# Random Walks Associated with Nonlinear Fokker–Planck Equations

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Usual Random Walk

## 3. Nonlinear Random Walk

## 4. A More General Perspective

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- Einstein, A. Investigations on the Theory of the Brownian Movement; Dover: New York, NY, USA, 1956. [Google Scholar]
- Langevin, P. Sur la théorie du mouvement brownien. C. R. Acad. Sci.
**1908**, 146, 530–533. (In French) [Google Scholar] - Fokker, A.D. Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld. Ann. Phys.
**1914**, 348, 810–820. (In German) [Google Scholar] [CrossRef] - Planck, M. Über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie. Sitzungsber. Preuss. Akad. Wiss.
**1917**, 24, 324–341. (In German) [Google Scholar] - Gardiner, C.W. Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences; Springer: Berlin/Heidelberg, Germany, 1996. [Google Scholar]
- Risken, H. The Fokker–Planck Equation; Springer: Berlin/Heidelberg, Germany, 1984. [Google Scholar]
- Durrett, R. Probability: Theory and Examples, 4th ed.; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Gnedenko, B.V.; Kolmogorov, A.N. Limit Distributions for Sums of Independent Random Variables; Addison-Wesley: Boston, MA, USA, 1967. [Google Scholar]
- Reif, F. Fundamentals of Thermal and Statistical Physics; McGraw-Hill: New York, NY, USA, 1965. [Google Scholar]
- Blum, J.R.; Chernoff, H.; Rosenblatt, M.; Teicher, H. Central limit theorems for interchangeable processes. Canad. J. Math.
**1958**, 10, 222–229. [Google Scholar] [CrossRef] - Nze, P.A.; Doukhan, P. Weak Dependence: Models and Applications to Econometrics. Economet. Theor.
**2004**, 20, 995–1045. [Google Scholar] [CrossRef] - Umarov, S.; Tsallis, C.; Steinberg, S. On a q-central limit theorem consistent with nonextensive statistical mechanics. Milan J. Math.
**2008**, 76, 307–328. [Google Scholar] [CrossRef] - Umarov, S.; Tsallis, C.; Gell-Mann, M.; Steinberg, S. Generalization of symmetric α-stable Lévy distributions for q > 1. J. Math. Phys.
**2010**, 51, 033502. [Google Scholar] [CrossRef] [PubMed] - Umarov, S.; Tsallis, C. The limit distribution in the q-CLT for q ≥ 1 is unique and can not have a compact support. J. Phys. A
**2016**, 49, 415204. [Google Scholar] [CrossRef] - Hilhorst, H.J. Note on a q-modified central limit theorem. J. Stat. Mech.
**2010**, 2010, P10023. [Google Scholar] [CrossRef] - Jauregui, M.; Tsallis, C. q-generalization of the inverse Fourier transform. Phys. Lett. A
**2011**, 375, 2085–2088. [Google Scholar] [CrossRef] - Jauregui, M.; Tsallis, C.; Curado, E.M.F. q-moments remove the degeneracy associated with the inversion of the q-Fourier transform. J. Stat. Mech.
**2011**, 2011, P10016. [Google Scholar] [CrossRef] - Tsallis, C. Possible generalization of the Boltzmann–Gibbs statistics. J. Stat. Phys.
**1988**, 52, 479–487. [Google Scholar] [CrossRef] - Tsallis, C. Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World; Springer: New York, NY, USA, 2009. [Google Scholar]
- Tirnakli, U.; Beck, C.; Tsallis, C. Central limit behavior of deterministic dynamical systems. Phys. Rev. E
**2007**, 75, 040106. [Google Scholar] [CrossRef] [PubMed] - Pluchino, A.; Rapisarda, A.; Tsallis, C. Nonergodicity and central-limit behavior for long-range Hamiltonians. EPL
**2007**, 80, 26002. [Google Scholar] [CrossRef] - Cirto, L.J.L.; Assis, V.R.V.; Tsallis, C. Influence of the interaction range on the thermostatistics of a classical many-body system. Physica A
**2014**, 393, 286–296. [Google Scholar] [CrossRef] - Christodoulidi, H.; Tsallis, C.; Bountis, T. Fermi-Pasta-Ulam model with long-range interactions: Dynamics and thermostatistics. EPL
**2014**, 108, 40006. [Google Scholar] [CrossRef] - Richardson, L.F. Atmospheric diffusion shown on a distance-neighbour graph. Proc. R. Soc. Lond.
**1926**, 110, 709–737. [Google Scholar] [CrossRef] - Shlesinger, M.F.; Zaslavsky, G.M.; Frisch, U. Lévy Flights and Related Topics in Physics, Lecture Notes in Physics; Springer: Berlin/Heidelberg, Germany, 1994. [Google Scholar]
- 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
