Reactions and Transport: Diffusion, Inertia, and Subdiffusion

Chapter 2Reactions and Transport: Diffusion, Inertia,and Subdiffusion

Particles, such as molecules, atoms, or ions, and individuals, such as cells or ani-mals, move in space driven by various forces or cues. In particular, particles orindividuals can move randomly, undergo velocity jump processes or spatial jumpprocesses [333]. The steps of the random walk can be independent or correlated,unbiased or biased. The probability density function (PDF) for the jump lengthcan decay rapidly or exhibit a heavy tail. Similarly, the PDF for the waiting timebetween successive jumps can decay rapidly or exhibit a heavy tail. We will discussthese various possibilities in detail in Chap. 3. Below we provide an introduction tothree transport processes: standard diffusion, transport with inertia, and anomalousdiffusion.

2.1 Reaction–Diffusion Equation

The classical and simplest model for spatial spread or dispersal is the diffusion equa-tion or Fick’s second law, which in spatially one-dimensional systems reads

∂ρ

∂t= D

∂2ρ

∂x2, (2.1)

where D is the diffusion coefficient. As is well known and shown explicitly inChap. 3, particles that perform a simple random walk or Brownian motion at themicroscopic level display diffusive dispersal at the macroscopic level. The diffusionequation preserves positivity, and the fundamental solution of (2.1) with a pointsource at x = 0 and t = 0, ρ(x, 0) = δ(x), is given by

ρ(x, t) = 1√4πDt

exp

(

− x2

4Dt

)

, t > 0. (2.2)

If the particles or individuals react or interact according to some rate law F(ρ)

and at the same time undergo diffusion, it is legitimate to combine the diffusionequation and the rate equation ρ̇ = F(ρ) [178]. The result is the well-knownreaction–diffusion (RD) equation:

V. Méndez et al., Reaction–Transport Systems, Springer Series in Synergetics,DOI 10.1007/978-3-642-11443-4_2, C© Springer-Verlag Berlin Heidelberg 2010

33

34 2 Reactions and Transport: Diffusion, Inertia, and Subdiffusion

∂ρ

∂t= D

∂2ρ

∂x2+ F(ρ), (2.3)

which provides a theoretical framework for the spatiotemporal dynamics not onlyof chemical systems but also of systems in other areas, such as biology [231, 264,343, 393, 310], ecology [328], physics [11], and materials science [464]. In thismonograph, we refer to (2.3) as the standard reaction–diffusion equation or simplyas the reaction–diffusion equation. This equation preserves positivity, if the rate lawF satisfies condition (1.2).

Besides the simple mathematical approach of combining the rate equation andthe diffusion equation, two fundamental approaches exist to derive the reaction–diffusion equation (2.3), namely a phenomenological approach based on the lawof conservation and a mesoscopic approach based on a description of the under-lying random motion. While it is fairly straightforward to show that the standardreaction–diffusion equation preserves positivity, the problem is much harder, not tosay intractable, for other reaction–transport equations. In this context, a mesoscopicapproach has definite merit. If that approach is done correctly and accounts for allreaction and transport events that particles can undergo, then by construction theresulting evolution equation preserves positivity and represents a valid reaction–transport equation. For this reason, we prefer equations based on a solid mesoscopicfoundation, see Chap. 3.

In the context of chemical systems, the type of reaction–transport equations westudy in this monograph describe reactions in the activation-controlled or activation-limited regime. We do not consider the opposite regime of diffusion-controlled ordiffusion-limited reactions. Diffusion-controlled reactions are very fast; a reactiveevent occurs as soon as the reactants encounter each other. The reaction rate isessentially given by the rate of transport of the reactants through the medium. Inactivation-controlled reactions, the reactants must overcome a substantial energybarrier after they encounter each other before a reactive event can occur. Cross-ing the energy barrier is the rate-limiting step for these reactions, and activation-controlled reactions are significantly slower than diffusion-controlled reactions.For the spatiotemporal dynamics of diffusion-limited reactions see for example[233, 56, 238, 255, 96, 361, 33].

2.1.1 Phenomenological Derivation of the Reaction–DiffusionEquation

Let S be an arbitrary surface enclosing a time-independent volume V . The generallaw of conservation for the particle density states that the rate of change of theamount of particles in V is due to the flow of particles across the surface S plus thenet production of particles in the volume V :

2.1 Reaction–Diffusion Equation 35

∂

∂t

∫

Vρ(x, t)dV = −

∫

SJ · dS+

∫

VF(ρ, x, t)dV, (2.4)

where ρ(x, t) represents the density of particles at x at time t , J the particle flux,and F the net rate of production of ρ. Application of the divergence theorem,

∫

SJ · dS =

∫

V∇ · J dV, (2.5)

to (2.4) yields

∫

VdV

[∂ρ

∂t+∇ · J− F

]

= 0. (2.6)

Since the integration volume is arbitrary, we obtain the conservation equation, orcontinuity equation, for ρ:

∂ρ

∂t= −∇ · J+ F(ρ). (2.7)

The continuity equation (2.7) needs to be closed via a constitutive equation forthe flux J. If the transport process corresponds to classical diffusion, the constitutiveequation is given by Fick’s first law,

J = −D∇ρ. (2.8)

Substituting this expression for the flux in (2.7), we obtain

∂ρ

∂t= ∇ · (D∇ρ)+ F(ρ). (2.9)

If D is constant, (2.9) reduces to (2.3) in the one-dimensional case. In some modelsfor insect populations, models for bacterial chemotaxis, or for animal grouping dueto social aggregation, the diffusion coefficient can be an explicit function of theparticle density or a function of other chemical components.

