1.1 Scaling laws and the renormalization group

Scaling laws, also known as power laws, describe the scale invariance found in many natural phenomena. There are some notable examples that need to be mentioned, since they are drawn from everyday experience. These include Pareto’s Law of income distribution [97], whose inverse power law (IPL) form describes the last few percents of a population as controlling most of the income within a Western country; Zipf’s IPL of the relative frequency of word usage within a language [84]; and finally the generic example of the IPL form of structural self-similarity of fractals [109, 141].

Kadanoff and Wilson established that the self-similarity of dynamic systems near critical points could be described by the RG method. This discovery not only provides a rational and practical theoretical expression for continuous phase transition but also provides a direct physical basis for the RG method. Subsequently, this method proved to be a powerful tool for studying physical phenomena through its treatment of the scaling properties of complex dynamics and chaos [144].

1.2 Complexity and the renormalization group

There is a long history of the inability to describe complex phenomena accurately, with existing contemporary theory leading to empirical paradoxes and confusion [131]. The complexity of these phenomena was derived from theory underlying real–world data and they can be seen almost everywhere, such as global climate, fully developed turbulence, financial systems, and the human brain, which makes the study of complexity fascinating.

Although a considerable amount of research has been devoted to the study of complexity, there has not appeared a clear and unified definition of complexity. Therefore, we propose the following two questions to be addressed in this article, and to find at least partial answers:

1. How can we characterize complexity?

2. What methods should be used to study complexity in order to understand real–world phenomena better?

There are many definitions of complexity, one being implicit in a phenomenon that is neither entirely ordered nor completely random [126]. Simple ordered systems have patterns that repeat themselves periodically in time or space, and they possess a highly organized structure, that is, a system that can be described by classical Newtonian mechanics. On the other hand, completely disordered systems have a random structure in space or time, described by stochastic dynamics or the random spatial organization of components. Such systems are complicated but simple because they can be easily described using probability density functions (PDFs) so that the main characteristics of the system can be reproduced. The single dynamic trajectory of the former simple system is replaced by an ensemble of trajectories in the later simple system and the behavior by that of the PDF and averages.

However, truly complex systems not only have a large number of small components that interact with non-identical components nearby and their environment, making their behavior inherently challenging to model, but the dynamics are not linear. The resulting chaotic dynamics make the individual trajectories chaotic. Consequently, the dynamics of the ensemble PDF cannot be described by the usual partial differential equations in phase space. The equations for the PDF have been determined to be fractional generalizations of the traditional phase space equations [144], as we subsequently discuss. For example, the stock market is a complex system with many interactive properties. Billions of shares and transactions interact through a finite chain of cause and effect, and even if we knew all relevant information, the stock market would still produce unpredictable behavior.

Therefore, the property of complexity in complex phenomena has observable characteristics determined by the interactions, such as nonlinear response to perturbation, long–term memory, scaling laws, and adaptive properties, to name a few. Based on these characteristics, the study of complexity has attracted contributions from many different fields, such as self–organization in physics [55], the spontaneous transition between disorder and order in social sciences [116, 131], the mathematics of nonlinear dynamics (such as chaos) [52, 130], and adaptation in biology [86]. Nevertheless, how to define, much less to measure, a system’s complexity remains a controversial topic and is one of the key challenges in studying complex phenomena. Up to now, many quantitative studies of complexity have given rise to a number of measures including, temporal measures, spatial measures, and entropy measures [31, 42, 90].

Inspired by the works of Galanter [42] and West [129], we recreated the trend chart labeled with known measures of complexity. As shown in Fig. 1, when the number of variables in a system is low, certainty, and the dynamics of single–particle trajectories play a leading role, resulting in low levels of complexity. In this few–particle case, behavior can usually be described by well–known techniques, such as celestial dynamics [10], fractals [72], and nonlinear dynamics, including chaos theory [74] and strange attractors [103]. When the number of system variables is high the dynamics of single–particle are replaced by the calculus of PDFs. However, due to the emergence of collective variables, the resulting level of complexity is again low. The treatment of higher levels of complexity arises, moving leftward from the extreme right, using models such as random walks [122], stochastic fractals [77], and the factional probability calculus [132] applied to describe complexity.

