MMixing time of A + B − > C P HaynesSchool of Mathematics and Physics,The University of Queensland,St Lucia, Queensland, 4072, Australia.
Abstract
A mixing time density of A + B → A and B diffuse at the same rate. The density is a measure of the number of A and B particles that mix through the center of the reaction zone. It alsocorresponds to the reaction density for the special case in which A and B annihilate upon contact. An exact expression is found for the generatingfunction of the mixing time. The analysis is extended to multiple reactionfronts and finitely ramified fractals. The full method involves using thekernel of the Laplace transform integral operator to map and analyze amoving homogeneous Dirichlet interior point condition. Comprehension of diffusion-reaction schemes in disordered media is often marredby the symbiotic nature between the diffusive aspect and the reactive aspect ofthe scheme. If the underlying diffusion is not understood then it is difficultto define the reactivity, whereas the reaction component naturally impedes thediffusivity of the reactants. Consequently, a major goal in experimental andtheoretical studies is to elucidate the diffusive or reactive aspects of a schemewhich requires understanding, for example; the effect of crowded intracellularconditions on the reduced mobility of a reactant [1, 2], the partial differentialequation description of a system [3], or the diffusion limit in inhomogeneousenvironments [4].For the irreversible bimolecular reaction between two different initially sepa-rated diffusing species A + B →
0, a significant study of the diffusive and reactiveaspects has occurred over the past 30 years [5, 6, 7, 8, 9]. Even so, given that A + B → a r X i v : . [ m a t h - ph ] A ug et of equations [14] D A ∂ xx C A ( x, t ) − ∂ t C A ( x, t ) = R ( x, t ) ,D B ∂ xx C B ( x, t ) − ∂ t C B ( x, t ) = R ( x, t ) , where C A and C B are the concentrations of each species remaining within thesystem at time t , D A and D B being the diffusion coefficients of the reactants, R the macroscopic reaction rate and ∂ x C A ( X, t ) is the partial derivative of C A with respect to x evaluated at x = X . As time evolves, the A and B specieswill mix forming a reaction zone whose evolution depends on R .Studies of A + B → R in order tosolve the underlying model [15]; typically, a mean field density of the form R = k C A C B ( k constant) is introduced thereby making the system nonlinear. Thevalidation of such an approximation has been confirmed through experimentalstudies [16, 17], although deviations [18] from the mean-field behavior are knownto occur and the exact form for R remains unresolved. Alternatively, by focusingon the diffusive aspect, it is possible to study the encounter rate of A and B ,which becomes useful in determining an upper bound on the rates of reaction.At the locations where A and B encounter one another they begin to mix.The way in which they mix is affected by the initial placement of the species [19].If it happens that both species exhaust themselves in the mixing process, thenthe mixing is homogeneous (or efficient), otherwise it is inhomogeneous. For ex-ample, on an infinite one dimensional domain, if the total number of A speciesin a particular region is greater than (and surrounds) the number of B speciesthen a rapid disappearance of the B species will eventuate from in-homogeneousmixing thus resulting in segregated island-type phenomenon [20]. For finite do-mains, the boundary of the domain will accentuate this mixing behavior. Pro-viding a general condition for when this behavior occurs is essential to the reac-tive aspect of the A + B → A + B → C ).It is possible to infer results pertaining to the mixing of the species, irre-spective of R , when D A = D B . This is done by studying a fluctuation densityof the system , C = C A − C B . The point at which C = 0, x = M ( t ) (whichis unique, say) represents the point where the A and B species mix. Studyingthe flux at x = M ( t ) will obtain the reaction density in the special case that A and B annihilate upon contact, however, there appears to lack a comprehensivetheoretical