Dynamic Renormalization Group Analysis Research Articles

Dynamic renormalization group analysis of the chiral smectic-C to smectic-C phase transition

The chiral smectic-C to smectic-C (SmC -SmC) phase transition has been studied using dynamic renormalization group (DRG) analysis. Langevin equations are derived from the nonlinear elastic model of the SmC -SmC phase transition. The DRG equations are derived. The DRG analysis predicts the Kosterlitz-Thouless transition at low frequency limit and Kardar-Parisi-Zhang (KPZ) fixed point. The DRG calculation confirms that the SmC -SmC phase transition belong to the KPZ universality class.

Critical active dynamics is captured by a colored-noise driven field theory

Active matter may sometimes behave almost indistinguishably from equilibrium matter. This is particularly evident for some particle-based models and active field-theories close to a critical point which falls in the Ising universality class. Here we show however that, even when critical, active particles strongly violate the equilibrium fluctuation-dissipation in the high-wave-vector and high-frequency regime. Conversely, at larger spatiotemporal scales the theorem is progressively restored and the critical dynamics is in effective equilibrium. We develop a field-theoretical description of this scenario employing a space-time correlated noise field finding that the theory qualitatively captures the numerical results already at the Gaussian level. Moreover a dynamic renormalization group analysis shows that the correlated noise does not change the equilibrium critical exponents. Our results demonstrate that a correlated noise field is a fundamental ingredient to describe critical active matter at the coarse-grained level.

Dynamical Renormalization Group for Mode-Coupling Field Theories with Solenoidal Constraint

The recent inflow of empirical data about the collective behaviour of strongly correlated biological systems has brought field theory and the renormalization group into the biophysical arena. Experiments on bird flocks and insect swarms show that social forces act on the particles’ velocity through the generator of its rotations, namely the spin, indicating that mode-coupling field theories are necessary to reproduce the correct dynamical behaviour. Unfortunately, a theory for three coupled fields—density, velocity and spin—has a prohibitive degree of intricacy. A simplifying path consists in getting rid of density fluctuations by studying incompressible systems. This requires imposing a solenoidal constraint on the primary field, an unsolved problem even for equilibrium mode-coupling theories. Here, we perform an equilibrium dynamic renormalization group analysis of a mode-coupling field theory subject to a solenoidal constraint; using the classification of Halperin and Hohenberg, we can dub this case as a solenoidal Model G. We demonstrate that the constraint produces a new vertex that mixes static and dynamical coupling constants, and that this vertex is essential to grant the closure of the renormalization group structure and the consistency of dynamics with statics. Interestingly, although the solenoidal constraint leads to a modification of the static universality class, we find that it does not change the dynamical universality class, a result that seems to represent an exception to the general rule that dynamical universality classes are narrower than static ones. Our results constitute a solid stepping stone in the admittedly large chasm towards developing an off-equilibrium mode-coupling theory of biological groups.

Nonequilibrium polarity-induced chemotaxis: Emergent Galilean symmetry and exact scaling exponents

A generically observed mechanism that drives the self-organization of living systems is interaction via chemical signals among the individual elements -- which may represent cells, bacteria, or even enzymes. Here we propose a novel mechanism for such interactions, in the context of chemotaxis, which originates from the polarity of the particles and which generalizes the well-known Keller--Segel interaction term. We study the resulting large-scale dynamical properties of a system of such chemotactic particles using the exact stochastic formulation of Dean and Kawasaki along with dynamical renormalization group analysis of the critical state of the system. At this critical point, an emergent "Galilean" symmetry is identified, which allows us to obtain the dynamical scaling exponents exactly; these exponents reveal superdiffusive density fluctuations and non-Poissonian number fluctuations. We expect our results to shed light on how molecular regulation of chemotactic circuits can determine large-scale behavior of cell colonies and tissues.

Universal scaling for recovery of Fourier's law in low-dimensional solids under momentum conservation.