2.1.2 n-Variable Reaction–Diffusion Equations

The extension of (2.3) to n-variable systems is fairly straightforward:

∂ρ

∂t= ∂

∂x

(

D∂ρ

∂x

)

+ F(ρ). (2.10)


In component form, (2.10) reads

∂ρi

∂t= ∂

∂x

(

Dii∂ρi

∂x

)

+n∑

j, j �=i

∂

∂x

(

Di j

∂ρ j

∂x

)

+ Fi (ρ), i = 1, . . . , n. (2.11)

The diagonal elements of D are called the “main-term” diffusion coefficients andthe off-diagonal elements are called the “cross-term” diffusion coefficients or cross-

diffusion terms. The cross-diffusion term ∂∂x

(

Di j∂ρ j∂x

)

links the gradient of species

j to the flux of species i . If the cross-diffusion term is positive, Di j > 0, then theflux of species i is directed toward decreasing values of the concentration of speciesj , whereas Di j < 0 implies that the flux is directed toward increasing values ofthe concentration of species j . The cross-diffusion terms Di j ( j �= i) must go tozero as the concentration of species i goes to zero, Di j (ρ) → 0 as ρi → 0, sincethere can be no flux of species i if ρi = 0. For chemical systems, thermodynamicsimposes the constraint that all eigenvalues of the diffusion matrix D must be realand positive, which implies that tr D > 0 and det D > 0 [454]. Some studies haveinvestigated reaction–diffusion equations with constant cross-diffusion coefficients.Such equations represent approximations with a limited range of validity. Further,such equations do not preserve positivity [68, 69].

In this monograph, we consider only reaction–diffusion systems where thecross-diffusion terms are negligible, i.e., the diffusion matrix is a diagonal matrix,D = diag(D1, . . . , Dn), and the diffusion coefficients Di , which must be positive,do not depend on ρ,

∂ρi

∂t= Di

∂2ρi

∂x2+ Fi (ρ), i = 1, . . . , n. (2.12)

2.2 Reaction–Transport Equations with Inertia

The diffusion equation has the well-known unrealistic feature that localized distur-bances spread infinitely fast, though with heavy attenuation, through the system. Tosee this, consider the fundamental solution (2.2). No matter how small t and howlarge x , the density ρ will be nonzero, though exponentially small. In many cases,this pathology of the diffusion equation and the reaction–diffusion reaction has neg-ligible consequences, and (2.1) and (2.3) have proven to be satisfactory descriptionsin numerous circumstances and systems.

The origin of the unphysical behavior of the diffusion equation and the reaction–diffusion equation can be understood from three different viewpoints: (i) the math-ematical viewpoint, (ii) the macroscopic or phenomenological viewpoint, and (iii)the mesoscopic viewpoint.

2.2 Reaction–Transport Equations with Inertia 37

2.2.1 Hyperbolic Reaction–Diffusion Equations

From a mathematical viewpoint, the origin of the infinitely fast spreading of localdisturbances in the diffusion equation can be traced to its parabolic character. Thiscan be addressed in an ad hoc manner by adding a small term τ∂t tρ to the diffu-sion equation or the reaction–diffusion equation to make it hyperbolic. From thediffusion equation (2.1) we obtain the telegraph equation, a damped wave equation,

τ∂

2ρ

∂t2+ ∂ρ

∂t= D

∂2ρ

∂x2. (2.13)

The fundamental solution of this equation with a point source at x = 0 and t = 0 isgiven by

ρ(x, t) =

⎧

⎪⎨

⎪⎩

1

N exp

[

− t

2τ

]

I0

[1

N√

ξ

]

, for |x | <√

D

τt,

0, otherwise,(2.14)

and converges to the solution (2.2) of the diffusion equation as τ → 0, see [494,p. 388]. Here I0 is the modified Bessel function, ξ = Dt2

/τ − x2, and N = √4Dτ .

Equation (2.14) also shows explicitly that perturbations governed by the telegraphequation spread with a finite speed

√D/τ , as expected for a damped wave equation.

Adding the term τ∂t tρ to (2.3), we obtain hyperbolic reaction–diffusion equa-tions (HRDEs):

τ∂

2ρ

∂t2+ ∂ρ

∂t= D

∂2ρ

∂x2+ F(ρ). (2.15)

This type of equation is also encountered in other areas, such as nonlinear waves,nucleation theory, and phase field models of phase transitions, where it is known asthe damped nonlinear Klein–Gordon equation, see for example [165, 355, 366].In the (singular) limit τ → 0, (2.15) goes to the reaction–diffusion equation(2.3). Front propagation in HRDEs has been studied analytically and numericallyin [149, 150, 152, 151, 374]. The use of HRDEs in applications is problematic.Such equations are obtained indeed very much in an ad hoc manner for reacting anddispersing particle systems, and they can be derived neither from phenomenologicalthermodynamic equations nor from more microscopic equations, see below.

For n-variable systems, the HRDEs are given by, i = 1, . . . , n,

τi∂

2ρi

∂t2+ ∂ρi

∂t= Di

∂2ρi

∂x2+ Fi (ρ). (2.16)


2.2.2 Reaction-Cattaneo Systems and Reaction-TelegraphEquations

From a macroscopic or phenomenological viewpoint, the reaction–diffusion equa-tion follows from the continuity equation

∂ρ

∂t= −∂ J

∂x+ F(ρ) (2.17)

and the use of Fick’s first law as the constitutive equation, see Sect. 2.1.1. Fick’s firstlaw implies that the flux adjusts instantaneously to the gradient of the density. Thisis physically unrealistic, and it gives rise to the pathological feature of infinitely fastspreading of local disturbances in the diffusion equation. Cattaneo and others, fora review see [222], have argued that the flux adjusts to the gradient with a smallbut nonzero relaxation time τ . Fick’s first law should be replaced as the constitutiveequation by the Cattaneo equation

τ∂ J

∂t+ J = −D

∂ρ

∂x. (2.18)

We call the hyperbolic system (2.17) and (2.18) a reaction-Cattaneo system. Euand Al-Ghoul have derived such systems from generalized hydrodynamic theory [9,7, 8, 6]. Reaction-Cattaneo systems can also be obtained from extended irreversiblethermodynamics [223], see for example [282]. If we differentiate (2.17) with respectto t and (2.18) with respect to x and eliminate mixed second derivatives, we obtainthe so-called reaction-telegraph equation,

τ∂

2ρ

∂t2+ [1− τ F ′

(ρ)]∂ρ∂t

= D∂

2ρ

∂x2+ F(ρ). (2.19)

Remark 2.1 The reaction-telegraph equation can also be derived as the kinetic equa-tion for a branching random evolution, see [101].

