Abstract
We study random velocity effects on a two-species reaction–diffusion system consisting of three reaction processes \(A+A\to(\varnothing,A)\), \(A+B\to A\). Using the field theory perturbative renormalization group, we analyze this system in the vicinity of its upper critical dimension \(d_{\mathrm c}=2\). A velocity ensemble is generated by means of stochastic Navier–Stokes equations. In particular, we investigate the effect of thermal fluctuations on the reaction kinetics. The overall analysis is performed in the one-loop approximation and possible macroscopic regimes are identified.
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References
P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics, Cambridge Univ. Press, Cambridge (2010).
G. Ódor, “Universality classes in nonequilibrium lattice systems,” Rev. Modern Phys., 76, 663–724 (2004); arXiv: cond-mat/0205644.
U. C. Täuber, Critical Dynamics: A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior, Cambridge Univ. Press, Cambridge (2014).
U. C. Täuber, M. Howard, and B. P. Vollmayr-Lee, “Applications of field-theoretic renormalization group methods to reaction-diffusion problems,” J. Phys. A: Math. Gen., 38, R79–R131 (2005).
A. A. Ovchinnikov, S. F. Timashev, A. A. Belyy, Kinetics of Diffusion Controlled Chemical Processes, Nova Sci. Publ., Commack, NY (1989).
R. Rajesh and O. Zaboronski, “Survival probability of a diffusing test particle in a system of coagulating and annihilating random walkers,” Phys. Rev. E, 70, 036111, 9 pp. (2004).
B. Vollmayr-Lee, J. Hanson, R. S. McIsaac, and J. D. Hellerick, “Anomalous dimension in a two-species reaction-diffusion system,” J. Phys. A: Math. Theor., 51, 034002, 16 pp. (2018).
J. D. Hellerick, R. C. Rhoades, and B. P. Vollmayr-Lee, “Numerical simulation of the trapping reaction with mobile and reacting traps,” Phys. Rev. E, 101, 042112, 10 pp. (2020).
D. Shapoval, V. Blavatska, and M. Dudka, “Survival in two-species reaction-diffusion system with Lévy flights: renormalization group treatment and numerical simulations,” J. Phys. A: Math. Theor., 55, 455002, 22 pp. (2022).
D. Forster, D. R. Nelson, and M. J. Stephen, “Large-distance and long-time properties of a randomly stirred fluid,” Phys. Rev. A, 16, 732–740 (1977).
L. Ts. Adzhemyan, N. V. Antonov, and A. N. Vasil’ev, The Field Theoretic Renormalization Group in Fully Developed Turbulence, Gordon and Breach, Amsterdam (1999).
A. N. Vasilev, The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics, Chapman and Hall/CRC, Boca Raton, USA (2004).
J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, Oxford Univ. Press, Oxford (2002).
M. Doi, “Second quantization representation for classical many-particle system,” J. Phys. A: Math. Gen., 9, 1465–1478 (1976).
L. Peliti, “Path integral approach to birth-death processes on a lattice,” J. Phys. France, 46, 1469–1484 (1985).
B. P. Lee, “Renormalization group calculation for the reaction \(kA to \varnothing\),” J. Phys. A: Math. Gen., 27, 2633–2652 (1994); arXiv: cond-mat/9311064.
L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 6: Fluid Mechanics, Elsevier, Amsterdam (2013).
U. Frisch, Turbulence. The legacy of 0 N. Kolmogorov, Cambridge Univ. Press, Cambridge (1995).
D. Forster, D. R. Nelson, and M. J. Stephen, “Long-time tails and the large-eddy behavior of a randomly stirred fluid,” Phys. Rev. Lett., 36, 867–870 (1976).
M. Hnatič, J. Honkonen, and T. Lučivjanský, “Symmetry breaking in stochastic dynamics and turbulence,” Symmetry, 11, 1193, 52 pp. (2019).
J.-M. Park and M. W. Deem, “Disorder-induced anomalous kinetics in the \(A+A\to\varnothing\) reaction,” Phys. Rev. E, 57, 3618–3621 (1998).
M. J. E. Richardson and J. Cardy, “The reaction process \(A+A\to\varnothing\) in Sinai disorder,” J. Phys. A: Math. Gen., 32, 4035–4045 (1999).
M. W. Deem and J.-M. Park, “Reactive turbulent flow in low-dimensional, disordered media,” Phys. Rev. E, 58, 3233–3228 (1998).
M. Hnatich and J. Honkonen, “Velocity-fluctuation-induced anomalous kinetics of the \(A+\vec{A}\to \varnothing\) reaction,” Phys. Rev. E, 61, 3904–3911 (2000).
J. Honkonen, “Anomalous transport processes in chemically active random environment,” Acta Phys. Slovaca, 52, 533–540 (2002); arXiv: cond-mat/0207151.
M. Gnatich, J. Honkonen, and T. Lučivjanský, “Study of anomalous kinetics of the annihilation reaction \(A+A\to\varnothing\),” Theoret. and Math. Phys., 169, 1481–1488 (2011).
Funding
The work was supported by VEGA grant No. 1/0535/21 of the Ministry of Education, Science, Research and Sport of the Slovak Republic.
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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2023, Vol. 217, pp. 19–29 https://doi.org/10.4213/tmf10464.
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Hnatič, M., Kecer, M. & Lučivjanský, T. Two-species reaction–diffusion system in the presence of random velocity fluctuations. Theor Math Phys 217, 1437–1445 (2023). https://doi.org/10.1134/S0040577923100021
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DOI: https://doi.org/10.1134/S0040577923100021