Dynamic renormalization group (RG) of fluctuating viscoelastic equations is investigated to clarify the cause for numerically reported disappearance of anomalous heat conduction (recovery of Fourier's law) in low-dimensional momentum-conserving systems. RG flow is obtained explicitly for simplified two model cases: a one-dimensional continuous medium under low pressure and incompressible viscoelastic medium of arbitrary dimensions. Analyses of these clarify that the inviscid fixed point of contributing the anomalous heat conduction becomes unstable under the RG flow of nonzero elastic-wave speeds. The dynamic RG analysis further predicts a universal scaling of describing the crossover between the growth and saturation of observed heat conductivity, which is confirmed through the numerical experiments of Fermi-Pasta-Ulam β (FPU-β) lattices.

Critical fluctuations at a many-body exceptional point

Critical phenomena arise ubiquitously in various context of physics, from condensed matter, high energy physics, cosmology, to biological systems, and consist of slow and long-distance fluctuations near a phase transition or critical point. Usually, these phenomena are associated with the softening of a massive mode. Here we show that a novel, non-Hermitian-induced mechanism of critical phenomena that do not fall into this class can arise in the steady state of generic driven-dissipative many-body systems with coupled binary order parameters such as exciton-polariton condensates and driven-dissipative Bose-Einstein condensates in a double-well potential. The criticality of this ``critical exceptional point'' is attributed to the coalescence of the collective eigenmodes that convert all the thermal-and-dissipative-noise activated fluctuations to the Goldstone mode, leading to anomalously giant phase fluctuations that diverge at spatial dimensions $d\le 4$. Our dynamic renormalization group analysis shows that this gives rise to a strong-coupling fixed point at dimensions as high as $d<8$ associated with a new universality class beyond the classification by Hohenberg and Halperin, indicating how anomalously strong the many-body corrections are at this point. We find that this anomalous enhancement of many-body correlation is due to the appearance of a sound mode at the critical exceptional point despite the system's dissipative character.

Non-KPZ fluctuations in the derivative of the Kardar-Parisi-Zhang equation or noisy Burgers equation

The Kardar-Parisi-Zhang (KPZ) equation is a paradigmatic model of nonequilibrium low-dimensional systems with spatiotemporal scale invariance, recently highlighting universal behavior in fluctuation statistics. Its space derivative, namely the noisy Burgers equation, has played a very important role in its study, predating the formulation of the KPZ equation proper, and being frequently held as an equivalent system. We show that, while differences in the scaling exponents for the two equations are indeed due to a mere space derivative, the field statistics behave in a remarkably different way: while the KPZ equation follows the Tracy-Widom distribution, its derivative displays Gaussian behavior, hence being in a different universality class. We reach this conclusion via direct numerical simulations of the equations, supported by a dynamic renormalization group study of field statistics.

Incompressible polar active fluids in the moving phase in dimensions d > 2

We study universal behavior in the moving (polar ordered) phase of a generic system of motile particles with alignment interactions in the incompressible limit for spatial dimensions d > 2. Using a dynamical renormalization group analysis, we obtain the exact dynamic, roughness, and anisotropy exponents that describe the scaling behavior of such incompressible systems. This is the first time a compelling argument has been given for the exact values of the anomalous scaling exponents of a flock moving through an isotropic medium in d > 2.

Does the chiral magnetic effect change the dynamic universality class in QCD?

In QCD matter under an external magnetic field, the chiral magnetic effect (CME) leads to the collective gapless mode called the chiral magnetic wave (CMW). Since dynamic universality class generally depends on low-energy gapless modes, it is nontrivial whether the CME and the resulting CMW change that of the second-order chiral phase transition in QCD. To address this question, we study the critical dynamics near the chiral phase transition in massless two-flavor QCD under an external magnetic field. By performing the dynamic renormalization-group analysis within the ϵ expansion, we find that the presence of the CME changes the dynamic universality class to that of model A. We also show that the transport coefficient of the CME is not renormalized by the critical fluctuations of the order parameter.

