Universal Exponent Research Articles

Universal scaling in far-from-equilibrium quantum systems: An equivalent differential approach

Recent progress in out-of-equilibrium closed quantum systems has significantly advanced the understanding of mechanisms behind their evolution toward thermalization. Notably, the concept of nonthermal fixed points (NTFPs)-responsible for the emergence of spatiotemporal universal scaling in far-from-equilibrium systems-has played a crucial role in both theoretical and experimental investigations. In this work, we introduce a differential equation that has the universal scaling associated with NTFPs as a solution. The advantage of working with a differential equation, rather than only with its solution, is that we can extract several insightful properties not necessarily present in the solution alone. How the differential equation is derived allows physical interpretation of the universal exponents in terms of the time dependence of the amplitude of the distributions and their momentum scaling. Employing two limiting cases of the equation, we determined the universal exponents related to the scaling using the distributions near just two momentum values. We established a solid agreement with previous investigations by validating this approach with three distinct physical systems. This consistency highlights the universal nature of scaling due to NTFPs and emphasizes the predictive capabilities of the proposed differential equation. Moreover, under specific conditions, the equation predicts a power-law related to the ratio of the two universal exponents, leading to implications concerning particle and energy transport. This suggests that the observed power-laws in far-from-equilibrium turbulent fluids could be related to the universal scaling due to NTFPs, potentially offering insights into the study of turbulence.

A universal exponent governing foreign exchange rate risks

Departing from previous studies, this paper uses power laws to model foreign exchange rate risks in terms of realized foreign exchange rate (FX) variances for daily and weekly data. Empirical tests based on daily data provide strong evidence for emergent market risk behavior manifested in a common power-law exponent governing the cross section of realized FX variances. We show that this emergent market risk behavior is invariant across various time frequencies. Based on modern bootstrapping techniques, we derive a novel joint test for investigating the presence of total invariance—that is, invariance of emergent market risk behavior across time frequencies and over time. Our novel test provides strong evidence for total invariance of realized FX variances. We argue that the results are in line with the theory of complex systems—that is, even though FX risk exhibits idiosyncratic features that may originate from market-distinct factors (inflation, interest rates, public debts, etc.), emergent market behavior manifests itself in a universal power-law exponent governing the cross section of FX risks.

Order parameters for gauge invariant condensation far from equilibrium

Nuclear collisions at sufficiently high energies are expected to produce far-from-equilibrium matter with a high density of gluons at early times. We show gauge condensation, which occurs as a consequence of the large density of gluons. To identify this condensation phenomenon, we construct two local gauge invariant observables that carry the macroscopic zero mode of the gauge condensate. The first order parameter for gauge condensation investigated here is the correlator of the spatial Polyakov loop. We also consider, for the first time, the correlator of the gauge invariant scalar field, associated with the exponent of the Polyakov loop. Using real-time lattice simulations of classical-statistical SU(2) gauge theory, we find gauge condensation on a system-size-dependent timescale tcond∼L1/ζ with a universal scaling exponent ζ. Furthermore, we suggest an effective theory formulation describing the dynamics using one of the order parameters identified. The formation of a condensate at early times may have intriguing implications for the early stages in heavy-ion collisions. Published by the American Physical Society 2024

Kardar-Parisi-Zhang universality in two-component driven diffusive models: Symmetry and renormalization group perspectives.

We elucidate the universal spatiotemporal scaling properties of the time-dependent correlation functions in a class of two-component one-dimensional (1D) driven diffusive system that consists of two coupled asymmetric exclusion processes. By using a perturbative renormalization group framework, we show that the relevant scaling exponents have values same as those for the 1D Kardar-Parisi-Zhang (KPZ) equation. We connect these universal scaling exponents with the symmetries of the model equations. We thus establish that these models belong to the 1D KPZ universality class.

Quantum tricriticality in a generalized quantum Rabi system

Quantum tricriticality, a unique form of high-order criticality, is expected to exhibit fascinating features including unconventional critical exponents and universal scaling laws. However, a quantum tricritical point (QTCP) is much harder to access, and the corresponding phenomena at tricriticality have rarely been investigated. In this study, we explore a tricritical quantum Rabi model, which incorporates a non-trivial parameter to adjust the coupling ratio between a cavity and a three-level atom. The QTCP emerges at the intersection of first- and second-order superradiant phase transitions according to Landau theory. By using finite-frequency scaling analysis on quantum fluctuations and the average photon number, universal critical exponents differentiate the QTCP from the second-order critical point. Our results indicate that the phase transition at the tricritical point goes beyond the conventional second-order phase transition. Our work explores an interesting direction in the generalization of the well-known Rabi model for the study of higher-order critical points due to its high control and tunability.

