# 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

## References

- Couder, Y.; Fort, E. Single-Particle Diffraction and Interference at a Macroscopic Scale. Phys. Rev. Lett.
**2006**, 97, 15410. [Google Scholar] [CrossRef] [PubMed] - Couder, Y.; Boudaoud, A.; Protière, S.; Fort, E. Walking droplets, a form of wave-particle duality at macroscopic scale? Europhys. News
**2010**, 41, 14–18. [Google Scholar] [CrossRef][Green Version] - Couder, Y.; Protière, S.; Fort, E.; Boudaoud, A. Dynamical phenomena: Walking and orbiting droplets. Nature
**2005**, 437, 208. [Google Scholar] [CrossRef] [PubMed] - Harris, D.M.; Moukhtar, J.; Fort, E.; Couder, Y.; Bush, J.W. Wavelike statistics from pilot-wave dynamics in a circular corral. Phys. Rev. E
**2013**, 88, 011001. [Google Scholar] [CrossRef] [PubMed] - Bush, J.W.M. Pilot-wave hydrodynamics. Annu. Rev. Fluid Mech.
**2015**, 47, 269–292. [Google Scholar] [CrossRef] - Couder, Y.; Fort, E. Probabilities and trajectories in a classical wave-particle duality. J. Phys. Conf. Ser.
**2012**, 361, 012001. [Google Scholar] [CrossRef][Green Version] - Bush, J.W.M. The new wave of pilot-wave theory. Phys. Today
**2015**, 68, 47–53. [Google Scholar] [CrossRef][Green Version] - Eddi, A.; Sultan, E.; Moukhtar, J.; Fort, E.; Rossi, M.; Couder, Y. Information stored in Faraday waves: The origin of a path memory. J. Fluid Mech.
**2011**, 674, 433–463. [Google Scholar] [CrossRef] - Perrard, S.; Labousse, M.; Miskin, M.; Fort, E.; Couder, Y. Self-organization into quantized eigenstates of a classical wave-driven particle. Nat. Commun.
**2014**, 5, 3219. [Google Scholar] [CrossRef] [PubMed][Green Version] - Durt, T. Do dice remember? Int. J. Theor. Phys.
**1999**, 38, 457–473. [Google Scholar] [CrossRef] - Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. I. Phys. Rev.
**1952**, 85, 166–179. [Google Scholar] [CrossRef] - Bohm, D. A Suggested Interpretation of the Quantum Theory in Terms of “Hidden” Variables. II. Phys. Rev.
**1952**, 85, 180–193. [Google Scholar] [CrossRef] - Nelson, E. Dynamical Theories of Brownian Motion; Princeton University Press: Princeton, NJ, USA, 1967; Volume 131, pp. 2381–2396. [Google Scholar]
- Labousse, M. Etude D’une Dynamique à Mémoire de Chemin: une Expérimentation Théorique. Ph.D. Thesis, Université Pierre et Marie Curie UPMC Paris VI, Paris, France, 2014. (In French). [Google Scholar]
- Fort, E.; Eddi, A.; Boudaoud, A.; Moukhtar, J.; Couder, Y. Path-memory induced quantization of classical orbits. Proc. Natl. Acad. Sci. USA
**2010**, 107, 17515–17520. [Google Scholar] [CrossRef][Green Version] - Dubertrand, R.; Hubert, M.; Schlagheck, P.; Vandewalle, N.; Bastin, T.; Martin, J. Scattering theory of walking droplets in the presence of obstacles. New J. Phys.
**2016**, 18, 113037. [Google Scholar] [CrossRef][Green Version] - Tadrist, L.; Shim, J.B.; Gilet, T.; Schlagheck, P. Faraday instability and subthreshold Faraday waves: surface waves emitted by walkers. arXiv, 2017; arXiv:1711.06791. [Google Scholar] [CrossRef]
- Bohm, D.; Vigier, J.P. Model of the causal interpretation of quantum theory in terms of a fluid with irregular fluctuations. Phys. Rev.
