Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory

Oscillons from Q-Balls through Renormalization

Using a renormalization-inspired perturbation expansion we show that oscillons in a generic field theory in (1+1) dimensions arise as dressed Q-balls of a universal (up to the leading nonlinear order) complex field theory. This theory reveals a close similarity to the integrable complex sine-Gordon model, which possesses exact multi-Q-balls. We show that excited oscillons, with characteristic modulations of their amplitude, are two-oscillons bound states generated from a two Q-ball solution.

Self-consistent expansion and field-theoretic renormalization group for a singular nonlinear diffusion equation with anomalous scaling

The method of self-consistent expansions is a powerful tool for handling strong coupling problems that might otherwise be beyond the reach of perturbation theory, providing surprisingly accurate approximations even at low order. First applied in its embryonic form to fully-developed turbulence, it has subsequently been successfully applied to a variety of problems that include polymer statistics, interface dynamics, and high-order perturbation theory for the anharmonic oscillator. Here, we show that the self-consistent expansion can be applied to singular perturbation problems arising in the theory of partial differential equations in conjunction with renormalization group methods. We demonstrate its application to Barenblatt's nonlinear diffusion equation for porous media filtration, where the long-time asymptotics exhibits anomalous dimensions that can be systematically calculated using the perturbative renormalization group. We find that even the first-order self-consistent expansion, when combined with the Callan-Symanzik equation, improves the approximation of the anomalous dimension obtained by the first-order perturbative renormalization group, especially in the strong coupling regime. We also develop a field-theoretic framework for deterministic partial differential equations to facilitate the application of self-consistent expansions to other dynamic systems and illustrate its application using the example of Barenblatt's equation. The scope of our results on the application of renormalization group and self-consistent expansions is limited to partial differential equations whose long-time asymptotics is controlled by incomplete similarity. However, our work suggests that these methods could be applied to a broader suite of singular perturbation problems such as boundary layer theory, multiple scales analysis, and matched asymptotic expansions, for which excellent approximations using renormalization group methods alone are already available.

Kinetic models reveal the interplay of protein production and aggregation

Protein aggregation is a key process in the development of many neurodegenerative disorders, including dementias such as Alzheimer's disease. Significant progress has been made in understanding the molecular mechanisms of aggregate formation in pure buffer systems, much of which was enabled by the development of integrated rate laws that allowed for mechanistic analysis of aggregation kinetics. However, in order to translate these findings into disease-relevant conclusions and to make predictions about the effect of potential alterations to the aggregation reactions by the addition of putative inhibitors, the current models need to be extended to account for the altered situation encountered in living systems. In particular, in vivo, the total protein concentrations typically do not remain constant and aggregation-prone monomers are constantly being produced but also degraded by cells. Here, we build a theoretical model that explicitly takes into account monomer production, derive integrated rate laws and discuss the resulting scaling laws and limiting behaviours. We demonstrate that our models are suited for the aggregation-prone Huntington's disease-associated peptide HttQ45 utilizing a system for continuous in situ monomer production and the aggregation of the tumour suppressor protein P53. The aggregation-prone HttQ45 monomer was produced through enzymatic cleavage of a larger construct in which a fused protein domain served as an internal inhibitor. For P53, only the unfolded monomers form aggregates, making the unfolding a rate-limiting step which constitutes a source of aggregation-prone monomers. The new model opens up possibilities for a quantitative description of aggregation in living systems, allowing for example the modelling of inhibitors of aggregation in a dynamic environment of continuous protein synthesis.

Inertia suppresses signatures of activity of active Brownian particles in a harmonic potential

A harmonically trapped active Brownian particle exhibits two types of positional distributions-one has a single peak and the other has a single well-that signify steady-state dynamics with low and high activity, respectively. Adding inertia to the translational motion preserves this strict classification of either single-peak or single-well densities but shifts the dividing boundary between the states in the parameter space. We characterize this shift for the dynamics in one spatial dimension using the static Fokker-Planck equation for the full joint distribution of the state space. We derive local results analytically with a perturbation method for a small rotational velocity and then extend them globally with a numerical approach.

