# Bouncing Oil Droplets, de Broglie’s Quantum Thermostat, and Convergence to Equilibrium

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## Abstract

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## 1. Introduction

## 2. dBB and Nelson Dynamics

#### 2.1. The dBB Theory

#### 2.2. A Simple Realization of de Broglie’s Quantum Thermostat—Nelson Dynamics

## 3. Relaxation to Quantum Equilibrium in the de Broglie–Bohm Theory

## 4. An H-Theorem for Nelson Dynamics

## 5. Relaxation to Quantum Equilibrium and Nelson Dynamics: The Static Case

#### 5.1. Fokker–Planck Operator and a Formal Connection to a Schrödinger Equation

#### 5.2. Superposition Ansatz

#### 5.3. One-Dimensional Oscillator and the Evolution of Gaussian Distributions for the Ground State

#### 5.4. Ergodicity in the Relaxation to Quantum Equilibrium for the Ground State of the Harmonic Oscillator

## 6. Nelson Dynamics: A Phenomenological Dynamical Model for Walkers?

#### 6.1. 2D Harmonic Oscillator

#### 6.2. Presence of Zeros in the Interference Pattern

## 7. Relaxation to Quantum Equilibrium with dBB and Nelson Dynamics: The Non-Static Case

#### 7.1. Nelson Dynamics and Asymptotic Coherent States

#### 7.2. Onset of Equilibrium with a Dynamical Attractor in dBB Dynamics and Nelson Dynamics

## 8. Dynamical Model for Droplets and Double Quantization of the 2-D Harmonic Oscillator

## 9. Conclusions and Open Questions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Numerical Simulations

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**Figure 2.**Simulations of 10,000 trajectories (calculated from the Ito equation, Equation (16), for the ground state (Equation (55)) of the 1D harmonic oscillator), whose initial positions are normally distributed, for 5 different choices of distribution width (for $a=0.5$ and $\alpha =1$). We observe, in each case, convergence to the equilibrium of Equation (55) as predicted by the theory.

**Figure 3.**Time evolution of ${H}_{V}$, ${L}_{f}$ and the ${L}_{1}$ norm, for a uniform initial probability distribution, calculated from the Ito equation, Equation (16), for the ground state of the 1D harmonic oscillator. Relaxation towards the distribution of the ground state $|{\Psi}_{st}{|}^{2}$ of Equation (55) is clearly visible. The simulation was performed for $\alpha =1$, $a=0.5$, and $\Delta t=0.01$, for 20,000 uniformly distributed initial conditions.

**Figure 4.**Histograms of the positions of a single particle, subject to Nelson dynamics for the ground state of the 1D harmonic oscillator. The full (red) curve corresponds to the quantum probability $|{\Psi}_{st}{|}^{2}$. Here, $a=0.5$, $\alpha =1$, and the total simulation time (t = 10,000) is sampled with $\Delta t=0.01$. ($\mathbf{a}$) The initial particle position is ${x}_{0}=2.5$, and the number of bins ${N}_{b}=100$ (each with spatial size $\Delta x=0.0635$); ($\mathbf{b}$) Same as (

**a**) but with ${N}_{b}=50$ and $\Delta x=0.1270$; ($\mathbf{c}$) Same as (

**a**) but with $t=200$; ($\mathbf{d}$) Same as (

**a**) but for ${x}_{0}=-0.85$.

**Figure 5.**(

**a**) A point-particle (the dot near the center) subject to the osmotic velocity field $-2a\alpha \phantom{\rule{0.166667em}{0ex}}\left(x\right(t),y(t\left)\right)$, due to the ground state of the 2D harmonic oscillator at time t; (

**b**) Color plot of the velocities along a trajectory for the evolution under Nelson dynamics, for the ground state of the 2D harmonic oscillator. The simulation (for $a=0.5$ and $\alpha =1$) started from the initial position $(-2,1)$ and was sampled up to $t=1000$ with step $\Delta t=0.01$.

**Figure 6.**Histogram of the positions in x of a single particle, in the case of the first Fock state given by Equation (68). The full curve (red) corresponds to the quantum probability $|{\Psi}_{1}{|}^{2}$. Here, $a=0.5$ and $\alpha =1$. The total simulation time t is $t=1000$, and the sampling time step is $\Delta t=0.01$. The initial position is ${x}_{i}=1$, and the number of bins ${N}_{b}=75$, each with width $\Delta x=0.08$.

**Figure 7.**Evolution in time of ${H}_{V}$ (27), ${L}_{f}$ (28), and of the ${L}_{1}$ (39) norm, for a uniform initial probability distribution, showing the relaxation towards the distribution of the first excited state $|{\Psi}_{1}{|}^{2}$ (68). The simulation was performed for $\alpha =1$, $a=0.5$, and $\Delta t=0.01$ and from 20,000 uniformly distributed initial conditions.

**Figure 8.**(

**a**) The quantum probability associated to the Bessel function of the first kind ${J}_{0}$; (

**b**) Color plot of the velocities reached along the trajectory for an evolution corresponding to (69). The initial position was $(1,1)$, the simulation time $t=5000$, and the sampling time step $\Delta t=0.005$. We chose $\alpha =0.1$ and $\u03f5=0.2$, and the size of the domain is $L=2$. On the boundary we impose a harmonic field force of the form $-2a\alpha \phantom{\rule{0.166667em}{0ex}}\mathit{r}$.

