Relativistic analysis of stochastic kinematics

The relativistic analysis of stochastic kinematics is developed in order to determine the transformation of the effective diffusivity tensor in inertial frames. Poisson-Kac stochastic processes are initially considered. For one-dimensional spatial models, the effective diffusion coefficient measured in a frame $\mathrm{\Sigma }$ moving with velocity $w$ with respect to the rest frame of the stochastic process is inversely proportional to the third power of the Lorentz factor $\gamma \left(w\right)={(1-w^2/c^2)}^{-1/2}$. Subsequently, higher-dimensional processes are analyzed and it is shown that the diffusivity tensor in a moving frame becomes nonisotropic: The diffusivities parallel and orthogonal to the velocity of the moving frame scale differently with respect to $\gamma \left(w\right)$. The analysis of discrete space-time diffusion processes permits one to obtain a general transformation theory of the tensor diffusivity, confirmed by several different simulation experiments. Several implications of the theory are also addressed and discussed.

Figure 1

Graph of the evolution of a realization of a Poisson-Kac process ($b_0^{\prime }=c=1$ and $a_0^{\prime }=1$) in the rest frame ${\mathrm{\Sigma }}^{\prime }$ [line a, i.e., $x^{\prime }(t^{\prime }$)] and in the inertial system $\mathrm{\Sigma }$ moving with constant velocity $w/c=0.8$ with respect to ${\mathrm{\Sigma }}^{\prime }$ [line b, i.e., $x\left(t\right)]$.

Figure 2

Overall probability density function $p(x,t)$ vs $x$ of a Poisson-Kac process ($b_0^{\prime }=c=1$ and $a_0^{\prime }=1$) in the frame $\mathrm{\Sigma }$ at several time instants for (a) $w/c=0.8$ and (b) $w/c=0.95$. Lines a–d refer to $t=30,50,70,100$, respectively.

Figure 3

Plot of ${\sigma }^2\left(t\right)$ vs $t$ for a 1D Poisson-Kac process ($b_0^{\prime }=c=1$ and $a_0^{\prime }=1$) in the frame $\mathrm{\Sigma }$ for different values of the frame velocity $w$. The arrow indicates increasing values of $w/c=0,0.2,0.4,0.6,0.8$.

Figure 4

Plot of $D/D_0$ vs $w$ ($c=1$) for a 1D Poisson-Kac process ($b_0^{\prime }=c=1$ and $a_0^{\prime }=1$). Circles represent the results of stochastic simulations and the solid line is the curve $D/D_0={\gamma }^{-3}\left(w\right)={(1-w^2)}^{3/2}$.

Figure 5

(a) Plot of $D_{x,x}/D_0$ ($•$ and line a) and $D_{y,y}/D_0$ ($\square$ and line b) vs the relative frame velocity $w$ for a two-dimensional Kolesnik-Kac process ($N=5, b_0^{\prime }=c=1$, and $a_0^{\prime }=1$). Line a represents the function ${\gamma }^{-3}\left(w\right)$ [Eq. (75)] and line b the function ${\gamma }^{-1}\left(w\right)$ [Eq. (81)]. Symbols $•$ and $\square$ correspond to the results of stochastic simulations for $D_{x,x}$ and $D_{y,y}$, respectively. (b) Diagonal diffusivities for the three-dimensional Poisson-Kac process (93) with $b_0^{\prime }=c=1$ and $a_0=1$. Symbols represent the results of stochastic simulations: $•, D_{x_1,x_1}$; $\square , D_{x_2,x_2}$; and $\blacksquare , D_{x_3,x_3}$. Lines a and b are the same as in (a).

Figure 6

Behavior of the second-order central moments ${\sigma }_{h,k}^{\prime 2}\left(t^{\prime }\right)$ measured in ${\mathrm{\Sigma }}^{\prime }$ vs $t^{\prime }$ for the two-dimensional spatial model described in the text. Symbols are the results of numerical simulation of the stochastic dynamics and solid lines are the theoretical predictions based on the Einstein scaling ${\sigma }_{h,k}^{\prime 2}\left(t^{\prime }\right)=2D_{h,k}^{\prime }{t}^{\prime }$, where $D_{h,k}^{\prime }$ are given by Eqs. (117), (121), and (125): line a and $\square , {\sigma }_{1,1}^{\prime 2}$; line b and $\circ , {\sigma }_{1,2}^{\prime 2}$; and line c and $•, {\sigma }_{2,2}^{\prime 2}$.

Figure 7

Effective transport parameters of the spatial two-dimensional STD model described in the text measured in ${\mathrm{\Sigma }}^{\prime }$ as a function of the relative frame velocity $w$. (a) Effective velocity entries: line a and $\square , v_1^{\prime }$; line b and $\circ , v_2^{\prime }$. (b) Effective tensor diffusivities: line a and $\square , D_{1,1}^{\prime }$; line b and $\circ , D_{1,2}^{\prime }$; and line c and $•, D_{2,2}^{\prime }$.

Figure 8

Effective tensor diffusivities $D_{h,k}^{\prime }$ vs the relative frame velocity $w$ for the spatial three-dimensional model described in the text: (a) $D_{1,1}^{\prime }$ vs $w$ and (b) $D_{h,k}^{\prime }$ vs $w$. Line a and $\square$ denote $D_{1,2}^{\prime }$; line b and $\blacksquare , D_{1,3}^{\prime }$; line c and $\circ , D_{2,2}^{\prime }$; line d and $•, D_{2,3}^{\prime }$; and line e and $▵, D_{3,3}^{\prime }$.

Figure 9

Plot of $d_{h,k}\left(w\right)$ vs $w$ for the filtered Wong-Zakai mollification of the process (128) sampled at $T=20$: (a) $d_{1,1}\left(w\right)$, (b) $d_{2,2}\left(w\right)$, (c) $d_{3,3}\left(w\right)$, (d) $d_{1,2}\left(w\right)=d_{2,1}\left(w\right)$, (e) $d_{1,3}\left(w\right)=d_{3,1}\left(w\right)$, and (f) $d_{2,3}\left(w\right)=d_{3,2}\left(w\right)$. Lines refer to Eqs. (126) and (127) and symbols to numerical simulations: line a, $a=0$; line b, $a=0.2$; line c, $a=0.4$; and line d, $a=0.6$.

Figure 10

Diffusivities $D_{1,2}^{\prime }$ and $D_{2,2}^{\prime }$ measured in ${\mathrm{\Sigma }}^{\prime }$ vs the relative frame velocity $w$ for model I discussed in the text: line a and $\circ , D_{1,2}^{\prime }$; line b and $•, D_{2,2}^{\prime }$. Solid lines represent the theoretical predictions (126) and (127) and symbols the results of the numerical simulations of the stochastic STD dynamics (130).

Figure 11

Diffusivities $D_{h,k}^{\prime }$ vs the relative frame velocity $w$ measured in ${\mathrm{\Sigma }}^{\prime }$ for model II discussed in the text: line a, $D_{1,1}^{\prime }$; line b, $D_{1,2}^{\prime }$; and line c, $D_{2,2}^{\prime }$.