Field theory of branching and annihilating random walks

被引:178
作者
Cardy, JL
Tauber, UC
机构
[1] Univ Oxford, Dept Phys Theoret Phys, Oxford OX1 3NP, England
[2] Univ Oxford All Souls Coll, Oxford OX1 4AL, England
[3] Univ Oxford Linacre Coll, Oxford OX1 1SY, England
关键词
stochastic processes; reaction-diffusion systems; dynamic critical phenomena; directed percolation;
D O I
10.1023/A:1023233431588
中图分类号
O4 [物理学];
学科分类号
0702 ;
摘要
We develop a systematic analytic approach to the problem of branching and annihilating random walks, equivalent to the diffusion-limited reaction processes 2A --> circle divide and A --> (m + 1) A, where m greater than or equal to 1. Starting from the master equation, a field-theoretic representation of the problem is derived, and fluctuation effects are taken into account via diagrammatic and renormalization group methods. For d > 2, the mean-field rate equation, which predicts an active phase as soon as the branching process is switched on, applies qualitatively for both even and odd m, but the behavior in lower dimensions is shown to be quite different for these two cases. For even m, and d near 2, the active phase still appears immediately, but with nontrivial crossover exponents which we compute in an expansion in epsilon = 2 - d and with logarithmic corrections in d = 2. However, there exists a second critical dimension d(c)' approximate to 4/3 below which a nontrivial inactive phase emerges, with asymptotic behavior characteristic of the pure annihilation process. This is confirmed by an exact calculation in d = 1. The subsequent transition to the active phase, which represents a new nontrivial dynamic universality class, is then investigated within a truncated loop expansion, which appears to give st correct qualitative picture. The model with m = 2 is also generalized to N species of particles, which provides yet another universality class and which is exactly solvable in the limit N --> infinity. For odd m, we show that the fluctuations of the annihilation process are strong enough to create a nontrivial inactive phase for all d less than or equal to 2. In this case, the transition to the active phase is in the directed percolation universality class. Finally, we study the modification when the annihilation reaction is 3A --> circle divide. When m = 0 (mod 3) the system is always in its active phase, but with logarithmic crossover corrections for d = 1, while the other cases should exhibit a directed percolation transition out of a fluctuation-driven inactive phase.
引用
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页码:1 / 56
页数:56
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