analysis of the flux at M ( t ), its implications in classifying the mixingbehavior of the species and its implications for the reactive part of the scheme.The flux at the mixing point x = M ( t ), F ( t ), in some sense solves thediffusional aspect of the diffusion-reaction process given that it represents thenumber of new A and B species that are mixed and are able to react. If themixing point is defined for all time, then A and B are mixing homogeneously.A study involving the author [21] found criterion for when homogeneous mixingoccurred on a finite one dimensional domain with no flux boundaries, althoughthe extension to arbitrary boundary conditions and one dimensional networks(i.e. finitely ramified fractals) has not been done.2his study finds the generating function of F for general initial and boundaryconditions on one dimensional domains, for the case in which A and B areinitially separated, diffuse at the same rate and for which the mixing point isdefined for all time. To the best of the author’s knowledge no such in-depthstudy has been performed. The paper is structured as follows: we define themathematical problem for a finite one dimensional domain in § § § M ( t ) is unique are provided in § § §
4. We conclude the study in § The analysis involves working with C = C A − C B , where C ( x, t ) : Ω → R ,Ω = [0 , ‘ ] × [0 , ∞ ), is the solution of D∂ xx C = ∂ t C, C ( x,
0) = ρ A ( x ) − v ρ B ( x ) ,∂ x C (0 , t ) = J ( t ) , ∂ x C ( ‘, t ) = J ( t ) , k J i k = Z ∞ | J i | dt. (1)Here, ρ A and ρ B are the initial continuous density functions of the A and B species whose support lies in [ I − A , I + A ] and [ I − B , I + B ] (respectively), where I − A > , I + B < ‘ , I − B > I + A and v ∈ [0 ,
1] is a parameter that sets the initial ratioof A and B . J ( t ) and J ( t ) are taken to be within the space of continuousfunctions. For this type of C ( x, x = M ( t ), such that C ( M ( t ) , t ) = 0, where themovement of M ( t ) is fully determined by the flux at x = M ( t ), F ( t ), since F ( t ) = F ( t ) − = lim x → M ( t ) − − D∂ x C ≡ lim x → M ( t ) + | D∂ x C | = F ( t ) + = F ( t ) . (2)Here M ( t ), which we also refer to as the mixing point, will always occur regard-less of the form of J i , however it may not be defined for all time. An illustrationof the problem is given in Fig. 1 a) and c). The goal is to find the generating function of F ( t ), f ( s ), which we definethrough the Laplace transform f ( s ) = L [ F ( t )] = R ∞ F ( t ) e − st dt, ( s > L S , where3 S [ F ] = R ∞ e − st d [ F ( t )] . The procedure involves first solving the Laplacetransformed system of (1), D ∆ c − s c = − ρ A + vρ B , ∂ x c (0 , s ) = j ( s ) , ∂ x c ( ‘, s ) = j ( s ) , (3)where c ( x, s ) = L [ C ( x, t )], j ( s ) = L [ J ( t )] and j ( s ) = L [ J ( t )], to obtain c = ch( ‘ − x )sh( ‘ ) √ sD ( j + Z [ x, C ( x, x )sh( ‘ ) √ sD ( j + Z [ x, C ( x, . (4)Here sh( x ) = sinh( x p sD ), ch( x ) = cosh( x p sD ), Z [ x, C ( x, R x ch( x ) C ( x , dx , Z [ x, C ( x, R ‘x ch( ‘ − x ) C ( x , dx , with the above representation beingin the Green’s function form [22, Pp253]. In § x = y ( s ) such that f ( s ) = D∂ x c ( y ( s ) , s ) = L [ D∂ x C ( M ( t ) , t )] = L [ F ( t )] , where y ( s ) = 1 √ s arctanh sh( ‘ ) − ch( ‘ ) − vZ (cid:2) I − B , ρ B (cid:3) − j j + Z (cid:2) I + A , ρ A (cid:3) !! , (5)and f = vuut(cid:0) Z (cid:2) I + A , ρ A (cid:3) + j (cid:1) − (cid:0) Z (cid:2) I + A , ρ A (cid:3) + j (cid:1) ch( ‘ ) − vZ (cid:2) I − B , ρ B (cid:3) + j sh( ‘ ) ! . (6)These results are valid provided M ( t ) is defined for all time, and is the uniquepoint in which C ( M ( t ) , t ) = 0. A uniqueness criteria for M ( t ) is given in § M ( t ) is not defined for all time, an alternative method is required. Notethat j i can be set to consider Dirichlet boundary conditions, given the inter-relation of j , j , c (0 , s ) and c ( ‘, s ) found by substituting x = 0 and x = ‘ inEq. (4); c ( ‘, s ) = ch( ‘ ) j + j + Z [ ‘, C ( x, √ sD sh( ‘ ) , c (0 , s ) = ch( ‘ ) j + j + Z [0 , C ( x, √ sD sh( ‘ ) . The results have been numerically validated for an example (see Fig. 1 b)).
