Exactly solvable reaction diffusion models on a Bethe Lattice through the empty-interval method

Matin, Laleh Farhang

doi:10.1007/s40094-015-0163-y

Cited by 4 publications

(6 citation statements)

References 30 publications

(43 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…and which would reproduce the asymptotic mean-field behaviour [11,6,35,43]. Indeed, the mean-field scaling discussed for the reset in the previous section 2 can be taken over almost unchanged, where now s plays the role previously taken by the reset rate r. The only change comes from the initial condition (3.3), such that now…”

Section: Densitymentioning

confidence: 91%

“…A well-known theorem [11,61] states that on the Bethe lattice, a connected cluster of n sites has n(q − 2) + 2 neighbours, independently of the shape of the cluster. Both the Cayley tree and the Bethe lattice are widely used in the context of analytical studies of spin systems and of different chemical reactions [66,31,41,4,11,42,6,12,35,43,67,16], random and cooperative sequential adsorption [13,14,12,45] or branched polymers [28,58].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Cross-over between diffusion-limited and reaction-limited regimes in the coagulation-diffusion process

Shapoval,

Dudka,

Durang

et al. 2018

Preprint

View full text Add to dashboard Cite

The change from the diffusion-limited to the reaction-limited cooperative behaviour in reaction-diffusion systems is analysed by comparing the universal long-time behaviour of the coagulation-diffusion process on a chain and on the Bethe lattice. On a chain, this model is exactly solvable through the empty-interval method. This method can be extended to the Bethe lattice, in the ben-Avraham-Glasser approximation. On the Bethe lattice, the analysis of the Laplace-transformed time-dependent particle-density is analogous to the study of the stationary state, if a stochastic reset to a configuration of uncorrelated particles is added. In this stationary state logarithmic corrections to scaling are found, as expected for systems at the upper critical dimension. Analogous results hold true for the time-integrated particle-density. The crossover scaling functions and the associated effective exponents between the chain and the Bethe lattice are derived.

show abstract

Section: Densitymentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Cross-over between diffusion-limited and reaction-limited regimes in the coagulation-diffusion process

Shapoval,

Dudka,

Durang

et al. 2018

Preprint

View full text Add to dashboard Cite

show abstract

“…In Fig. 1 we compare the analytic result (28) with the outcome of numerical simulations. The intrinsic motion of the particles has been implemented by means of an uncoupled Continuous-Time-Random-Walk (CTRW) model with an exponentially decaying waiting time pdf ψ(t) ∝ exp (−t/ t ) and a Gaussian jump length pdf λ(∆y) ∝ exp (−|∆y| 2 /2Σ 2 ).…”

Section: Fokker-planck Equationmentioning

confidence: 99%

Reaction–Diffusion Kinetics in Growing Domains

Escudero

Yuste

Abad

et al. 2018

Handbook of Statistics

View full text Add to dashboard Cite

Reaction-diffusion models have been used over decades to study biological systems. In this context, evolution equations for probability distribution functions and the associated stochastic differential equations have nowadays become indispensable tools. In population dynamics, say, such approaches are utilized to study many systems, e.g., colonies of microorganisms or ecological systems. While the majority of studies focus on the case of a static domain, the time-dependent case is also important, as it allows one to deal with situations where the domain growth takes place over time scales that are relevant for the computation of reaction rates and of the associated reactant distributions. Such situations are indeed frequently encountered in the field of developmental biology, notably in connection with pattern formation, embryo growth or morphogen gradient formation. In this chapter, we review some recent advances in the study of pure diffusion processes in growing domains. These results are subsequently taken as a starting point to study the kinetics of a simple reaction-diffusion process, i.e., the encounter-controlled annihilation reaction. The outcome of the present work is expected to pave the way for the study of more complex reaction-diffusion systems of possible relevance in various fields of research.

show abstract

“…The above reactions have been comprehensively studied in static media, especially in the decades around the last turn of the century, and are still a subject of active research [5][6][7][8][9][10]. A plethora of different methods have been developed to investigate the behavior, notably, scaling arguments [2], the method of the interparticle distribution function (IPDF) [2,11], the even/odd interval method [12][13][14], renormalization group techniques [4,5,15], hierarchies of n-point distribution functions [16,17], as well as other alternative methods [9,[18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Encounter-controlled coalescence and annihilation on a one-dimensional growing domain

et al. 2018

View full text Add to dashboard Cite

The kinetics of encounter-controlled processes in growing domains is markedly different from that in a static domain. Here, we consider the specific example of diffusion limited coalescence and annihilation reactions in one-dimensional space. In the static case, such reactions are among the few systems amenable to exact solution, which can be obtained by means of a well-known method of intervals. In the case of a uniformly growing domain, we show that a double transformation in time and space allows one to extend this method to compute the main quantities characterizing the spatial and temporal behavior. We show that a sufficiently fast domain growth brings about drastic changes in the behavior. In this case, the reactions stop prematurely, as a result of which the survival probability of the reacting particles tends to a finite value at long times and their spatial distribution freezes before reaching the fully self-ordered state. We obtain exact results for the survival probability and for key properties characterizing the degree of self-ordering induced by the chemical reactions, i.e., the interparticle distribution function and the pair correlation function.These results are confirmed by numerical simulations.

show abstract

Exactly solvable reaction diffusion models on a Bethe Lattice through the empty-interval method

Cited by 4 publications

References 30 publications

Cross-over between diffusion-limited and reaction-limited regimes in the coagulation-diffusion process

Cross-over between diffusion-limited and reaction-limited regimes in the coagulation-diffusion process

Reaction–Diffusion Kinetics in Growing Domains

Encounter-controlled coalescence and annihilation on a one-dimensional growing domain

Contact Info

Product

Resources

About