# Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation

^{†}

## Abstract

**:**

## 1. Introduction

#### 1.1. Stochastic Mechanics (SM)

#### 1.2. Scale Relativity Theory (SR)

## 2. Stochastic Representation of Trajectories

- divergence of the velocity, that is divergence of the difference quotient,
- oscillatory singularity or
- difference between forward and backward velocities.

**equivalence class of stochastic paths**having the same expectation as the given deterministic signal. Mathematical notation and preliminaries for the subsequent treatment are presented in Appendix A. A possibly non-differentiable continuous trajectory is represented by a continuous Markov stochastic process evaluated in the virtual space of paths as follows:

**Definition**

**1**(Markov Stochastic Embedding)

**.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**1**(Gaussian stochastic embedding)

**.**

- 1.
- $\Omega \subseteq {\chi}_{\beta}$,
- 2.
- ${X}_{t}$ has i. i. d. Gaussian increments,
- 3.
- $\mathbb{E}{X}_{t}=x\left(t\right)$ and
- 4.
- ${\upsilon}_{+}^{\beta}\mathbb{E}\left(\right)open="("\; close=")">|{X}_{t}\left|\right|{X}_{t}=x$ hold almost sure.

**consistent stochastic embedding**. This theorem allows for Nelson’s characterization of the Langevin diffusion process.

## 3. Nelson’s Characterization

## 4. The Complex Velocity Operator in SR and SM Theories

**absolutely continuous**part of the trajectory, while the fractal part corresponds to the

**singular**and possibly

**oscillatory**parts. Since, at the level of physical description, there is no way to favor the forward rather than the backward velocity, the description should incorporate them on equal grounds, i.e., forming a bivariate vector field $\mathbb{R}\otimes \mathbb{R}$

**Remark**

**1.**

## 5. The Real Stochastic Geodesic Equations

#### 5.1. Path-Wise Separable Solutions

#### 5.2. Stationary Drift Fields

#### 5.3. Stationary Density Solutions

#### 5.4. Transient Drift Fields

#### 5.5. Asymptotic Density Solutions

## 6. The Complex Stochastic Geodesic Equations

## 7. Real-Valued and Complex Cole–Hopf Transformations

## 8. Numerical Results

#### 8.1. Exact Simulations

#### 8.2. Free Diffusion

#### 8.3. Particle in a Box

## 9. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Notations, General Definitions and Properties of Fractional Velocity

**Definition**

**A1**

**(Asymptotic**

**O**

**notation)**

**Definition**

**A2.**

**Definition**

**A3.**

**Definition**

**A4.**

**Definition**

**A5**

**(Fractional**

**order**

**velocity)**

**Definition**

**A6.**

**set of change**${\chi}_{\pm}^{\beta}\left(f\right):=\left(\right)open="\{"\; close="\}">x:{\upsilon}_{\pm}^{\beta}f\left(x\right)\ne 0$.

**Definition**

**A7.**

- Product rule$$\begin{array}{cc}\hfill {\upsilon}_{+}^{\beta}\left[f\phantom{\rule{0.166667em}{0ex}}g\right]\left(x\right)& ={\upsilon}_{+}^{\beta}f\left(x\right)g\left(x\right)+{\upsilon}_{+}^{\beta}g\left(x\right)f\left(x\right)+{[f,g]}_{\beta}^{+}\left(x\right),\hfill \\ \hfill {\upsilon}_{-}^{\beta}\left[f\phantom{\rule{0.166667em}{0ex}}g\right]\left(x\right)& ={\upsilon}_{-}^{\beta}f\left(x\right)g\left(x\right)+{\upsilon}_{-}^{\beta}g\left(x\right)f\left(x\right)-{[f,g]}_{\beta}^{-}\left(x\right),\hfill \end{array}$$
- Quotient rule$$\begin{array}{cc}\hfill {\upsilon}_{+}^{\beta}[f/g]\left(x\right)& =\frac{{\upsilon}_{+}^{\beta}f\left(x\right)g\left(x\right)-{\upsilon}_{+}^{\beta}g\left(x\right)f\left(x\right)-{[f,g]}_{\beta}^{+}}{{g}^{2}\left(x\right)},\hfill \\ \hfill {\upsilon}_{-}^{\beta}[f/g]\left(x\right)& =\frac{{\upsilon}_{-}^{\beta}f\left(x\right)g\left(x\right)-{\upsilon}_{-}^{\beta}g\left(x\right)f\left(x\right)+{[f,g]}_{\beta}^{-}}{{g}^{2}\left(x\right)},\hfill \end{array}$$