The purpose of this section is to prove that ∂ x C ( M ( t ) , t ) = L − [ ∂ x c ( y ( s ) , s )]. C ( x, t ) is a scalar function of time and space, but can alternatively be definedby letting x assume a function of time; C ( X ( t ) , t ), where X belongs to a set B t of moving points defined for all t ≥
0. If some κ t ⊂ B t is to span Ω such thatevery X ( t ) ∈ κ t is distinct for each t , we require X ( t ) to be a translation of adesignated origin point X ( t ) within κ t : X ( t ) = [( X ( t ) + ξ ) mod ‘ ] mod 0 , ξ ∈ [0 , ‘ ] . (7)4his implies that there are moving points which possess discontinuities in time asthey cross the boundary (i.e. at some time t c , lim t → t − c X ( t ) = ‘ , lim t → t + c X ( t ) =0) and that Z ‘ C ( x, t ) dx = Z ‘ C ([( X ( t ) + ξ ) mod ‘ ] mod 0 , t ) dξ ∀ t ≥ . Analogously, c ( x, s ) can be considered in terms of c ( Y ( s ) , s ). Here Y ∈ B s ,which is a set of moving points defined for all s ≥ κ s ⊂ B s spans Ω, with every Y ( s ) ∈ κ s being a translation of a designated origin point Y ( s ), then Z ‘ c ( x, s ) dx = Z ‘ c ([( Y ( s ) + ξ ) mod ‘ ] mod 0 , s ) dξ ∀ s ≥ . The aim is to study a zero solution of Eq. (4) to obtain properties of thezero solution of Eq. (1). To do this we study the Laplace-Stieltjes transform L S [ | C ( X ( t ) , t ) | ] for X ( t ) ∈ ¯ κ t , where¯ κ t = { X ( t ) ∈ κ t | X (0) ∈ ( I + A , I − B ) , ∀ t > C ( X ( t ) , t ) > , C ( X ( t ) , t ) < C ( X ( t ) , t ) = 0 } . The quantity L S [ | C ( X ( t ) , t ) | ] exists for every X ( t ) ∈ ¯ κ t ; given M ( t ) is smoothlydefined through Eq. (2), we set X ( t ) = M ( t ) in Eq. (7) and note that C ([( M ( t ) + ξ ) mod ‘ ] mod 0 , t ) is differentiable a.e. in [0 , ∞ ) and therefore is ofbounded variation. Provided ∀ t ≥ , M ( t ) ∈ (0 , ‘ ) is the unique point in which C ( M ( t ) , t ) = 0 , (8)the kernel of L S , ker[ L S ] = {| C ( X ( t ) , t ) | = k | X ( t ) ∈ ¯ κ t , k ≥ } = C ( M ( t ) , t ) =0. That is, if Eq. (8) is true, mass is exhausted out of the system untillim t →∞ C ( x, t ) = 0 (since k J i k < ∞ ), thereby ensuring that the only so-lution to | C ( X ( t ) , t ) | = k is X ( t ) = M ( t ) when k = 0. If we take ¯ κ s = { Y ( s ) | c ( Y ( s ) , s ) = L [ C ( X ( t ) , t )] , X ( t ) ∈ ¯ κ t } , then there exists a bijective map φ : ¯ κ t → ¯ κ s such that if Eq. (8) is true, then for X ( t ) ∈ ¯ κ t , L S [ | C ( X ( t ) , t ) | ] = | s c ( φ ( X ( t )) , s ) − C ( X (0) , | and ∀ s ≥ , ∃ ! y ( s ) : s c ( y ( s ) , s ) = C ( M (0) , , where φ ( M ( t )) = y ( s ) . (9)As M (0) ∈ ( I + A , I − B ), it follows that c ( y ( s ) , s ) = 0. This result establishes thelink between the two zero solutions c ( y ( s ) , s ) and C ( M ( t ) , t ). Note that sucha link can be established through direct analysis of the L operator, althoughthe above clearly shows how the mapping fails in the case in which the initialdistributions are mixed (i.e. if I + A = I − B , then C ( M (0) , = 0 or is not defined).It remains to show that ∂ x C ( M ( t ) , t ) = L − [ ∂ x c ( y ( s ) , s )]. Re-define X ( t ) ∈ ¯ κ t by X ( t ) = M ( t ) + ξ , so that ∀ ξ , ( M (0) + ξ ) ∈ ( I + A , I − B ). For some | ξ | small, L : C ( M ( t ) + ξ, t ) → c ( φ ( M ( t ) + ξ ) , s ) is a continuous map which ensures that φ is continuous and hence thatlim ξ → c ( φ ( M ( t )+ ξ ) , s ) = c ( y ( s ) , s ) = 0 = L [ C ( M ( t ) , t )] = lim ξ → L [ C ( M ( t )+ ξ, t )] . ( x ) M ( t ) ρ ( x ) ‘ J J t R t F ( t ) d t t M ( t ) a) b) c) Figure 1: a) Illustration of problem. b) R t F dt and c) M ( t ) for the case J = 0, J = − e − t , ρ A = ( u ( x − ) − u ( x − )), ρ B = k sin( πx )( u ( x − ) − u ( x − )), where u ( x ) is the Heaviside step function. M ( t ) is numericallydetermined using a Crank-Nicolson method [24] and R t F ( t ) dt is compared tothe numerically inversed-Laplace transform of s − f ( s ) (using [25]) as given byEq. (6).From this, and Eq. (2) being well defined, we have f ( s ) D = ∂ x c ( y ( s ) , s ) = lim ξ → ∂ ξ c ( φ ( M ( t )+ ξ ) , s ) = L [lim ξ → ∂ ξ C ( M ( t )+ ξ, t )] = L (cid:20) F ( t ) D (cid:21) . It is therefore possible to define f = f ( s ) − = lim x → y ( s ) − − D∂ x c ≡ lim x → y ( s ) + | D∂ x c | = f ( s ) + = f. (10)This relation implies that ∂ xx c ( y ( s ) , s ) = 0 and consequently, from Eq. (9),that the left hands side of Eq. (3) is zero at x = y ( s ). This is true provided ∀ s ≥ , y ( s ) / ∈ [ I − A , I + A ] or y ( s ) / ∈ [ I − B , I + B ]. Conversely, if for any s = s ∗ , y ( s ) ∈ [ I − A , I + A ] or y ( s ) ∈ [ I − B , I + B ] then Eq. (2) (and hence Eq. (8)) would befalse. By using this fact, the fact that c ( I + A , s ) > c ( I − B , s ) < s → ∞ ,and a strong minimum principle [23, Pp 260], it can be shown that ∀ s ≥ , y ( s ) ∈ ( I + A , I − B ) . Through this condition, we are able to find a solution for y ( s ) such that L [ M ( t )] = y ( s ); evaluating x = y ( s ) in Eq. (4), setting c ( y ( s ) , s ) = 0, solving for y ( s ) (byusing the fact that y ( s ) ∈ ( I + A , I − B )) results in Eq. (5). This can be used to findthe required f ( s ) through Eq. (10), the result being Eq. (6). M ( t ) So far, it has been shown how to find the generating function of F ( t ), the resultsbeing valid irrespective of the boundary conditions, provided M ( t ) is unique.6n this section, we define criteria (i) in t or (i ∗ ) and (ii ∗ ) in s for Eq. (8) to betrue.For uniqueness of M ( t ), we require C ( x, t ) ≶ x ≷ M ( t ) respectively,which implies that C (0 , t ) ≥ C ( ‘, t ) ≤ C (0 , t ) = 0 (say) onsome time interval T (with C ( x, t ) < x >