**1954**, 96, 208–216. [Google Scholar] [CrossRef] - Bohm, D.; Hiley, B. Non-locality and locality in the stochastic interpretation of quantum mechanics. Phys. Rep.
**1989**, 172, 93–122. [Google Scholar] [CrossRef] - Kyprianidis, P. The Principles of a Stochastic Formulation of Quantum Theory. Found. Phys.
**1992**, 22, 1449–1483. [Google Scholar] [CrossRef] - Valentini, A. Signal locality, uncertainty and the subquantum H-theorem. I. Phys. Lett. A
**1991**, 156, 5–11. [Google Scholar] [CrossRef] - Petroni, N.C. Asymptotic behaviour of densities for Nelson processes. In Quantum Communications and Measurement; Springer: New York, NY, USA, 1995; pp. 43–52. [Google Scholar]
- Petroni, N.C.; Guerra, F. Quantum Mechanical States as Attractors for Nelson Processes. Found. Phys.
**1995**, 25, 297–315. [Google Scholar] [CrossRef] - Guerra, F. Introduction to Nelson Stochastic Mechanics as a Model for Quantum Mechanics. In The Foundations of Quantum Mechanics—Historical Analysis and Open Questions; Garola, C., Rossi, A., Eds.; Springer: Dordrecht, The Netherlands, 1995; pp. 339–355. [Google Scholar]
- Efthymiopoulos, C.; Contopoulos, G.; Tzemos, A.C. Chaos in de Broglie–Bohm quantum mechanics and the dynamics of quantum relaxation. Ann. Fond. Louis Broglie
**2017**, 42, 133–160. [Google Scholar] - Valentini, A.; Westman, H. Dynamical origin of quantum probabilities. Proc. R. Soc. A
**2005**, 461, 253–272. [Google Scholar] [CrossRef][Green Version] - Colin, S.; Struyve, W. Quantum non-equilibrium and relaxation to quantum equilibrium for a class of de Broglie–Bohm-type theories. New J. Phys.
**2010**, 12, 043008. [Google Scholar] [CrossRef] - Towler, M.; Russell, N.J.; Valentini, A. Time scales for dynamical relaxation to the Born rule. Proc. R. Soc. A
**2011**, 468, 990–1013. [Google Scholar] [CrossRef][Green Version] - Colin, S. Relaxation to quantum equilibrium for Dirac fermions in the de Broglie–Bohm pilot-wave theory. Proc. R. Soc. A
**2012**, 468, 1116–1135. [Google Scholar] [CrossRef] - Contopoulos, G.; Delis, N.; Efthymiopoulos, C. Order in de Broglie–Bohm quantum mechanics. J. Phys. A Math. Theor.
**2012**, 45, 165301. [Google Scholar] [CrossRef][Green Version] - Abraham, E.; Colin, S.; Valentini, A. Long-time relaxation in the pilot-wave theory. J. Phys. A Math. Theor.
**2014**, 47, 395306. [Google Scholar] [CrossRef] - De la Peña, L.; Cetto, A.M. The Quantum Dice: An Introduction to Stochastic Electrodynamics; Springer: Dordrecht, The Netherlands, 2013. [Google Scholar]
- De la Peña, L.; Cetto, A.M.; Valdés Hernández, A. The Emerging Quantum; Springer: Basel, Switzerland, 2015. [Google Scholar]
- Deotto, E.; Ghirardi, G.C. Bohmian mechanics revisited. Found. Phys.