- $f\in {\mathbb{H}}^{\phantom{\rule{0.166667em}{0ex}}\beta}$ and $g\in {\mathbb{C}}^{\phantom{\rule{0.166667em}{0ex}}1}$$$\begin{array}{cc}\hfill {\upsilon}_{+}^{\beta}f\circ g\left(x\right)& ={\upsilon}_{+}^{\beta}f\left(g\right){\left(\right)}^{{g}^{\prime}}\beta ,\hfill \end{array}$$
- $f\in {\mathbb{C}}^{\phantom{\rule{0.166667em}{0ex}}1}$ and $g\in {\mathbb{H}}^{\phantom{\rule{0.166667em}{0ex}}\beta}$$$\begin{array}{cc}\hfill {\upsilon}_{+}^{\beta}f\circ g\left(x\right)& ={f}^{\prime}\left(g\right)\phantom{\rule{0.166667em}{0ex}}{\upsilon}_{+}^{\beta}g\left(x\right),\hfill \\ \hfill {\upsilon}_{-}^{\beta}f\circ g\left(x\right)& ={f}^{\prime}\left(g\right)\phantom{\rule{0.166667em}{0ex}}{\upsilon}_{-}^{\beta}g\left(x\right).\hfill \end{array}$$

## Appendix B. The Stochastic Variation Problem

**Remark**

**A1.**

## Appendix C. Matlab Simulation Code

## References

- Langevin, P. Sur la theórie du mouvement Brownien. C. R. Acad. Sci.
**1908**, 146, 530–533. (in French). [Google Scholar] - Nelson, E. Derivation of the Schrödinger equation from Newtonian mechanics. Phys. Rev.
**1966**, 150, 1079. [Google Scholar] [CrossRef] - Nottale, L. Fractals in the Quantum Theory of Spacetime. Int. J. Mod. Phys. A
**1989**, 4, 5047–5117. [Google Scholar] [CrossRef] - ksendal, B. Stochastic Differential Equations, 6th ed.; Springer: Berlin/Heidelberg, Germany, 2003. [Google Scholar]
- Beck, C. Dynamical systems of Langevin type. Phys. A Stat. Mech. Appl.
**1996**, 233, 419–440. [Google Scholar] [CrossRef] - Mackey, M.; Tyran-Kaminska, M. Deterministic Brownian motion: The effects of perturbing a dynamical system by a chaotic semi-dynamical system. Phys. Rep.
**2006**, 422, 167–222. [Google Scholar] [CrossRef][Green Version] - Tyran-Kamińska, M. Diffusion and Deterministic Systems. Math. Model. Nat. Phenom.
**2014**, 9, 139–150. [Google Scholar] [CrossRef] - Gillespie, D.T. The mathematics of Brownian motion and Johnson noise. Am. J. Phys.
**1996**, 64, 225–240. [Google Scholar] [CrossRef] - Klages, R. Deterministic Diffusion in One-Dimensional Chaotic Dynamical Systems; Wissenschaft und Technik Verlag: Berlin, Germany, 1996. [Google Scholar]
- Knight, G.; Klages, R. Linear and fractal diffusion coefficients in a family of one-dimensional chaotic maps. Nonlinearity
**2010**, 24, 227–241. [Google Scholar] [CrossRef][Green Version] - Zwanzig, R. Nonequilibrium Statistical Mechanics; Oxford University Press: Oxford, UK, 2001. [Google Scholar]
- Bateman, H. Some recent researches in the motion of fluids. Mon. Weather Rev.
**1915**, 43, 163–167. [Google Scholar] [CrossRef] - Burgers, J.M. The Nonlinear Diffusion Equation; Springer: Dordrecht, The Netherlands, 1974. [Google Scholar]
- Gurbatov, S.; Malakhov, A.; Saichev, A.; Saichev, A.I.; Malakhov, A.N.; Gurbatov, S.N. Nonlinear Random Waves and Turbulence in Nondispersive Media: Waves, Rays, Particles (Nonlinear Science: Theory & Applications); John Wiley & Sons Ltd.: Hoboken, NJ, USA, 1992. [Google Scholar]
- Busnello, B. A Probabilistic Approach to the Two-Dimensional Navier-Stokes Equations. Ann. Probab.
**1999**, 27, 1750–1780. [Google Scholar] [CrossRef] - Busnello, B.; Flandoli, F.; Romito, M. A probabilistic representation for the vorticity of a three-dimensional viscous fluid and for general systems of parabolic equations. Proc. Edinb. Math. Soc.
**2005**, 48, 295–336. [Google Scholar] [CrossRef] - Constantin, P.; Iyer, G. A stochastic Lagrangian representation of the three-dimensional incompressible Navier-Stokes equations. Commun. Pure Appl. Math.
**2007**, 61, 330–345. [Google Scholar] [CrossRef][Green Version] - Constantin, P.; Iyer, G. A stochastic-Lagrangian approach to the Navier-Stokes equations in domains with boundary. Ann. Appl. Probab.
**2011**, 21, 1466–1492. [Google Scholar] [CrossRef] - Fényes, I. Eine wahrscheinlichkeitstheoretische Begründung und Interpretation der Quantenmechanik. Z. Phys.
**1952**, 132, 81–106. [Google Scholar] [CrossRef] - Weizel, W. Ableitung der Quantentheorie aus einem klassischen, kausal determinierten Modell. Z. Phys.
**1953**, 134, 264–285. [Google Scholar] [CrossRef] - Pavon, M. Hamilton’s principle in stochastic mechanics. J. Math. Phys.
**1995**, 36, 6774–6800. [Google Scholar] [CrossRef] - Nottale, L. Scale Relativity and Schrödinger’s equation. Chaos Solitons Fractals
**1998**, 9, 1051–1061. [Google Scholar] [CrossRef] - Nottale, L. Macroscopic Quantum-Type Potentials in Theoretical Systems Biology. Cells
**2013**, 3, 1–35. [Google Scholar] [CrossRef] [PubMed] - Mandelbrot, B.B. Intermittent turbulence in self-similar cascades: divergence of high moments and dimension of the carrier. In Multifractals and 1/f Noise: Wild Self-Affinity in Physics (1963–1976); Springer: New York, NY, USA, 1999; pp. 317–357. [Google Scholar]
- Salem, R. On some singular monotonic functions which are strictly increasing. Trans. Am. Math. Soc.