0) implies that the limiting F ( t ) − in Eq. (2) is not defined. We therefore require that1. Neither C (0 , t ) or C ( ‘, t ) be identically zero for some T , unless T = (0 , ∞ ).For either case j and j cannot change signs.Here, the case T = (0 , ∞ ) is considerable as homogeneous boundary conditionsin t are mapped to homogeneous boundary conditions in s .Denote C to be the set of completely monotonic functions [26]. A function h ( s ) ∈ C , if h ( s ) = L S [ H ( t )] , ( − n d n hds n > , n ≥ , s > , (11)where H is bounded and non-decreasing.It is possible to pose conditions in s , using the theory of completely mono-tonic functions. We require(i*) ∀ s ≥ y ( s ) ∈ ( I + A , I − B ),(ii*) c (0 , s ), − c ( ‘, s ) ∈ C unless T = (0 , ∞ ), in which case j , − j ∈ C .Further conditions can be derived from (i ∗ ); taking s = 0 in y ( s ) in Eq. (5) gives − j (0) − j (0) = 1 − v which implies that R ‘ C ( x, dx = − R ∞ J + J dt . Thelatter is a mass balancing condition stating that the total mass leakage overtime occurring at either side of the mixing point M ( t ) must be equal. If this isnot true, M ( t ) converges to the boundary. Example: Homogeneous Dirichlet boundary conditions. If C (0 , t ) = C ( ‘, t ) = 0, v = 1, C ( x, = C ( ‘ − x,
0) then (i) is not true. To seethis, consider C = C = C − C , where C ( x,
0) = ρ A , C ( x,
0) = ρ B and∆ C i = ∂C i ∂t { C i (0 , t ) = C i ( ‘, t ) = 0 } , i = 1 , . If mass Q , Q , and Q = Q − Q exits the system 1,2 and 3 respectively, then R ∞ Q dt = 0 and Q must change sign unless Q = Q which occurs when C ( x,
0) = C ( ‘ − x, C ( x, = C ( ‘ − x, ∗ ) is satisfied.7 xample: Homogeneous Neumann boundary conditions. If v = 1, j = j = 0, then (i) is true and the theory in § v < M ( t ) takes some time t c to travelstowards the x = ‘ boundary, such that C ( ‘, t c ) = 0 and C ( x, t ) > x ∈ [0 , ‘ ] for t > t c . Here t c represents the time in which all species are mixedor the time at which there are no more B species for the case in which A and B annihilate upon contact. Unphysical scenario.
When a boundary flux changes signs, an unphysical situation occurs (i.e. J ( t )changing signs would imply a sudden introduction of A species directly after aninjection of B species). For such a case the absolute mass in the system M ( t )is not necessarily conserved. By considering the amount of mass in the systemon either side of M ( t ) it is found that d M dt (cid:12)(cid:12)(cid:12) x
0, the above reduces to the result reported inRef. [21].