Figure 1

The measure of complexity and the number of interacting entities are the axes for rugged landscape of scientific knowledge. The extreme left is described by simple system dynamics, but even here nonlinear dynamics has chaos with more than three degrees of freedom, strange attractors for dissipative nonlinear dynamic systems and non-integrable Hamiltonians. On the extreme right things are again simple with equilibrium thermodynamics and random walks. Moving from right to left the many degrees of freedom are replaced by low dimensional emergent variables and their mutual interactions which dominate the system

The middle region of the mountain range contains a vast region where all the science and mathematics remain to be done. It represents a robust domain where complexity and both the successful and maladaptive networks of society and nature involving both order and randomness, with neither dominating, reside. The delicate balance between order and randomness is observed all around us in living systems, in social organizations, as well as in complex physical networks. For example, the intermittency in the thunderous sound of water from a leaky faucet crashing into the sink as you try to get to sleep; the unpredictable tremors that occur in many pathologies of motor control networks; and disruptive interaction between states on the international stage [128]. Note that the trend in the figure implies a nonlinear increase in complexity when moving away from order (extreme left to right) or from disordered (extreme right to left) and is suggested by the rugged terrain of the mountain.

It is the FC and RG theory that provide entry into the central region of the unknown. Hopefully, this review will facilitate the passage.

At the present stage, the research into the nature of complexity divides into two basic strategies [85]. The first strategy is to create and study relatively simple mathematical models, which although they may not accurately capture the behavior of real systems they can abstract the most significant qualitative elements into a solvable framework from which we can extract scientific insights. The research tools used to gain insight into complexity have been nearly as varied as the disciplines to which they have been applied, including FC theory [141], RG theory [132], dynamical systems theory [8], game theory [5], information theory [34], and computational complexity theory [85].

In particular, in the process of analyzing complex systems, nonextensive statistical mechanics (NSM) [118, 120, 121] plays an important role in bridging the transitional regions between scale-free and exponential distributions. Nonlinear statistical coupling [82, 83] explains the distinction between scale and the shape of distributions, and that the shape quantifies a nonlinear source. The important distribution families in this paradigm are generalized Pareto distribution and the Student’s t-distribution.

The second strategy attempts to create more comprehensive, detailed, and realistic models, usually in the form that enables computer simulations, since analytic solutions are often beyond reach. This modelling approach simulates the interacting parts of a complex dynamic network, is accurate in capturing small details, and motivates observation and measurement. For example, Monte Carlo simulation was introduced into the complexity context by [119], especially agent-based simulation [147, 148] in financial markets and other technologies.

1.3 Inverse power laws and fractional calculus

One of the first analytic treatments of the generation and spreading of complexity was the movement of heavy particles in an ambient fluid of lighter particles. Today we would classify this behavior as complicated but not complex. Consider the standard diffusion equation:

where D denotes a positive constant, x and t are the space and time variables, respectively, and ρ(x, t) is the phase space density function. The diffusion equation is used to describe the change of the matter density in diffusive phenomena. In physics, Eq. (1.1) describes the macroscopic behavior of many large particles undergoing Brownian motion in an ambient fluid and is often replace by the PDF P(x, t). In mathematics, the PDF is related to Markov processes, and is often modeled using simple random walks. The diffusion model can also be applied in many other fields, such as the movement of alleles in population genetics [80], or more generically the flow of information in complicated networks [129].

One of the basic boundary-value problems for diffusion is the Cauchy problem:

where δ(x) is the Dirac delta function. This initial value problem describes a point source for a tracer released at the origin at time t = 0, which then diffuses throughout the ambient fluid. The formal solution to the Cauchy problem is:

where the Green’s function G(x,t), also known as the propagator, is obtained by taking the Fourier transform of Eq. (1.1), solving the resulting linear rate equation for the characteristic function and inverse Fourier transforming the resulting solution to obtain:

which when inserted into Eq. (1.2) yields the well-known Gaussian or normal PDF.