Remark 2.2 Nomenclature in this field is unfortunately not uniform, and someauthors use the term hyperbolic reaction–diffusion equations for reaction-telegraphequations.

Note that the reaction-telegraph equation (2.19) differs from the ad hoc HRDE(2.15) by the additional term−τ F ′

(ρ)(∂ρ/∂t) on the left-hand side. It can be shownthat solutions of (2.19) converge to solutions of the reaction–diffusion equation (2.3)as τ → 0 [494]. Traveling wave front solutions for the reaction-telegraph equationhave been investigated by several authors [201, 176, 282, 291, 285, 136, 288, 137,114, 116, 115, 117].

The hyperbolic systems derived from a mathematical or macroscopic viewpointovercome the pathological feature of the reaction–diffusion equation, but they sufferfrom other drawbacks: (i) Hyperbolic equations typically do not preserve positivity.


Even if ρ(x, 0) ≥ 0, the solution ρ(x, t) of (2.19) will in general assume also nega-tive values [178], which is unacceptable for a true density. (ii) In order to ensure thedissipative character of the reaction-telegraph equation (2.19), the damping coeffi-cient 1− τ F ′

(ρ) must be positive, i.e.,

F ′(ρ) <

1

τfor all ρ. (2.20)

This relation between the relaxation time τ of the flux and the time scale 1/F ′(ρ)

of the reaction appears to be a purely mathematical requirement. The followingmesoscopic approach will shed light on the foundational problems of the reaction-Cattaneo system (2.17) and (2.18) and the reaction-telegraph equation (2.19) hintedat by points (i) and (ii).

For n-variable systems, the reaction-Cattaneo systems and reaction-telegraphequations read, i = 1, . . . , n,

∂ρi

∂t= −∂ Ji

∂x+ Fi (ρ), (2.21)

τi∂ Ji

∂t+ Ji = −Di

∂ρi

∂x, (2.22)

and

τi∂

2ρi

∂t2+ ∂ρi

∂t− τi

n∑

j=1

∂Fj

∂ρ j

∂ρ j

∂t= Di

∂2ρi

∂x2+ Fi (ρ). (2.23)

2.2.3 Persistent Random Walks and Reactions

From a mesoscopic viewpoint, the pathology of the diffusion equation can be tracedto the lack of inertia of Brownian particles; their direction of motion in succes-sive time intervals is uncorrelated. This lack of correlation has two consequences:(i) The particles move with infinite velocity. There is some probability, thoughexponentially small, that a dispersing particle will travel arbitrarily far from itscurrent position in any small but nonzero amount of time. Clearly, this cannot betrue for molecules or organisms. (ii) The motion of the dispersing individuals isunpredictable even on the smallest time scales. Again, this cannot be true, eitherfor molecules or organisms. It is therefore desirable to adopt a model for dispersionthat leads to more predictable motion with finite speed at smaller time scales andapproaches diffusive motion on larger time scales. The natural choice is a persistentrandom walk, also known as a correlated random walk. It is the simplest velocityjump process and was introduced by Fürth [146] and further studied by Taylor [434]and Goldstein [163], as the simplest generalization of the ordinary random walk. Inthe persistent random walk the particles have a well-defined finite speed. However,


the average velocity of the particles vanishes, and no convective flow occurs in thesystem.

In the correlated or persistent random walk [474], a particle or individual takessteps of length �x and duration �t . The particle continues in its previous directionwith probability α = 1− μ�t and reverses direction with probability β = μ�t . Inthe continuum limit �x → 0 and �t → 0, such that

lim�x,�t→0

�x

�t= γ = constant, (2.24)

we obtain the following set of equations for the density of particles going to theright, ρ+(x, t), and the density of particles going to the left, ρ−(x, t):

∂ρ+∂t

+ γ∂ρ+∂x

= μ(ρ− − ρ+), (2.25)

∂ρ−∂t

− γ∂ρ−∂x

= μ(ρ+ − ρ−). (2.26)

The particles travel with speed γ and turn with frequency μ. The persistent randomwalk is characterized by two parameters, in contrast to the ordinary random walkor Brownian motion, which is completely characterized by the diffusion coefficientD. The persistent random walk spans the whole range of dispersal, from ballisticmotion, in the limit μ → 0, to diffusive motion, in the limit γ → ∞, μ → ∞,such that lim γ

2/2μ = D = constant. The total density of the dispersing particles is

given by

ρ(x, t) = ρ+(x, t)+ ρ−(x, t), (2.27)

and the flux J of particles is given by J = γ j , where the “flow” j is defined as

j (x, t) = ρ+(x, t)− ρ−(x, t). (2.28)

Adding (2.25) and (2.26), we obtain the continuity equation

∂ρ

∂t+ γ

∂ j

∂x= 0. (2.29)

Subtracting (2.26) from (2.25), we recover the Cattaneo equation

∂ j

∂t+ γ

∂ρ

∂x= −2μj. (2.30)

Differentiating (2.29) with respect to t and (2.30) with respect to x and eliminatingthe mixed second derivatives, we obtain the telegraph equation


τ∂

2ρ

∂t2+ ∂ρ

∂t= D

∂2ρ

∂x2, (2.31)

where

τ = 1

2μ(2.32)

is the correlation time of the particle turning process, and

D = γ2

2μ. (2.33)

