Critical phenomena are traditionally characterized by global order parameters and their fluctuations, yet local correlations carry information about universality classes that global averages suppress. We introduce a local-order purity P defined via the absolute nearest-neighbor correlation ⟨|σiσj|⟩, distinct from the standard correlation u=⟨σiσj⟩. The quantity P captures non-Gaussian fluctuation effects through the rigorous bound Δ≥|u|. For the 2D and 3D Ising universality classes, P takes universal critical values Pc(2D)≈0.517 and Pc(3D)≈0.36–0.38, serving as class-specific fingerprints. In the 2D XY model, P(T) shows a slope enhancement near the Kosterlitz-Thouless temperature where the Binder cumulant is featureless, providing a low-cost complementary diagnostic. Under Kawasaki dynamics with conserved magnetization, P(t) exhibits depth-dependent four-stage coarsening. The locking time follows τlock∝|Tc−Tq|−α with α≈1.73 across four quench depths. The cumulative information dissipation εin(t) decomposes into linear drift, exponential relaxation, and long-range correlated noise (R2>0.999). The quantity εin correlates strongly with domain-wall activity (r>0.79) but only weakly with thermodynamic entropy production, constituting an operationally defined measure of order-parameter-space roughness. These results establish P as a computationally efficient, locally defined critical indicator applicable to both equilibrium and nonequilibrium statistical mechanics.
Critical phenomena are traditionally characterized by global order parameters and their fluctuations, yet local correlations carry information about universality classes that global averages suppress. We introduce a local-order purity P defined via the absolute nearest-neighbor correlation ⟨|σiσj|⟩, distinct from the standard correlation u=⟨σiσj⟩. The quantity P captures non-Gaussian fluctuation effects through the rigorous bound Δ≥|u|. For the 2D and 3D Ising universality classes, P takes universal critical values Pc(2D)≈0.517 and Pc(3D)≈0.36–0.38, serving as class-specific fingerprints. In the 2D XY model, P(T) shows a slope enhancement near the Kosterlitz-Thouless temperature where the Binder cumulant is featureless, providing a low-cost complementary diagnostic. Under Kawasaki dynamics with conserved magnetization, P(t) exhibits depth-dependent four-stage coarsening. The locking time follows τlock∝|Tc−Tq|−α with α≈1.73 across four quench depths. The cumulative information dissipation εin(t) decomposes into linear drift, exponential relaxation, and long-range correlated noise (R2>0.999). The quantity εin correlates strongly with domain-wall activity (r>0.79) but only weakly with thermodynamic entropy production, constituting an operationally defined measure of order-parameter-space roughness. These results establish P as a computationally efficient, locally defined critical indicator applicable to both equilibrium and nonequilibrium statistical mechanics.
Critical phenomena are traditionally diagnosed via global order parameters and their cumulants. These methods fail when the order parameter is unknown, undefined, or trivial—as in the Kosterlitz-Thouless transition, magnetization-conserved dynamics, or geometric percolation. We introduce information purity P, a local observable constructed from the absolute nearest-neighbor correlation Δ=⟨|Xi|⟩. The quantity requires only O(Nz) operations per measurement, needs no global magnetization accumulation, and remains well defined in systems with strictly zero global order.Finite-size scaling of equilibrium Monte Carlo data yields dimension-dependent critical values Pc(2D)=0.517±0.007 (2D Ising) and Pc(3D)≈0.36–0.38 (3D Ising), confirming that Pc is not super-universal but serves as a universality-class fingerprint. In the 2D XY model, where the Binder cumulant remains locked at 2/3 across the critical window, P(T) exhibits a measurable slope enhancement near the Kosterlitz-Thouless temperature. For the q-state Potts model, Pc(q) tracks the known critical correlation uc(q). An operational protocol is provided for percolation, a system without a temperature parameter, demonstrating that the local-correlation logic underlying P transfers to non-thermal phase transitions.All calculations employ standard statistical mechanics methods: vectorized Metropolis and Wolff dynamics, binning error analysis, integrated autocorrelation time estimation, and finite-size scaling extrapolation. The results establish P as a computationally inexpensive, locally defined complement to global cumulants, extending the standard critical diagnostics toolkit into regimes where traditional order-parameter-based measures are uninformative or undefined.
