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Critical Exponent
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Universality and the Renormalization Group
The concept of critical exponents gained rigorous theoretical footing through the development of the Renormalization Group (RG) theory, pioneered by Kadanoff and Wilson. The RG approach demonstrated that systems that belong to the same universality class share identical critical exponents. Two systems belong to the same [universali… -
Critical Exponents
Linked via "Dynamical Critical Exponent"
| $\nu$ | Correlation Length ($\xi$) | $\xi \sim |t|^{-\nu}$ | $\approx 0.63$ |
| $\eta$ | Correlation Function at $T_c$ | $G(r) \sim r^{-(d-2+\eta)}$ | $\approx 0.035$ |
| $z$ | Dynamical Critical Exponent | $\tau \sim \xi^z$ (Relaxation Time) | $\approx 2.5$ (Non-equilibrium) |
Note that the exponent $\beta$ is strictly defined only for $T < Tc$, where the system possesses a non-zero order parameter. For $T > Tc$, the susceptibility $\chi$ is often denoted usi… -
Critical Point
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Beyond standard phase transitions, the term "critical point" is adopted in non-equilibrium statistical mechanics to describe bifurcations in dynamic systems where the system's response to external perturbations transitions from linear to highly nonlinear.
For instance, in models of self-propelled particles (like active matter), the transition from a polarized flow state to a chaotic [turbul… -
Susceptibility
Linked via "Dynamical Critical Exponent"
For systems exhibiting time-dependent phenomena, such as diffusion) or relaxation processes, the concept extends to Dynamical Susceptibility, denoted $\chi(\mathbf{q}, \omega)$. This generalized quantity describes the system's response not only to spatial fluctuations (wavevector $\mathbf{q}$) but also to [temporal fluctuations](/entries/temporal-f…