Exact Results for a Three-Body Reaction-Diffusion System

Jul 2, 1992

Vladimir Privman

In the realm of theoretical physics, diffusion limited reactions in low dimensions have increasingly piqued the interest of scientists[1-11]. These fluctuation effects hold significant sway in lower dimensions, resulting in deviations from the predictable mean-field rate-equation behavior. From a mathematical point of view, only the simplest reactions offer exact solutions. For instance, the two-body annihilation, A+A→inert, has yielded the most detailed exact results using methods typically used for the Glauber-dynamics Ising models. Here, the particles A act as interfaces on a lattice dual to the Ising spin lattice[5,9,13]. Exact results have also been obtained for cellular-automata type, simultaneous-update, discrete-time dynamics[11].

Three-Body Reaction-Diffusion System

In this study, we will arrive at exact results for a specific three-body reaction scheme. This scheme involves considering Zk-symmetric, k-state “spins” on a linear lattice. Interestingly, a Potts-spin d= 1 lattice has been previously examined[14] as a model of mixed pairwise annihilation and coagulation of interfaces between fully ordered (T= 0) Potts phases. The approach here, however, is fundamentally different. We will utilize discrete-time and space cellular-automaton dynamics, with the case k= 3 displaying certain simplified characteristics that allow for an exact solution.

The average density instead of adhering to the predicted ρ(t)∼1/t form for large times, shows a shift to ρ(t)∼1/√t, as seen in two-body annihilation, reflecting a deviation from mean-field behaviour, increasingly evident with time.

Employing Spin Variables

The application of spin variables leads to intriguing results. The simplest, yet most effective, initial conditions for spin variables are fully random: spins σj(0) are uncorrelated at different sites j, indicating subtle correlations in the initial distribution of the reacting-diffusing particles on the dual lattice.

This study presents a compelling case for how a three-body reaction-diffusion system behaves under specific conditions and offers exact results for these phenomena. By examining how particle density changes in correlation with 2-spin relationality, we are able to provide an exact model that closely mirrors the fluctuation-modified density in two-body annihilation scenarios. Moreover, this exact approach opens up new avenues for study in both theoretical and applied physics, particularly when it comes to multi-body reaction-diffusion systems.

Conclusion

The exploration of three-body reaction-diffusion systems brings a new perspective to theoretical physics. As we continually uncover the complexity of these systems, comprehensive understanding eventually paves the way for significant scientific and technological advancements. In this context, the presented analysis provides a vital stepping stone, shedding light on an intriguing aspect of theoretical physics—offering a peek into the fascinating world of particle interactions.

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