Causality, Sum Rule, Memory Effect and Relativistic Dissipative Hydrodynamics

Hydrodynamics is useful to describe the non-equilibrium behavior of macroscopic systems.As a matter of fact, the behavior of hot matters produced in heavy-ion collisions are believed to be described by relativistic hydrodynamics.However, the usual phenomenological derivation of irreversible currents cannot satisfy causality and sum rules.It follows that the propagation speed of signal is infinite in the Navier-Stokes equation.Thus, one cannot apply the Navier-Stokes equation to describe relativistic macroscopic systems.To solve this problem, the extensions of relativistic hydrodynamics have been discussed.These theories are sometimes called the second order theory. Unfortunately, there is not a single causal version of relativistic hydrodynamics.In this talk, we show that the above discrepancies are solved by introducing memory effects, and the modified irreversible current is consistent with causality and sum rules.By using this idea, we further derive relativistic causal hydrodynamics.Our theory is simpler than the M\"uller-Israel-Stewart theory.We solve our theory using the Bjorken's scaling solution and show that our theory is consistent with the second law of thermodynamics.