**1943**, 53, 427–439. [Google Scholar] [CrossRef] - Prodanov, D. Conditions for continuity of fractional velocity and existence of fractional Taylor expansions. Chaos Solitons Fractals
**2017**, 102, 236–244. [Google Scholar] [CrossRef] - Carlen, E.A. Conservative diffusions. Commun. Math. Phys.
**1984**, 94, 293–315. [Google Scholar] [CrossRef] - Guerra, F. The Foundations of Quantum Mechanics—Historical Analysis and Open Questions. In The Foundations of Quantum Mechanics—Historical Analysis and Open Questions: Lecce, 1993; Garola, C., Rossi, A., Eds.; Springer: Dordrecht, The Netherlands, 1995; pp. 339–355. [Google Scholar]
- Nottale, L. Fractal Space-Time And Microphysics: Towards A Theory Of Scale Relativity; World Scientific: Singapore, 1993. [Google Scholar]
- Zambrini, J.C. Variational processes and stochastic versions of mechanics. J. Math. Phys.
**1986**, 27, 2307–2330. [Google Scholar] [CrossRef] - Yasue, K. Stochastic calculus of variations. J. Funct. Anal.
**1981**, 41, 327–340. [Google Scholar] [CrossRef] - Hermann, R. Numerical simulation of a quantum particle in a box. J. Phys. A Math. Gen.
**1997**, 30, 3967. [Google Scholar] [CrossRef] - Hopf, E. The partial differential equation u
_{t}+uu_{x}=μu_{xx}. Commun. Pure Appl. Math.**1950**, 3, 201–230. [Google Scholar] [CrossRef] - Cole, J.D. On a quasi-linear parabolic equation occurring in aerodynamics. Quart. Appl. Math.
**1951**, 9, 225–236. [Google Scholar] [CrossRef][Green Version] - Dunkel, J. Relativistic Brownian Motion and Diffusion Processes. Ph.D. Thesis, Augsburg University, Augsburg, Germany, 2008. [Google Scholar]
- Lage, J.L.; Kulish, V.V. On the Relationship between Fluid Velocity and de Broglie’s Wave Function and the Implications to the Navier–Stokes Equation. Int. J. Fluid Mech. Res.
**2002**, 29, 13. [Google Scholar] [CrossRef] - Tsekov, R. On the Stochastic Origin of Quantum Mechanics. Rep. Adv. Phys. Sci.
**2017**, 1, 1750008. [Google Scholar] [CrossRef][Green Version] - Cole Hopf Transformations in Maxima; MDPI: Switzerland, 2018. [CrossRef]
- Cresson, J. Scale calculus and the Schrödinger equation. J. Math. Phys.
**2003**, 44, 4907–4938. [Google Scholar] [CrossRef] - Sornette, D. Brownian representation of fractal quantum paths. Eur. J. Phys.
**1990**, 11, 334. [Google Scholar] [CrossRef] - McClendon, M.; Rabitz, H. Numerical simulations in stochastic mechanics. Phys. Rev. A
**1988**, 37, 3479–3492. [Google Scholar] [CrossRef] - Al-Rashid, S.N.T.; Habeeb, M.A.Z.; Ahmad, K.A. Application of Scale Relativity (ScR) Theory to the Problem of a Particle in a Finite One-Dimensional Square Well (FODSW) Potential. J. Quantum Inf. Sci.
**2011**, 1, 7–17. [Google Scholar] [CrossRef] - Al-Rashid, S.N.T.; Habeeb, M.A.Z.; Ahmed, K.A. Application of Scale Relativity to the Problem of a Particle in a Simple Harmonic Oscillator Potential. J. Quantum Inf. Sci.
**2017**, 7, 77–88. [Google Scholar] [CrossRef] - Botez, I.C.; Agop, M. Order-Disorder Transition in Nanostructures via Non-Differentiability. J. Comp. Theor. Nanosci.
**2015**, 12, 1746–1755. [Google Scholar] [CrossRef] - Ganguly, S.; Chakraborty, S. Sedimentation of nanoparticles in nanoscale colloidal suspensions. Phys. Lett. A
**2011**, 375, 2394–2399. [Google Scholar] [CrossRef] - Metzler, R.; Klafter, J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A Math. Gen.
**2004**, 37, R161. [Google Scholar] [CrossRef] - Puente, C.; López, M.; Pinzón, J.; Angulo, J. The Gaussian Distribution Revisited. Adv. Appl. Probab.
**1996**, 28, 500. [Google Scholar] [CrossRef] - Puente, C. The exquisite geometric structure of a Central Limit Theorem. Fractals
**2003**, 11, 39–52. [Google Scholar] [CrossRef] - Flack, R.; Hiley, B. Feynman Paths and Weak Values. Entropy
**2018**, 20, 367. [Google Scholar] [CrossRef] - Prodanov, D. Fractional Velocity as a Tool for the Study of Non-Linear Problems. Fractal Fract.
**2018**, 2, 4. [Google Scholar] [CrossRef] - Prodanov, D. Characterization of strongly nonlinear and singular functions by scale space analysis. Chaos Solitons Fractals
**2016**, 93, 14–19. [Google Scholar] [CrossRef] - Prodanov, D. Regularization of derivatives on non-differentiable points. J. Phys. Conf. Ser.
**2016**, 701, 012031. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Virtual trajectories of the separable process. (