Schrödinger's Range-Shifting Cat: How Skewed Temperature Dependence Impacts Persistence with Climate Change

AbstractThe majority of species display strongly asymmetric responses to climatic variables, yet most analytic models used to investigate how species will respond to climate change assume symmetric responses, with largely unknown consequences. Applying a known mapping of population dynamical equations onto corresponding well-studied problems from quantum mechanics, we extend analytical results to incorporate this asymmetry. We derive expressions in terms of parameters representing climate velocity, dispersal rate, maximum growth rate, niche width, high-frequency climate variability, and environmental performance curve skew for three key responses: (1) population persistence, (2) lag between range displacement and climate displacement, and (3) location of maximum population sensitivity. We find that asymmetry impacts these climate change responses, but surprisingly, under our model assumptions, the direction (i.e., warm skewed or cool skewed) of performance curve asymmetry does not strongly contribute to either persistence or lags. Conservation measures to support range-shifting populations may have most benefit near their environmental optimum or where the environmental dependence is shallow, irrespective of whether this is the leading or trailing edge. A metapopulation simulation corroborates our results. Our results shed fresh light on how key features of a species' environmental performance curve can impact its response to climate change.

Geometric phase for nonlinear oscillators from perturbative renormalization group

We formulate a renormalization-group approach to a general nonlinear oscillator problem. The approach is based on the exact group law obeyed by solutions of the corresponding ordinary differential equation. We consider both the autonomous models with time-independent parameters, as well as nonautonomous models with slowly varying parameters. We show that the renormalization-group equations for the nonautonomous case can be used to determine the geometric phase acquired by the oscillator during the change of its parameters. We illustrate the obtained results by applying them to the Van der Pol and Van der Pol-Duffing models.

Impact of Dissipation on Universal Fluctuation Dynamics in Open Quantum Systems

Recent theoretical and experimental works have explored universal dynamics related to surface growth physics in isolated quantum systems. In this Letter, we theoretically elucidate that dissipation drastically alters universal particle-number-fluctuation dynamics associated with surface-roughness growth in one-dimensional free fermions and bosons. In a system under dephasing that causes loss of spatial coherence, we numerically find that a universality class of surface-roughness dynamics changes from the ballistic class to a class with the Edwards-Wilkinson scaling exponents and an unconventional scaling function. We provide the analytical derivation of the diffusion equation from the dephasing Lindblad equation via a renormalization-group technique and succeed in explaining the drastic change. Furthermore, we numerically find the same change of the universality class under a more nontrivial dissipation, i.e., symmetric incoherent hopping.

Kinetic profiling of therapeutic strategies for inhibiting the formation of amyloid oligomers

Protein self-assembly into amyloid fibrils underlies several neurodegenerative conditions, including Alzheimer's and Parkinson's diseases. It has become apparent that the small oligomers formed during this process constitute neurotoxic molecular species associated with amyloid aggregation. Targeting the formation of oligomers represents, therefore, a possible therapeutic avenue to combat these diseases. However, it remains challenging to establish which microscopic steps should be targeted to suppress most effectively the generation of oligomeric aggregates. Recently, we have developed a kinetic model of oligomer dynamics during amyloid aggregation. Here, we use this approach to derive explicit scaling relationships that reveal how key features of the time evolution of oligomers, including oligomer peak concentration and lifetime, are controlled by the different rate parameters. We discuss the therapeutic implications of our framework by predicting changes in oligomer concentrations when the rates of the individual microscopic events are varied. Our results identify the kinetic parameters that control most effectively the generation of oligomers, thus opening a new path for the systematic rational design of therapeutic strategies against amyloid-related diseases.