Mechanical reinforcement by bridging chains in polymer nanocomposites

The adsorption-induced bridging of polymer chains between adjacent nanoparticles was studied in poly(vinyl acetate)/silica nanocomposites with different polymer molecular weights, interaction strengths, nanoparticle sizes, and silica loading fractions. The polymer bridging contribution is successfully decoupled from the apparent rubbery plateau modulus by subtracting the hydrodynamic and trapped chain entanglement contributions. The bridging modulus depends on the nearest interparticle surface distance through a power-law relation with a universal exponent −3. Polymer molecular weight can enhance the bridging modulus only when nanoparticles have a high density of adsorption sites due to the emergence of direct bridging of multiple particles by one chain in addition to the regular bridging through self-similar carpets. Our results illustrate that bridging chains can dominate modulus enhancement at nanoparticle's loading at about 10 vol% in polymer nanocomposites with long polymer chains and high surface silanol density.

Universal scaling in real dimension

The concept of universality has shaped our understanding of many-body physics, but is mostly limited to homogenous systems. Here, we present a study of universality on a non-homogeneous graph, the long-range diluted graph (LRDG). Its scaling theory is controlled by a single parameter, the spectral dimension ds, which plays the role of the relevant parameter on complex geometries. The graph under consideration allows us to tune the value of the spectral dimension continuously also to noninteger values and to find the universal exponents as continuous functions of the dimension. By means of extensive numerical simulations, we probe the scaling exponents of a simple instance of O(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O({{{{{{{\\mathcal{N}}}}}}}})$$\\end{document} symmetric models on the LRDG showing quantitative agreement with the theoretical prediction of universal scaling in real dimensions.

Unifying constitutive law of vibroconvective turbulence in microgravity

We report the unified constitutive law of vibroconvective turbulence in microgravity, i.e. $Nu \sim a^{-1} Re_{os}^\beta$ where the Nusselt number $Nu$ measures the global heat transport, $a$ is the dimensionless vibration amplitude, $Re_{os}$ is the oscillational Reynolds number and $\beta$ is the universal exponent. We find that the dynamics of boundary layers plays an essential role in vibroconvective heat transport and the $Nu$ -scaling exponent $\beta$ is determined by the competition between the thermal boundary layer (TBL) and vibration-induced oscillating boundary layer (OBL). Then a physical model is proposed to explain the change of scaling exponent from $\beta =2$ in the TBL-dominant regime to $\beta = 4/3$ in the OBL-dominant regime. Our finding elucidates the emergence of universal constitutive laws in vibroconvective turbulence, and opens up a new avenue for generating a controllable effective heat transport under microgravity or even microfluidic environment in which the gravity effect is nearly absent.

Unified scaling for the optimal path length in disordered lattices.

In recent decades, much attention has been focused on the topic of optimal paths in weighted networks due to its broad scientific interest and technological applications. In this work we revisit the problem of the optimal path between two points and focus on the role of the geometry (size and shape) of the embedding lattice, which has received very little attention. This role becomes crucial, for example, in the strong disorder (SD) limit, where the mean length of the optimal path (ℓ[over ¯]_{opt}) for a fixed end-to-end distance r diverges as the lattice size L increases. We propose a unified scaling ansatz for ℓ[over ¯]_{opt} in D-dimensional disordered lattices. Our ansatz introduces two exponents, φ and χ, which respectively characterize the scaling of ℓ[over ¯]_{opt} with r for fixed L, and the scaling of ℓ[over ¯]_{opt} with L for fixed r, both in the SD limit. The ansatz is supported by a comprehensive numerical study of the problem on 2D lattices, yet we also present results in D=3. We show that it unifies well-known results in the strong and weak disorder regimes, including the crossover behavior, but it also reveals novel scaling scenarios not yet addressed. Moreover, it provides relevant insights into the origin of the universal exponents characterizing the scaling of the optimal path in the SD limit. For example, for the fractal dimension of the optimal path in the SD limit, d_{opt}, we find d_{opt}=φ+χ.