**2004**, 37. [Google Scholar] [CrossRef] - Dubkov, A.A.; Spagnolo, B.; Uchaikin, V.V. Lévy flight superdiffusion: An introduction. Int. J. Bifurcat. Chaos
**2008**, 18, 2649–2672. [Google Scholar] [CrossRef] - Lenzi, E.K.; Mendes, R.S.; Andrade, J.S., Jr.; da Silva, L.R.; Lucena, L.S. N-dimensional fractional diffusion equation and Green function approach: Spatially dependent diffusion coefficient and external force. Phys. Rev. E
**2005**, 71, 052101. [Google Scholar] [CrossRef] [PubMed] - Srokowski, T. Non-Markovian Lévy diffusion in nonhomogeneous media. Phys. Rev. E
**2007**, 75, 051105. [Google Scholar] [CrossRef] [PubMed] - Spohn, H. Surface dynamics below the roughening transition. J. Phys. I
**1993**, 3, 69–81. [Google Scholar] [CrossRef] - Borland, L. Option pricing formulas based on a non-Gaussian stock price model. Phys. Rev. Lett.
**2002**, 89, 098701. [Google Scholar] [CrossRef] [PubMed] - Schwammle, V.; Curado, E.M.F.; Nobre, F.D. A general nonlinear Fokker–Planck equation and its associated entropy. Eur. Phys. J. B
**2007**, 58, 159–165. [Google Scholar] [CrossRef] - Schwammle, V.; Nobre, F.D.; Curado, E.M.F. Consequences of the H theorem from nonlinear Fokker–Planck equations. Phys. Rev. E
**2007**, 76, 041123. [Google Scholar] [CrossRef] [PubMed] - Casas, G.A.; Nobre, F.D.; Curado, E.M.F. Entropy production and nonlinear Fokker–Planck equations. Phys. Rev. E
**2012**, 86, 061136. [Google Scholar] [CrossRef] [PubMed] - Mendes, G.A.; Ribeiro, M.S.; Mendes, R.S.; Lenzi, E.K.; Nobre, F.D. Nonlinear Kramers equation associated with nonextensive statistical mechanics. Phys. Rev. E
**2015**, 91, 052106. [Google Scholar] [CrossRef] [PubMed] - Sicuro, G.; Rapcan, P.; Tsallis, C. Nonlinear inhomogeneous Fokker–Planck equations: Entropy and free-energy time evolution. Phys. Rev. E
**2016**, 94, 062117. [Google Scholar] [CrossRef] [PubMed] - Muskat, M. The Flow of Homegeneous Fluids Through Porous Media; McGraw-Hill: New York, NY, USA, 1937. [Google Scholar]
- Polunarinova-Kochina, P.Y. Theory of Ground Water Movement; Princeton University Press: Princeton, NJ, USA, 1962. [Google Scholar]
- Buckmaster, J. Viscous sheets advancing over dry beds. J. Fluid Mech.
**1977**, 81, 735–756. [Google Scholar] [CrossRef] - Larsen, E.W.; Pomraning, G.C. Asymptotic analysis of nonlinear Marshak waves. SIAM J. Appl. Math.
**1980**, 39, 201–212. [Google Scholar] [CrossRef] - Kath, W.L. Waiting and propagating fronts in nonlinear diffusion. Physica D
**1984**, 12, 375–381. [Google Scholar] [CrossRef] - Plastino, A.R.; Plastino, A. Non-extensive statistical mechanics and generalized Fokker–Planck equation. Physica A
**1995**, 222, 347–354. [Google Scholar] [CrossRef] - Shiino, H. Free energies based on generalized entropies and H-theorems for nonlinear Fokker–Planck equations. J. Math. Phys.
**2001**, 42, 2540–2553. [Google Scholar] [CrossRef] - Frank, T.D. Nonlinear Fokker–Planck Equations: Fundamentals and Applications; Springer: New York, NY, USA, 2005. [Google Scholar]
- Curado, E.M.F.; Nobre, F.D. Derivation of nonlinear Fokker–Planck equations by means of approximations to the master equation. Phys. Rev. E
**2003**, 67, 021107. [Google Scholar] [CrossRef] [PubMed] - Lenzi, E.K.; Malacarne, L.C.; Mendes, R.S. Path integral approach to the nonextensive canonical density matrix. Physica A
**2000**, 278, 201–213. [Google Scholar] [CrossRef] - Vidiella-Barranco, A.; Moya-Cessa, H. Nonextensive approach to decoherence in quantum mechanics. Phys. Lett. A
**2001**, 279, 56–60. [Google Scholar] [CrossRef] - Pedron, I.T.; Mendes, R.S.; Buratta, T.J.; Malacarne, L.C.; Lenzi, E.K. Logarithmic diffusion and porous media equations: A unified description. Phys. Rev. E
**2005**, 72, 031106. [Google Scholar] [CrossRef] [PubMed] - Anteneodo, C. Non-extensive random walks. Physica A
**2005**, 358, 289–298. [Google Scholar] [CrossRef]