The Onsager theory of wall-bounded turbulence and Taylor's momentum anomaly

We discuss the Onsager theory of wall-bounded turbulence, analysing the momentum dissipation anomaly hypothesized by Taylor. Turbulent drag laws observed with both smooth and rough walls imply ultraviolet divergences of velocity gradients. These are eliminated by a coarse-graining operation, filtering out small-scale eddies and windowing out near-wall eddies, thus introducing two arbitrary regularization length-scales. The regularized equations for resolved eddies correspond to the weak formulation of the Navier-Stokes equation and contain, in addition to the usual turbulent stress, also an inertial drag force modelling momentum exchange with unresolved near-wall eddies. Using an Onsager-type argument based on the principle of renormalization group invariance, we derive an upper bound on wall friction by a function of Reynolds number determined by the modulus of continuity of the velocity at the wall. Our main result is a deterministic version of Prandtl's relation between the Blasius [Formula: see text] drag law and the 1/7 power-law profile of the mean streamwise velocity. At higher Reynolds, the von Kármán-Prandtl drag law requires instead a slow logarithmic approach of velocity to zero at the wall. We discuss briefly also the large-eddy simulation of wall-bounded flows and use of iterative renormalization group methods to establish universal statistics in the inertial sublayer. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

Instabilities of Scalar Fields around Oscillating Stars

The behavior of fundamental fields in strong gravity or nontrivial environments is important for our understanding of nature. This problem has interesting applications in the context of dark matter, of dark energy physics, or of quantum field theory. The dynamics of fundamental fields has been studied mainly in static or stationary backgrounds, whereas most of our Universe is dynamic. In this Letter we investigate "blueshift" and parametric instabilities of scalar fields in dynamical backgrounds, which can be triggered (for instance) by oscillating stars in scalar-tensor theories of gravity. We discuss possible implications of our results, which include constraints on an otherwise hard-to-access parameter space of scalar-tensor theories.

Numerical renormalization-group-based approach to secular perturbation theory

Perturbation theory is a crucial tool for many physical systems, when exact solutions are not available, or nonperturbative numerical solutions are intractable. Naive perturbation theory often fails on long timescales, leading to secularly growing solutions. These divergences have been treated with a variety of techniques, including the powerful dynamical renormalization group (DRG). Most of the existing DRG approaches rely on having analytic solutions up to some order in perturbation theory. However, sometimes the equations can only be solved numerically. We reformulate the DRG in the language of differential geometry, which allows us to apply it to numerical solutions of the background and perturbation equations. This formulation also enables us to use the DRG in systems with background parameter flows and, therefore, extend our results to any order in perturbation theory. As an example, we apply this method to calculate the soliton-like solutions of the Korteweg-de Vries equation deformed by adding a small damping term. We numerically construct DRG solutions which are valid on secular timescales, long after naive perturbation theory has broken down.

Anharmonic spectral features via trajectory-based quantum dynamics: A perturbative analysis of the interplay between dynamics and sampling

The performance of different approximate algorithms for computing anharmonic features in vibrational spectra is analyzed and compared on model and more realistic systems that present relevant nuclear quantum effects. The methods considered combine approximate sampling of the quantum thermal distribution with classical time propagation and include Matsubara dynamics, path integral dynamics approaches, linearized initial value representation, and the recently introduced adaptive quantum thermal bath. A perturbative analysis of these different methods enables us to account for the observed numerical performance on prototypes for overtones and combination bands and to draw qualitatively correct trends for the numerical results obtained for Fermi resonances. Our results prove that the unequal performances of these approaches often derive from the method employed to sample initial conditions and not, as usually assumed, from the lack of coherence in the time propagation. Furthermore, as confirmed by the analysis reported in Benson and Althorpe, J. Chem. Phys. 130, 194510 (2021), we demonstrate, both via the perturbative approach and numerically, that path integral dynamics methods fail to reproduce the intensities of these anharmonic features and follow purely classical trends with respect to their temperature behavior. Finally, the remarkably accurate performance of the adaptive quantum thermal bath approach is documented and motivated.

Renormalization group as a Koopman operator

Koopman operator theory is shown to be directly related to the renormalization group. This observation allows us, with no assumption of translational invariance, to compute the critical exponents η and δ, as well as ratios of critical exponents, of classical spin systems from single observables alone. This broadens the types of problems that the renormalization group framework can be applied to and establish universality classes of. In addition, this connection may allow for a new, data-driven way in which to find the renormalization group fixed point(s), and their relevant and irrelevant directions.