**A**) virtual trajectories; (

**B**) empirical vs. theoretical density. (

**A**) exact simulation of separable process is compared with the Euler–Maruyama algorithm. E—Exact simulation, E–M—Euler–Maruyama simulation; An offset is added to the exact solution for appreciation. Time is given in arbitrary units; (

**B**) the empirical transition density is estimated from $n={log}^{2}\left({N}_{s}\phantom{\rule{0.166667em}{0ex}}N\right)$ bins. Pearson’s correlation is given as an inset—r = 0.9976.

**Figure 3.**Simulations of free particles virtual trajectories. (

**A**) virtual trajectories: free particles; (

**B**) empirical vs. theoretical density. Simulations are based on $N=10,000$ points in ${N}_{s}=1000$ simulations. (

**A**,

**B**) width of potential well is $2L=100$ units. The empirical pdf is estimated from $n={log}^{2}\left({N}_{s}\phantom{\rule{0.166667em}{0ex}}N\right)$ bins. Pearson’s correlations are given as inset—r = 0.9883. Norming of the free particle transient results in Rayleigh density.

**Figure 4.**Simulations of particles in a box for two quantum numbers. (

**A**) virtual trajectories: particle in a box, n = 2; (

**B**) empirical vs. theoretical density; (

**C**) test particles in a box, n = 8; (

**D**) empirical vs. theoretical density. (

**A**,

**B**) width of potential well is $2L=100$ units.

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Prodanov, D.
Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation. *Entropy* **2018**, *20*, 492.
https://doi.org/10.3390/e20070492

**AMA Style**

Prodanov D.
Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation. *Entropy*. 2018; 20(7):492.
https://doi.org/10.3390/e20070492

**Chicago/Turabian Style**

Prodanov, Dimiter.
2018. "Analytical and Numerical Treatments of Conservative Diffusions and the Burgers Equation" *Entropy* 20, no. 7: 492.
https://doi.org/10.3390/e20070492