Brownian motion, or the diffusion equation, ceases to be a good model for dis-persal at scales where particles or individuals have a well-defined velocity. In mostphysical or chemical applications, the limiting scale is determined by the mean freepath. In liquids, the mean free path is a fraction of the molecular diameter, and per-sistence or inertia effects are negligible even on mesoscopic scales. Velocity is not arelevant variable in these situations, and the position of the particle is determined bymany independent effects. Dispersal has therefore a strongly diffusive character, andreaction–diffusion equations are an appropriate description for chemical reactionsin aqueous solutions. The persistent random walk provides a better description forparticles dispersing in dilute gases, where the mean free path can be several ordersof magnitude larger than the molecular diameter, depending on gas pressure. TheMcKean discrete velocity model for dilute gases is identical with a persistent ran-dom walk [273]. Turbulent diffusion and dispersal of animals, especially bacteria,are two other areas where the velocity of particles or organisms is well defined, andpersistence effects are not negligible, on macroscopic scales. Section 10.6 of [303]presents the persistent random walk as a model for turbulent diffusion and discussesthe inadequacy of the classical diffusion equation in this context. The persistent ran-dom walk also provides a better description for spatial spread in population dynam-ics than the often used diffusion equation [201, 195, 178]. Microorganisms andanimals tend to continue moving in the same direction in successive time intervals.Velocity is well defined and persistence effects are important on macroscopic scales.In fact, Fürth [146] applied his theory to experiments on the motion of bacteria.

Besides these practical considerations, describing the motion of particles or indi-viduals by a persistent random walk has several advantages from a theoretical view-point: (i) The persistent random walk is a generalization of Brownian motion; itcontains the latter as a limiting case, see above. (ii) The persistent random walkovercomes the pathological feature of Brownian motion or the diffusion equationdiscussed above; it fulfills the physical requirement of bounded velocity. (iii) Thepersistent random walk provides a unified treatment that covers the whole range oftransport, from the diffusive limit to the ballistic limit.

If the particles moving according to a persistent random walk react with eachother, the evolution equations for the densities, (2.25) and (2.26), must be modified


to include a kinetic rate term. The contributions from different processes, such asreactions and transport, to the evolution of a system are additive, if all the processesare Markovian [178]. Since the persistent random walk is a Markovian process, it islegitimate to add kinetic terms to the transport equation for (ρ+, ρ−):

∂ρ+∂t

+ γ∂ρ+∂x

= μ(ρ− − ρ+)+ F+(ρ+, ρ−), (2.34a)

∂ρ−∂t

− γ∂ρ−∂x

= μ(ρ+ − ρ−)+ F−(ρ+, ρ−). (2.34b)

Remark 2.3 If the state space is reduced from two variables, (ρ+, ρ−), to one vari-able, ρ, the process ceases to be Markovian. It is not legitimate to simply add akinetic term to (2.31).

The problem arises as to how to “distribute” the kinetic term F(ρ) of thereaction–diffusion equation to the left- and right-going densities ρ+ and ρ−. Thechoice most commonly made in the literature [201, 195, 178, 176, 177] is the so-called isotropic reaction walk (IRW):

F+(ρ+, ρ−) = F−(ρ+, ρ−) =1

2F(ρ). (2.35)

This choice is based on the assumption that F(ρ) is a source term for the particles,that the reaction does not depend on the direction of motion, and that new particleschoose either direction with equal probability. With (2.35) we obtain from (2.34a)and (2.34b) the reaction-Cattaneo system

∂ρ

∂t+ γ

∂ j

∂x= F(ρ), (2.36)

∂ j

∂t+ γ

∂ρ

∂x= −2μj. (2.37)

Traveling waves for isotropic reaction walks have been studied in [201, 176]. How-ever, isotropic reaction walks are unsound; they violate a basic principle of kinetics[178, 205, 204], namely that the rate of removal or death of particles of a given typemust go to zero as the density of those particles goes to zero, see Sect. 1.1. Other-wise, the concentration of those particles can become negative, which is unphysi-cal. We consider here a class of reaction random walks (RRWs) that are free fromthis drawback. The kinetic terms are based on the following assumptions: (i) Theparticles undergo a birth and death process with “fertilities” and “mortalities” thatare independent of the direction of motion of the particles. (ii) The direction of“daughter” particles is correlated with that of the “mother” particle. The degree ofcorrelation is given by κ . The value κ = 1/2 corresponds to no correlation, κ = 1 to

2.3 Reactions and Anomalous Diffusion 43

complete correlation, and κ = 0 to complete anticorrelation. In light of assumption(i), it is appropriate to adopt the production–loss form for F(ρ), see (1.3), (1.4), and(1.5). Then

F+(ρ+, ρ−) = [κρ+ + (1− κ)ρ−] f +(ρ)− f −(ρ)ρ+, (2.38a)

F−(ρ+, ρ−) = [(1− κ)ρ+ + κρ−] f +(ρ)− f −(ρ)ρ−, (2.38b)

where f +(ρ) and f −(ρ) are defined by (1.5). Note that if F+(ρ) contains a con-

stant term a, as occurs in the Brusselator, the Lengyel–Epstein model, and manyother chemical schemes, then f +(ρ) contains the term a/ρ. Other valid choices forthe kinetic terms are possible. A discussion of this aspect from the viewpoint ofchemical kinetics and population dynamics can be found in Chaps. 5 and 10.