Scale-invariant topology and bursty branching of evolutionary trees emerge from niche construction

Phylogenetic trees describe both the evolutionary process and community diversity. Recent work has established that they exhibit scale-invariant topology, which quantifies the fact that their branching lies in between the two extreme cases of balanced binary trees and maximally unbalanced ones. In addition, the backbones of phylogenetic trees exhibit bursts of diversification on all timescales. Here, we present a simple, coarse-grained statistical model of niche construction coupled to speciation. Finite-size scaling analysis of the dynamics shows that the resultant phylogenetic tree topology is scale-invariant due to a singularity arising from large niche construction fluctuations that follow extinction events. The same model recapitulates the bursty pattern of diversification in time. These results show how dynamical scaling laws of phylogenetic trees on long timescales can reflect the indelible imprint of the interplay between ecological and evolutionary processes.

Universality of filamentous aggregation phenomena

We use perturbative renormalization group theory to study the kinetics of protein aggregation phenomena in a unified manner across multiple timescales. Using this approach, we find that, irrespective of the specific molecular details or experimental conditions, filamentous assembly systems display universal behavior in time. Moreover, we show that the universality classes for protein aggregation correspond to simple autocatalytic processes and that the diversity of behavior in these systems is determined solely by the reaction order for secondary nucleation with respect to the protein concentration, which labels all possible universality classes. We validate these predictions on experimental data for the aggregation of several different proteins at several different initial concentrations, which by appropriate coordinate transformations we are able to collapse onto universal kinetic growth curves. These results establish the power of the perturbative renormalization group in distilling the ultimately simple temporal behavior of complex protein aggregation systems, creating the possibility to study the kinetics of general self-assembly phenomena in a unified fashion.

Global Density Profile For Particle Non-Conserving One Dimensional Transport From Renormalization Group Flows

The totally asymmetric simple exclusion process along with particle adsorption and evaporation kinetics is a model of boundary-induced nonequilibrium phase transition. In the continuum limit, the average particle density across the system is described by a singular differential equation involving multiple scales which lead to the formation of boundary layers (BL) or shocks. A renormalization group analysis is developed here by using the location and the width of the BL as the renormalization parameters. It not only allows us to cure the large distance divergences in the perturbative solution for the BL but also generates, from the BL solution, an analytical form for the global density profile. The predicted scaling form is checked against numerical solutions for finite systems.

An alternative approach to Michaelis-Menten kinetics that is based on the renormalization group

We apply to Michaelis-Menten kinetics an alternative approach to the study of Singularly Perturbed Differential Equations, that is based on the Renormalization Group (SPDERG). To this aim, we first rebuild the perturbation expansion for Michaelis-Menten kinetics, beyond the standard Quasi-Steady-State Approximation (sQSSA), determining the 2nd order contributions to the inner solutions, that are presented here for the first time to our knowledge. Our main result is that the SPDERG 2nd order uniform approximations reproduce the numerical solutions of the original problem in a better way than the known results of the perturbation expansion, even in the critical matching region. Indeed, we obtain analytical results nearly indistinguishable from the numerical solutions of the original problem in a large part of the whole relevant time window, even in the case in which the kinetic constants produce an expansion parameter value as large as ɛ=0.5.

Perturbative dynamics of open quantum systems by renormalization group method

We analyze perturbative dynamics of a composite system consisting of a quantum mechanical system and an environment by the renormalization group (RG) method. The solution obtained from the RG method has no secular terms and approximates the exact solution for a long time interval. Moreover, the RG method causes a reduction of the dynamics of the composite system under some assumptions. We show that this reduced dynamics is closely related to a quantum master equation for the quantum mechanical system. We compare this dynamics with the exact dynamics in an exactly solvable spin-boson model.

