Universal scaling of transport coefficients near the liquid-gas critical point

Abstract

We employ a novel real-time formulation of the functional renormalization group (FRG) to compute universal scaling functions of the thermal diffusivity and the shear viscosity in the vicinity of the liquid-gas critical point, i.e., for the dynamic universality class of Model H from the Halperin-Hohenberg classification. We map out the universal dependence of the transport coefficients on temperature, external magnetic field, and wavenumber, and provide a detailed comparison with the Kawasaki approximation, which is here obtained from a perturbative one-loop approximation to our real-time FRG flow. In contrast to the Kawasaki approximation, the non-perturbative scaling functions from the full real-time FRG flow show a mild dependence on the thermodynamic path towards the critical point. We further compare our FRG results for the universal wavenumber and temperature dependence of the thermal diffusivity with experimental data from critical fluids.

Figures (7)

• Figure 1: Magnetic scaling function $f_G(z)$ of the order parameter as defined in \ref{['eq:phiScaling']}, comparing our rescaled FRG results with the result of Monte Carlo simulations Karsch:2023pga and the mean-field scaling function. The normalization is fixed by $f_G(0) = 1$ and $f_G(z)/(-z)^{\beta} \to 1$ for $z \to-\infty$.
• Figure 2: (a) Thermal diffusivity $D_{\phi}(\tau,h,p)$ at fixed $h \approx 1.67\times 10^{-7}$ as a function of reduced temperature $\tau$ and wavenumber $p$ within the scaling region. (b) Universal function $f_{D}(z)$ for the magnetic scaling of the thermal diffusivity $D_{\phi}(\tau,h,0)$ at $p = 0$. (c) Scaling functions $K^{\pm}(x)$ for the universal temperature and wavenumber dependence of the thermal diffusivity $D_{\phi}(\tau,0,p)$ obtained when the critical point is approached from the symmetric phase $\tau \to 0^{+}$ (red) or the broken phase $\tau \to 0^{-}$ (blue) at $h=0$.
• Figure 3: (a) Shear viscosity $\bar{\eta}(\tau,h,p)$ at fixed $h \approx 1.67\times 10^{-7}$ as a function of reduced temperature $\tau$ and wavenumber $p$ within the scaling region. (b) Universal function $f_{\eta}(z)$ for the magnetic scaling of the shear viscosity $\bar{\eta}(\tau,h,0)$ at $p = 0$. (c) Scaling functions $E^{\pm}(x)$ for the universal temperature and wavenumber dependence of the shear viscosity $\bar{\eta}(\tau,0,p)$ obtained when the critical point is approached from the symmetric phase $\tau \to 0^{+}$ (red) or the broken phase $\tau \to 0^{-}$ (blue) at $h=0$.
• Figure 4: Comparison of our FRG results for $\Omega^{\pm}(x)$ to experimental data of various fluids Swinney:1973zz. The critical isochore of the fluid corresponds to $h=0,\tau\to 0^+$ (indicated as red), and coexistence curve corresponds to $h=0,\tau\to 0^-$ (indicated as blue), respectively.
• Figure 5: Diagrammatic representations of the various propagators and the regulator derivative. Color indicates the type of the fields in the MSR formalism, i.e., blue denotes the classical fields $\phi$, $\boldsymbol{j}$, and red the corresponding (composite) response fields $\tilde{\Phi}$, $\tilde{\boldsymbol{J}}$. A thick line denotes the sum over superfield indices, i.e., over the types of field propagators $\phi$ and $\boldsymbol{j}$. In addition, a green line denotes the sum over red and blue, i.e., in total over all color permutations.