For n-variable systems, the evolution equations for persistent random walks withreaction read, i = 1, . . . , n,

∂ρ+,i

∂t+ γi

∂ρ+,i

∂x= μi

(

ρ−,i − ρ+,i)+ F+,i (ρ+, ρ−), (2.39a)

∂ρ−,i

∂t− γi

∂ρ−,i

∂x= μi

(

ρ+,i − ρ−,i)+ F−,i (ρ+, ρ−). (2.39b)

2.3 Reactions and Anomalous Diffusion

For a large variety of applications, simple Brownian motion or Fickian diffusion isnot a satisfactory model for spatial dispersal of particles or individuals. Physical,chemical, biological, and ecological systems often display anomalous diffusion,where the mean square displacement (MSD) of a particle does not grow linearlywith time:

〈x(t)2〉 ∝ tγ . (2.40)

If 0 < γ < 1, the process is subdiffusive; if γ > 1, it is superdiffusive. Superdif-fusion is encountered, for example, in turbulent fluids [407], in chaotic systems[51], in rotating flows [418, 472], in oceanic gyres [44], for nanorods at viscousinterfaces [93], and for surfactant diffusion in living polymers [14]. Subdiffusionis observed in disordered ionic chains [45], in porous systems [100], in amorphoussemiconductors [383, 174], in disordered materials [307], in subsurface hydrology[43, 38, 23, 42, 382, 91], and for proteins and lipids in plasma membranes of variouscells [380, 477, 387], for mRNA molecules in Escherichia coli cells [162], and forproteins in the nucleus [463].

Motor proteins can lead to superdiffusive transport of engulfed microbeads inliving eukaryotic cells with γ = 1.47 ± 0.07 for short times, up to the order of


1 s, with a crossover to subdiffusive or ordinary diffusion at longer times [70]. Acommon cause of subdiffusive transport in living cells is macromolecular crowding,which generates an environment where diffusion is hindered by obstacles and traps.For example, in the cell cytoplasm diffusion processes with values of γ ranging from0.5 to about 0.85, depending on the mass of the diffusing molecule, are observed[476]. Subdiffusive motion, γ = 0.737±0.003, was also observed for lipid granulesin the cytoplasm of yeast cells [438]. The value of γ increased slightly to 0.755 ±0.006, less subdiffusive behavior, when the actin filaments were disrupted.

Anomalous diffusion is often caused by memory effects and Lévy-type statistics[185, 53]. Specifically, superdiffusion is observed for random walks with heavy-tailed jump length distributions and subdiffusion for heavy-tailed waiting time dis-tributions, see Sect. 3.4. The latter type of distribution can be caused by “traps” thathave an infinite mean waiting time [185]. For reviews of anomalous diffusion see,e.g., [298, 299, 229].

2.3.1 Continuous-Time Random Walks

Anomalous diffusion is often modeled by a continuous-time random walk (CTRW)[304, 213, 298, 299, 102], though other approaches have been explored, such asstochastic switching generating superdiffusion [123], fractional Brownian motion[266, 258, 254], and generalized Langevin equations with a memory kernel [465–467]. In a CTRW, the length of a given jump and the waiting time between twosuccessive jumps are random variables characterized by the jump probability dis-tribution function (PDF) ψ(x, t). For a more detailed discussion of CTRWs seeChap. 3. The spatial jump length PDF is given by w(x) = ∫∞

0 ψ(x, t)dt and thewaiting time PDF by φ(t) = ∫∞−∞ ψ(x, t)dx . CTRWs can be characterized by themean waiting time,

T =∫ ∞

0tφ(t)dt, (2.41)

and the second moment of the jump length PDF,

σ2 =

∫ ∞

−∞x2

w(x)dx . (2.42)

A CTRW can be described by the evolution equation for the probability densityp(x, t) of the particle being at site x at time t , given that it was at x = 0 at t = 0[381]:

p(x, t) = δ(x)�(t)+∫ ∞

0

∫ ∞

−∞ψ(x − x ′, t − t ′)p(x ′, t ′)dx ′dt ′, (2.43)


where �(t) is the survival probability given by

�(t) = 1−∫ t

0φ(t ′)dt ′. (2.44)

The Laplace transform of a function f (x, t) is denoted either by L[ f (x, t)] =∫∞

0 dt f (x, t) exp(−st) or by f̂ (x, s). Similarly the Fourier transform of a functionf (x, t) is denoted either by F[ f (x, t)] = ∫∞

−∞ dx f (x, t) exp(ikx) or by f̃ (k, t).Equation (2.43) can be solved in Laplace–Fourier space,

ˆ̃p(k, s) = 1− φ̂(s)

s

p̃0(k)

1− ˆ̃ψ(k, s)

, (2.45)

where p̃0(k) = 1 is the Fourier transform of the initial condition p0(x) = δ(x). Weconsider the large-scale, long-time limit of CTRWs with independent jump lengthand waiting time PDFs, i.e., ψ(x, t) = w(x)φ(t). If the waiting time PDF doesnot have heavy tails, then the mean waiting time T is finite, and the long-time limitcorresponds to

φ̂(s)→ 1− T s, as s → 0. (2.46)

If the CTRW also has a short-range jump length PDF w(x), then the variance σ2 is

finite, and the large-scale limit corresponds to

w̃(k)→ 1− σ2k2

2, as k → 0. (2.47)

This results in

ˆ̃p(k, s) = p̃0(k)

s + Dk2, (2.48)

with D ≡ σ2/(2T ), or

(

s + Dk2) ˆ̃p(k, s) = p̃0(k). (2.49)

Inverse Laplace and Fourier transforming (2.49), we find that p(x, t) obeys the dif-fusion equation (2.1) by making use of the fact that

F[

∂2 p(x, t)

∂x2

]

= −k2 p̃(k, t) (2.50)

and that


L[∂p(x, t)

∂t

]

= s p̂(x, s)− p0(x). (2.51)