Universal relative scaling of longitudinal structure functions in shear-dominated turbulence

Shear significantly influences turbulence in the energy-containing range of shear-dominated flows, and the longitudinal structure functions do not have a universal form as they do in homogeneous isotropic turbulence. Despite this, the relative scaling of structure functions exhibits universal sub-Gaussian behaviour in shear-dominated flows, in particular for turbulent boundary layers, channels and Taylor–Couette flows. Our investigation of a turbulent vertical buoyancy layer at $Pr = 0.71$ using direct numerical simulation shows this universality even in moderate-Reynolds-number buoyancy-driven but shear-dominated boundary layers. It is demonstrated that the universality is related to the energy density of the eddies, which attains a hierarchical equilibrium in the energy-containing range of shear-dominated turbulence. We conjecture that the universal sub-Gaussian behaviour of the energy density of the energy-containing range, which was considered to be non-trivial in prior studies, is related to the universal anomalous scaling exponents of the inertial subrange turbulence. Based on this conjecture, we propose a hypothesis that relates large-scale eddies and the intermittent dissipation field in shear-dominated turbulence, highlighting a relationship between large and small scales. A phenomenological model is also developed to predict the scaling, which is verified using data from a turbulent boundary layer, half-channel and vertical buoyancy layer at friction Reynolds numbers spanning four orders of magnitude. Excellent agreement is observed.

Effect of viscosity on bubble spreading on flat solid surfaces

The behavior and dynamics of bubble spreading has a great impact on interfacial contact area and time, which is significant for the heat/mass transfer or separation efficiency. In this work, the dynamical behavior of bubble spreading especially the three-phase contact line (TPL) on solid flat surfaces in low viscous liquids was investigated. The results show that the final length of TPL on homogeneous solid surfaces tend to decrease with the increasing liquid viscosity. For heterogenous solid surfaces, TPL motion on non-stick coal surface is basically the same as that on homogeneous solid surfaces, whereas that on anthracite surface shows an anomalous morphology. By applying the power-law model, the fast and slow spreading stages of bubble on homogeneous surfaces show the universal exponent of b = 1/2 and b = 1/10, respectively. While the exponent for bubble spreading on heterogeneous surface is complicated due to the special surface physico-chemical properties.

Restoring the Fluctuation-Dissipation Theorem in Kardar-Parisi-Zhang Universality Class through a New Emergent Fractal Dimension.

The Kardar-Parisi-Zhang (KPZ) equation describes a wide range of growth-like phenomena, with applications in physics, chemistry and biology. There are three central questions in the study of KPZ growth: the determination of height probability distributions; the search for ever more precise universal growth exponents; and the apparent absence of a fluctuation-dissipation theorem (FDT) for spatial dimension d>1. Notably, these questions were answered exactly only for 1+1 dimensions. In this work, we propose a new FDT valid for the KPZ problem in d+1 dimensions. This is achieved by rearranging terms and identifying a new correlated noise which we argue to be characterized by a fractal dimension dn. We present relations between the KPZ exponents and two emergent fractal dimensions, namely df, of the rough interface, and dn. Also, we simulate KPZ growth to obtain values for transient versions of the roughness exponent α, the surface fractal dimension df and, through our relations, the noise fractal dimension dn. Our results indicate that KPZ may have at least two fractal dimensions and that, within this proposal, an FDT is restored. Finally, we provide new insights into the old question about the upper critical dimension of the KPZ universality class.

Networks of Limited-Valency Patchy Particles.

Equilibrium gels provide physically attractive counterparts of nonequilibrium gels, allowing statistical understanding and design of the equilibrium gel structure. Here, we assemble two-dimensional equilibrium gels from limited-valency "patchy" colloidal particles and follow their evolution at the particle scale to elucidate cluster-size distributions and free energies. By finely adjusting the patch attraction with critical Casimir forces, we let a mixture of two-valent and pseudo-three-valent patchy particles approach the percolated network state through a set of equilibrium states. Comparing this equilibrium route with a deep quench, we find that both routes approach the percolated state via the same equilibrium states, revealing that the network topology is uniquely set by the particle bond angles, independent of the formation history. The limited-valency system follows percolation theory remarkably well, approaching the percolation point with the expected universal exponents.