Physical principles of filamentous protein self-assembly kinetics

The polymerization of proteins and peptides into filamentous supramolecular structures is an elementary form of self-organization of key importance to the functioning biological systems, as in the case of actin biofilaments that compose the cellular cytoskeleton. Aberrant filamentous protein self-assembly, however, is associated with undesired effects and severe clinical disorders, such as Alzheimer's and Parkinson's diseases, which, at the molecular level, are associated with the formation of certain forms of filamentous protein aggregates known as amyloids. Moreover, due to their unique physicochemical properties, protein filaments are finding extensive applications as biomaterials for nanotechnology. With all these different factors at play, the field of filamentous protein self-assembly has experienced tremendous activity in recent years. A key question in this area has been to elucidate the microscopic mechanisms through which filamentous aggregates emerge from dispersed proteins with the goal of uncovering the underlying physical principles. With the latest developments in the mathematical modeling of protein aggregation kinetics as well as the improvement of the available experimental techniques it is now possible to tackle many of these complex systems and carry out detailed analyses of the underlying microscopic steps involved in protein filament formation. In this paper, we review some classical and modern kinetic theories of protein filament formation, highlighting their use as a general strategy for quantifying the molecular-level mechanisms and transition states involved in these processes.

Renormalization group analysis for an asymmetric simple exclusion process

A perturbative renormalization group method is used to obtain steady-state density profiles of a totally asymmetric simple exclusion process with particle adsorption and evaporation. This method allows us to obtain a globally valid solution for the density profile without the asymptotic matching of bulk and boundary layer solutions. In addition, we show a nontrivial scaling of the boundary layer width with the system size close to specific phase boundaries.

Charged plate in asymmetric electrolytes: One-loop renormalization of surface charge density and Debye length due to ionic correlations

Self-consistent field theory (SCFT) is used to study the mean potential near a charged plate inside a m:-n electrolyte. A perturbation series is developed in terms of g=4πκb, where band1/κ are Bjerrum length and bare Debye length, respectively. To the zeroth order, we obtain the nonlinear Poisson-Boltzmann theory. For asymmetric electrolytes (m≠n), the first order (one-loop) correction to mean potential contains a secular term, which indicates the breakdown of the regular perturbation method. Using a renormalizaton group transformation, we remove the secular term and obtain a globally well-behaved one-loop approximation with a renormalized Debye length and a renormalized surface charge density. Furthermore, we find that if the counterions are multivalent, the surface charge density is renormalized substantially downwards and may undergo a change of sign, if the bare surface charge density is sufficiently large. Our results agrees with large MC simulation even when the density of electrolytes is relatively high.

Fluid flow enhances the effectiveness of toxin export by aquatic microorganisms: a first-passage perspective on microvilli and the concentration boundary layer

A central challenge for organisms during development is determining a means to efficiently export toxic molecules from inside the developing embryo. For aquatic microorganisms, the strategies employed should be robust with respect to the variable ocean environment and limit the chances that exported toxins are reabsorbed. As a result, the problem of toxin export is closely related to the physics of mass transport in a fluid. In this paper, we consider a model first-passage problem for the uptake of exported toxins by a spherical embryo. By considering how macroscale fluid turbulence manifests itself on the microscale of the embryo, we determine that fluid flow enhances the effectiveness of toxin export as compared to the case of diffusion-limited transport. In the regime of a large Péclet number, a perturbative solution of the advection-diffusion equation reveals that a concentration boundary layer forms at the surface of the embryo. The model results suggest a functional role for cell surface roughness in the export process, with the thickness of the concentration boundary layer setting the length scale for cell membrane protrusions known as microvilli. We highlight connections between the model results and experiments on the development of sea urchin embryos.

Asymptotic solutions of the Oosawa model for the length distribution of biofilaments

Nucleated polymerisation phenomena are general linear growth processes that underlie the formation of a range of biofilaments in nature, including actin and tubulin that are key components of the cellular cytoskeleton. The conventional theoretical framework for describing this process is the Oosawa model that takes into account homogeneous nucleation coupled to linear growth. In his original work, Oosawa provided an analytical solution to the total mass concentration of filaments; the time evolution of the full length distribution has, however, been challenging to access, in large part due to the nonlinear nature of the rate equations inherent in the description of such phenomena and to date analytical solutions for the filament distribution are known only in certain special cases. Here, by exploiting a technique based on the method of matched asymptotics, we present an analytical treatment of the Oosawa model that describes the shape of the length distribution of biofilaments reversibly growing through primary nucleation and filament elongation. Our work highlights the power of matched asymptotics for obtaining closed-form analytical solutions to nonlinear master equations in biophysics and allows us to identify the key time scales that characterize biological polymerization processes.