CTRWs display subdiffusive behavior if the variance of the jump length PDFremains finite, but the waiting time PDF is heavy-tailed, such that the mean waitingtime T is infinite. An example is a waiting time PDF derived from a Mittag–Lefflerfunction for the survival probability, �(t) = Eγ (−tγ ) with 0 < γ < 1 [381]. The

asymptotic behavior of a heavy-tailed waiting time PDF is given by φ(t) ∼ t−(1+γ )

as t →∞. Consequently, the long-time limit corresponds to

φ̂(s)→ 1− (τ0s)γ , as s → 0, (2.52)

where τ0 is a parameter with units of time. For this type of CTRW, the long-time,large-scale limit of the evolution equation in Fourier–Laplace space reads

ˆ̃p(k, s) = p̃0(k)

s + Dγ s1−γ k2, (2.53)

where

Dγ ≡σ

2

2τγ0(2.54)

is a generalized diffusion constant. To inverse Laplace transform this equation, weexploit the fact that the Grünwald–Letnikov fractional derivative, defined for 0 <

p < 1 by [353],

GLD pt f (t) ≡ lim

h→0nh=t

h−pn∑

r=0

(−1)r(

p

r

)

f (t − rh), (2.55)

has the Laplace transform [353]:

L[

GLD pt f (t)

]

= s p f̂ (s). (2.56)

Consequently, inverse Laplace and Fourier transforming (2.53) we find that p(x, t)obeys the fractional diffusion equation

∂p

∂t= Dγ

GLD1−γt

∂2 p(x, t)

∂x2. (2.57)

Working with definition (2.55) is not very convenient. For sufficiently smoothfunctions f (t), the Grünwald–Letnikov fractional derivative is equivalent to theRiemann–Liouville fractional derivative [353, p. 75]. The latter is defined by


D1−γt f (t) = 1

�(γ )

∂

∂t

∫ t

0

f (t ′)(t − t ′)1−γ

dt ′, (2.58)

for 0 < γ < 1, and the fractional diffusion equation reads

∂p

∂t= Dγ D1−γ

t∂

2 p(x, t)

∂x2. (2.59)

For the mean square displacement we obtain

〈x2(t)〉 = 2

�(1+ γ )Dγ tγ , (2.60)

confirming that CTRWs with short-range jump length PDFs and heavy-tailed wait-ing time PDFs display subdiffusion. The solution of the fractional diffusion equation(2.59) can be written in terms of the Fox H-function [298]:

p(x, t) = 1√

4πDγ tγH2,0

1,2

[

x2

4Dγ tγ

∣∣∣∣

(1− γ /2, 2)

(0, 1), (1/2, 1)

]

. (2.61)

The mean density ρ(x, t) of a system of independent particles that undergo aCTRW obeys the same equation as the PDF p(x, t), which can be written in theform of a generalized Master equation. The Montroll–Weiss equation (2.45) can berewritten for ˆ̃ρ(k, s) and rearranged as

s ˆ̃ρ(k, s)− ρ̃0(k) =s

1− φ̂(s)

( ˆ̃ψ(k, s)− φ̂(s)

) ˆ̃ρ(k, s), (2.62)

where the left-hand side is the Fourier–Laplace transform of the derivative ∂ρ/∂t .If we apply the Fourier–Laplace transform inversion, then we obtain an integro-differential equation, the generalized Master equation,

∂ρ

∂t=∫ t

0

∫

R

K (x − x ′, t − t ′)ρ(z, t ′)dx ′dt ′, (2.63)

where the kernel K (x, t) is defined in terms of its Fourier–Laplace transform

ˆ̃K (k, s) = s

1− φ̂(s)

( ˆ̃ψ(k, s)− φ̂(s)

)

. (2.64)

In the uncoupled case, ψ(x, t) = w(x)φ(t), for which ˆ̃ψ(k, s) = w̃(k)φ̂(s), the

generalized Master equation takes the form


∂ρ

∂t=∫ t

0K (t − t ′)

[∫

R

ρ(x − x ′, t ′)w(x ′)dx ′ − ρ(x, t ′)]

dt ′, (2.65)

where the memory kernel K (t) is defined in terms of its Laplace transform

K̂ (s) = sφ̂(s)

1− φ̂(s). (2.66)

2.3.2 Reaction–Subdiffusion Equations

As shown above, the standard diffusion equation (2.1) has a fractional diffusionequation (2.59) as its analog in the subdiffusive case. As in the case of reaction–transport equation with inertia, see Sect. 2.2, the question arises how to combinereactions and subdiffusion in the activation-controlled regime. (For a discussionof the subdiffusion-limited case, which is outside the scope of this monograph asmentioned on page 34, see for example [491–493, 369, 391, 392, 389, 409, 410,390, 411, 203, 187].) In some schemes, [188, 189, 186, 187], reactions terms aresimply added to the fractional diffusion equation, in a manner similar to the ad hocHRDEs (2.16), assuming at the outset that the effects of subdiffusion and reactionsare separable as in the standard reaction–diffusion (2.11). However, it is easy to

show that already for the simple case of linear decay, Uk−→ P,

∂ρ

∂t= Dγ D1−γ

t∂

2ρ(x, t)

∂x2− kρ, (2.67)

this approach cannot be correct. Equation (2.67) does not preserve positivity [187].The memory kernel in (2.59), recall that D1−γ

t represents a nonlocal-in-timeintegral operator, is a clear indication that subdiffusive transport is non-Markovian.Incorporating kinetic terms into a non-Markovian transport equation requires greatcare and is best carried out at the mesoscopic level. We show in Sect. 3.4 how toproceed directly at the level of the mesoscopic balance equations for non-MarkovianCTRWs. Here we pursue a different approach. As stated above, if all processes areMarkovian, then contributions from different processes are indeed separable andsimply additive. As is well known, processes often become Markovian if a suffi-ciently large and appropriate state space is chosen. For the case of reactions andsubdiffusion, the goal of a Markovian description can be achieved by taking the agestructure of the system explicitly into account as done by Vlad and Ross [460, 461].This approach is equivalent to Model B, see Sect. 3.4.

