Concept Of Renormalization Research Articles

On critical renormalization of complex polynomials

Holomorphic renormalization plays an important role in complex polynomial dynamics. We consider invariant continua that are not polynomial-like Julia sets because of extra critical points. However, under certain assumptions, these invariant continua can be identified with Julia sets of lower degree polynomials up to a topological conjugacy. Thus we extend the concept of renormalization.

Steady Euler Flows on $${\mathbb {R}}^3$$ with Wild and Universal Dynamics

Understanding complexity in fluid mechanics is a major problem that has attracted the attention of physicists and mathematicians during the last decades. Using the concept of renormalization in dynamics, we show the existence of a locally dense set $\mathscr G$ of stationary solutions to the Euler equations in $\mathbb R^3$ such that each vector field $X\in \mathscr G$ is universal in the sense that any area preserving diffeomorphism of the disk can be approximated (with arbitrary precision) by the Poincar\'e map of $X$ at some transverse section. We remark that this universality is approximate but occurs at all scales. In particular, our results establish that a steady Euler flow may exhibit any conservative finite codimensional dynamical phenomenon; this includes the existence of horseshoes accumulated by elliptic islands, increasing union of horseshoes of Hausdorff dimension $3$ or homoclinic tangencies of arbitrarily high multiplicity. The steady solutions we construct are Beltrami fields with sharp decay at infinity. To prove these results we introduce new perturbation methods in the context of Beltrami fields that allow us to import deep techniques from bifurcation theory: the Gonchenko-Shilnikov-Turaev universality theory and the Newhouse and Duarte theorems on the geometry of wild hyperbolic sets. These perturbation methods rely on two tools from linear PDEs: global approximation and Cauchy-Kovalevskaya theorems. These results imply a strong version of V.I. Arnold's vision on the complexity of Beltrami fields in Euclidean space.

Rigidity of Newton dynamics

We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit can be distinguished in combinatorial terms from all other orbits), or the orbit of this point eventually lands in the filled-in Julia set of a polynomial-like restriction of the original map. As a corollary, we show that the Julia sets of Newton maps in many non-trivial cases are locally connected; in particular, every cubic Newton map without Siegel points has locally connected Julia set.In the parameter space of Newton maps of arbitrary degree we obtain the following rigidity result: any two combinatorially equivalent Newton maps are quasiconformally conjugate in a neighborhood of their Julia sets provided that they either non-renormalizable, or they are both renormalizable “in the same way”.Our main tool is a generalized renormalization concept called “complex box mappings” for which we extend a dynamical rigidity result by Kozlovski and van Strien so as to include irrationally indifferent and renormalizable situations.

From Ramanujan to renormalization: the art of doing away with divergences and arriving at physical results

A century ago Srinivasa Ramanujan --- the great self-taught Indian genius of mathematics --- died, shortly after returning from Cambridge, UK, where he had collaborated with Godfrey Hardy. Ramanujan contributed numerous outstanding results to different branches of mathematics, like analysis and number theory, with a focus on special functions and series. Here we refer to apparently weird values which he assigned to two simple divergent series, $\sum_{n \geq 1} n$ and $\sum_{n \geq 1} n^{3}$. These values are sensible, however, as analytic continuations, which correspond to Riemann's $\zeta$-function. Moreover, they have applications in physics: we discuss the vacuum energy of the photon field, from which one can derive the Casimir force, which has been experimentally measured. We discuss its interpretation, which remains controversial. This is a simple way to illustrate the concept of renormalization, which is vital in quantum field theory.