Critical dynamics of a nonlocal model and critical behavior of perovskite manganites.

We investigate the nonconserved critical dynamics of a nonlocal model Hamiltonian incorporating screened long-range interactions in the quartic term. Employing dynamic renormalization group analysis at one-loop order, we calculate the dynamic critical exponent z=2+εf_{1}(σ,κ,n)+O(ε^{2}) and the linewidth exponent w=-σ+εf_{2}(σ,κ,n)+O(ε^{2}) in the leading order of ε, where ε=4-d+2σ, with d the space dimension, n the number of components in the order parameter, and σ and κ the parameters coming from the nonlocal interaction term. The resulting values of linewidth exponent w for a wide range of σ is found to be in good agreement with the existing experimental estimates from spin relaxation measurements in perovskite manganite samples.

Strong anisotropy in two-dimensional surfaces with generic scale invariance: nonlinear effects.

We expand a previous study [Phys. Rev. E 86, 051611 (2012)] on the conditions for occurrence of strong anisotropy in the scaling properties of two-dimensional surfaces displaying generic scale invariance. In that study, a natural scaling ansatz was proposed for strongly anisotropic systems, which arises naturally when analyzing data from, e.g., thin-film production experiments. The ansatz was tested in Gaussian (linear) models of surface dynamics and in nonlinear models, like the Hwa-Kardar (HK) equation [Phys. Rev. Lett. 62, 1813 (1989)], which are susceptible of accurate approximations through the former. In contrast, here we analyze nonlinear equations for which such approximations fail. Working within generically scale-invariant situations, and as representative case studies, we formulate and study a generalization of the HK equation for conserved dynamics and reconsider well-known systems, such as the conserved and the nonconserved anisotropic Kardar-Parisi-Zhang equations. Through the combined use of dynamic renormalization group analysis and direct numerical simulations, we conclude that the occurrence of strong anisotropy in two-dimensional surfaces requires dynamics to be conserved. We find that, moreover, strong anisotropy is not generic in parameter space but requires, rather, specific forms of the terms appearing in the equation of motion, whose justification needs detailed information on the dynamical process that is being modeled in each particular case.

Invited review: KPZ. Recent developments via a variational formulation

Recently, a variational approach has been introduced for the paradigmatic Kardar-Parisi-Zhang (KPZ) equation. Here we review that approach, together with the functional Taylor expansion that the KPZ nonequilibrium potential (NEP) admits. Such expansion becomes naturally truncated at third order, giving rise to a nonlinear stochastic partial differential equation to be regarded as a gradient-flow counterpart to the KPZ equation. A dynamic renormalization group analysis at one-loop order of this new mesoscopic model yields the KPZ scaling relation $\alpha + z = 2$, as a consequence of the exact cancelation of the different contributions to vertex renormalization. This result is quite remarkable, considering the lower degree of symmetry of this equation, which is in particular not Galilean invariant. In addition, this scheme is exploited to inquire about the dynamical behavior of the KPZ equation through a path-integral approach. Each of these aspects offers novel points of view and sheds light on particular aspects of the dynamics of the KPZ equation. Received: 25 June 2013, Accepted: 10 December 2013; Reviewed by: F. Reis, Instituto de Fisica, Univ. Fed. Fluminense, Brazil; Edited by: R. Dickman; DOI: http://dx.doi.org/10.4279/PIP.050010Cite as: H S Wio, R R Deza, C Escudero, J A Revelli, Papers in Physics 5, 050010 (2013)

Renormalization Group Analysis of Weakly Rotating Turbulent Flows

Dynamic renormalization group (RNG) analysis is applied to the investigation of the behavior of the infrared limits of weakly rotating turbulence. For turbulent flow subject to weak rotation, the anisotropic part in the renormalized propagation is considered to be a perturbation of the isotropic part. Then, with a low-order approximation, the coarsening procedure of RNG transformation is performed. After implementing the coarsening and rescaling procedures, the RNG analysis suggests that the spherically averaged energy spectrum has the scaling behavior E(k) ∝ k−11/5 for weakly rotating turbulence. It is also shown that the Coriolis force will disturb the stability of the Kolmogorov −5/3 energy spectrum and will change the scaling behavior even in the case of weak rotation.

