# Unstable Points, Ergodicity and Born’s Rule in 2d Bohmian Systems

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## Abstract

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## 1. Introduction

## 2. One Nodal Point

## 3. Two Nodal Points

## 4. Multiple Nodal Points

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Detection of Order and Chaos

## References

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**Figure 1.**(

**a**) A nodal point-X-point complex and the deviations of the trajectories approaching the X-point. (

**b**) The total potential close to a nodal point and its corresponding X-point (red dot) and Y-point (black dot). Both figures are drawn in the system $(u,v)$ of the moving nodal point.

**Figure 2.**(

**a**) A snapshot of the time-dependent Bohmian flow and the invariant curves of the Y-point at $t=1$ along with various Bohmian trajectories integrated from $t=0$ (green dots) up to $t=3$ (red dots) in the case of Equation (6) ($a=b=1,c=\sqrt{2}/2$). The black dots correspond to $t=1$, i.e., to the flow snapshot. We note that the flow changes in $t\in [0,3]$, but we still understand the form of the trajectories by reading the coordinates of the nodal point (${x}_{N}$ red) and the Y-point (${x}_{Y}$ green and ${y}_{N}={y}_{Y}$ blue) for $t\in [0,3]$, as shown in (

**b**). There, we see that ${y}_{N}={y}_{Y}$ passes from $-\infty $ to ∞ (at $t\simeq 1.84$), ${x}_{N}$ changes its sign from negative to positive. (

**c**) The Bohmian flow and the invariant curves of the Y-point at $t=2.5$, where ${x}_{N}>0$. The stable/unstable invariant curves are calculated in positive/negative time s.

**Figure 3.**The distance between chaotic Bohmian trajectory and the nodal point (blue curve of the upper part) and the Y-point (red curve of the upper part) and the corresponding stretching number a for $t\in [300,400]$. We observe that most of the significant scattering events correspond to the close approaches to the nodal points (and their associated X-points).

**Figure 4.**The distribution of the points (at every $\Delta t=0.05$) of 3000 trajectories when the initial distribution satisfies BR, up to $t=3000$.

**Figure 5.**The colorplots of two locally ergodic–chaotic trajectories separately up to $t=2\times {10}^{6}$ (

**a**) on the left and (

**b**) on the right of y-axis in the single node case.

**Figure 6.**5000 initial conditions, in the case of a single nodal point, distributed according to BR at $t=0$ (

**a**) on the $x-y$ plane and (

**b**) projected on ${P}_{0}={\left|{\Psi}_{0}\right|}^{2}$. Blue/red initial conditions produce chaotic/ordered Bohmian trajectories.

**Figure 7.**Colorplot of 5000 trajectories (in the case of a single nodal point) with ${P}_{0}\ne {\left|{\Psi}_{0}\right|}^{2}$, at $t=3000$. It is very different from that of the BR distribution in Figure 4.

**Figure 8.**The stable (blue) and unstable (red) asymptotic curves of the Y-point in the case of two nodal points for (

**a**) t = 1.5 and (

**b**) t = 1.8 in the case of Equation (19) ($a=1.23,b=1,c=\sqrt{2}/2$). We observe the change in the behavior of the nodal points from attractors to repellers.

**Figure 9.**5000 initial conditions in the case of 2 nodal points distributed according to BR at $t=0$ (

**a**) on the $x-y$ plane and (

**b**) projected on ${P}_{0}={\left|{\Psi}_{0}\right|}^{2}$. Blue/red initial conditions produce chaotic/ordered Bohmian trajectories.

**Figure 10.**The colorplot of the points of 5000 trajectories (in the case of two nodal points) initially satisfying BR, up to $t=3000$.

**Figure 11.**The colorplots of (

**a**) the ordered and (

**b**) of the chaotic trajectories in an initial BR distribution (in the case of two nodal points).

**Figure 12.**The colorplots of 5000 trajectories in two initial distributions with ${P}_{0}\ne {\left|{\Psi}_{0}\right|}^{2}$, up to $t=3000$. The shape of (

**a**) is different from that of (

**b**) and they both are very different from that of the BR distribution (Figure 10).

**Figure 13.**The Bohmian flow along with the stationary nodal points (red dots), the moving nodal points (black dots) and the Y-points (green dots) at $t=0.1$.

**Figure 14.**Details of the collision between the moving nodal point 19 and the fixed nodal point 19 at $y=0$. (

**a**) Before the collision ($t=1.8$), the nodal point 19 and a nearby green point ${Y}_{19}$ move toward the fixed point 16 together with the green point ${Y}_{19}$ on the left of 19 (see the arrows). (

**b**) At $t={t}_{col}=1.8403$ we observe the collision. (

**c**) After the collision ($t=1.87$), the moving nodal point 19 and ${Y}_{19}$ are above point 16 but ${Y}_{19}$ is now on the right of the nodal point 19 and the green point ${Y}_{16}$ has moved to the left of 16.

**Figure 15.**The asymptotic curves of some Y-points of the upper left corner of Figure 13, stable (blue) and unstable (red).

**Figure 16.**Colorplots of two different chaotic trajectories in the case of multiple nodal points up to $t=5\times {10}^{6}$: (

**a**) $x\left(0\right)=2.8,y\left(0\right)=0.8$ and (

**b**) $x\left(0\right)=-3,y\left(0\right)=-1$. They are very similar, i.e., they are approximately ergodic.

**Figure 17.**10,000 initial conditions in the case of multiple nodal points distributed according to Born’s distribution at $t=0$, chaotic (blue) and ordered (red).

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**MDPI and ACS Style**

Tzemos, A.C.; Contopoulos, G.
Unstable Points, Ergodicity and Born’s Rule in 2d Bohmian Systems. *Entropy* **2023**, *25*, 1089.
https://doi.org/10.3390/e25071089

**AMA Style**

Tzemos AC, Contopoulos G.
Unstable Points, Ergodicity and Born’s Rule in 2d Bohmian Systems. *Entropy*. 2023; 25(7):1089.
https://doi.org/10.3390/e25071089

**Chicago/Turabian Style**

Tzemos, Athanasios C., and George Contopoulos.
2023. "Unstable Points, Ergodicity and Born’s Rule in 2d Bohmian Systems" *Entropy* 25, no. 7: 1089.
https://doi.org/10.3390/e25071089