Title: Random walks with homotopic spatial inhomogeneities Show Chinese title

Title: 带有同伦空间非均匀性的随机游走

Abstract: In this work we study a generalization of the standard random walk, an homotopic random walk (HRW), using a deformed translation unitary step that arises from a homotopy of the position-dependent masses associated to the Tsallis and Kaniadakis nonexensive statistics. The HRW implies an associated homotopic Fokker-Planck equation (HFPE) provided with a bi-parameterized inhomogeneous diffusion. The trajectories of the HRW exhibit convergence to a position, randomness as well as divergence, according to deformation and homotopic parameters. The HFPE obtained from associated master equation to the HRW presents the features: a) it results an special case of the van Kampen diffusion equation (5) of Ref. [N. G. van Kampen, \emph{Z. Phys. B Condensed Matter} \textbf{68}, 135 (1987)]; b) it exhibits a superdiffusion in function of deformation and homotopic parameters; c) Tsallis and Kaniadakis deformed FPE are recovered as special cases; d) a homotopic mixtured diffusion is observed; and e) it has a stationary entropic density, characterizing a inhomogeneous screening of the medium, obtained from a homotopic version of the H-Theorem. Show Chinese abstract

Abstract: 在本工作中，我们研究了标准随机游走的一个推广，即同伦随机游走（HRW），使用了一个从与Tsallis和Kaniadakis非广延统计相关的位置依赖质量的同伦中产生的变形平移幺正步骤。HRW 伴随着一个双参数化的非均匀扩散的同伦福克-普朗克方程（HFPE）。根据变形和同伦参数，HRW 的轨迹表现出收敛到一个位置、随机性以及发散性。从 HRW 相关的主方程得到的 HFPE 具有以下特点：a) 它是参考文献 [N. G. van Kampen,\emph{Z. Phys. B 凝聚态} \textbf{68} , 135 (1987)] 中 van Kampen 扩散方程（5）的一个特例；b) 它表现出关于变形和同伦参数的超扩散；c) Tsallis 和 Kaniadakis 变形的 FPE 作为特例被恢复；d) 观察到一种同伦混合扩散；e) 它具有一个稳态熵密度，表征了介质的非均匀屏蔽，这是从同伦版本的 H 定理中获得的。