Geometric Principles of Surface Growth

I introduce a new equation describing epitaxial growth processes. This equation is derived from a simple variational geometric principle and it has a straightforward interpretation in terms of continuum and microscopic physics. It is also able to reproduce the critical behavior already observed, mound formation and mass conservation, but however does not fit a divergence form as the most commonly spread models of conserved surface growth. This formulation allows me to connect the results of the dynamic renormalization group analysis with intuitive geometric principles, whose generic character may well allow them to describe surface growth and other phenomena in different areas of physics.

Dynamic renormalization-group analysis of the d+1 dimensional Kuramoto-Sivashinsky equation with both conservative and nonconservative noises

The long-wavelength properties of the (d + 1)-dimensional Kuramoto-Sivashinsky (KS) equation with both conservative and nonconservative noises are investigated by use of the dynamic renormalization-group (DRG) theory. The dynamic exponent z and roughness exponent α are calculated for substrate dimensions d = 1 and d = 2, respectively. In the case of d = 1, we arrive at the critical exponents z = 1.5 and α = 0.5 , which are consistent with the results obtained by Ueno et al. in the discussion of the same noisy KS equation in 1+1 dimensions [Phys. Rev. E 71, 046138 (2005)] and are believed to be identical with the dynamic scaling of the Kardar-Parisi-Zhang (KPZ) in 1+1 dimensions. In the case of d = 2, we find a fixed point with the dynamic exponents z = 2.866 and α = -0.866 , which show that, as in the 1 + 1 dimensions situation, the existence of the conservative noise in 2 + 1 or higher dimensional KS equation can also lead to new fixed points with different dynamic scaling exponents. In addition, since a higher order approximation is adopted, our calculations in this paper have improved the results obtained previously by Cuerno and Lauritsen [Phys. Rev. E 52, 4853 (1995)] in the DRG analysis of the noisy KS equation, where the conservative noise is not taken into account.

Dynamic renormalization group analysis of propagation of elastic waves in two-dimensional heterogeneous media

We study localization of elastic waves in two-dimensional heterogeneous solids with randomly distributed Lam\'e coefficients, as well as those with long-range correlations with a power-law correlation function. The Matin-Siggia-Rose method is used, and the one-loop renormalization group (RG) equations for the the coupling constants are derived in the limit of long wavelengths. The various phases of the coupling constants space, which depend on the value $\rho$, the exponent that characterizes the power-law correlation function, are determined and described. Qualitatively different behaviors emerge for $\rho<1$ and $\rho>1$. The Gaussian fixed point (FP) is stable (unstable) for $\rho<1$ ($\rho>1$). For $\rho<1$ there is a region of the coupling constants space in which the RG flows are toward the Gaussian FP, implying that the disorder is irrelevant and the waves are delocalized. In the rest of the disorder space the elastic waves are localized. We compare the results with those obtained previously for acoustic wave propagation in the same type of heterogeneous media, and describe the similarities and differences between the two phenomena.