**1998**, 28, 1–30. [Google Scholar] [CrossRef] - De Broglie, L. Interpretation of quantum mechanics by the double solution theory. Ann. Fond. Louis Broglie
**1987**, 12, 399–421. [Google Scholar] - Bacciagaluppi, G. Nelsonian mechanics revisited. Found. Phys. Lett.
**1999**, 12, 1–16. [Google Scholar] [CrossRef][Green Version] - de Broglie, L. La mécanique ondulatoire et la structure atomique de la matière et du rayonnement. J. Phys. Radium
**1927**, 8, 225–241. (In French) [Google Scholar] [CrossRef] - Valentini, A. On the Pilot-Wave Theory of Classical, Quantum and Subquantum Physics. Ph.D Thesis, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy, 1992. [Google Scholar]
- Valentini, A. Signal locality, uncertainty and the subquantum H-theorem. II. Phys. Lett. A
**1991**, 158, 1–8. [Google Scholar] [CrossRef] - Efthymiopoulos, C.; Kalapotharakos, C.; Contopoulos, G. Origin of chaos near critical points of quantum flow. Phys. Rev. E
**2009**, 79, 036203. [Google Scholar] [CrossRef] [PubMed] - Valentini, A. Inflationary cosmology as a probe of primordial quantum mechanics. Phys. Rev. D
**2010**, 82, 063513. [Google Scholar] [CrossRef] - Underwood, N.G.; Valentini, A. Quantum field theory of relic nonequilibrium systems. Phys. Rev. D
**2015**, 92, 063531. [Google Scholar] [CrossRef] - Jüngel, A. Entropy Methods for Partial Differential Equations; Springer: New York, NY, USA, 2016. [Google Scholar]
- Bricmont, J. Bayes, Boltzmann and Bohm: Probabilities in Physics. In Chances in Physics; Bricmont, J., Ghirardi, G., Dürr, D., Petruccione, F., Galavotti, M.C., Zanghi, N., Eds.; Springer: Berlin/Heidelberg, Germany, 2001; pp. 3–21. [Google Scholar]
- Gardiner, C.W. Handbook of Stochastic Processes; Springer: New York, NY, USA, 1985. [Google Scholar]
- Risken, H. Fokker–Planck Equation. In The Fokker–Planck Equation; Risken, H., Frank, T., Eds.; Springer: Berlin/Heidelberg, Germany, 1996; pp. 63–95. [Google Scholar]
- Petroni, N.C.; De Martino, S.; De Siena, S. Exact solutions of Fokker–Planck equations associated to quantum wave functions. Phys. Lett. A
**1998**, 245, 1–10. [Google Scholar] [CrossRef] - Brics, M.; Kaupuzs, J.; Mahnke, R. How to solve Fokker–Planck equation treating mixed eigenvalue spectrum? Cond. Matter Phys.
**2013**, 16, 13002. [Google Scholar] [CrossRef] - Hatifi, M.; Willox, R.; Colin, S.; Durt, T. Bouncing oil droplets, de Broglie’s quantum thermostat and convergence to equilibrium. arXiv, 2018; arXiv:quant-ph/1807.00569v3. [Google Scholar]
- Gray, R.M. Probability, Random Processes, and Ergodic Properties; Springer: New York, NY, USA, 2009. [Google Scholar]
- Arnold, V.; Avez, A. Problèmes Ergodiques de la Méchanique Classique; Gauthier-Villars: Paris, France, 1967. (In French) [Google Scholar]
- Labousse, M.; Oza, A.U.; Perrard, S.; Bush, J.W. Pilot-wave dynamics in a harmonic potential: Quantization and stability of circular orbits. Phys. Rev. E
**2016**, 93, 033122. [Google Scholar] [CrossRef] [PubMed] - Grössing, G. Sub-quantum thermodynamics as a basis of emergent quantum mechanics. Entropy
**2010**, 12, 1975–2044. [Google Scholar] [CrossRef] - Hestenes, D. The Zitterbewegung Interpretation of Quantum Mechanics. Founds. Phys.
**1990**, 20, 1213–1232. [Google Scholar] [CrossRef] - Colin, S.; Wiseman, H.M. The zig-zag road to reality. J. Phys. A Math. Theor.
**2011**, 44, 345304. [Google Scholar] [CrossRef] - Gilet, T. Quantumlike statistics of deterministic wave-particle interactions in a circular cavity. Phys. Rev. E
**2016**, 93, 042202. [Google Scholar] [CrossRef] [PubMed] - Bechhoefer, J.; Ego, V.; Manneville, S.; Johnson, B. An experimental study of the onset of parametrically pumped surface waves in viscous fluids. J. Fluid Mech.
