Lévy Walk with Multiple Internal States

Abstract

Lévy walk with multiple internal states can effectively model the motion of particles that don’t immediately move back to the directions or areas which they come from. When the Lévy walk behaves superdiffusion, it is discovered that the non-immediately-repeating property, characterized by the constructed transition matrix, has no influence on the particle’s mean square displacement (MSD) or Pearson coefficient. This is a kind of stable property of Lévy walk. However, if the Lévy walk shows the dynamical behaviors of normal diffusion, then the effect of non-immediately-repeating emerges. For the Lévy walk with some particular transition matrices, it may display nonsymmetric dynamics; in these cases, the behaviors of their variances are detailedly discussed, especially some comparisons with the ones of the continuous time random walks are made (a striking difference is the changes of the exponents of the variances). The first passage time distribution and its average of Lévy walks are simulated, the results of which turn out that the first passage time can distinguish Lévy walks with different transition matrices, while the MSD can not.

A Calculations of \(\big <x(t)\big>\), \(\big <x^2(t)\big>\) and \(Var\big (x(t)\big )\) of Subdiffusion with Multiple Internal States and \(M_6\) Transition Matrix

From [18], we have

$$\begin{aligned} \big |P(\mathbf {k},s)\big>=\frac{{I}-{\varPhi }(s)}{s}\big [I-M^T H(\mathbf {k},s)\big ]\big |\mathrm{init}\big >, \end{aligned}$$

(23)

where $H(\mathbf {x},t)=\varLambda (\mathbf {x})\varPhi (t)$ and $\varLambda (\mathbf {x})$, $\varPhi (t)$ are matrices of jump length and waiting time, respectively. In this section, we still consider the same internal states shown in the third section. Thus if we let the jump length distribution matrix as

$$\begin{aligned} \varLambda (x,y)=\mathrm{diag}\big (\gamma ^{+}(x)\gamma ^{+}(y), \gamma ^{+}(x)\gamma ^{-}(y), \gamma ^{-}(x)\gamma ^{+}(y), \gamma ^{-}(x)\gamma ^{-}(y)\big ), \end{aligned}$$

where $\gamma ^{+}(l)={\left\{ \begin{array}{ll} \sqrt{\frac{2}{\pi \sigma ^2}}\exp \left( -\frac{l^2}{2\sigma ^2}\right) &{} {l\geqslant 0}\\ 0 &{} {l<0} \end{array}\right. },$ $\gamma ^{-}(l)={\left\{ \begin{array}{ll} 0 &{} {l\geqslant 0}\\ \sqrt{\frac{2}{\pi \sigma ^2}}\exp \left( -\frac{l^2}{2\sigma ^2}\right) &{} {l<0} \end{array}\right. }.$ And the Fourier transform of $\varLambda (x,y)$ w.r.t. x and y is

$$\begin{aligned} \begin{aligned}&{\varLambda }(k_x,k_y)\\&=\mathrm{diag}\left( \exp \left( -\frac{k_x^2+k_y^2}{2}\right) \left( 1+\sqrt{\frac{2}{\pi }}i k_x\right) \left( 1+\sqrt{\frac{2}{\pi }}i k_y\right) ,\right. \\&~~~~~~~~~~~~~~~\exp \left( -\frac{k_x^2+k_y^2}{2}\right) \left( 1+\sqrt{\frac{2}{\pi }}i k_x\right) \left( 1-\sqrt{\frac{2}{\pi }}i k_y\right) ,\\&~~~~~~~~~~~~~~~\exp \left( -\frac{k_x^2+k_y^2}{2}\right) \left( 1-\sqrt{\frac{2}{\pi }}i k_x\right) \left( 1+\sqrt{\frac{2}{\pi }}i k_y\right) ,\\&~~~~~~~~~~~~~~~\left. \exp \left( -\frac{k_x^2+k_y^2}{2}\right) \left( 1-\sqrt{\frac{2}{\pi }}i k_x\right) \left( 1-\sqrt{\frac{2}{\pi }}i k_y\right) \right) . \end{aligned} \end{aligned}$$

(24)

And the matrix of distribution of waiting time asymptotically behaves as $\varPhi (s)\sim (1-s^\gamma )I$ in the Laplace space, where $0<\gamma <1$. By utilizing Eqs. (23) and (24) and $\varPhi (s)$, one can get the corresponding $\big <x(t)\big>$ and $\big <x^2(t)\big>$ with the same method in Sect. 3. That is

$$\begin{aligned} \big <x(t)\big >\sim \frac{2}{19}\sqrt{\frac{2}{\pi }}\frac{1}{\varGamma (1+\gamma )}t^\gamma , \end{aligned}$$

and

$$\begin{aligned} \big <x^2(t)\big >\sim \frac{16}{361\pi }\frac{1}{\varGamma (1+2\gamma )}t^{2\gamma }. \end{aligned}$$

Thus we obtain the variance

$$\begin{aligned} \mathrm{Var}\big (x(t)\big )\sim \frac{8}{361\pi }\left[ \frac{2}{\varGamma (1+2\gamma )}-\frac{1}{\varGamma (1+\gamma )^2}\right] t^{2\gamma }. \end{aligned}$$