Kinematic dynamo, supersymmetry breaking, and chaos

The kinematic dynamo (KD) describes the growth of magnetic fields generated by the flow of a conducting medium in the limit of vanishing backaction of the fields onto the flow. The KD is therefore an important model system for understanding astrophysical magnetism. Here, the mathematical correspondence between the KD and a specific stochastic differential equation (SDE) viewed from the perspective of the supersymmetric theory of stochastics (STS) is discussed. The STS is a novel, approximation-free framework to investigate SDEs. The correspondence reported here permits insights from the STS to be applied to the theory of KD and vice versa. It was previously known that the fast KD in the idealistic limit of no magnetic diffusion requires chaotic flows. The KD-STS correspondence shows that this is also true for the diffusive KD. From the STS perspective, the KD possesses a topological supersymmetry, and the dynamo effect can be viewed as its spontaneous breakdown. This supersymmetry breaking can be regarded as the stochastic generalization of the concept of dynamical chaos. As this supersymmetry breaking happens in both the diffusive and the nondiffusive cases, the necessity of the underlying SDE being chaotic is given in either case. The observed exponentially growing and oscillating KD modes prove physically that dynamical spectra of the STS evolution operator that break the topological supersymmetry exist with both real and complex ground state eigenvalues. Finally, we comment on the nonexistence of dynamos for scalar quantities.

Figure 1

The three possible spectra of the stochastic evolution operator of the STS of a model with the phase space being a three-dimentional compact sphere. Such models have only two supersymmetric eigenstates (big dots at the origin) in one-to-one correspondence with the two De Rahm cohomology classes of the 3D sphere in the zeroth ($\mathrm{k}=0$) and the third ($\mathrm{k}=3$) degrees. Spectrum (a) corresponds to the unbroken topological supersymmetry because both ground states are supersymmetric with one of them (for $k=3$) being the state of thermodynamic equilibrium. The other two spectra (b and c) correspond to the “chaotic” dynamics, i.e., to the situations when the ground states (big leftmost dots) are nonsupersymmetric because they have nonzero eigenvalues. The eigenstates of the stochastic evolution operator that represent the modes of the magnetic field in the KD theory are the non-$\widehat{d}$-symmetric pairs of 1- and 2-forms connected by the dashed lines representing the action of the d operator. Note that the SEO spectra for 1- and 2-forms may also have other eigenstates (with positive real parts of their eigenvalues) that cannot be associated with the magnetic field. Those from ${\mathrm{\Omega }}^0$ can be though of as the eigenstates of the gauge potential $\phi \in {\mathrm{\Omega }}^0$ with which one can perform the gauge-invariance transformation of the vector potential, $A\to A+\widehat{d}\phi$, without changing the magnetic field. The additional non-$\widehat{d}$-closed eigenmodes in ${\mathrm{\Omega }}^2$ are unphysical because they correspond to the magnetic field configurations such that $\widehat{d}F=(\text{div}B)d^{3}x\ne 0$; i.e., they contain magnetic monopoles.