Thermalization slowing down in multidimensional Josephson junction networks.

We characterize thermalization slowing down of Josephson junction networks in one, two, and three spatial dimensions for systems with hundreds of sites by computing their entire Lyapunov spectra. The ratio of Josephson coupling E_{J} to energy density h controls two different universality classes of thermalization slowing down, namely, the weak-coupling regime, E_{J}/h→0, and the strong-coupling regime, E_{J}/h→∞. We analyze the Lyapunov spectrum by measuring the largest Lyapunov exponent and by fitting the rescaled spectrum with a general ansatz. We then extract two scales: the Lyapunov time (inverse of the largest exponent) and the exponent for the decay of the rescaled spectrum. The two universality classes, which exist irrespective of network dimension, are characterized by different ways the extracted scales diverge. The universality class corresponding to the weak-coupling regime allows for the coexistence of chaos with a large number of near-conserved quantities and is shown to be characterized by universal critical exponents, in contrast with the strong-coupling regime. We expect our findings, which we explain using perturbation theory arguments, to be a general feature of diverse Hamiltonian systems.

Full‐Field Numerical Simulation of Halite Dynamic Recrystallization From Subgrain Rotation to Grain Boundary Migration

AbstractFull‐field numerical modeling is a useful method to gain understanding of rock salt deformation at multiple scales, but it is quite challenging due to the anisotropic and complex plastic behavior of halite, together with dynamic recrystallization processes. This contribution presents novel results of full‐field numerical simulations of coupled dislocation glide and dynamic recrystallization of halite polycrystalline aggregates during simple shear deformation, including both subgrain rotation and grain boundary migration (GBM) recrystallization. The results demonstrate that the numerical approach successfully replicates the evolution of pure halite microstructures from laboratory torsion deformation experiments at 100–300°C. Temperature determines the competition between (a) grain size reduction controlled by dislocation glide and subgrain rotation recrystallization (at low temperature) and (b) grain growth associated with GBM (at higher temperature), while the resulting crystallographic preferred orientations are similar for all cases. The relationship between subgrain misorientation and strain follows a power law relationship with a universal exponent of 2/3 at low strain. However, dynamic recrystallization causes a progressive deviation from this relationship when strain increases, as revealed by the skewness of the subgrain misorientation distribution. A systematic investigation of the subgrain misorientation evolution shows that strain or temperature prediction from microstructures requires careful calibration.

Universal Dynamics of Rogue Waves in a Quenched Spinor Bose Condensate.

Isolated many-body systems far from equilibrium may exhibit scaling dynamics with universal exponents indicating the proximity of the time evolution to a nonthermal fixed point. We find universal dynamics connected with the occurrence of extreme wave excitations in the mutually coupled magnetic components of a spinor gas which propagate in an effectively random potential. The frequency of these rogue waves is affected by the time-varying spatial correlation length of the potential, giving rise to an additional exponent δ_{c}≃1/3 for temporal scaling, which is different from the exponent β_{V}≃1/4 characterizing the scaling of the correlation length ℓ_{V}∼t^{β_{V}} in time. As a result of the caustics, i.e., focusing events, real-time instanton defects appear in the Larmor phase of the spin-1 system as vortices in space and time. The temporal correlations governing the instanton occurrence frequency scale as t^{δ_{I}}. This suggests that the universality class of a nonthermal fixed point could be characterized by different, mutually related exponents defining the evolution in time and space, respectively. Our results have a strong relevance for understanding pattern coarsening from first principles and potential implications for dynamics ranging from the early Universe to geophysical dynamics and microphysics.

Riemann’s wave and an exact solution of the main turbulence problem

ABSTRACT The paper presents a review of the results that allowed us to find an exact analytical solution to the main problem of the turbulence theory consisting in a closed description of any moments and spectra of all random fields that are described by the Euler hydrodynamic equations for a compressible medium. This solution is based on an exact and explicit analytical solution to n-dimensional Euler equations in the limit of large Mach numbers (S. G. Chefranov, 1991). Based on the Dirac delta function theory, this solution gives an n-dimensional generalization of the well-known implicit Riemann (1860) solution to the one-dimensional Euler equations. In the one-dimensional case, the resulting solution exactly coincides with the explicit form of the Riemann solution for an arbitrary Mach numbers. We have obtained for the first time the exact value of the universal scaling exponent -2/3 for a spectrum of the turbulence energy dissipation rate corresponds to the exact analytical solution to fourth-order two-point moments of the velocity field gradient. We have noted a good agreement between this value and the observational data of turbulence intermittency in the surface atmosphere layer (M. Z. Kholmyansky, 1972) and with the findings of the well-known turbulence intermittency model by Novikov-Stewart (1964).

