Uwe C Täuber et al 2005 J. Phys. A: Math. Gen. 38 R79 doi:10.1088/0305-4470/38/17/R01
Applications of field-theoretic renormalization group methods to reaction–diffusion problems
REVIEW ARTICLEUwe C Täuber1, Martin Howard2 and Benjamin P Vollmayr-Lee3
Show affiliationsTOPICAL REVIEW
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 → 
A( < 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 
= dc − d expansions with respect to the upper critical dimension dc. 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 analyse field-theoretic approaches to non-equilibrium 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: even-offspring branching and annihilating random walks. Finally, we summarize the state of the field and present our perspective on outstanding problems for the future.
- PACS
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05.40.Fb Random walks and Levy flights
64.60.A- Specific approaches applied to studies of phase transitions
64.60.Ht Dynamic critical phenomena
- MSC
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82C43 Time-dependent percolation (See also 60K35)
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) (See also 60H10)
82C28 Dynamic renormalization group methods (See also 81T17)
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. (See also 60G50)
82C27 Dynamic critical phenomena
82C26 Dynamic and nonequilibrium phase transitions (general)
- Subjects
- Dates
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Issue 17 (29 April 2005)
Received 31 January 2005
Published 13 April 2005
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Applications of field-theoretic renormalization group methods to reaction–diffusion problems
Uwe C Täuber et al 2005 J. Phys. A: Math. Gen. 38 R79
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