The BEST framework for the search for the QCD critical point and the chiral magnetic effect

Starting from the kinetic theory description of massive spin-$\frac{1}{2}$ particles in the presence of a magnetic field, equations for relativistic dissipative nonresistive magnetohydrodynamics are obtained in the small polarization limit. We use a relaxation-time approximation for the collision kernel in the relativistic Boltzmann equation and calculate nonequilibrium corrections to the phase-space distribution function of spin-polarizable particles. We demonstrate that our framework naturally leads to emergence of the well-known Einstein--de Haas and Barnett effects. We obtain multiple transport coefficients and show, for the first time, that the coupling between spin and magnetic field appear at gradient order in the hydrodynamic equation. 

/pdf/relativistic-spin-magnetohydrodynamics-35eisvt6.pdf

Relativistic Spin Magnetohydrodynamics

We study relativistic magnetohydrodynamics with longitudinal boost invariance in the presence of chiral magnetic effects and finite electric conductivity. With initial magnetic fields parallel or antiparallel to electric fields, we derive the analytic solutions of electromagnetic fields and the chiral number and energy density in an expansion of several parameters determined by initial conditions. The numerical solutions show that such analytic solutions work well in weak fields or large chiral fluctuations. We also discuss the properties of electromagnetic fields in the laboratory frame.

Anomalous magnetohydrodynamics with longitudinal boost invariance and chiral magnetic effect

Hydrodynamic attractors have recently gained prominence in the context of early stages of ultrarelativistic heavy-ion collisions at the RHIC and LHC. We critically examine the existing ideas on this subject from a phase space point of view. In this picture the hydrodynamic attractor can be seen as a special case of the more general phenomenon of dynamical dimensionality reduction of phase space regions. We quantify this using principal component analysis. Furthermore, we adapt the well known slow-roll approximation to this setting. These techniques generalize easily to higher dimensional phase spaces, which we illustrate by a preliminary analysis of a dataset describing the evolution of a five-dimensional manifold of initial conditions immersed in a 16-dimensional representation of the phase space of the Boltzmann kinetic equation in the relaxation time approximation.

Hydrodynamic attractors in phase space

We have studied analytically the longitudinally boost-invariant motion of a relativistic dissipative fluid with spin. We have derived the analytic solutions of spin density and spin chemical potential as a function of proper time $\tau$ in the presence of viscous tensor and the second order relaxation time corrections for spin. Interestingly, analogous to the ordinary particle number density and chemical potential, we find that the spin density and spin chemical potential decay as $\sim\tau^{-1}$ and $\sim\tau^{-1/3}$, respectively. It implies that the initial spin density may not survive at the freezeout hyper-surface. These solutions can serve both to gain insight on the dynamics of spin polarization in relativistic heavy-ion collisions and as testbeds for further numerical codes.

Analytic solutions of relativistic dissipative spin hydrodynamics with Bjorken expansion

Motivated by the recently found realization of the 1 + 1 dimensional Bjorken flow in ideal and nonideal relativistic magnetohydrodynamics (MHD), we use appropriate symmetry arguments, and determine the evolution of magnetic fields arising from the 3 + 1 dimensional self-similar and Gubser flows in an infinitely conductive relativistic fluid (ideal MHD). In the case of the 3 + 1 dimensional self-similar flow, we arrive at a family of solutions, that are related through a differential equation arising from the corresponding Euler equation. To find the magnetic field evolution from the Gubser flow, we solve the MHD equations of a stationary fluid in a conformally flat dS3 × E1 spacetime. The results are then Weyl transformed back into the Minkowski spacetime. In this case, the temporal evolution of the resulting magnetic field is shown to exhibit a transition between an early time 1/t decay to a 1/t3 decay at a late time. Here, t is the time coordinate. Transverse and longitudinal components of the magnetic fields arising from these flows are also found. The latter turns out to be sensitive to the transverse size of the fluid. In contrast to the result arising from the Gubser flow, the radial domain of validity of the magnetic field arising from the self-similar flow is highly restricted. A comparison of the results suggests that the (conformal) Gubser MHD may give a more appropriate qualitative picture of the magnetic field decay in the plasma of quarks and gluons created in heavy ion collisions.