Uniqueness of Landau-Lifshitz energy frame in relativistic dissipative hydrodynamics

We show that the relativistic dissipative hydrodynamic equation derived from the relativistic Boltzmann equation by the renormalization-group method uniquely leads to the one in the energy frame proposed by Landau and Lifshitz, provided that the macroscopic-frame vector, which defines the local rest frame of the flow velocity, is independent of the momenta of constituent particles, as it should. We argue that the relativistic hydrodynamic equations for viscous fluids must be defined on the energy frame if consistent with the underlying relativistic kinetic equation.

Bridging steady states with renormalization group analysis

Transitions between different condensed phases, molecular conformations, chemical compositions, or spatiotemporal patterns play important roles in many branches of natural science and at the same time incur serious challenges in their precise characterization. We design an approach for computing connecting orbits bridging steady states based on the renormalization group analysis. The technique is successfully applied to several interesting examples and good analytic results are obtained in a systematic and unified way.

Singular perturbation analysis of a reduced model for collective motion: a renormalization group approach

In a system of noisy self-propelled particles with interactions that favor directional alignment, collective motion will appear if the density of particles is beyond a critical density. Starting with a reduced model for collective motion, we determine how the critical density depends on the form of the initial perturbation. Specifically, we employ a renormalization-group improved perturbative method to analyze the model equations and show analytically, up to first order in the perturbation parameter, how the critical density is modified by the strength of the initial angular perturbation in the system.

Renormalization group approach to interfacial motion in incompressible Richtmyer-Meshkov instability

Nonlinear interfacial motion in incompressible Richtmyer-Meshkov instability is theoretically investigated using the renormalization group approach. The amplitude equation describing the asymptotic interfacial motion is derived using this approach. A comparison with calculations carried out by the weakly nonlinear analysis is performed for various Atwood numbers and the validity of the renormalization group approach is discussed. We show that this approach suppresses the divergence in the perturbative solutions obtained by the weakly nonlinear analysis and provides better approximations for the growth rate of bubbles and spikes and interfacial profiles at the asymptotic nonlinear stage without requiring the use of Padé approximants.

Construction of coarse-grained order parameters in nonequilibrium systems

We develop a renormalization-group (RG) procedure that includes important system-specific features. The key ingredient is to systematize the coarse-graining procedure that generates the RG flow. The coarse-graining technology comes from the control and operator theoretic model reduction. The resulting "generalized" RG is a consistent generalization of the Wilsonian RG. We apply the procedure to a deterministic nonlinear wave equation (NLWE) with probabilistic initial conditions. We derive the form of the projection operator from the dynamics of the NLWE and then use it to generate the RG flow for the distribution of initial conditions. The probability density of the initial conditions is chosen to be a Boltzmann weight that is quartic in the field variables. In our calculation, we find that in contrast to conventional implementations of the RG, naïve power counting breaks down. We also show that the resulting RG equations are different from those derived from the conventional RG.

Comment on "Renormalization-group theory for the phase-field crystal equation"

Athreya, Goldenfeld, and Dantzig [Phys. Rev. E 74, 011601 (2006)] claim that the current implementation of the renormalization-group method neglects the proper ordering of renormalization and differentiation. Their analysis is, however, based on the wrong multiple-scales method results.

Solution of reduced equations derived with singular perturbation methods

For singular perturbation problems in dynamical systems, various appropriate singular perturbation methods have been proposed to eliminate secular terms appearing in the naive expansion. For example, the method of multiple time scales, the normal form method, center manifold theory, and the renormalization group method are well known. It is shown that all of the solutions of the reduced equations constructed with those methods are exactly equal to the sum of the most divergent secular terms appearing in the naive expansion. For the proof, a method to construct a perturbation solution which differs from the conventional one is presented, where we make use of the theory of the Lie symmetry group.