A three-dimensional mesoscale model for progressive time-dependent deformation and fracturing of brittle rock with application to slope stability

An understanding of progressive time-dependent deformation is essential for determining appropriate measures to ensure the long-term integrity of rock-masses surrounding engineering structures. We propose a 3D numerical model, that uses the Norton-Bailey creep law and a time-independent damage evolution law, to investigate the progressive time-dependent deformation and fracturing of brittle rock. The model considers material heterogeneity and the concept of mesoscopic renormalization at the mesoscale. The cooperative interaction between microcrack distribution and damage evolution leads to local material degeneration as a function of increasing time in the model. First, the input parameters for the model were calibrated and the model was validated using laboratory experiments. Numerical creep simulations were then performed for a range of constant stresses. Our model can accurately replicate the evolution of strain and strain rate as a function of time, the output of acoustic emission energy, and the emergence of a macroscopic failure plane seen in laboratory experiments. Our simulations also show that the minimum creep strain rate and the time-to-failure increase and decrease, respectively, as stress is increased, also seen in laboratory experiments. For example, increasing the differential stress by 10 MPa increased the minimum creep strain rate increased by an order of magnitude. Finally, we use the proposed 3D model to investigate the time-dependent stability of an engineering-scale rock slope containing faults at the Fushun West Open Pit coal mine. A small increase in fault-adjacent damage was observed after 40 days and, between 60 and 120 days, the damage on the side of the slope increased. After 140 days, the localized growth of the damaged elements split the slope into two segments, resulting in slope failure. Our 3D numerical model highlights potential slope instability in the Fushun West Open Pit coal mine and can be used to investigate slope stability in engineering projects worldwide.

Gamma-Limit of the Onsager--Machlup Functional on the Space of Curves

The Onsager--Machlup (OM) and Freidlin--Wentzell (FW) functionals are both widely used in seeking the most probable transition path between two states for a diffusion process. We study the relation between these two functionals on the space of curves. We prove that the $\Gamma$-limit of the OM functional on the space of curves is the geometric form of the FW functional in a proper time scale $T=T(\epsilon)$ as $\epsilon\to 0$. For other time scales, the limit of the OM functional is infinite in general. We then introduce the concept of renormalization for the OM functional and prove that the $\Gamma$-limit of the renormalized OM functional is the geometric FW functional in any time scale on the space of curves.

Automatic differentiation for second renormalization of tensor networks

Tensor renormalization group (TRG) constitutes an important methodology for accurate simulations of strongly correlated lattice models. Facilitated by the automatic differentiation technique widely used in deep learning, we propose a uniform framework of differentiable TRG ($\partial$TRG) that can be applied to improve various TRG methods, in an automatic fashion. Essentially, $\partial$TRG systematically extends the concept of second renormalization [PRL 103, 160601 (2009)] where the tensor environment is computed recursively in the backward iteration, in the sense that given the forward process of TRG, $\partial$TRG automatically finds the gradient through backpropagation, with which one can deeply "train" the tensor networks. We benchmark $\partial$TRG in solving the square-lattice Ising model, and demonstrate its power by simulating one- and two-dimensional quantum systems at finite temperature. The deep optimization as well as GPU acceleration renders $\partial$TRG manybody simulations with high efficiency and accuracy.

How to renormalize integral equations with singular potentials in effective field theory

We briefly review general concepts of renormalization in quantum field theory and discuss their application to solutions of integral equations with singular potentials in the few-nucleon sector of the low-energy effective field theory of QCD. We also describe a particular subtractive renormalization scheme and consider a specific application to a toy-model with a singular potential serving as its effective field theoretical leading-order approximation.

A three-dimensional numerical meso-approach to modeling time-independent deformation and fracturing of brittle rocks

A new 3D numerical mesoscale model is proposed to describe the time-independent deformation and fracturing of brittle rocks that accounts for material heterogeneity and local material degradation. The concept of renormalization at the mesoscale was introduced to capture the co-operative interaction of microcracks. The maximum tensile stress criterion and the Drucker-Prager criterion (which considers all three principal stresses) were used to evaluate the damage of specimen-forming elements in tension and shear, respectively. Simulations are presented alongside experimental data to show the capabilities of the model in tackling the failure process of intact rock, with an emphasis on microcrack initiation and propagation. We demonstrate that our model is capable of simulating the initiation and propagation of microcracks in deforming rocks and captures the mechanical behavior, strength, and failure patterns seen in laboratory experiments. Our 3D model therefore emerges as a powerful tool to study the evolution of failure in brittle rocks.