An example of the FC is introduced in this context by replacing the second–order space derivative with a symmetric space–fractional derivative of order β with 0 < β ⩽ 2. We can write the symmetric space–fractional diffusion equation [50]:

where D₀ is a positive coefficient and the fractional derivative is taken in the sense of Riesz (for details, see Appendix A). Here again the Fourier transform of the PDF yields the equation for the characteristic function with the solution from the Fourier transform of Eq. (1.3) given by the inverse Fourier transform of the characteristic function:

This is a symmetric Lévy PDF, the solution to the Cauchy initial value problem for the fractional diffusion equation given for Eq. (1.3). Note that the only closed–form solutions given by Eq. (1.4) arise for a limited number of values of the Lévy index in the interval 0 < β ⩽ 2, including that of Gauss β = 2. The Gauss PDF is the only case having finite second moments, all other second moments of the L évy PDF diverge including the Cauchy with β = 1. The diverging second moment violates a fundamental condition necessary for the proof of the central limit theorem (CLT) to converge. Such strange behavior is common in our complex world.

It is worth mentioning here that as |x| → ∞, the analytic solution of the symmetric Lévy PDF can be obtained by taking the |k| → 0 limit of the characteristic function within the integrand of Eq. (1.4):

Thus, the asymptotic form of the symmetric Lévy PDF is an IPL [25, 127]. Using the IPL the failure of the second moment to converge for 0 < β < 2 is obvious.

1.4 Renormalization group and fractional calculus

From the perspective of scaling laws or IPLs, the present review seeks connections between the RG and the FC. Furthermore, we find that the two research areas are not independent of one another, but have a particular relationship, in that they both emerged from studies seeking to understand complex phenomena. As mentioned above, these two mathematical strategies provide systematic ways to research complexity, and have the potential of providing deep understanding of actual phenomena. Based on this, the purposes of this review are as follows:

1. Introduce the RG and FC methods and point out how they are connected.

2. Provide new ideas for the future improvement and application of these methods.

Section 2 methods briefly introduces the RG method, which although derived from mathematical physics has applications in a broad spectrum of disciplines. We touch on a number of these non-physical application in this section. Section 3, we discuss the origin and developments of the RG method in SPT and its applications in various fields of disciplinary research. In Section 4, the connection between RG and FC is discussed. In the last section, conclusions and prospects are given.

2.1 Basic ideas

The physical phenomena studied by the RG method have one common feature: the correlation length diverges ξ → ∞, that is, there is no characteristic scale with which to characterize the process. This is observed during a phase transition of water to ice as the temperature T is lowered to its critical value T_c. The water molecules have a short–range exponential correlation function and a constant correlation length when T > T_c, but as T approaches T_c the correlation function becomes a long–range IPL and the correlation length diverges as the water solidifies.

The purpose of the RG method is to derive typical scaling relations among the physical parameters controlling the phenomenon’s behavior:

where K₁, K₂ are coupling constants within the system and F (·) is an unknown analytic function. For example, for ferromagnetic systems, K1−1 could be the system scale size and K₂ is the system temperature; for hydrodynamic systems, K₁ could be the wave numbers and K₂ the frequency. The scaling indices α and τ are the critical exponents of the system. When the correlation length diverges ξ → ∞, it is found that the above scaling relation only depends on the symmetry of the system but is independent of the micro details, and possesses universality. This emergence of universal macroscopic behavior was the theme of the classic Science article “More is different” by P. Anderson [4] who received the Nobel Prize in Physics.

As mentioned in Section 1.1, the scaling laws refers to the fact that systems at different scales have the same properties. At the same time, due to the correlation length ξ diverging to infinity, the universality signifying the individuality of different systems retreats to secondary importance, and its common behavior of universality emerges center stage.

In addition to the theoretical analysis alluded to above, experimental observations [53, 104] found that different physical systems do have similar characteristics near the critical points. The phenomenon that occurs near the continuous phase transition points (critical points) is called a critical phenomenon. The characteristics near these points are summarized in scaling laws and in the universality hypothesis (see Fig. 2).