Understanding diffusion phenomenon through finitely ramified deterministicfractals continues to be of modeling importance to disordered media [28, 29, 30].To further our understanding of the mixing point on one dimensional domains,we consider the fluctuation density on a one dimensional path through-out agiven fractal.The analysis involves working with C ( x, t ) : Ω → R , Ω = [0 , ‘ f ] × [0 , ∞ ), isfound to satisfy D∂ xx C = ∂ t C + D N X i =0 δ ( x − x i ) ∂ x C ( x i , t ) and C ( x,
0) = ρ A ( x ) − ρ B ( x ) . (14)Here ‘ f is the length of the path throughout the fractal, ρ A and ρ B are theinitial concentrations of A and B species on the path (other initial conditionscan be specified throughout the fractal) and x i are points on the path wheresource fluctuation of the species can occur. For the example problem illustratedin Fig. 3 a), ‘ f = 4 ‘ , and fluctuations can occur at points x , x , x , x or x .Note that boundary conditions at x = 0 and x = x f can be also be appliedif required. Observe the difference between Eqs. (1) and (14); they both areone dimension problems, except Eq. (14) has additional source terms at x = x i .These source terms will affect the relation in Eq. (2). That is, an expression F ( t ) = F ( t ) − = lim x → M ( t ) − − D∂ x C ≡ lim x → M ( t ) + | D∂ x C | = F ( t ) + = F ( t ) , (15)is not necessarily true given that fluctuations occurring at x i can make sucha result invalid. To further emphasis this point, we consider two examples to10xplain the behavior of the mixing point along a one dimensional path within afractal.Case i) There is a unique mixing point that remains between two source pointson the path for all time; i.e. ∀ t > , M ( t ) ∈ [ x i , x i +1 ]. For example, when thereis an instantaneous release of an equal number of A and B species at x and x (respectively) at time t = 0 on the T-tree illustrated in Fig. 3 b), with no-fluxboundary conditions throughout the structure, M ( t ) is always between x and x . In general, the required f ( s ) is derived as in § T + P = f produced in the interval ( x i , x i +1 ). It is found that p P + T = √ s q p x i + p x i +1 − p x i p x i +1 ch( ‘ )sh( ‘ ) = f, (16)where p x i and p x i +1 are the Laplace transformed concentrations at x i and x i +1 (and can be found through Ref. [31]). This provides a means of testing resultson fractals and developing asymptotic scaling laws.Case ii) There is a unique mixing point that does not remain between twosource points on the path for all time. In this case, it is not possible to calculatethe mixing density. To see this, consider the instantaneous release of an equalnumber of A and B species at x and x at time t = 0 on the T-tree illustratedin Fig. 3 b), with no-flux boundary conditions throughout the structure. Forthe system in t , a unique mixing point M ( t ) will originate at x at t = 0 andwill then move towards x . When it arrives at x , M ( t ) will branch into twomixing points, one that moves up into the domain [ x , x ] and the other whichcontinue towards x . This violates the uniqueness of M ( t ). For the system in s ,there is a unique moving homogeneous Dirichlet point, y ( s ), that originates at x as s → ∞ and converges between x and x as s →
0. When y ( s ) is betweenpoints x and x or x and x , its value is correctly defined and is such thatthere exists some s ∗ such that lim s → s −∗ y ( s ) = lim s → s + ∗ y ( s ) = x . However,there are two different formulations for y ( s ) depending on the interval y ( s ) liesin and the theory breaks down. Note it is possible that through a particularchoice of initial conditions the above problem involves a unique formulation for y ( s ) over both intervals for all s ≥
0. If this occurs, then the analysis is similarto case i) and exact results can be found for f ( s ).Even if there are no exact results for this problem, it is hypothesized that ifa unique homogeneous Dirichlet point y ( s ) converges to y f ( s ) defined in somedomain [ x f , x f +1 ] as s → f f ( s ) should give an approxi-mation to the long time mixing behavior of A and B i.e. F ( t ) ≈ L − [ f f ( s )],where F ( t ) is the true mixing density. Indeed, the difference between the twoexamples is the additional mixing point that originate in the side branch incase ii). This mixing point dies off rather quickly and the asymptotic mixingbehavior of A and B within the structure for large t should be dominated bythe same mechanism as Eq. (16). 11 Summary and Discussion
