Clustering and Dispersion Rates for Some Interacting Particle Systems on $\mathbb{Z}$

Bramson, Maury; Griffeath, David

doi:10.1214/aop/1176994771

Cited by 127 publications

(112 citation statements)

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“…For example, by exploiting a duality with the voter model, Bramson and Griffeath solved a particular version of the A + A → 0 model in one and two dimensions [32], with the result…”

Section: Relation Between Rg and Other Methodsmentioning

confidence: 99%

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Applications of field-theoretic renormalization group methods to reaction–diffusion problems

Täuber¹,

Howard²,

Vollmayr-Lee³

2005

J. Phys. A: Math. Gen.

274

506

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Abstract. We review the application of field-theoretic renormalization group (RG) methods to the study of fluctuations in reaction-diffusion problems. We first investigate the physical origin of universality in these systems, before comparing RG methods to other available analytic techniques, including exact solutions and Smoluchowski-type approximations. Starting from the microscopic reaction-diffusion master equation, we then pedagogically detail the mapping to a field theory for the single-species reaction kA → ℓA (ℓ < k). We employ this particularly simple but non-trivial system to introduce the field-theoretic RG tools, including the diagrammatic perturbation expansion, renormalization, and Callan-Symanzik RG flow equation. We demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative ǫ = d c − d expansions with respect to the upper critical dimension d c . With these basics established, we then provide an overview of more sophisticated applications to multiple species reactions, disorder effects, Lévy flights, persistence problems, and the influence of spatial boundaries. We also analyze field-theoretic approaches to nonequilibrium phase transitions separating active from absorbing states. We focus particularly on the generic directed percolation universality class, as well as on the most prominent exception to this class: evenoffspring branching and annihilating random walks. Finally, we summarize the state of the field and present our perspective on outstanding problems for the future.

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“…For example, by exploiting a duality with the voter model, Bramson and Griffeath solved a particular version of the A + A → 0 model in one and two dimensions [32], with the result…”

Section: Relation Between Rg and Other Methodsmentioning

confidence: 99%

“…Before specifying H, let us discuss the initial and final terms in the action (32) in more detail. The initial terms are of the form exp(φ…”

Section: Coherent State Representation and Path Integralsmentioning

confidence: 99%

Applications of field-theoretic renormalization group methods to reaction–diffusion problems

Täuber¹,

Howard²,

Vollmayr-Lee³

2005

J. Phys. A: Math. Gen.

274

506

View full text Add to dashboard Cite

show abstract

“…This is plausible since the exponent in one dimension [43] and to the perturbative result 1 2π + 2 ln(8π )−5 8π of [16] (solid line). The latter is an expansion in the deviation from the upper critical dimension,…”

mentioning

confidence: 86%

Long-range and many-body effects in coagulation processes

Winkler

Frey

2013

Phys. Rev. E

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We study the problem of diffusing particles which coalesce upon contact. With the aid of a nonperturbative renormalization group, we first analyze the dynamics emerging below the critical dimension two, where strong fluctuations imply anomalously slow decay. Above two dimensions, the long-time, low-density behavior is known to conform with the law of mass action. For this case, we establish an exact mapping between the physics at the microscopic scale (lattice structure, particle shape and size) and the macroscopic decay rate in the law of mass action. In addition, we identify a term violating this classical law. It originates in long-range and many-particle fluctuations and is a simple, universal function of the macroscopic decay rate.

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“…In this case, explicit expressions for the correlation functions (valid at all times) can be obtained by using the full solution of Eqs. (8), (9),…”

mentioning

confidence: 99%

Correlation functions for diffusion-limited annihilation,A+A→0

Masser

ben‐Avraham

2001

Phys. Rev. E

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The full hierarchy of multiple-point correlation functions for diffusion-limited annihilation, A + A → 0, is obtained analytically and explicitly, following the method of intervals. In the long time asymptotic limit, the correlation functions of annihilation are identical to those of coalescence, A + A → A, despite differences between the two models in other statistical measures, such as the interparticle distribution function.02.50. Ey, 05.50.+q, 05.70.Ln, The kinetics of nonequilibrium processes, in particular diffusion-limited reactions, have attracted much recent interest [1][2][3][4][5][6][7][8]. Because of the lack of a comprehensive approach for the study of such systems, models that yield to an exact analysis are of prime importance. In this respect, none have been studied more than diffusion-limited annihilation, A + A → 0 [9-28], and coalescence, A + A → A [9, [13][14][15]17,21,23,26,[28][29][30][31][32][33]. Known exact results include the time dependence of the particle concentration and the two-point correlation function (for finding two particles at two different points, simultaneously). It has also been shown that the full hierarchy of n-point correlation functions for the two processes is identical [21,23,34,35], but explicit expressions for n > 3 are unavailable.Here we attack the problem of correlation functions for annihilation, using the method of parity intervals (or even/odd intervals) [20,26,27,36]. We recover the identity relation of the n-point correlation functions for annihilation and coalescence, and we derive explicit expressions, valid in the long time asymptotic limit, for all n.Consider the annihilation model, defined on the line −∞ < x < ∞. Particles A are represented by points which perform unbiased diffusion with a diffusion constant D. When two particles meet they annihilate instantly. Since the reaction step is infinitely fast, the system models the diffusion-limited annihilation process A + A → 0.An exact treatment of the problem is possible through the method of parity intervals [20,26,27,36]. The key parameter is G(x, y; t)-the probability that the interval [x, y] contains an even number of particles at time t [37]. Particles near the edges of an interval may diffuse into or out of the interval, affecting the probability G. (On the other hand, reactions inside the interval cannot affect its parity.) With this observation in mind, one derives a rate equation for the probability G(x, y; t) [26]:

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Clustering and Dispersion Rates for Some Interacting Particle Systems on $\mathbb{Z}$

Cited by 127 publications

References 0 publications

Applications of field-theoretic renormalization group methods to reaction–diffusion problems

Applications of field-theoretic renormalization group methods to reaction–diffusion problems

Long-range and many-body effects in coagulation processes

Correlation functions for diffusion-limited annihilation,A+A→0

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