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Bjorken flow in one-dimensional relativistic magnetohydrodynamics with magnetization

We study the one-dimensional, longitudinally boost-invariant motion of an ideal fluid with infinite conductivity in the presence of a transverse magnetic field, i.e., in the ideal transverse magnetohydrodynamical limit. In an extension of our previous work Roy et al., [Phys. Lett. B 750, 45 (2015)], we consider the fluid to have a non-zero magnetization. First, we assume a constant magnetic susceptibility $χ_{m}$ and consider an ultrarelativistic ideal gas equation of state. For a paramagnetic fluid (i.e., with $χ_{m}>0$), the decay of the energy density slows down since the fluid gains energy from the magnetic field. For a diamagnetic fluid (i.e., with $χ_{m}<0$), the energy density decays faster because it feeds energy into the magnetic field. Furthermore, when the magnetic field is taken to be external and to decay in proper time $τ$ with a power law $\simτ^{-a}$, two distinct solutions can be found depending on the values of $a$ and $χ_m$. Finally, we also solve the ideal magnetohydrodynamical equations for one-dimensional Bjorken flow with a temperature-dependent magnetic susceptibility and a realistic equation of state given by lattice-QCD data. We find that the temperature and energy density decay more slowly because of the non-vanishing magnetization. For values of the magnetic field typical for heavy-ion collisions, this effect is, however, rather small. Only for magnetic fields which are about an order of magnitude larger than expected for heavy-ion collisions, the system is substantially reheated and the lifetime of the quark phase might be extended.