Departamento de Ciências Exatas e Naturais, Universidade Estadual do Sudoeste da Bahia, Rodovia BR 415, km 03, s/n, Itapetinga, BA 45700-000, Brazil.
Phys Rev E. 2023 Mar;107(3-1):034113. doi: 10.1103/PhysRevE.107.034113.
We study a generalization of the random walk (RW) based on a deformed translation of the unitary step, inherited by the q algebra, a mathematical structure underlying nonextensive statistics. The RW with deformed step implies an associated deformed random walk (DRW) provided with a deformed Pascal triangle along with an inhomogeneous diffusion. The paths of the RW in deformed space are divergent, while those corresponding to the DRW converge to a fixed point. Standard random walk is recovered for q→1 and a suppression of randomness is manifested for the DRW with -1<γ_{q}<1 and γ_{q}=1-q. The passage to the continuum of the master equation associated to the DRW led to a van Kampen inhomogeneous diffusion equation when the mobility and the temperature are proportional to 1+γ_{q}x, and provided with an exponential hyperdiffusion that exhibits a localization of the particle at x=-1/γ_{q} consistent with the fixed point of the DRW. Complementarily, a comparison with the Plastino-Plastino Fokker-Planck equation is discussed. The two-dimensional case is also studied, by obtaining a 2D deformed random walk and its associated deformed 2D Fokker-Planck equation, which give place to a convergence of the 2D paths for -1<γ_{q_{1}},γ_{q_{2}}<1 and a diffusion with inhomogeneities controlled by two deformation parameters γ_{q_{1}},γ_{q_{2}} in the directions x and y. In both the one-dimensional and the two-dimensional cases, the transformation γ_{q}→-γ_{q} implies a change of sign of the corresponding limits of the random walk paths, as a property of the deformation employed.
我们研究了一种基于 q 代数中继承的单位步长的变形平移的随机游走（RW）的推广，q 代数是一种非广延统计的数学结构。具有变形步长的 RW 意味着具有变形帕斯卡三角形和非均匀扩散的相关变形随机游走（DRW）。RW 在变形空间中的路径是发散的，而对应于 DRW 的路径则收敛到一个固定点。当 q→1 时，标准随机游走得以恢复，并且对于-1<γ_{q}<1 和 γ_{q}=1-q 的 DRW，表现出随机性的抑制。当与 DRW 相关的主方程的连续体过渡时，当迁移率和温度与 1+γ_{q}x 成比例时，会导致 van Kampen 非均匀扩散方程，并提供了一个指数超扩散，该扩散表现出粒子在 x=-1/γ_{q}处的局域化，这与 DRW 的固定点一致。此外，还讨论了与 Plastino-Plastino Fokker-Planck 方程的比较。还研究了二维情况，通过获得二维变形随机游走及其相关的二维变形 Fokker-Planck 方程，这导致了-1<γ_{q_{1}},γ_{q_{2}}<1 时的二维路径收敛，并且在 x 和 y 方向上由两个变形参数 γ_{q_{1}},γ_{q_{2}}控制的非均匀扩散。在一维和二维情况下，当 γ_{q}→-γ_{q}时，随机游走路径的相应极限的符号发生变化，这是所采用的变形的属性。