**1995**, 288, 325–350. [Google Scholar] [CrossRef] - Cohen-Tannoudji, C.; Diu, B.; Laloe, F. Quantum Mechanics; Wiley: New York, NY, USA, 1977. [Google Scholar]
- Hatifi, M.; Lopez-Fortin, C.; Durt, T. de Broglie’s double solution: Limitations of the self-gravity approach. Ann. Fond. Louis Broglie
**2018**, 43, 63–90. [Google Scholar] - Pucci, G.; Harris, D.M.; Faria, L.M.; Bush, J.W.M. Walking droplets interacting with single and double slits. J. Fluid Mech.
**2018**, 835, 1136–1156. [Google Scholar] [CrossRef] - Sáenz, P.J.; Cristea-Platon, T.; Bush, J. Statistical projection effects in a hydrodynamic pilot-wave system. Nature
**2017**, 14, 3. [Google Scholar] - Durt, T. L. de Broglie’s double solution and gravitation. Ann. Fond. Louis Broglie
**2017**, 42, 73–102. [Google Scholar] - Nelson, E. Review of stochastic mechanics. J. Phys. Conf. Ser.
**2012**, 361, 012011. [Google Scholar] [CrossRef][Green Version] - Bell, J. On the Einstein–Podolsky–Rosen paradox. Physics
**1964**, 1, 195–200. [Google Scholar] [CrossRef] - Durt, T. Characterisation of an entanglement-free evolution. arXiv, 2001; arXiv:quant-ph/0109112. [Google Scholar]
- Kostin, M. On the Schrödinger-Langevin Equation. J. Chem. Phys.
**1972**, 57, 3589–3591. [Google Scholar] [CrossRef] - Nassar, A. Fluid formulation of a generalised Schrodinger-Langevin equation. J. Phys. A Math. General
**1985**, 18, L509. [Google Scholar] [CrossRef] - Olavo, L.; Lapas, L.; Figueiredo, A. Foundations of quantum mechanics: The Langevin equations for QM. Ann. Phys.
**2012**, 327, 1391–1407. [Google Scholar] [CrossRef] - Nassar, A.B.; Miret-Artés, S. Bohmian Mechanics, Open Quantum Systems and Continuous Measurements; Springer: Basel, Switzerland, 2017. [Google Scholar]
- Bacciagaluppi, G. A Conceptual Introduction to Nelson’s Mechanics. In Endophysics, Time, Quantum and the Subjective; Buccheri, R., Saniga, M., Elitzur, A., Eds.; World Scientific: Singapore, 2005; pp. 367–388. [Google Scholar]
- Wallstrom, T. Inequivalence between the Schrödinger equation and the Madelung hydrodynamic equations. Phys. Rev. A
**1994**, 49, 1613–1617. [Google Scholar] [CrossRef] [PubMed] - Derakhshani, M. A Suggested Answer to Wallstrom’s Criticism (I): Zitterbewegung Stochastic Mechanics. arXiv, 2015; arXiv:quant-ph/1510.06391. [Google Scholar]
- Bacciagaluppi, G.; Valentini, A. Quantum Theory at the Crossroads: Reconsidering the 1927 Solvay Conference; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Colin, S.; Durt, T.; Willox, R. de Broglie’s double solution program: 90 years later. Ann. Fond. Louis Broglie
**2017**, 42, 19–71. [Google Scholar] - Borghesi, C. Equivalent quantum equations with effective gravity in a system inspired by bouncing droplets experiments. arXiv, 2017; arXiv:1706.05640. [Google Scholar]
- Press, W.H.; Teukolsky, S.A.; Vetterling, W.T.; Flannery, B.P. Numerical Recipes 3rd Edition: The Art of Scientific Computing, 3rd ed.; Cambridge University Press: New York, NY, USA, 2007. [Google Scholar]
- Higham, D.J. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev.
**2001**, 43, 525–546. [Google Scholar] [CrossRef]

**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