**Figure 1.**(

**A**) Scaled probability distribution, ${P}^{\prime}\left({m}^{\prime}\right)={n}^{2}{\tilde{P}}_{n}\left(m\right)$, versus scaled position, ${m}^{\prime}={n}^{-2}m$, for the usual random walk $(\nu =1)$. The continuous line is the Gaussian distribution given in Equation (6), and the black squares represent the probabilities ${\tilde{P}}_{n}\left(m\right)$ after 30 times steps (i.e., $n=30$); (

**B**) The dependence on n of a power of the standard deviation, ${\sigma}^{2}$, for the usual case $(\nu =1)$.

**Figure 2.**(

**A**) Scaled probability distribution, ${P}^{\prime}\left({m}^{\prime}\right)={n}^{1+\nu}{\tilde{P}}_{n}\left(m\right)$, versus scaled position, ${m}^{\prime}={n}^{-1-\nu}m$, for the nonlinear random walk with $\nu =1.5$. The continuous line is the q-Gaussian distribution given in Equation (14) with $q=2-\nu $, and the black circles represent the probabilities ${\tilde{P}}_{n}\left(m\right)$ after 30 times steps (i.e., $n=30$); (

**B**) The dependence on n of a power of the standard deviation, ${\sigma}^{1+\nu}$, for the nonlinear case with $\nu =1.5$.

© 2017 by the authors. 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**

Dos Santos Mendes, R.; Lenzi, E.K.; Malacarne, L.C.; Picoli, S.; Jauregui, M.
Random Walks Associated with Nonlinear Fokker–Planck Equations. *Entropy* **2017**, *19*, 155.
https://doi.org/10.3390/e19040155

**AMA Style**

Dos Santos Mendes R, Lenzi EK, Malacarne LC, Picoli S, Jauregui M.
Random Walks Associated with Nonlinear Fokker–Planck Equations. *Entropy*. 2017; 19(4):155.
https://doi.org/10.3390/e19040155

**Chicago/Turabian Style**

Dos Santos Mendes, Renio, Ervin Kaminski Lenzi, Luis Carlos Malacarne, Sergio Picoli, and Max Jauregui.
2017. "Random Walks Associated with Nonlinear Fokker–Planck Equations" *Entropy* 19, no. 4: 155.
https://doi.org/10.3390/e19040155