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Long-range hops in a two-species reaction-diffusion system: renormalization group and numerical simulations\\
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\underline{Dmytro Shapoval}$^{*}$, Viktoria Blavatska, Maxym Dudka \\
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Institute of Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitskii Street, UA -- 79011 Lviv, Ukraine\\
${\mathbb L}^4$ Collaboration \& Doctoral College for the Statistical Physics of Complex Systems, Leipzig-Lorraine-Lviv-Coventry\\
$^*$ shapoval@icmp.lviv.ua
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We consider the reaction-diffusion system of two-species with an anomalous superdiffusion undergoing the annihilation and coagulation reactions $A + A \to (0, A)$ as well as the trapping reaction $A + B \to A$. The superdiffusion is modeled via long-ranged L\'evy flights represented by random walks with step-lengths obeying a L\'evy distribution $P(r) = r^{- d - \sigma}$ (a heavy-tailed probability distribution), with control parameter $0< \sigma \leq 2$. This system for the case of the ordinary diffusing Brownian particles is known to demonstrate scaling of
particle density and density correlation function of target particles $B$ with nontrivial universal exponents including anomalous dimension for $d \leq d_{c}$ (fluctuation-dominated kinetics), where $d_c = 2$ is the upper critical dimension [1, 2].
It is well-known, that replacing the diffusive propagation with long-ranged L\'evy flights modifies the dynamics of reactive systems [3]. Moreover, when the diffusion is anomalous as when the particles perform such L\'evy flights, the upper critical dimension depends on the L\'evy index $\sigma$ [3, 4]. We are interested in the question how superdiffusion modifies the scaling of observable quantities below the upper critical dimension $d \leq d_{c} = \sigma$. We have applied the renormalization group formalism [5] for the system under study and calculated the decay exponents of the particle density as well as the density correlation function in the case of the L\'evy flights. We have performed the numerical simulations of the studied system as well, obtained quantitative estimates for the decay exponents are in good agreement with our analytical results [6].
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[1] R. Rajesh, O. Zaboronski, {\it Phys. Rev. E} {\bf 70}, 036111 (2004)
[2] B. Vollmayr-Lee, J. Hanson, R.S. McIsaac, J.D. Hellerick, {\it J. Phys. A: Math. Theor}. {\bf 51},
034002 (2018); ibid. 53, 179501 (2020)
[3] H. Hinrichsen and M. Howard, {\it Eur. Phys. J. B} {\bf 7} 635 (1999)
[4] D. C. Vernon, {\it Phys. Rev. E} {\bf 68} 041103 (2003)
[5] U. C. T{\"a}uber, Critical Dynamics: {\it A Field Theory Approach to Equilibrium and Non-Equilibrium Scaling Behavior} (Cambridge University Press, Cambridge, 2014)
[6] D. Shapoval, V. Blavatska, M. Dudka, {\it in preparation} (2021)
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