Whether an Enormously Large Energy Density of the Quantum Vacuum Is Catastrophic

The problem of an enormously large energy density of the quantum vacuum is discussed in connection with the concept of renormalization of physical parameters in quantum field theory. Using the method of dimensional regularization, it is recalled that the normal ordering procedure of creation and annihilation operators is equivalent to a renormalization of the cosmological constant leading to its zero and nonzero values in Minkowski space-time and in the standard cosmological model, respectively. It is argued that a frequently discussed gravitational effect, resulting from an enormously large energy density described by the nonrenormalized (bare) cosmological constant, might be nonobservable much like some other bare quantities introduced in the formalism of quantum field theory.

Acceleration due to buoyancy and mass renormalization

The acceleration of a light buoyant object in a fluid is analyzed. Misconceptions about the magnitude of that acceleration are briefly described and refuted. The notion of the added mass is explained and the added mass is computed for an ellipsoid of revolution. A simple approximation scheme is employed to derive the added mass of a slender body. The slender-body limit is non-analytic, indicating a singular character of the perturbation due to the thickness of the body. An experimental determination of the acceleration is presented and found to agree well with the theoretical prediction. The added mass illustrates the concept of mass renormalization in an accessible manner.

Differential representation of the delta function in two-dimensional quantum mechanics

The exactly solvable model of a two-dimensional system consisting of a particle in a Dirac delta function potential is studied in position space in lieu of momentum space. This study is facilitated by representing the delta function as the second derivative of a radial function, which depends on a length scale. Such a representation of the delta function provides an instructive approach to understanding the role of an arbitrary length scale in the concepts of regularization and renormalization.

Numerical Investigation of Damage Evolution and Localized Fracturing of Brittle Rock in Compression

In the present paper, a numerical model for deformation and fracturing of brittle rocks is used to study the damage evolution and localized fracturing of brittle rocks based on statistical mesoscopic damage mechanics. In the model, material heterogeneity and the mesoscopic renormalization concept are introduced to consider the interaction among microcracks in rock. The temporal and spatial evolution of acoustic emissions in the stressed rock are also described. Correspondingly, numerical tests on rock specimens with different heterogeneities, porosities, and scales are performed to investigate the damage evolution and localized fracturing of brittle rocks. The localized damage and fracturing of brittle rocks induced by microstructural damage is investigated. The study shows that heterogeneity, porosities, and scales of rock specimen have significant impact on damage evolution and localized fracturing mode. In addition, the study reveals that the postpeak response of stress-strain curve is due to the deformation of the structural elements, and localization means that the descending branch of the stress-strain curve of the rock specimen is size dependent, and the stress-strain curve can therefore not be regarded as a pure material property.

Matrix product density operators: Renormalization fixed points and boundary theories

We consider the tensors generating matrix product states and density operators in a spin chain. For pure states, we revise the renormalization procedure introduced in (Verstraete et al., 2005) and characterize the tensors corresponding to the fixed points. We relate them to the states possessing zero correlation length, saturation of the area law, as well as to those which generate ground states of local and commuting Hamiltonians. For mixed states, we introduce the concept of renormalization fixed points and characterize the corresponding tensors. We also relate them to concepts like finite correlation length, saturation of the area law, as well as to those which generate Gibbs states of local and commuting Hamiltonians. One of the main result of this work is that the resulting fixed points can be associated to the boundary theories of two-dimensional topological states, through the bulk-boundary correspondence introduced in (Cirac et al., 2011).

Perturbative Yang–Mills ground state in the temporal gauge

Second order corrections to the perturbative ground state wave functional and vacuum energy of a Yang-Mills theory are calculated in the temporal gauge. Using dimensional regularization, the concepts of renormalization and a running coupling constant are motivated in a spatially gauge invariant way, before the introduction of a scalar product. After definition of a scalar product, it is shown that the renormalization constants of this approach are just those of the Axial gauge.

