Multiscaling of correlation functions in single species reaction-diffusion systems

Area of Science:

• Statistical Physics
• Chemical Kinetics
• Reaction-Diffusion Systems

Background:

• Understanding particle distribution in reaction-diffusion systems is crucial for various scientific fields.
• Previous studies have explored scaling behaviors but lacked detailed multi-particle analysis.
• The behavior of systems like A + A --> O and 3A --> O requires advanced theoretical frameworks.

Purpose of the Study:

• To derive the multi-scaling of probability distributions for multi-particle configurations.
• To analyze binary (A + A --> O) and ternary (3A --> O) reaction-diffusion systems in different dimensions.
• To investigate the validity and corrections of epsilon expansions against exact solutions.

Main Methods:

• Utilized the dynamical renormalization group (RG) as the principal analytical tool.
• Derived probability distributions Pt(N, Delta V) for N particles in a volume Delta V at time t.
• Employed epsilon expansions (epsilon = 2-d) and compared results with exact solutions for d=1.

Main Results:

• For binary reactions (A + A --> O), derived decay laws for Pt(N, Delta V) in d<=2 dimensions.
• For d=2, decay is (ln t/t)^N (ln t)^(-N(N-1)/2). For d<2, decay is t^(-Nd/2) t^(-N(N-1)epsilon/4).
• For ternary reactions (3A --> O) in d=1, Pt(N, delta V) approximately (ln t/t)^(N/2) (ln t)^(-N(N-1)(N-2)/6).

Conclusions:

• The dynamical renormalization group provides accurate multi-scaling predictions for these systems.
• Epsilon corrections of order two and higher were found to be absent for binary reactions in d=1 for N=1, 2, 3, 4.
• Conjectured the absence of epsilon^2 corrections for all N in the binary reaction case.