Let ξi (x, t, τ ) be the density of particles of type i , i = 1, . . . , n, whose waitingtime (age) at position x and time t lies in the range (τ, τ + dτ). The concentrationof species i , ρi (x, t), at position x and time t , is then given by


ρi (x, t) =∫ ∞

0ξi (x, t, τ )dτ . (2.68)

In terms of chemical and related systems, reactions typically create and destroyparticles. In terms of ecological systems and populations dynamics, individuals areborn and die. In other words, kinetic events affect the waiting times of particles. Weassume that new particles are created with zero age. The same holds for newbornindividuals. Our assumption implies that all processes resulting in the arrival of aparticle or individual at a given site x are treated equally. We do not distinguishbetween arrival via a jump to x from another site x ′ or arrival by a reactive or birthevent at x . Any arrival event sets the waiting time τ at x equal to zero. We assumethat locally the reactions obey classical kinetic laws, as is the case in porous mediafor instance, and that the local kinetics of particles or individuals can be written inproduction–loss form, Fi (ρ) = F+

i (ρ)−F−i (ρ), see (1.3). As discussed in Sect. 1.1,

F−i (ρ)→ 0 as ρi → 0. To ensure the nonnegativity of the age-dependent densities

ξi (x, t, τ ), it is sufficient to require that F−i (ρ)/ρi remains bounded from above as

ρi → 0. Define Wi (x′ → x, τ ) to be the rate at which an individual of species i

with an age between τ and τ + dτ moves from position x ′ to x . The evolution ofξi (x, t, τ ) is governed by the balance equation [461]

(∂

∂t+ ∂

∂τ

)

ξi (x, t, τ ) = − ξi (x, t, τ )∫

x ′Wi (x → x ′, τ )dx ′

− ξi (x, t, τ )

ρi (x, t)F−

i (ρ(x, t)), (2.69)

with the boundary condition

ξi (x, t, τ = 0) = F+i (ρ(x, t))+

∫

x ′

∫

τ′ ξi (x

′, t, τ ′)× Wi (x

′ → x, τ ′)dx ′dτ ′.(2.70)

This boundary condition implies that entities with zero age at a particular positionare either created there with a rate F+

i (ρ(x, t)) or arrive there from other positions,as discussed above.

Let the jump PDF of the CTRW of species i be ψi (x → x ′, τ ), and

�i (x, τ ) =∫

x ′

∫ ∞

τ

ψi (x → x ′, τ ′)dx ′dτ ′ (2.71)

be the survival probability of a particle of type i at position x . The connectionbetween Wi (x → x ′, τ ′) and ψi (x → x ′, τ ) is given by the following relation[460]:

ψi (x → x ′, τ ) = �i (x, τ )Wi (x → x ′, τ ). (2.72)


The solution to (2.69) with boundary condition (2.70) reads [461]

Zi (x, t) = F+i (ρ(x, t))+

∫ t

0

∫

x ′Zi (x

′, t − τ

′)ψi (x

′ → x, τ ′)

× exp

[

−∫ t

t−τ′

F−i (ρ(x ′, t ′′)ρi (x

′, t ′′)

dt ′′]

dx ′dτ ′

+∫ ∞

t

∫

x ′ξi (x

′, t = 0, τ ′ − t)

ψi (x′ → x, τ ′)

�i (x′, τ

′ − t)

× exp

[

−∫ t

0

F−i (ρ(x ′, t ′′))ρi (x

′, t ′′)

dt ′′]

dx ′dτ ′, (2.73)

ρi (x, t) =∫ t

0�i (x, τ )Zi (x, t − τ) exp

[

−∫ t

t−τ

F−i (ρ(x, t ′′)ρi (x, t ′′)

dt ′′]

dτ

+∫ ∞

tξi (x, t = 0, τ − t)

�i (x, τ )

�i (x, τ − t)

× exp

[

−∫ t

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′]

dτ, (2.74)

where Zi (x, t) is defined to be the zero-age density, Zi (x, t) ≡ ξi (x, t, τ = 0).Equations (2.73) and (2.74) extend the usual linear CTRW formalism to includegeneral nonlinear birth and death processes.

In the following, we consider the usual case of spatially homogeneous CTRWswith independent jump and waiting time PDFs, i.e., ψi (x → x ′, τ ) = ψi(x ′ − x, τ ) = wi (x

′ − x)φi (τ ). The survival probability then does not dependon position, �i (x, τ ) = �i (τ ). We choose the initial condition as ξi (x, t =0, τ ) = ρi (x, 0)δ(τ ), i.e., at time zero all individuals are at the beginning of awaiting period. To obtain kinetic equations for reaction–transport systems withanomalous diffusion, we need to take the long-time and the large-scale limit, i.e.,w̃i (k) → 1− σ

2i k2

/2, see (2.47). As shown in [484], in this limit (2.73) and (2.74)lead to

∂ρi (x, t)

∂t= F+

i (ρ(x, t))− F−i (ρ(x, t))

+σ2i

2∇2{∫ t

0φi (t − t ′)Zi (x, t ′) exp

[

−∫ t

t ′F−

i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′]

dt ′}

,

(2.75)


ρi (x, t) =∫ t

0�i (t − t ′)Zi (x, t ′) exp

[

−∫ t

t ′F−

i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′]

dt ′. (2.76)

We need to eliminate Zi (x, t) from the system (2.75) and (2.76). We rewrite (2.76)as

ρi (x, t) exp

[∫ t

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′]

=∫ t

0�i (t − t ′)Zi (x, t ′) exp

[∫ t ′

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′]

dt ′. (2.77)

We Laplace transform this equation, use �̂(s) = [1− φ̂(s)]/s, and obtain

sφ̂i (s)

1− φ̂i (s)L[

ρi (x, t) exp

(∫ t

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′)]

