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Time-evolution of magnetic field in hot nuclear matter with fluctuating topological charge

The time-evolution of the magnetic field in hot homogeneous nuclear matter has two qualitatively different stages separated by the sphaleron transition time $τ_c$. At early times the axial chemical potential and the corresponding chiral conductivity $σ_χ$ are slow functions of time. The soft chiral modes $k<σ_χ$ of the magnetic field grow exponentially with time, which is known as the chiral instability. At later times $σ_χ$ fluctuates due to the sphaleron transitions and can be regarded as a random process. It is argued that the average magnetic field is exponentially damped at later times. The time-evolution of the average magnetic energy is more complicated and depends on the electrical conductivity of the chiral matter but does not depend on chirality. It exhibits instability only if the matter is a poor electrical conductor, such as the quark-gluon plasma near the critical temperature. The precise conditions for the instability and the growth rate of the unstable modes are derived.