Figure 2

Ising model simulations of a dynamic system at critical and non-critical temperatures. (a) Binary 256 × 256 lattices showing the configuration of spins after 2,000 timesteps at low temperature T = 0.5 T_c; (b) at critical temperature T = T_c; (c) at high temperature T = 1.28 T_c. At high temperature the spins are randomly configured, at low temperature they have an alignment of spins, and at critical temperature they have a fractal configuration

Link: https://github.com/mattbierbaum/ising.js

The basic idea of the universality hypothesis is that the long–range correlations near the critical points determine all the singularity properties of the underlying process. When the state of the system is close to its critical points, where the correlation length diverges (ξ → ∞), we can obtain the same results by describing physical phenomena on any finite scale near the critical points. Similar to isometric spirals of different scales by zooming in and out, the textures can be superimposed together.

Due to the above scaling characteristics, the RG method can be applied to the analysis of critical phenomena. The small–scale fluctuations are smoothed by the scale transformation, and the effective action at the large–scales dominate. Since the similarity between global and local behavior has been achieved, this transformation will not affect the global properties.

The main steps of the RG method are as follows:

1. Use the renormalization transformation to obtain a coarse–grained operator by changing the length scale and reducing the number of degrees of freedom acting on the state variable.

2. Induce the transformation of the model parameters. Generally, use the approximate method to keep the partition function of the system consistent. In this process, we can obtain the renormalization flow (RG flow).

3. Analyze the fixed points of this transformation, as well as the behavior of the system near the fixed points.

2.2 Renormalization group transformations

Much of the analytic work in physics is done using conservative Hamiltonian systems, in large part because such systems reproduce the mechanical forces of Newton. Let H = H(K) be the Hamiltonian of a system of interest, where K = (K₁, K₂, ⋯) is the vector of coupling constants. Each set of these constants defines a point in the “Hamiltonian space”, that being the space of coupling constants. The RG transformations operate on this space:

The RG operator R reduces the number of degrees of freedom from N to N′:

where b is the rescaling factor, and d is the spatial dimension.

The essential condition to be satisfied by the RG transformation is that the form of the partition function remains invariant under the transformation:

Therefore the total free energy does not change, but the free energy per unit cell (spin) increases as:

All lengths, measured in units of the new lattice spacing, are reduced by the factor b. Thus, the correlation length scale is:

Fixed points. The iterations of the RG transform R traces a trajectory in the Hamiltonian space, this “Hamiltonian flow” is identified as RG flow. The most important information regarding RG flow concerns its fixed points. The trajectories end at the fixed points K^∗, defined by the RG transform becoming equivalent to the identity operator:

which occurs, if and only if:

The set of fixed points provides the possible macroscopic state of the system at a large scale. At these points the system is invariant under subsequent scale changes and thus the correlation length remains constant:

and does not change.

We can regard the RG flow as evolution such as that of a dynamical system. In particular, there are some fixed points in space, which represent the particular theory of scale invariance. These fixed points can be divided into two categories: stable and unstable critical points. When a fixed point has no unstable directions they are stable critical points and are called “sinks”. If we start from a point in space and approach such an attractive fixed point, the RG flow will be brought closer to the fixed point by the behavior described using effective long-wave theory. When a fixed point has unstable directions in space they are unstable critical points and are called “sources”.

Therefore, the asymptotic behavior of all correlation functions in the original system are determined by the nature of the fixed points. These points divide the space into different regions and describe the transition between different phases. This is the bare mathematics of critical phenomena.

An example: To illustrate the main ideas of the RG method, we use a classical model from statistical physics–the two–dimensional Ising model. The model consists of a square lattice with N sites and E nearest neighbor bonds, the i-th site of the lattice is assigned a spin s_i, 1 ⩽ i ⩽ N, which can be +1 (spin up) or −1 (spin down). When neighboring spins are oriented in the same (different) direction, there is an interaction energy −J(J), where J is the spin–spin coupling constant. The spin s_i also couples with an external magnetic field h with an energy −hs_i. In this case, both the height and width of the blocks of spins should have the same length ba, where b is the rescaling factor, and a is the lattice spacing (see Fig. 3).

Thus the Hamiltonian (total energy) of the spin system can be written as:

where s represents the configuration of N spins, i.e., s = (s₁, ⋯, s_N), the first summation is over all E nearest-neighbor pairs of spins and the second summation is over all N sites of the lattice. According to statistical mechanics, at a given absolute temperature T, the probability for s to appear p(s) is proportional to the Boltzmann factor exp(−βH(s)), and thus is given by:

where β = 1/(k_BT), k_B is the Boltzmann constant, K = βJ, B = βh, and:

is the partition function of the system.

