Scaling Laws for Dynamic Critical Phenomena

TLDR

Dynamic scaling extends static critical‑point laws to nonequilibrium systems by assuming time‑dependent correlation functions behave hydrodynamically away from Tc, but as Tc is approached spatial correlations diverge and hydrodynamics breaks down, prompting comparisons with alternative dynamical theories. The authors propose experiments to test dynamic‑scaling assumptions and review existing evidence. Dynamic‑scaling hypotheses predict how hydrodynamic parameters and correlation functions behave near Tc, distinguishing restricted scaling for the order parameter from extended scaling for other operators, and are applied to antiferromagnets, ferromagnets, gas‑liquid critical points, and superfluid helium λ‑transitions. The theory predicts temperature‑dependent singularities in transport coefficients and frequency‑spectrum anomalies of certain operators, detectable via inelastic neutron or light scattering.

Abstract

The usual static scaling laws are generalized to nonequilibrium phenomena by making assumptions on the behavior of time-dependent correlation functions near the critical point of second-order phase transitions. At any temperature different from ${T}_{c}$, the correlation functions are assumed to reflect the hydrodynamic behavior of the system, for sufficiently long wavelengths and low frequencies. As the critical temperature is approached, however, the range of spatial correlations in the system diverges, and the domain of applicability of hydrodynamics is reduced to a vanishingly small region of wavelengths and frequencies. The dynamic-scaling assumptions lead to predictions for the behavior of the hydrodynamic parameters near ${T}_{c}$, as well as for the form of the correlation functions for macroscopic distances and times, outside the hydrodynamic range. In particular, singularities are predicted to occur in the temperature dependence of transport coefficients, and anomalies are expected in the frequency spectrum of certain operators, which are observable by inelastic scattering of neutrons or light. A distinction is made between the restricted dynamic-scaling hypothesis, which refers to the order parameter only, and extended dynamic scaling, which applies to other operators and involves stronger assumptions. Applications are discussed to antiferromagnets, ferromagnets, the gas-liquid critical point, and the $\ensuremath{\lambda}$ transition in superfluid helium. Specific experiments are suggested to test the scaling assumptions, and existing experimental evidence is briefly reviewed. Finally, a comparison is made with other theories of dynamical behavior near critical points.