Deformed random walk: Suppression of randomness and inhomogeneous diffusion

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 $q\to 1$ and a suppression of randomness is manifested for the DRW with $-1<{\gamma }_q<1$ and ${\gamma }_q=1-q$. 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 $1+{\gamma }_{q}x$, and provided with an exponential hyperdiffusion that exhibits a localization of the particle at $x=-1/{\gamma }_q$ 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 $-1<{\gamma }_{q_1},{\gamma }_{q_2}<1$ and a diffusion with inhomogeneities controlled by two deformation parameters ${\gamma }_{q_1},{\gamma }_{q_2}$ in the directions $x$ and $y$. In both the one-dimensional and the two-dimensional cases, the transformation ${\gamma }_q\to -{\gamma }_q$ implies a change of sign of the corresponding limits of the random walk paths, as a property of the deformation employed.

Figure 1

Path of the walker after $n=100$ steps starting at $x=0$ with $p=1-p=1/2$.

Figure 2

Path of the walker after $n=100$ steps starting at $x=0$ with $p=1-p=1/2$ for the deformed random walk with ${\gamma }_q=0.5$ (black dashed line) and ${\gamma }_q=-0.5$ (gray dashed line). We see that the convergence to $-{\gamma }_q^{-1}$ is very fast, as a consequence of the corrections up to the order ${\gamma }_q^{100}$ expressed by (16) for $n=100$. Solid lines indicate their corresponding exponential asymptotic behavior $X\left(t\right)=(e^{-|{\gamma }_q|t}-1)/{\gamma }_q$.

Figure 3

Path of the walker after $n=100$ steps starting at $x=0$ provided with deformed steps $\{{(-1)}_q,{(+1)}_q\}$ and with $p=1-p=1/2$. We see the divergence to $-\infty$ for ${\gamma }_q=0.5$ (black) and to $+\infty$ for ${\gamma }_q=-0.5$ (gray).

Figure 4

Profiles of the probability distribution $P(x,t)$ of the DFPE [Eq. (29)] for times $t/\tau =0.01,1,4,8$ with ${\gamma }_q=0.5$ (left panel) and ${\gamma }_q=-0.5$ (right panel) with the initial condition $P(x,t=0)=\delta \left(x\right)$. The gray dashed lines illustrate the asymptotic behavior of $P(x,t)\to \delta (x+1/{\gamma }_q)$, exhibiting asymmetry and a localization at $x=-1/{\gamma }_q$ due to the deformation.

Figure 5

Time evolution of the mean standard deviation for ${\gamma }_q{l}_0=0,0.2,0.4,0.8$ of the deformed Fokker-Planck eqaution (24). The black straight line indicates the normal diffusion case $\langle {\left(\mathrm{\Delta }x\right)}^2\left(t\right)\rangle =\mathrm{\Gamma }t$, corresponding to null deformation ${\gamma }_q{l}_0=0$. As the deformation parameter increases the exponential hyper-diffusion is more pronounced, which is consistent with the localization of the particle at $x=-1/{\gamma }_q$ and with the deformed random walk (Fig. 2).

Figure 6

Paths of the walker after $n=100$ steps starting at $(x,y)=(0,0)$ provided $p_1=p_2=p_3=1/4$. Top left shows the two-dimensional RW and top right shows the RW provided with deformed steps $\{({(+1)}_{q_1},{(+1)}_{q_2}),({(+1)}_{q_1},{(-1)}_{q_2}),({(-1)}_{q_1},{(+1)}_{q_2}),({(-1)}_{q_1},{(-1)}_{q_2})\}$ for ${\gamma }_{q_1}={\gamma }_{q_2}=0.5$ (black) and ${\gamma }_{q_1}={\gamma }_{q_2}=0.5$ (gray). Again, divergent paths are manifested. The plot below illustrates the two-dimensional DRW with a mixture of deformations ${\gamma }_{q_1}=0.5$ and ${\gamma }_{q_2}=-0.5$, whose convergence to the fixed point $(-2,2)$ is observed and the gray straight line indicates the exponential asymptotic behavior $(X\left(t\right),Y\left(t\right))=((e^{-|{\gamma }_{q_1}|t}-1)/{\gamma }_{q_1},(e^{-|{\gamma }_{q_2}|t}-1)/{\gamma }_{q_2})$.

Figure 7

Contours of the probability distribution $P(x,y,t)$ of the two-dimensional DFPE [Eq. (50)] for times $t/\tau =0.1,0.5,5,10$ with ${\gamma }_{q_1}=0.5$ and ${\gamma }_{q_2}=-0.5$ with the initial condition $P(x,y,t=0)=\delta \left(x\right)\delta \left(y\right)$. In white and black are indicated the regions with low and high probability density.