Figure 3

Block spins transformation. The blocks (dotted blue box) behave as if they were atoms, but with a reduced number. • represents the lattice spin, represents the block spin

One way of defining the spin of a block is to use the majority rule:

where I refers to the index of the blocks. Consequently, the Hamiltonian for the coarse–gained problem is given approximately by:

Then the coarse–gained operator for the spin configuration induces a transformation in the space of model parameters:

where K = (K, B). This example constitutes the block renormalization transformation.

Lattice spin models are not restricted to the understanding of phase transitions and critical phenomena in physical systems. Versions of this model have been applied to fractal geometry, such as the criticality and phase transitions in Koch curves [113] and Sierpinski gaskets [44], as well as to the molecular theory of biological evolution and the origin of life [29, 37] and finally in the use of network science to the decision making behavior of humans [128].

The above briefly describes the main ideas in the derivation of RG theory, so let us now turn our attention toward applications. The applications of the RG method in various disciplines both past and present has led to extensions and nuanced interpretations of the RG.

2.3 Applications of renormalization group

3.1 The origin

From the perspective of the applications of the RG method in perturbed quantum field theory and critical phenomena, Chen et al. in the years from 1991 to 1996 [20, 21, 22, 23, 24] demonstrated a unified understanding of various singular perturbation methods and reductive perturbation methods using several examples. Firstly, they discussed the general relation between the RG method and multiple-scale analysis by means of examples. Secondly, for a class of singular perturbation problems, they showed that these problems could also be solved by the RG method. Thirdly, they demonstrated with several switchback problems that the RG approach has technical advantages over conventional asymptotic methods. Lastly, they demonstrated that the RG method is a general and systematic method to derive slow-motion equations using the Newell-Whitehead equation.

To better understand their work, and in keeping with historical precedent we use an example to illustrate the efficiency of this method.

Rayleigh equation. Lord Rayleigh supplemented the weak viscous force acting on a harmonic oscillator with a cubic nonlinearity, which in standard notation can be expressed as:

where 0< ε ≪ 1 is known as the small perturbation parameter. The solution to this equation is obtained by assuming a naive expansion in powers of ε:

which when substituted into the Rayleigh equation, and comparing coefficients of the same power of ε yields:

This subsidiary set of equations is solved in ascending orders of the perturbation parameter. The solution to the oscillator equation is inserted as the driver to the order ε equation and solved. In this way we obtain the straightforward perturbation solution to O(ε) of the Rayleigh equation:

where θ₀ is a constant determined by the initial conditions at t = t₀.

Since this direct expansion possesses R02(1−R024)(t−t0) which is the so called secular term [81]. The secular terms make y_n(t)/y_{n − 1}(t) unbounded as t approaches to infinity, such as tⁿ cos t and tⁿ sin t. Therefore, the solution (3.1) is divergent and not uniformly valid. To regularize the series, which is a formal way of suppressing the divergence in the solution, we introduce an arbitrary intermediate time τ that partitions t − t₀ into t − τ and τ − t₀. We absorb the terms containing τ − t₀ into the renormalized counterparts R and θ as follows:

This partitioning enables us to rewrite Eq.(3.1) after some algebra as the following renormalized perturbation result:

where R and θ are functions of τ. Since τ does not appear in the original problem, the solution should not depend on it. Therefore, requiring that the solution be independent of τ yields:

for any t, so that the RG equation is:

Thus, the final uniformly valid result is

where the time-dependent amplitude R(t) is given by the solution to the nonlinear rate equation Eq. (3.2).

The idea behind the RG method applied in this way is that the resonant part of the equation dominates the dynamics of an system. By decomposing the nonlinear term into resonant and non-resonant parts, the former is constant in time and yields a secular term, which leads to the RG equation. The solution to the RG equation is the main piece of an approximate solution to the initial perturbed equation. Such a decomposition is very useful in studying the long–time behavior of solutions. The RG method became a powerful technique with which to solve a sequence of perturbation problems. The advantage of this method over other more algebraically intensive methods is that the starting point is a straightforward naive perturbation expansion, for which no prior knowledge is required. In contrast to conventional methods, this approach neither requires assumptions regarding the perturbation series’ structure nor does it require matching of the solution pieces in an asymptotic domain.