The mixing time generating function for the diffusion reaction system A + B → A and B annihilate upon contact. Note that the study was restrictedto continuous initial and boundary conditions as it realistically coincides withexperimental conditions, although it is possible to relax these conditions.The generalization and application of this method to modified diffusion-reaction schemes involving more general evolution equations would be of in-terest. This does not make reference to formulations A + B → B where B isa moving trap, or the reaction A + B → C whose results can analogously beinterpreted from the result presented within, but to modified problems such asthe reaction- sub-diffusion problem [32], or cases which require further interiorpoint conditions (e.g. semi-permeable cellulose membrane [33]).From this study, a natural question to ask is whether F can be used todefine the reaction rate through a form R ( ξ, t ) = R t F ( τ ) G ( ξ, t − τ ) dτ , forsome specified G ( ξ, t ) whose physical significance remains unknown. Indeedfor the case in which A and B annihilate upon contact at x = M ( t ) ( ξ = 0), G (0 , t ) = δ ( t ) and R ( M ( t ) , t ) = F ( t ). Further motivation for defining R ( x, t ) interms of F ( t ) lies in the case in which D A = D B ; as the analysis presented inthe beginning of section § C ( x, t ) = D A C A − D B C B , the analysis of the homogeneous Dirichlet boundary conditionsis still possible, although F ( t ) will ultimately depend on R ( M ( t ) , t ). This placesemphasis on the fact that the particles mixing behavior will ultimately dependon how they react when they first encounter one another, a modeling aspectthat marries well with the introductory statement to this paper.The theory presented was extended to consider multiple mixing points andgeneral one dimensional systems (i.e. finitely ramified fractals). We derived anecessary condition to determine whether A and B were mixing homogeneouslythrough a given mixing point. It was shown that for finitely ramified fractals,the mixing density can be governed by two different mechanisms; depending onthe initial placement of the A and B species, a purely one dimensional mixingmechanism can occur. This becomes significant when dealing with fractals withno loops. It is believed that further progress in deriving the mixing density infractal domains can be made using the respective propagator for fractals [34]. References [1] M. T. Klann, A. Lapin, and M Reuss. Agent-based simulation of reac-tions in the crowded and structured intracellular environment: Influence ofmobility and location of the reactants.
BMC Systems Biology , 5, 71, 2011.[2] I. Schoen, H. Krammer, and D. Braun. Hybridization kinetics is differentinside cells.
PNAS , 108:3473–3480, 2011.123] B. Franz, M. B. Flegg, S. J. Chapman, and R. Erban. Multiscale reaction-diffusion algorithms: Pde-assisted brownian dynamics.
SIAM J. Appl.Math. , 73:1224–1247, 2013.[4] R. Li, J. A. Fowler, and B. A. Todd. Calculated rates of diffusion-limitedreactions in a three-dimensional network of connected compartments: Ap-plication to porour catalysts and biological systems.
Phys. Rev. Lett. ,113:028303, 2014.[5] M. J. E. Richardson and M. R. Evans. Localization transition of a dynamicreaction front.
J. Stat. Phys. , 30:811–818, 1997.[6] G. T. Barkema, M. J. Howard, and J. L. Cardy. Reaction-diffusion frontfor A + B → ∅ in one dimension. Phys. Rev. E. Rapid Comm. , 53:R2017–R2020, 1996.[7] M. Araujo, H. Larralde, S. Havlin, and H. E. Stanley. Scaling anomaliesin reaction front dynamics of confined systems.
Phys. Rev. Lett. , 71:3592–3595, 1993.[8] S. Kisilevich, M. Sinder, J. Pellag, and V. Sokolovsky. Exponential temporalasymptotics of the A + B → Phys. Rev. E , 77:046103, 2008.[9] B. M. Shipilevsky. Diffusion-controlled death of A -particle and B -particleislands at propagation of the sharp annihilation front A + B → Phys.Rev. E. , 77:030101(R), 2008.[10] D. Toussaint and F. Wilczek Particle-antiparticle annihilation in diffusivemotion.
J. Chem. Phys. , 78:2642–2647, 1983.[11] S. Kwon, S. Y. Yoon, and Y. Kim. Continuously varying exponents in A + B → Phys. Rev. E ,74:021109, 2006.[12] L. Frachebourg, P. L. Krapivsky, and E. Ben-Naim. Segregation in a one-dimensional model of interacting species.