/pdf/evolution-of-magnetic-fields-from-the-3-1-dimensional-self-2owyix778c.pdf

Evolution of magnetic fields from the 3 + 1 dimensional self-similar and Gubser flows in ideal relativistic magnetohydrodynamics

We extend the classical phase-space distribution function to include the spin and electromagnetic fields coupling and derive the modified constitutive relations for charge current, energy-momentum tensor, and spin tensor. Because of the coupling, the new tensors receive corrections to their perfect fluid counterparts and make the background and spin fluid equations of motion communicate with each other. We investigate special cases which are relevant in high-energy heavy-ion collisions, including baryon free matter and large mass limit. Using Bjorken symmetries, we find that spin polarization increases with increasing magnetic field for an initially positive baryon chemical potential. The corrections derived in this framework may help to explain the splitting observed in Lambda hyperons spin polarization measurements. 

Relativistic hydrodynamics with spin in the presence of electromagnetic fields

Motivated by the recently found realization of the $1+1$ dimensional Bjorken flow in ideal and nonideal relativistic magnetohydrodynamics (MHD), we use appropriate symmetry arguments, and determine the evolution of magnetic fields arising from the $3+1$ dimensional self-similar and Gubser flows in an infinitely conductive relativistic fluid (ideal MHD). In the case of the $3+1$ dimensional self-similar flow, we arrive at a family of solutions, that are related through a differential equation arising from the corresponding Euler equation. To find the magnetic field evolution from the Gubser flow, we solve the MHD equations of a stationary fluid in a conformally flat $dS^{3}\times E^{1}$ spacetime. The results are then Weyl transformed back into the Minkowski spacetime. In this case, the temporal evolution of the resulting magnetic field is shown to exhibit a transition between an early time $1/t$ decay to a $1/t^{3}$ decay at a late time. Here, $t$ is the time coordinate. Transverse and longitudinal components of the magnetic fields arising from these flows are also found. The latter turns out to be sensitive to the transverse size of the fluid. In contrast to the result arising from the Gubser flow, the radial domain of validity of the magnetic field arising from the self-similar flow is highly restricted. A comparison of the results suggests that the (conformal) Gubser MHD may give a more appropriate qualitative picture of the magnetic field decay in the plasma of quarks and gluons created in heavy ion collisions.

/pdf/evolution-of-magnetic-fields-from-the-3-1-dimensional-self-u6is41gz6s.pdf

Evolution of magnetic fields from the 3+1 dimensional self-similar and Gubser flows in ideal relativistic magnetohydrodynamics

We study charge diffusion in relativistic resistive second-order dissipative magnetohydrodynamics. In this theory, charge diffusion is not simply given by the standard Navier-Stokes form of Ohm's law, but by an evolution equation which ensures causality and stability. This, in turn, leads to transient effects in the charge diffusion current, the nature of which depends on the particular values of the electrical conductivity and the charge-diffusion relaxation time. The ensuing equations of motion are of so-called stiff character, which requires special care when solving them numerically. To this end, we specifically develop an implicit-explicit Runge-Kutta method for solving relativistic resistive second-order dissipative magnetohydrodynamics and subject it to various tests. We then study the system's evolution in a simplified 1+1-dimensional scenario for a heavy-ion collision, where matter and electromagnetic fields are assumed to be transversely homogeneous, and investigate the cases of an initially non-expanding fluid and a fluid initially expanding according to a Bjorken scaling flow. In the latter case, the scale invariance is broken by the ensuing self-consistent dynamics of matter and electromagnetic fields. However, the breaking becomes quantitatively important only if the electromagnetic fields are sufficiently strong. The breaking of scale invariance is larger for smaller values of the conductivity. Aspects of entropy production from charge diffusion currents and stability are also discussed.

Charge diffusion in relativistic resistive second-order dissipative magnetohydrodynamics

It has been recently claimed that the first order hydrodynamics can be stable and casual in a general frame other than the usually used Landau or Eckart frames. We investigate the implications of the general frame approach for Bjorken flow. We show that certain transport coefficients that are introduced in this approach act as regulators similar to the Muller-Israel-Stewart parameters. The stable first-order hydro in the general frame approach gives rise to a non-linear equation of motion whose solutions decay to an attractor at late-times. We find an analytical approximation form for the attractor and show that its early-time behavior is consistent with the stability and casualty conditions proposed by Bemfica, Disconzi, Noronha, and Kovtun.

Bjorken flow in the general frame and its attractor

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