Infinite invariant density in a semi-Markov process with continuous state variables

We report on a fundamental role of a non-normalized formal steady state, i.e., an infinite invariant density, in a semi-Markov process where the state is determined by the interevent time of successive renewals. The state describes certain observables found in models of anomalous diffusion, e.g., the velocity in the generalized Lévy walk model and the energy of a particle in the trap model. In our model, the interevent-time distribution follows a fat-tailed distribution, which makes the state value more likely to be zero because long interevent times imply small state values. We find two scaling laws describing the density for the state value, which accumulates in the vicinity of zero in the long-time limit. These laws provide universal behaviors in the accumulation process and give the exact expression of the infinite invariant density. Moreover, we provide two distributional limit theorems for time-averaged observables in these nonstationary processes. We show that the infinite invariant density plays an important role in determining the distribution of time averages.

Figure 1

Time evolution of the propagator for different times ($\gamma =0.5$ and $\nu =0.2$). Symbols with lines are the results of numerical simulations. Dashed and dotted lines are the theories, i.e., Eqs. (37) and (34), respectively. Infinite density can be observed for $v>v_c\left(t\right)$ while the propagator follows a different scaling, i.e., Eq. (34), for $v<v_c\left(t\right)$. We used the PDF $\psi \left(\tau \right)=\gamma {\tau }^{-1-\gamma }$ for $\tau \ge 1$ and $\psi \left(\tau \right)=0$ for $\tau <1$ as the flight-duration PDF.

Figure 2

Rescaled propagators for (a) $\nu =0.2$, (b) $\nu =0.5$, and (c) $\nu =0.8$ ($\gamma =0.5$). Symbols with lines are the results of numerical simulations for different times $t$. Dashed lines represent the scaling functions, i.e., Eq. (40). The flight-duration PDF is the same as that in Fig. 1.

Figure 3

Trajectories of $v\left(t\right)$ and the integral of a function $f\left(v\right)=\left|v\right|$, i.e., $X\left(t\right)={\int }_0^t\left|v\left(t^{\prime }\right)\right|d{t}^{\prime }$. Because velocity $v\left(t\right)$ is a piecewise constant function, $X\left(t\right)$ is a piecewise linear function of $t$. Parameter sets $(\gamma ,\nu )$ are $(0.8,0.2)$ and $(0.5,0.8)$ for (a) and (b), respectively.

Figure 4

Ergodicity breaking parameter as a function of the measurement time for $\nu =0.4$, 0.5, and $\nu =0.8$ ($\gamma =0.3$). We note that $\nu =0.4$ and 0.5 satisfy $2\nu -1<\gamma$ while $\nu =0.8$ satisfies $2\nu -1>\gamma$. Symbols represent the results of numerical simulations. The long-dashed line represents Eq. (71), the two dotted lines represent Eq. (75), and the dotted line represent Eq. (78). The flight-duration PDF is the same as that in Fig. 1.

Figure 5

Phase diagram of the parameter space $(\gamma ,\nu )$ for (a) $f\left(v\right)=\left|v\right|$ and (b) $f\left(v\right)=v^2$. The solid line $\gamma =1$ describes the boundary of the infinite measure. The dotted line represents the boundary that the average of the observable $f\left(v\right)$ with respect to the infinite or probability measure diverges. For $\int f\left(v\right)p_{\mathrm{eq}}\left(v\right)dv<\infty$ and $\gamma >1$ (region III), the time average converges to a constant, implying the EB parameter goes to zero. For $\int f\left(v\right)I_{\infty }\left(v\right)dv<\infty$ and $\gamma <1$ (region II), the EB parameter becomes a nonzero constant given by Eq. (71), implying the time averages remain random variables. For $\int f\left(v\right)I_{\infty }\left(v\right)dv=\infty$ and $\gamma <1$ (region I), the EB parameter becomes a nonzero constant given by Eqs. (75) or (78), implying the time averages remain random variables and it depends on $\gamma$ as well as $\nu$, which is different from case $\int f\left(v\right)I_{\infty }\left(v\right)<\infty$ and $\gamma <1$.

Figure 6

Time evolution of the propagator for different times ($\gamma =0.5$ and $\nu =0.2$). Symbols with lines are the results of numerical simulations, which is the same as those in Fig. 1. Dashed and solid lines are the theories, i.e., Eqs. (37) and (A8), respectively. The flight-duration PDF is the same as that in Fig. 1.

Figure 7

Probability density function $P\left(x\right)$ of time averages for (a) $\gamma =0.5$ and $\nu =0.7$ and (b) $\gamma =0.7$ and $\nu =0.8$. The solid and the dotted histograms represent PDFs obtained by numerical simulations for $f\left(v\right)=\left|v\right|$ and $f\left(v\right)=v^2$, respectively. The solid line is the PDF of the Mittag-Leffler distribution. Note that the PDFs for $f\left(v\right)=v^2$ follow the Mittag-Leffler distribution with order $\gamma =0.5$ and 0.7 in cases (a) and (b), respectively. On the other hand, the PDFs for $f\left(v\right)=\left|v\right|$ depend on the exponent $\gamma$ as well as $\nu$, implying that the PDFs are different from the Mittag-Leffler distribution. A similar distribution was found in distributional limit theorems of time-averaged observables in infinite ergodic theory [31].