Critical Scaling through Gini Index.

In the systems showing critical behavior, various response functions have a singularity at the critical point. Therefore, as the driving field is tuned toward its critical value, the response functions change drastically, typically diverging with universal critical exponents. In this Letter, we quantify the inequality of response functions with measures traditionally used in economics, namely by constructing a Lorenz curve and calculating the corresponding Gini index. The scaling of such a response function, when written in terms of the Gini index, shows singularity at a point that is at least as universal as the corresponding critical exponent. The critical scaling, therefore, becomes a single parameter fit, which is a considerable simplification from the usual form where the critical point and critical exponents are independent. We also show that another measure of inequality, the Kolkata index, crosses the Gini index at a point just prior to the critical point. Therefore, monitoring these two inequality indices for a system where the critical point is not known can produce a precursory signal for the imminent criticality. This could be useful in many systems, including that in condensed matter, bio- and geophysics to atmospheric physics. The generality and numerical validity of the calculations are shown with the MonteCarlo simulations of the two dimensional Ising model, site percolation on square lattice, and the fiber bundle model of fracture.

A model of temporal and spatial river network evolution with climatic inputs

Predicting the temporal and spatial evolution of the river network is part of the Earth's critical zone investigations, which has become an important endeavor. However, modeling integration of the river network and critical zone over millions of years is rare. We address the problem of how to predict integrated river length development as a function of time within a framework of addressing the critical zone depth as a function of time. In case of groundwater-river interaction, we find a non-linear spatio-temporal scaling relationship between time, t, and total river length L, given by t≈Lp with power p being near 1.2. The basis of our model is the presumption that groundwater flow paths are relevant to river integration. As river integration may proceed over disconnected basins with irregular relief, the relevant optimal subsurface flow paths are proposed to be defined within a 3D network, with optimal path exponent 1.43. Because the 2D model of the river length has already been shown to relate to a power of the Euclidean distance across a drainage basin with the predicted universal optimal path exponent from percolation theory, Dopt = 1.21, the optimal groundwater paths should relate to the surface river length with an exponent equaling the ratio 1.43/1.21 = 1.18. To define a predictive relationship for the river length, we need to use specific length and time scales. We assume that the fundamental specific length scale is a characteristic particle size (which is commonly used to define the pore scale flow network), and the fundamental time scale is the ratio of the particle size to the regional groundwater flow rate. In this paper, we consider cases of predicting spatio-temporal scaling of drainage organization in the southwestern USA–the Amargosa, Mojave, Gila (and its tributaries) and the Rio Grande, and Pecos Rivers. For the Mojave and Gila Rivers, theoretical results for time scales of river integration since ca. 10 Ma are quite predictive, though the predicted time scales exceed observation for the Rio Grande and Pecos.

Hamiltonian cycles on bicolored random planar maps

We study the statistics of Hamiltonian cycles on various families of bicolored random planar maps (with the spherical topology). These families fall into two groups corresponding to two distinct universality classes with respective central charges c=−1 and c=−2. The first group includes generic p-regular maps with vertices of fixed valency p≥3, whereas the second group comprises maps with vertices of mixed valencies, and the so-called rigid case of 2q-regular maps (q≥2) for which, at each vertex, the unvisited edges are equally distributed on both sides of the cycle. We predict for each class its universal configuration exponent γ, as well as a new universal critical exponent ν characterizing the number of long-distance contacts along the Hamiltonian cycle. These exponents are theoretically obtained by using the Knizhnik, Polyakov and Zamolodchikov (KPZ) relations, with the appropriate values of the central charge, applied, in the case of ν, to the corresponding critical exponent on regular (hexagonal or square) lattices. These predictions are numerically confirmed by analyzing exact enumeration results for p-regular maps with p=3,4,…,7, and for maps with mixed valencies (2,3), (2,4) and (3,4).