**Figure 10.**The time evolution of a non-equilibrium ensemble, illustrated with position histograms at six different times ((

**a**): $t=0$, (

**b**): $t=1.2$, (

**c**): $t=2.4$, (

**d**): $t=3.6$, (

**e**): $t=4.8$, (

**f**): $t=6$). The continuous curve is the squared modulus ${|\Psi |}^{2}$ for the coherent state of Equation (70). As can be seen in (

**d**–

**f**), once equilibrium is reached, the distribution clings to the coherent state and follows its oscillation faithfully. The center of the wave packet moves between $-2$ and 2 with a period $2\pi $. We started from a uniform distribution of initial conditions and chose $a=0.5$, $\alpha =1$, and ${x}_{0}=2$. The sampling time step is $\Delta t=0.01$, and the number of bins is ${N}_{b}=50$, each with width $\Delta x=0.0461$.

**Figure 11.**Time evolution of ${H}_{V}$ (27), ${L}_{f}$ (28), and ${L}_{1}$ (39), for a uniform initial probability distribution, showing the relaxation towards the distribution ${|\Psi |}^{2}$ of the coherent state of Equation (70). The simulation was performed for $\alpha =1$, $a=0.5$, and $\Delta t=0.01$ and from 20,000 uniformly distributed initial conditions.

**Figure 12.**Histogram of the positions for a unique trajectory satisfying the Ito equation, Equation (16), for Equation (70). The full curve corresponds to the integration of ${|\Psi |}^{2}$ over one period. The center of the wave packet moves between $-2$ and 2 with a period $2\pi $. Here, $a=0.5$ and $\alpha =1$. Total simulation time t is $t=30,000$ and the sampling time step is $\Delta t=0.01$. The initial position is ${x}_{i}=1$, and the number of bins ${N}_{b}=100$, each with width $\Delta x=0.1$.

**Figure 13.**Plots showing three possible de Broglie–Bohm (dBB) trajectories for a single point particle in the case of (73) with $M={2}^{2}=4$. Each plot (

**a**–

**c**) is associated with different initial random phases (${\theta}_{{n}_{x},{n}_{y}}$ with ${n}_{x}$ (${n}_{y}$) taking the values 0, 1) and different initial positions.

**Figure 14.**Plots of the evolution in time of the coarse-grained H-functions ${H}_{V}$ (

**a**) and ${L}_{1}$ (

**b**) for the Nelson and dBB dynamics. The full line corresponds to the dBB dynamics and the dashed line corresponds to the quantum thermostat. We started from 10,000 initial positions uniformly distributed in a box of size 10 × 10; we chose $a=0.5$, $\alpha =0.1$, and $M={4}^{2}=16$ energy states.

**Figure 15.**dBB trajectories obtained for a single point particle in a superposition of eigenstates (75). Each plot is associated with a different combination $(n,m)$, as indicated. In the (

**a,b**) graphs, we imposed $a=1$ and, respectively, $\omega =1,\frac{{\xi}_{0}}{{\xi}_{2}}=0.05$ and $\omega =0.5,\frac{{\xi}_{0}}{{\xi}_{3}}=0.05$; for (

**c**), we imposed $a=3,\omega =0.5$, $\frac{{\xi}_{0}}{{\xi}_{3}}=0.0708,\frac{{\xi}_{0}}{{\xi}_{2}}=0.0456$, and $\frac{{\xi}_{0}}{{\xi}_{1}}=0.0773$.

**Figure 16.**Plots of three quantities associated to the lemniscate in Figure 15c. ($\mathbf{a}$) shows the ${L}_{z}$-component of the angular momentum, and the polar plots ($\mathbf{b}$,$\mathbf{c}$) show the probability density ${\left|\psi \right|}^{2}$ (

**b**) and the $\theta $-component of the probability current (5) along the trajectory (

**c**).

**Figure 17.**dBB trajectories obtained for a single point particle in a superposition of eigenstates (75). Plots (

**a**,

**b**) correspond to $\omega =0,7$ and $\omega =1$, respectively. Case (

**c**) is obtained after multiplying the amplitude of the $(n,m)=(4,2)$ state by a complex phase (${e}^{\left(0.3i\right)}$). We took $a=1$ in all cases.

**Figure 18.**dBB trajectories obtained for a single point particle in a superposition of eigenstates (75) (with $m=-2,0,2$) showing intermittent transitions between two types of trajectories. The relevant parameter values are $\omega =0.2$, $a=1$, $\frac{{\xi}_{0}}{{\xi}_{3}}=0.0342,\frac{{\xi}_{0}}{{\xi}_{2}}=0.2547$, and $\frac{{\xi}_{0}}{{\xi}_{1}}=0.0505$.

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**MDPI and ACS Style**

Hatifi, M.; Willox, R.; Colin, S.; Durt, T.
Bouncing Oil Droplets, de Broglie’s Quantum Thermostat, and Convergence to Equilibrium. *Entropy* **2018**, *20*, 780.
https://doi.org/10.3390/e20100780

**AMA Style**

Hatifi M, Willox R, Colin S, Durt T.
Bouncing Oil Droplets, de Broglie’s Quantum Thermostat, and Convergence to Equilibrium. *Entropy*. 2018; 20(10):780.
https://doi.org/10.3390/e20100780

**Chicago/Turabian Style**

Hatifi, Mohamed, Ralph Willox, Samuel Colin, and Thomas Durt.
2018. "Bouncing Oil Droplets, de Broglie’s Quantum Thermostat, and Convergence to Equilibrium" *Entropy* 20, no. 10: 780.
https://doi.org/10.3390/e20100780