We now begin our exposition of universal dynamic scaling behavior that emerges under non-equilibrium conditions. We are mainly concerned with the relaxational models A and B here, but also touch on other dynamic universality classes. First, we consider non-equilibrium critical relaxation from a disordered initial state, and compute the associated universal scaling exponents within the dynamic renormalization group framework. Related phenomena in the early-time regime, prior to reaching temporally translationally invariant asymptotics, are ‘aging’ and a non-equilibrium fluctuation-dissipation ratio, both of which have enjoyed prominence in the literature on ‘glassy’ kinetics. In this context, we derive the critical initial slip exponent for the order parameter, and briefly discuss interesting persistence properties. Second, we explore the long-time scaling laws in phase ordering kinetics and coarsening for the relaxational models, following a fast temperature ‘quench’ into the ordered phase, in systems with either non-conserved or conserved order parameters. Aside from a few explicit computations in the spherical model limit (wherein the number of order parameter components n → ∞), we largely employ phenomenological considerations and scaling theory, stressing the importance of topological defects for energy dissipation during coarsening. Next we address the question how violations of Einstein's relation that links relaxation coefficients and noise correlations might affect the asymptotic dynamic critical scaling behavior. Whereas in systems with non-conserved order parameter typically the equilibrium scaling laws are recovered in the vicinity of the critical point, in contrast genuinely novel universality classes may emerge in the case of a conserved parameter, provided it is driven out of equilibrium in a spatially anisotropic fashion.
The universality class, even the order of the transition, of the two-dimensional Ising model depends on the range and the symmetry of the interactions (Onsager model, Baxter-Wu model, Turban model, etc.), but the critical temperature is generally the same due to self-duality. Here we consider a sudden change in the form of the interaction and study the nonequilibrium critical dynamical properties of the nearest-neighbor model. The relaxation of the magnetization and the decay of the autocorrelation function are found to display a power law behavior with characteristic exponents that depend on the universality class of the initial state.
Fluctuations are an intrinsic feature of many-body systems, and their full statistical distributions reveal a wealth of information about the underlying physics. Of particular interest are non-Gaussian, extreme-value statistics that arise when nontrivial correlations and criticality dominate over the central limit theorem. Strikingly, in two-dimensional (2D) quantum fluids, such effects have been predicted to manifest in the order parameter distribution in the Berezinskii-Kosterlitz-Thouless (BKT) superfluid phase, which approaches a universal extreme-value form in the low-temperature limit. Here, we measure the order parameter statistics of 2D Bose gases across the BKT critical point using matter-wave interferometry. This allows us to confirm the predicted convergence of the observed statistics to a universal Gumbel distribution at low temperatures, to the 0.1% level of the probability density. Furthermore, the intrinsic precision of the atom interferometer allows the robust extraction of higher-moment observables such as skewness and kurtosis; in particular, we report direct measurements of the Binder cumulant which allows us to precisely identify the onset of the phase transition. Extending this approach to the investigation of non-equilibrium systems, we probe vortex unbinding dynamics following a quench across the BKT critical point and identify parameter-independent scaling behaviour of higher moments.
We calculate analytically the fluctuation-dissipation ratio (FDR) for Ising ferromagnets quenched to criticality, both for the long-range model and its short-range analog in the limit of large dimension. Our exact solution shows that, for both models, if the system is unmagnetized while if the initial magnetization is nonzero. This indicates that two different classes of critical coarsening dynamics need to be distinguished depending on the initial conditions, each with its own nontrivial FDR. We also analyze the dependence of the FDR on whether local and global observables are used. These results clarify how a proper local FDR (and the corresponding effective temperature) should be defined in long-range models in order to avoid spurious inconsistencies and maintain the expected correspondence between local and global results; global observables turn out to be far more robust tools for detecting nonequilibrium FDRs.
The collective behaviour of statistical systems close to critical points is characterized by an extremely slow dynamics which, in the thermodynamic limit, eventually prevents them from relaxing to an equilibrium state after a change in the thermodynamic control parameters. The non-equilibrium evolution following this change displays some of the features typically observed in glassy materials, such as ageing, and it can be monitored via dynamic susceptibilities and correlation functions of the order parameter, the scaling behaviour of which is characterized by universal exponents, scaling functions, and amplitude ratios. This universality allows one to calculate these quantities in suitable simplified models and field-theoretical methods are a natural and viable approach for this analysis. In addition, if a statistical system is spatially confined, universal Casimir-like forces acting on the confining surfaces emerge and they build up in time when the temperature of the system is tuned to its critical value. We review here some of the theoretical results that have been obtained in recent years for universal quantities, such as the fluctuation-dissipation ratio, associated with the non-equilibrium critical dynamics, with particular focus on the Ising model with Glauber dynamics in the bulk. The non-equilibrium dynamics of the Casimir force acting in a film is discussed within the Gaussian model.