= φ̂i (s)L[

Zi (x, t ′) exp

(∫ t ′

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′)]

. (2.78)

The inverse Laplace transform of (2.78) leads to

∫ t

0Ki (t − t ′)ρi (x, t ′) exp

[∫ t ′

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′]

dt ′

=∫ t

0φi (t − t ′)Zi (x, t ′) exp

[∫ t ′

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′]

dt ′. (2.79)

We define the memory kernel Ki (t) in terms of its Laplace transform as in (2.66):

K̂i (s) ≡sφ̂i (s)

1− φ̂i (s). (2.80)

Equation (2.75) can be rewritten as

∂ρi (x, t)

∂t= F+

i (ρ(x, t))− F−i (ρ(x, t))

+σ2i

2∇2{

exp

[

−∫ t

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′]

×∫ t

0φi (t − t ′)Zi (x, t ′) exp

[∫ t ′

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′]

dt ′}

(2.81)


We substitute (2.79) into (2.81) and obtain

∂ρi (x, t)

∂t= F+

i (ρ(x, t))− F−i (ρ(x, t))

+σ2i

2∇2{∫ t

0Ki (t − t ′)ρi (x, t ′) exp

[

−∫ t

t ′F−

i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′]

dt ′}

,

(2.82)

which is the generalized reaction–diffusion equation for reacting systems withanomalous diffusion [484]. The reaction terms and the Laplacian operator in (2.82)are reminiscent of the standard reaction–diffusion equation (2.12). However, unlikein a standard reaction–diffusion equation, the Laplacian acts on a nonlocal memoryterm captured by a time integral. The presence of both the kernel Ki (t − t ′), relatedto the waiting time PDF of the CTRW, and the death rate F−

i (ρ(x, t)) in the memoryterm indicates that the effects of reaction and subdiffusion are, indeed, not separable.

Remark 2.4 In the derivation of the generalized reaction–diffusion equation (2.82)we do not explicitly refer to the particular form of the waiting time PDF. Equation(2.82) is valid for arbitrary waiting time PDFs φi (t) and has much wider applica-bility than subdiffusive transport.

Remark 2.5 It is easy to see that (2.82) simplifies to a standard reaction–diffusionsystem if the CTRW is Markovian, i.e., the waiting times are exponentially dis-tributed, φi (t) = (1/τ0,i )e

−t/τ0,i . In this case K̂i (s) = 1/τ0,i , and therefore,Ki (t) = δ(t)/τ0,i .

Remark 2.6 In the derivation of (2.82) we have assumed that the spatial jump lengthPDF w̃(k) is of the form w̃(k) = 1 − σ

2k2/2. It is straightforward to extend the

derivation to the case of long-range jumps or Lévy flights by choosing w̃(k) =1− σ

α |k|α /2, 1 < α < 2, see Chap. 3, and to the case of coupled jump length andwaiting time PDFs.

The generalized reaction–diffusion equation (2.82) can be written in a form usingfractional derivatives for subdiffusive transport, where the waiting PDF of species iis given in Laplace space by (2.52), φ̂i (s)→ 1− (τ0,i s)

γi . In that case

K̂i (s) =sφ̂i (s)

1− φ̂i (s)� τ

−γi0,i s1−γi , (2.83)

and (2.78) reads

Exercises 53

τ−γi0,i s1−γiL

[

ρi (x, t) exp

(∫ t

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′)]

= φ̂i (s)L[

Zi (x, t ′) exp

(∫ t ′

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′)]

. (2.84)

We inverse Laplace transform (2.84), using (2.56) and the equivalence of theGrünwald–Letnikov and Riemann–Liouville fractional derivatives, to obtain

τ−γi0,i D1−γi

t

(

ρi (x, t) exp

[∫ t

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′])

=∫ t

0φi (t − t ′)Zi (x, t ′) exp

[∫ t ′

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′]

dt ′. (2.85)

We substitute (2.85) into (2.81) and find [485]

∂ρi (x, t)

∂t= F+

i (ρ(x, t))− F−i (ρ(x, t))

+Di;γi∇2

{

exp

[

−∫ t

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′]

×D1−γit

(

ρi (x, t) exp

[∫ t

0

F−i (ρ(x, t ′′))ρi (x, t ′′)

dt ′′])}

. (2.86)

Exercises

2.1 Consider (2.1) with ρ(x, 0) = δ(x). Define

〈x(t)m〉 ≡∫∞−∞ xm

ρ(x, t)dx∫∞−∞ ρ(x, t)dx

. (2.87)

Obtain evolution equations for 〈x(t)〉 and 〈x(t)2〉 and solve them.

2.2 Solve (2.1) with ρ(x, 0) = δ(x) on the interval [0, L] with no-flux boundaryconditions, i.e., (∂ρ/∂x)(0) = (∂ρ/∂x)(L) = 0.

2.3 Solve the RD equation (2.3) for the pure death process F(ρ) = −rρ withρ(x, 0) = δ(x) on (−∞,∞).

2.4 Consider the nonlinear diffusion equation


∂ρ

∂t= D

∂

∂x

[(ρ

ρ0

)n∂ρ

∂x

]

(2.88)

on (−∞,∞) with ρ(x, 0) = δ(x), where D and ρ0 are positive constants.(a) Verify that

ρ(x, t) =

⎧

⎪⎪⎨

⎪⎪⎩

ρ0

A(t)

[

1−(

x

cA(t)

)2]1/n

, |x | ≤ cA(t),

0, |x | > cA(t),

(2.89)

where

A(t) =(

t

t0

) 12+n

, c =�(

1n + 3

2

)

[

π1/2

ρ0�(

1n + 1

)] , t0 =c2n

2D(n + 2). (2.90)

Here � is the Gamma function.(b) Determine 〈x(t)〉 and 〈x(t)2〉.

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Reactions and Transport: Diffusion, Inertia, and Subdiffusion