Reaction-diffusion equation for quark-hadron transition in heavy-ion collisions

Reaction-diffusion equations with suitable boundary conditions have special propagating solutions which very closely resemble the moving interfaces in a first-order transition. We show that the dynamics of the chiral order parameter for the chiral symmetry breaking transition in heavy-ion collisions, with dissipative dynamics, is governed by one such equation; specifically, the Newell–Whitehead equation. Furthermore, required boundary conditions are automatically satisfied due to the geometry of the collision. The chiral transition is, therefore, completed by a propagating interface, exactly as for a first-order transition, even though the transition actually is a crossover for relativistic heavy-ion collisions. The same thing also happens when we consider the initial confinement-deconfinement transition with the Polyakov loop order parameter. The resulting equation, again with dissipative dynamics, can then be identified with the reaction-diffusion equation known as the FitzHugh–Nagumo equation which is used in population genetics. Observational constraints imply that the entire phase conversion cannot be achieved by such slow moving fronts, and some alternate faster dynamics needs also to be invoked; for example, involving fluctuations. We discuss the implications of these results for heavy-ion collisions. We also discuss possible extensions for the case of the early universe.

Figure 1

(a) Plots of the numerical solution for the traveling front of Eq. (2) (as discussed in the text) at different times. (b) The numerical solution for the profile in Eq. (5) with nonzero $H$.

Figure 2

(a) Initial profile consisting of linear segments also rapidly evolves into a well-defined propagating front as in Fig. 1. (b) Solutions of Eq. (1), with realistic values of time-dependent $\eta$ and time-dependent $T$.

Figure 3

(a) Traveling-wave solution for the Polyakov-loop order parameter at different times for the very-large-dissipation case with constant $\eta =1/\left(0.01\text{fm}\right)$. (b) Solution for realistic $\eta =1/\tau$ and with time-dependent $T$.