To illustrate the effects of this method theoretically, different researchers provided rigorous mathematical analysis from different perspectives, and on that basis, improved the method.

3.2 Recent developments

3.2.2 Simplification and comparison to normal form theory

DeVille et al. [32, 33] not only examined the mathematical basis of the RG method but derived a simplified algorithm for this method up to O(ε²). In addition, the crucial step of the RG method is a near–identity change of coordinates. It is equivalent to the NF theory for systems with autonomous/non-autonomous perturbations to second–order and higher–order systems.

Consider the following weakly nonlinear autonomous ordinary differential equation (ODE):

dxdt=Ax+εf(x),

and weakly nonlinear non-autonomous ODE:

dxdt=Ax+εf(x,t),

where 0 < ε ≪ 1, A is a complex diagonal matrix with purely imaginary eigenvalues, and f(x), f(x, t) are both smooth functions.

Based on the previous studies, a simplified version that highlights the mathematical underpinnings of the RG method can be shown as the following steps:

1. Derive a naive perturbation expansion for the solution of the given differential equation with an arbitrary initial time t₀ and initial condition x(t₀).

2. Renormalize the initial condition by absorbing those terms in the naive expansion which are time–independent and bounded into x(t₀).

3. Apply the RG condition:

   dxdt0=0.

By showing the relationship between the RG method and the NF theory, the diagram can be summarized as the following form:

where V denotes the space of vector fields, S denotes the space of truncated asymptotic expansion. RG₁, RG₂ and RG₃ represent the above three steps, respectively. NF denotes the change of coordinates central to the NF method.

The essential reductive step in the RG method is the change of coordinates known as RG₂. This coordinate change is near–identity on time scales of O(1/ε). It absorbs the non–resonant terms from the asymptotic expansion into the initial condition, which is similar to the coordinate change involving the dependent variable in the NF theory. These two methods are used together to remove the non–resonance term. Holzer et al. [57] developed this theory and used three examples to illustrate that the RG method can well handle the perturbation problem with logarithmic switching terms.

The advantages of the RG method over the NF method are that the secular terms are easier to identify, and the form of the near–identity coordinate change is not necessarily known in advance so that the RG method is quite general.

3.3 Nine applications of the renormalization group method

4.1 Fractional Brownian motion

Quan [96] studied the connection between FBM and the RG in statistical physics, and determined statistical, geometric, and fractal properties of complex phenomena. The RG method is a powerful tool for qualitatively studying physical phenomena as well as pure mathematical problems with scaling properties. Meanwhile, some complex systems with IPL properties can also be studied using the FC theory.

As shown in Fig. 4, when a beam of white light enters a dispersive prism, the prism decomposes the white light into its constituent spectral colors (the colors of the rainbow). Complex systems are equivalent to such a beam of white light, which can be decomposed into its various spectra (observations) through a dispersive prism system. These observations are typical representations of complex systems: scale–free, self–similarity, long–range dependence, and long–term memory.

Figure 4

The connection between renormalization group and fractional calculus

So, what is this prism system (or lens) that connects complexity with its macroscopic observations? Through discussions in the previous sections, we tried to explore the essence of complex systems through phenomena from different angles and found that the “lens” is the IPL or scaling laws. Through this “lens”, RG and FC become the common ground method to reveal the essence of complexity. Therefore, it should emphasize that these two research areas are by nature connected. This connection is the IPL and scaling laws, both are essential for studying complex systems.

This point can be made directly by way of example wherein we consider a random walk process determined by a fractal function and called the Weierstrass random walk (WRW) [59] for reasons that will become obvious in due course. Consider the discrete probability described by the stepping PDF for the WRW on a one-dimensional lattice with sites indexed by q:

where a and b are dimensionless constants greater than one. δ_ij is the Kronecker delta function: δ_ij = 1 for i = j and δ_ij = 0 for i ≠ j. We follow the analysis of this discrete process given by West and Grigolini [130]. The first property of note is that the second moment of this diffusive process is:

which diverges for b² > a since the series is infinite. The lattice structure function P^(k) is the discrete Fourier transform of the PDF and consequently is the discrete version of the characteristic function:

This series was introduced by Weierstrass in 1872 [71] to represent a process that is continuous everywhere but is nowhere differentiable. Thanks to Mandelbrot we now know that this was the first consciously constructed fractal function and the divergence of the second moment is a consequence of its non-analytic properties.