Renormalization group interpretation of the Born and Rytov approximations

In this paper the method of renormalization group (RG) [Phys. Rev. E54, 376 (1996)] is related to the well-known approximations of Rytov and Born used in wave propagation in deterministic and random media. Certain problems in linear and nonlinear media are examined from the viewpoint of RG and compared with the literature on Born and Rytov approximations. It is found that the Rytov approximation forms a special case of the asymptotic expansion generated by the RG, and as such it gives a superior approximation to the exact solution compared with its Born counterpart. Analogous conclusions are reached for nonlinear equations with an intensity-dependent index of refraction where the RG recovers the exact solution.

Secular series and renormalization group for amplitude equations

We have developed a technique that circumvents the process of elimination of secular terms [L.-Y. Chen, N. Goldenfeld, and Y. Oono, Phys. Rev. E 54, 376 (1996)] and reproduces the uniformly valid approximations, amplitude equations, and first integrals. The technique is based on a rearrangement of secular terms and their grouping into the secular series that multiplies the constants of the asymptotic expansion. We illustrate the technique by deriving amplitude equations for standard nonlinear oscillator and boundary-layer problems.

Reduction of amplitude equations by the renormalization group approach

This article elucidates and analyzes the fundamental underlying structure of the renormalization group (RG) approach as it applies to the solution of any differential equation involving multiple scales. The amplitude equation derived through the elimination of secular terms arising from a naive perturbation expansion of the solution to these equations by the RG approach [L.-Y. Chen, N. Goldenfeld, and Y. Oono, Phys. Rev. E 54, 376 (1996)] is reduced to an algebraic equation which is expressed in terms of the Thiele semi-invariants or cumulants of the eliminant sequence {Zi}i=1infinity. Its use is illustrated through the solution of both linear and nonlinear perturbation problems and certain results from the literature are recovered as special cases. The fundamental structure that emerges from the application of the RG approach is not the amplitude equation but the aforementioned algebraic equation.

Adaptive mesh computation of polycrystalline pattern formation using a renormalization-group reduction of the phase-field crystal model

We implement an adaptive mesh algorithm for calculating the space and time dependence of the atomic density field in microscopic material processes. Our numerical approach uses the systematic renormalization-group formulation of a phase-field crystal model of a pure material to provide the underlying equations for the complex amplitude of the atomic density field--a quantity that is spatially uniform except near topological defects, grain boundaries, and other lattice imperfections. Our algorithm employs a hybrid formulation of the amplitude equations, combining Cartesian and polar decompositions of the complex amplitude. We show that this approach leads to an acceleration by three orders of magnitude in model calculations of polycrystalline grain growth in two dimensions.

Noise-induced phase transitions in field-dependent relaxational dynamics: the Gaussian ansatz

We present an analytic mean-field theory for relaxational dynamics in spatially extended systems that undergo purely noise-induced phase transitions to ordered states. The theory augments the usual mean-field approach with a Gaussian ansatz that yields quantitatively accurate results for strong coupling. We obtain analytic results not only for steady-state mean fields and distribution widths, but also for the dynamical approach to a steady state or to collective oscillatory behaviors in multifield systems. Because the theory yields dynamical information, it can also predict the initial-condition-dependent final state (disordered state, steady or oscillatory ordered state) in multistable arrays.

Return times for stochastic processes with power-law scaling

An analytical study of the return time distribution of extreme events for stochastic processes with power-law correlation has been carried out. The calculation is based on an epsilon expansion in the correlation exponent: C(t)=/t/-1+epsilon. The fixed point of the theory is associated with stretched exponential scaling of the distribution; analytical expressions have been provided in the preasymptotic regime. Also, the permanence time distribution appears to be characterized by stretched exponential scaling. The conditions for application of the theory to non-Gaussian processes have been analyzed and the relations with the issue of return times in the case of multifractal measures have been discussed.

