and statistics of real vector fields in flat (n+1)-dimensional space-time with an emphasis on the field topology and stochasticity in physical applications such as stochastic magnetic and velocity fields in cosmological systems. We show that the natural field topology defined by the metric, induced by the Euclidean vector norm, is physically implausible. However, any vector field corresponds to a dynamical system with a topology in the corresponding phase space. This phase space topology, unlike the natural topology in Euclidean space, is defined using open balls that contain nearby vectors in both vector space and real space and hence is more physical. In addition, it is preserved under time translation if certain conditions including time reversal invariance are satisfied by the field. If these mathematical conditions are not satisfied, therefore, the field’s topology can spontaneously change as the field evolves in time. In this context, similar to topological entropy, which measures the complexity of a dynamical system in the phase space, a simple quantity is defined for a vector field which measures its spatial complexity in real space. For stochastic fields, this spatial complexity can be taken as a measure of the field’s stochasticity level. Generalizing a previous work based on renormalization group invariance, we show that corresponding to any arbitrary vector field, there exists a scalar field whose properties provide a means to quantify the vector field’s spatial complexity, stochasticity level and dissipation rate.
A. Jafari and E. Vishniac
Thu, 12 Sep 19
62/84
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