As the WRW process proceeds the set of sites visited consists of localized clumps of sites, interspersed by gaps, nested within clusters of clumps over ever-larger spatial scales. The WRW generates a hierarchy of clumps that are statistically self–similar, as suggested by Fig. 5. The parameter a determines the number of subclusters within a cluster and the parameter b determines the scale size between groupings.

Figure 5

The landing sites for the WRW are depicted and the islands of clusters are radially seen

Let us now apply the RG method to the lattice structure function and determine the scaling properties of the WRW. Scaling the argument of the lattice structure function by b and reordering terms in the series yields:

The solution to this RG equation can be separated into homogeneous and singular parts and the singular part P^s(k) is obtained by solving the homogeneous scaling equation:

The formal solution to this equation is given by:

which when inserted into Eq. (4.1) yields:

The last equality implies that A(k) is periodic in the logarithm of k with period ln b, μ = ln a/ln b and the solution can be written:

with the complex power–law index:

The analytic forms of the Fourier coefficients in Eq. (4.2) are given in Hughes et al. [59].

It is possible to prove that the dominant behavior of the WRW is determined by the lowest-order term in the singular part of the solution for the lattice structure function but we do not do that here. We assume that the dominant behavior is given by the n = 0 term in the series:

whose inverse Fourier transform is the IPL:

and K(μ) is a known function of μ. Thus, the singular part of the WRW has an IPL stepping PDF and this behavior intuitively justifies ignoring all the other terms in the series.

We now write for the time-dependent form of the discrete PDF:

where we assume that each step n in WRW process occurs at equal time intervals. This equation was analyzed in 1970 by Gillis and Weiss [46], who determined that its solution is a Lévy PDF, thereby connecting the RG solution the WRW to discussion of the fractional diffusion equation given earlier.

4.2 Fractional kinetic equations

Zaslavsky [143] studied chaotic particle kinetics generated by a class of non-integrable Hamiltonians. He derived the fractional Fokker-Planck-Kolmogorov (FFPK) equation for the PDF to describe the ensemble behavior of the random walk in the fractal space–time system. Using the RG method, he was able to solve the FFPK equation and concluded that the critical exponents of the anomalous kinetics represented the dynamics of the Hamiltonian. That is, when the fixed points of the RG equation exist, they lead to the relationship between the space–time fractional exponents and the space–time scale expressed in the scaling form of the PDF solution. It is a pioneering work applying RG theory to fractional–order stochastic dynamical systems.

Consider a random walk generated by a Hamiltonian system whose dynamics are chaotic. The names Kolmogorov [66], Arnold [7] and Moser [78] are associated with laying out the boundaries of classical mechanics and as West and Grigolini [130] explain, KAM theory describes how the continuous trajectories of particles determined by a Hamiltonian break up into a chaotic sea of randomly disconnected points. This kind of “strange kinetics" produces the FFPK equations to describe the evolution of the PDFs in phase space. Schematically we write the FFPK equation for an ensemble of chaotic trajectories as:

where δ_t is the infinitesimal time–variation of the PDF and δ_l is the corresponding variational quantity in space.

The application of the RG transform R to this equation n − times yields for an ensemble of chaotic trajectories:

where on the left we obtain by scaling the time increment Δt^α:

and on the right we obtain by scaling the space increment Δl^β:

Thus, in the simplest case we can write:

which diverges as:

unless:

Eq. (4.3) defines a nontrivial fixed point of the RG relation and yields:

thereby yielding the fixed point kinetic equation:

The solution to FFPK equation is obtained by interpreting the fractional time derivative in the Caputo sense and taking the Fourier-Laplace transform of the equation from which after some algebra we obtain the generalization of Eq. (1.4):

After rescaling and some algebra we can rewrite this equation as:

thereby providing a generic form for the solution to the fixed point FFPK equation. Note that this is precisely the form of Eq. (2.2) discussed earlier.