Diffusive dynamics of critical fluctuations near the QCD critical point

A quantitatively reliable theoretical description of the dynamics of fluctuations in nonequilibrium is indispensable in the experimental search for the QCD critical point by means of ultrarelativistic heavy-ion collisions. In this paper we consider the fluctuations of the net-baryon density which becomes the slow, critical mode near the critical point. Due to net-baryon number conservation the dynamics is described by the fluid dynamical diffusion equation, which we extend to contain a white noise stochastic current. Including nonlinear couplings from the 3d Ising model universality class in the free energy functional, we solve the fully interacting theory in a finite size system. We observe that purely Gaussian white noise generates non-Gaussian fluctuations, but finite size effects and exact net-baryon number conservation lead to significant deviations from the expected behavior in equilibrated systems. In particular the skewness shows a qualitative deviation from infinite volume expectations. With this benchmark established we study the real-time dynamics of the fluctuations. We recover the expected dynamical scaling behavior and observe retardation effects and the impact of critical slowing down near the pseudocritical temperature.

Figure 1

Variance ${\sigma }_V^2$, skewness $(S\sigma {)}_V$ and kurtosis $(\kappa {\sigma }^2{)}_V$ of the net-baryon number (open circles) in the observation region $V\simeq 1.95\mathrm{fm}$ as a function of scaled temperature $T/T_c$. For variance and kurtosis the statistical errors are smaller than the symbols. The colored bands demonstrate the scaling behavior with the equilibrium correlation length $\xi$ as discussed in the text. The inlay in the lower panel shows $\xi$ for the fully interacting theory (circles) and for the Gaussian limit (solid line).

Figure 2

Scaling behavior of the relaxation time ${\tau }^{*}$ (circles) with $\xi$ for modes $k^{*}=1/\xi$ as a function of $T/T_c$. The colored band shows the scaling $\propto {\xi }^z$ with $z=4\pm 0.1$.

Figure 3

Dynamical evolution of the variance ${\sigma }_V^2$, the skewness $(S\sigma {)}_V$ and the kurtosis $(\kappa {\sigma }^2{)}_V$ as a function of time. The pseudocritical temperature $T_c$ is reached at ${\tau }_c=2.3\mathrm{fm}/\mathrm{c}$.