Scaling concepts in describing continuous phase transitions

Phase transitions, like the boiling of water upon increasing temperature, are a part of everyday experience and are yet, upon closer inspection, unusual phenomena, and reveal a host of fascinating features. Comprehending key aspects of phase transitions has lead to the uncovering of new ways of describing matter composed of large numbers of interacting elements, which form a dominant way of analysis in contemporary statistical mechanics and much else. An introductory discussion is presented here of the concepts of scaling, universality and renormalization, which forms the foundation of the study of continuous phase transitions, such as the spontaneous magnetization of ferromagnetic substances.

Renormalized scattering series for frequency-domain waveform modelling of strong velocity contrasts

An improved description of scattering and inverse scattering processes in reflection seismology may be obtained on the basis of a scattering series solution to the Helmoltz equation, which allows one to separately model primary and multiple reflections. However, the popular scattering series of Born is of limited seismic modelling value, since it is only guaranteed to converge if the global contrast is relatively small. For frequency-domain waveform modelling of realistic contrasts, some kind of renormalization may be required. The concept of renormalization is normally associated with quantum field theory, where it is absolutely essential for the treatment of infinities in connection with observable quantities. However, the renormalization program is also highly relevant for classical systems, especially when there are interaction effects that act across different length scales. In the scattering series of De Wolf, a renormalization of the Green's functions is achieved by a split of the scattering potential operator into fore- and backscattering parts; which leads to an effective reorganization and partially re-summation of the different terms in the Born series, so that their order better reflects the physics of reflection seismology. It has been demonstrated that the leading (single return) term in the De Wolf series (DWS) gives much more accurate results than the corresponding Born approximation, especially for models with high contrasts that lead to a large accumulation of phase changes in the forward direction. However, the higher order terms in the DWS that are associated with internal multiples have not been studied numerically before. In this paper, we report from a systematic numerical investigation of the convergence properties of the DWS which is based on two new operator representations of the DWS. The first operator representation is relatively similar to the original scattering potential formulation, but more global and explicit in nature. The second representation is based on the T-operator formalism from quantum scattering theory, that offers a different perspective on the interaction between up- and downgoing waves, as well as significant computational advantages (e.g. domain decomposition and fast recursive methods for one-way propagators). Our numerical results demonstrate the convergence properties of the DWS are indeed superior to those of the Born series.

Fractal fragmentation and small-angle scattering

The small-angle scattering form factor of a three-dimensional idealized fragmentation model based on the concept of renormalization is calculated. The system consists of randomly oriented microscopic fractal objects whose positions are uncorrelated. It is shown that in the fractal region, the monodisperse form factor is characterized by a succession of maxima and minima superimposed on a simple power-law decay, and whose scattering exponent coincide with the fractal dimension ofthe scatterer. The results could be used to obtain additional structural information about systems obtained through fragmentation processes at microscale.

Renormalized Mori-Zwanzig-reduced models for systems without scale separation.

Model reduction for complex systems is a rather active area of research. For many real-world systems, constructing an accurate reduced model is prohibitively expensive. The main difficulty stems from the tremendous range of spatial and temporal scales present in the solution of such systems. This leads to the need to develop reduced models where, inevitably, the resolved variables do not exhibit (spatial and/or temporal) scale separation from the unresolved ones. We present a brief survey of recent results on the construction of Mori-Zwanzig-reduced models for such systems. The construction is inspired by the concepts of scale dependence and renormalization which first appeared in the context of high-energy and statistical physics.

Microscale Fragmentation and Small-Angle Scattering from Mass Fractals

Using the small-angle scattering method, we calculate here the mono- and polydisperse structure factor from an idealized fragmentation model based on the concept of renormalization. The system consists of a large number of fractal microobjects which are randomly oriented and whose positions are uncorrelated. It is shown that, in the fractal region, the monodisperse form factor is characterized by a generalized power-law decay (i.e., a succession of maxima and minima superimposed on a simple power-law decay) and whose scattering exponent coincides with the fractal dimension of the scatterer. The present analysis of the scattering structure factor allows us to obtain the number of fragments resulted at a given iteration. The results could be used to obtain additional structural information about systems obtained through microscale fragmentation processes.