Phys. Rev. Lett. , 77:2125–2128,1996.[13] R. Kopelman. Fractal reaction kinetics.
Science , 241:1620–1626, 1988.[14] D. ben Avraham and S. Havlin.
Diffusion and Reactions in Fractals andDisordered systems . Cambridge University Press, Cambridge, 2000.[15] E. Ben-Naim and S. Redner. Inhomogeneous two-species annihilation inthe steady-state.
J. Phys. A: Math. Gen. , 25:L575–L583, 1992.[16] Y. L. Koo and R. Kopelman. Space-and time-resolved diffusion-limitedbinary reaction kinetics in capillaries: experimental observation of segre-gation, anomalous exponents, and depletion zone.
J. Stat. Phys. , 65:919 –924, 1991. 1317] E. Monson and R. Kopelman. Nonclassical kinetics of an elementary A + B → C reaction-diffusion system showing effects of a speckled initialreactant distribution and eventual self-segregataion: Experiments. Phys.Rev. E , 69:021103, 2004.[18] S. Cornell and M. Droz. Steady-state reaction-diffusion front scaling for mA + nB → [inert]. Phys. Rev. Lett. , 70(24):3824–3827, 1993.[19] K. Lindenberg, A. H. Romero, and J. M. Sancho. Nonclassical kinetics inconstrained geometries: initial distribution effects.
Int. J. Bifur. Chaos ,8:853–868, 1998.[20] B. M. Shipilevsky. Self-similar evolution of the A -particle island - semi-infinite B -particle sea reaction diffusion system. Phys. Rev. E , 88:012133,2013.[21] C. P. Haynes, R.Voituriez, and O. B´enichou. Reaction kinetics of A + B → J. Phys. A.: Math. Theor. , 45:415001, 2012.[22] J. K. Hunter and B. Nachtergaele.
Applied Analysis . World Scientific,Singapore, 2001.[23] W. Walter.
Ordinary differential equations . Springer-Verlag, New York,1998.[24] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery.
Nu-merical recipes . Cambridge University Press, Cambridge, 2007.[25] H. Stehfest. Numerical inversion of Laplace transforms algorithm 368.
Com-mun. ACM , 13:47–49, 1979.[26] C. Berg. Stieltjes-Pick-Bernstein-Schoenberg and their connection to com-plete monotonicity, in j. mateu and e. porcu, eds., positive definite func-tions: From schoenberg to space-time challenges (dept. of mathematics,universitat jaume i de castellÂťo, spain, 2008). 2008.[27] L. D. Landau and E. M. Lifshitz.
Mechanics (Third edition) . ButterworthHeinemann, Amsterdam, 1976.[28] A. S. Balankin and B. E. Elizarraraz. Map of fluid flow in fractal porousmedium into fractal continuum flow.
Phys. Rev. E , 85:056314, May 2012.[29] A. S. Balankin, B. Mena, J. Pati˜no, and D. Morales. Electromagnetic fieldsin fractal continua.
Phys. Lett. A , 377:783–788, 2013.[30] S. Condamin, O. B´enichou, V. Tejedor, R. Voituriez, and J. Klafter. Firstpassage times in complex scale invariant media.
Nature , 450:77–80, 2007.[31] C. P. Haynes and A. P. Roberts. Global first-passage times of fractal lat-tices.
Phys. Rev. E , 78:041111, 2008.1432] B. I. Henry and S. L. Wearne. Existence of turing instabilities in a two-species fractional reaction-diffusion system.
SIAM J. Appl. Math. , 62:870,2002.[33] S. H. Park, H. Peng, R. Kopelman, and H. Taitelbaum. Dynamicallocalization-delocalization transition of the reaction-diffusion front at asemipermeable cellulose membrane.
Phys. Rev. E , 75:026107, Feb 2007.[34] B. O’Shaughnessy and I. Procaccia. Analytical solutions for diffusion onfractal objects.