Renormalization-group theory for the phase-field crystal equation

We derive a set of rotationally covariant amplitude equations for use in multiscale simulation of the two-dimensional phase-field crystal model by a variety of renormalization-group (RG) methods. We show that the presence of a conservation law introduces an ambiguity in operator ordering in the RG procedure, which we show how to resolve. We compare our analysis with standard multiple-scale techniques, where identical results can be obtained with greater labor, by going to sixth order in perturbation theory, and by assuming the correct scaling of space and time.

Exact renormalization group approach to a nonlinear diffusion equation

The exact renormalization group is applied to a nonlinear diffusion equation with a discontinuous diffusion coefficient. The generating functional of the solution for the initial-value problem of nonlinear diffusion equations is first introduced, and next a regularization scheme is presented. It is shown that the renormalization of an action functional in the generating functional leads to an anomalous diffusion exponent in full order of the perturbation series with respect to a nonlinearity.

Photon emission from a driven single-molecule source: A renormalization group approach

The photon emission from a single molecule driven simultaneously by a laser and a slow electric radio frequency (rf) field is studied. We use a non-Hermitian Hamiltonian approach which accounts for the radiative decay of a two-level system modeling the single-molecule source. We apply the renormalization group method for differential equations to obtain long time solution of the corresponding Schrodinger equation, which allows us to calculate the average waiting time for the first photon emission. Then, we analyze the conditions for suppression and enhancement of photon emission in this dissipative two-level system. In particular we derive a transcendental equation, which yields the nontrivial rf field control parameters, for which enhancement and suppression of photon emission occurs. For finite values of radiative decay rate an abrupt transition to the state when both situations are indistinguishable is found for certain values of the rf field parameters. Our results are shown to be in agreement with the available experiments [Ch. Brunel et al., Phys. Rev. Lett. 81, 2679 (1998)].

Renormalization group approach to multiscale simulation of polycrystalline materials using the phase field crystal model

We propose a computationally efficient approach to multiscale simulation of polycrystalline materials, based on the phase field crystal model. The order parameter describing the density profile at the nanoscale is reconstructed from its slowly varying amplitude and phase, which satisfy rotationally covariant equations derivable from the renormalization group. We validate the approach using the example of two-dimensional grain nucleation and growth.

Deformation of a renormalization-group equation applied to infinite-order phase transitions

By adding a linear term to a renormalization-group equation in a system exhibiting infinite-order phase transitions, the asymptotic behavior of running coupling constants is derived in an algebraic manner. A benefit of this method is presented explicitly using several examples.

Renormalization group theory for perturbed evolution equations

We show that proto-RG (renormaliation group) theory can be used to give a systematic description of the evolution of soltion of perturbed equations. The equations describing the deformation of its shape as the effect of perturbation are proto-RG equations. The RG approach may be simpler than inverse scattering theory (IST) and another approaches, because it dose not rely on any knowledge of IST. It is very concise and easy to understand.

Renormalization group and nonequilibrium action in stochastic field theory

We investigate the renormalization group approach to nonequilibrium field theory. We show that it is possible to derive nontrivial renormalization group flow from iterative coarse graining of a closed-time-path action. This renormalization group is different from the usual in quantum field theory textbooks, in that it describes nontrivial noise and dissipation. We work out a specific example where the variation of the closed-time-path action leads to the so-called Kardar-Parisi-Zhang equation, and show that the renormalization group obtained by coarse graining this action, agrees with the dynamical renormalization group derived by directly coarse graining the equations of motion.

Asymptotic analysis and renormalized perturbation theory of the non-Hermitian dynamics of an inviscid vortex

An analysis of the non-Hermitian fluid systems described by the Rayleigh equation in an unbounded domain has been carried out in the regime of large wave numbers. The evolution of a special class of localized vorticities is also discussed. Asymptotic and perturbative approaches lead to the same final result. In the limit considered, the system is stable. The perturbation analysis reveals interesting pathologies of the non-Hermitian systems. Under specific conditions, the expansion is found to show secular growth. A discussion about the mechanism of insurgence of such singular behavior is presented. It is also shown that the divergent expansion is renormalizable by means of the renormalization group method-the renormalized results are in complete conformity with the asymptotic solutions.