Spectral Properties of One-Dimensional Diffusive Systems Subject to Stochastic Forcing

Abstract The vertical wavenumber and frequency spectra of horizontal wind and temperature in stochastically driven systems with diffusion, either due to uniform background eddy and molecular transport, or due to adjustment processes associated with shear or convective instability, are studied. Because of the dominating role of vertical transport in a stratified fluid, one-dimensional Langevin-type equations could be ascribed to such systems in the vertical direction. The linear equation with uniform diffusion is solved explicitly, and the spectra follow power-law distributions if the stochastic force is Gaussian. The nonlinear equations with gradient (either shear or lapse rate) dependent diffusion coefficients are shown to support scale invariance, and the power-law indices of the spectra are determined from dynamic renormalization group (DRG) analysis under rather general conditions. The exact power-law indices vary with the spectrum of the stochastic force and the nonlinearity of the systems. If the wavenumber spectrum of the force is moderately red (between k0 and k−2), the spectral indices of horizontal wind and temperature and the range of their variability are in general agreement with those inferred from wind and temperature measurements. The indices in both linear and nonlinear cases are confirmed by numerical simulations. This theory may suggest an alternative explanation to the universal vertical wavenumber and frequency spectra and their variability. By relating the universal spectra to systems characterized by stochastic forcing and background diffusion or diffusive adjustment due to shear or convective instability, which are ubiquitous in a stratified fluid, the difficulty to associate the time- and location-independent spectral features directly with the highly time- and location-dependent gravity waves or wave-breaking events is avoided. If such systems are suggestive of the real atmosphere, there is a need to be cautious in making assumptions regarding gravity waves solely based on the universal spectra when analyzing and interpreting wind and temperature observations.

Self-consistent expansion results for the nonlocal Kardar-Parisi-Zhang equation.

In this paper various predictions for the scaling exponents of the nonlocal Kardar-Parisi-Zhang (NKPZ) equation are discussed. I use the self-consistent expansion (SCE), and obtain results that are quite different from the result obtained in the past, using dynamic renormalization-group analysis, a scaling approach, and a self-consistent mode-coupling approach. It is shown that the results obtained using SCE recover an exact result for a subfamily of the NKPZ models in one dimension, while all the other methods fail to do so. It is also shown that the SCE result is the only one that is compatible with simple observations on the dependence of the dynamic exponent z in the NKPZ model on the exponent rho characterizing the decay of the nonlinear interaction. The reasons for the failure of other methods to deal with NKPZ are also discussed.

Nonlocal growth equations—a test case for dynamic renormalization group analysis

In this paper we discuss nonlocal growth equations such as the generalization of the Kardar–Parisi–Zhang (KPZ) equation that includes long-range interactions, also known as the Nonlocal-Kardar–Parisi–Zhang (NKPZ) equation, and the nonlocal version of the molecular-beam-epitaxy (NMBE) equation. We show that the steady-state strong coupling solution for nonlocal models such as NKPZ and NMBE can be obtained exactly in one dimension for some special cases, using the Fokker–Planck form of these equations. The exact results we derive do not agree with previous results obtained by Dynamic Renormalization Group (DRG) analysis. This discrepancy is important because DRG is a common method used extensively to deal with nonlinear field equations. While difficulties with this method for d>1 has been realized in the past, it has been believed so far that DRG is still safe in one dimension. Our result shows differently. The reasons for the failure of DRG to recover the exact one-dimensional results are also discussed.

Continuum analysis of an avalanche model for solar flares.

We investigate the continuum limit of a class of self-organized critical lattice models for solar flares. Such models differ from the classical numerical sandpile model in their formulation of stability criteria in terms of the curvature of the nodal field, and are known to belong to a different universality class. A fourth-order nonlinear hyperdiffusion equation is reverse engineered from the discrete model's redistribution rule. A dynamical renormalization-group analysis of the equation yields scaling exponents that compare favorably with those measured in the discrete lattice model within the relevant spectral range dictated by the sizes of the domain and the lattice grid. We argue that the fourth-order nonlinear diffusion equation that models the behavior of the discrete model in the continuum limit is, in fact, compatible with magnetohydrodynamics (MHD) of the flaring phenomenon in the regime of strong magnetic field and the effective magnetic diffusivity characteristic of strong MHD turbulence.