Lognormal-like statistics of a stochastic squeeze process

We analyze the full statistics of a stochastic squeeze process. The model's two parameters are the bare stretching rate $w$ and the angular diffusion coefficient $D$. We carry out an exact analysis to determine the drift and the diffusion coefficient of $log\left(r\right)$, where $r$ is the radial coordinate. The results go beyond the heuristic lognormal description that is implied by the central limit theorem. Contrary to the common “quantum Zeno” approximation, the radial diffusion is not simply $D_r=(1/8)w^2/D$ but has a nonmonotonic dependence on $w/D$. Furthermore, the calculation of the radial moments is dominated by the far non-Gaussian tails of the $log\left(r\right)$ distribution.

Figure 1

Scaled stretching rate $w_r/w$ versus $w/D$. Numerical results (black symbols) are based on simulations with 2000 realizations. Lines represent the naive result, Eq. (8) (dotted green line); the exact result, Eq. (23) (solid red line); and its practical approximation, Eq. (24) (dashed-dotted blue line). For large values of $w/D$ we get $w_r/w=1$, as for a pure stretch.

Figure 2

Scaled diffusion coefficient $D_r/w$ versus $w/D$. Numerical results (black symbols) are based on simulations with 2000 realizations. Lines represent the naive result, Eq. (9) (dotted green line); the exact result, Eq. (33) (solid red line); and the approximation, Eq. (28), with $\tau =1/\left(2D\right)$ (dashed-dotted blue line) and with Eq. (34) (dashed orange line).

Figure 3

(a) Phase distribution for $(w/D)=10/3$ after time $\left(wt\right)=6$, with initial conditions $\phi =0$ (yellow region) and $\phi =\pi /2$ (green bars) and 2000 realizations. For longer times, both reach the steady state of Eq. (16) (red line). (b) Distributions of $\phi$ modulo $\pi$.

Figure 4

(a) Values of $X_n$ versus $n$ for some values of $w/D$. From bottom to top: $w/D=1$, 2, 3, 4, 5. (b)Values of ${\mathrm{\Delta }}_n$ versus $n$ for the same values of $w/D$, the larger $w/D$, the smaller ${\mathrm{\Delta }}_1$. (c) ${\mathrm{\Delta }}_n$ versus $n$ for large $w/D$. Here $w/D=400$. The asymptotic approximation, Eq. (22), is indicated by the blue line.

Figure 5

Scaled moments versus $w/D$. Solid red lines are the exact results for the second and fourth moments, given by Eq. (48) and Eq. (60), and the large $w/D$ asymptotic values are at 2 and 4, respectively. These are compared with the numerical results (black symbols) and contrasted with the lognormal prediction (dashed orange lines). The latter provides an overestimate for intermediate values of $w/D$.

Figure 6

(a) Distribution of $ln\left(r\right)$ for $w/D=10/3$ after time $wt=2000$ with initial conditions $r=1$ and $\phi =0$. Numerical results (green histogram), which are based on 2000 realizations, are fitted to a Gaussian distribution (blue line). (b) Inverse cumulative probability of the same distribution. The dotted black line indicates the numerically determined value $ln{〈r^2〉}^{1/2}\approx 1323$. This value is predominated by the tail of the distribution. The Gaussian fit fails to reproduce this value and provides a gross overestimate, $ln{〈r^2〉}^{1/2}\approx 1701$.

Figure 7

We consider 2000 realizations of a stochastic squeeze process. For each realization trace $a=\text{trace}\left(\mathit{U}\right)$ is calculated. (a) Cumulative count of the $a$ values. Green circles represent positive values; blue rectangle, negative values. Here $(w/D)=10/3$ and $wt=40$. For simulations with longer times the distributions of positive and negative values become identical (not shown). (b) Scatterplots of $\left|a\right|$ versus the radial coordinate $r$. For simulations with longer times we get a full correlation.

Figure 8

$ln〈r^2〉$ versus $\sigma$ for a lognormal distribution. Without loss of generality $\mu =0$. The true result is represented by the red line. Numerical estimates based on ${10}^2$ and ${10}^5$ realizations are indicated by green crosses and blue rectangles, respectively. For the latter set of realizations we get a much better estimate using an optional procedure (black circles). Namely, we calculate the sample average and the sample variance of the $lnr$ values in order to determine $\mu$ and $\sigma$ and then use Eq. (46) to estimate the moments.