# Active and Purely Dissipative Nambu Systems in General Thermostatistical Settings Described by Nonlinear Partial Differential Equations Involving Generalized Entropy Measures

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## Abstract

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## 1. Introduction

## 2. Stochastic Nambu Systems in General Thermostatistic Settings

#### 2.1. Nambu Dynamics: Deterministic Case

#### 2.2. Nambu Dynamics: Stochastic Case

#### 2.3. Approach to Stationarity and Stationary Solutions

#### 2.4. Active Nambu Systems Exhibit Attractors Defined by Classical Nambu Systems

## 3. Examples of Active and Purely-Dissipative Systems

#### 3.1. Brownian Motion in a Potential Field

#### 3.2. Charged Particle in a Magnetic Field

#### 3.3. Active Spinning Top Featuring Non-Extensive Statistics: An Approach Involving Thermodynamic State Variables

#### 3.4. Numerics

## 4. Discussion

#### 4.1. Invariants of Nambu Dynamics as Pseudo-Invariants

#### 4.2. Active, Stochastic Systems and Generalized, Non-Extensive Entropic Measures

## 5. Conclusions

## Conflicts of Interest

## Appendix A. Derivation of Equations (14) and (15)

## References

- Nambu, Y. Generalized Hamiltonian dynamics. Phys. Rev. D
**1973**, 7, 2405–2412. [Google Scholar] [CrossRef] - Pandit, S.A.; Gangal, A.D. On generalized Nambu mechanics. J. Phys. A
**1998**, 31, 2899–2912. [Google Scholar] [CrossRef] - Plastino, A.R.; Plastino, A.; da Silva, L.R.; Casas, M. Dynamic thermostatting, divergenceless phase-space flows, and KBB systems. Physica A
**1999**, 271, 343–356. [Google Scholar] [CrossRef] - Roston, G.B.; Plastino, A.R.; Casas, M.; Plastino, A.; da Silva, L.R. Dynamic thermostatting and statistical ensemble. Eur. Phys. J. B
**2005**, 48, 87–93. [Google Scholar] [CrossRef] - Steeb, W.H.; Euler, N. A note on Nambu mechanics. Il Nuovo Cim. B
**1991**, 106, 263–272. [Google Scholar] [CrossRef] - Tegmen, A. Momentum map and action-angle variables for Nambu mechanics. Czechoslov. J. Phys.
**2004**, 54, 749–757. [Google Scholar] [CrossRef] - Frank, T.D. A Fokker-Planck approach to canonical-dissipative Nambu systems: With an application to human motor control during dynamic haptic perception. Phys. Lett. A
**2010**, 374, 3136–3142. [Google Scholar] [CrossRef] - Yamaleev, R.M. Generalized Newtonian equations of motion. Ann. Phys.
**1999**, 277, 1–18. [Google Scholar] [CrossRef] - Yamaleev, R.M. Relativistic equations of motion within Nambu’s formalism of dynamics. Ann. Phys.
**2000**, 285, 141–160. [Google Scholar] [CrossRef] - Molgado, A.; Rodriguez, A. Mapping between the dynamic and mechanical properties of the relativistic oscillator and Euler free rigid body. J. Nonlinear Math. Phys.
**2007**, 14, 534–547. [Google Scholar] [CrossRef] - Codriansky, S.; Bernardo, C.A.G.; Aglaee, A.; Carrillo, F.; Castellanos, J.; Pereira, G.; Perez, J. Developments in Nambu mechanics. J. Phys. A
**1994**, 27, 2565–2578. [Google Scholar] [CrossRef] - Plastino, A.R.; Plastino, A. Statistical treatment of autonomous systems with divergenceless flow. Physica A
**1996**, 232, 458–476. [Google Scholar] [CrossRef] - Chatterjee, R. Dynamical symmetries and Nambu mechanics. Lett. Math. Phys.
**1996**, 36, 117–126. [Google Scholar] [CrossRef] [Green Version] - Nutku, Y. Quantization with maximally degenerate Poisson brackets: The harmonic oscillator. J. Phys. A
**2003**, 36, 7559–7567. [Google Scholar] [CrossRef] - Baleanu, D. Angular momentum and Killing-Yano tensors. Proc. Inst. Math. NSA Ukraine
**2004**, 50, 611–616. [Google Scholar] [CrossRef] - Curtright, T.L.; Zachos, C. Deformation quantization of superintegrable systems and Nambu mechanics. New J. Phys.
**2002**, 4. [Google Scholar] [CrossRef] - Curtright, T.L.; Zachos, C. Classic and quantum Nambu mechanics. Phys. Rev. D
**2002**, 68, 085001. [Google Scholar] [CrossRef] - Zachos, C. Membranes and consistent quantization of Nambu dynamics. Phys. Lett. B
**2003**, 570, 82–88. [Google Scholar] [CrossRef] - Frank, T.D. Active systems with Nambu dynamics: With applications to rod wielding for haptic length perception and self-propagating systems on two-spheres. Eur. Phys. J. B
**2010**, 74, 195–203. [Google Scholar] [CrossRef] - Hirayama, M. Realization of Nambu mechanics: A particle interacting with an SU(2) monopole. Phys. Rev. D
**1977**, 16, 530. [Google Scholar] [CrossRef] - Yamaleev, R.M. Generalized Lorentz-force equations. Ann. Phys.
**2001**, 292, 157–178. [Google Scholar] [CrossRef] - Pletnev, N.G. Fillipov-Nambu n-algebras relevant to physics. Sib. Electron. Math. Rep.
**2009**, 6, 272–311. [Google Scholar] - Gonera, C.; Nuktu, Y. Super-integrable Calogero-type systems admit maximal number of Poisson structures. Phys. Lett. A
**2001**, 285, 301–306. [Google Scholar] [CrossRef] - Tegmen, A.; Vercin, A. Superintegrable systems, multi-Hamiltonian structures and Nambu mechanics in an arbitrary dimension. Int. J. Mod. Phys. B
**2004**, 19, 393–409. [Google Scholar] [CrossRef] - Guha, P. Application of Nambu mechanics to systems of hydrodynamical type II. J. Nonlinear Math. Phys.
**2004**, 11, 223–232. [Google Scholar] [CrossRef] - Müller, R.; Nevir, P. A geometric application of Nambu mechanics: The motion of three point vortices in the plane. J. Phys. A
**2014**, 47, 105201. [Google Scholar] [CrossRef] - Roupas, Z. Phase space geometry and chaotic attractors in dissipative Nambu mechanics. J. Phys. A
**2012**, 45, 195101. [Google Scholar] [CrossRef] - Mathis, W.; Stahl, D.; Mathis, R. Oscillator synthesis based on Nambu mechanics and canonical dissipative damping. In Proceedings of the 21st European Conference on Circuit Theory and Design (ECCTD 2013), Dresden, Germany, 8–12 September 2013.
- Mathis, W.; Mathis, R. Dissipative Nambu systems and oscillator circuit design. Nonlinear Theory Appl. IEICE
**2014**, 5, 259–271. [Google Scholar] [CrossRef] - Frank, T.D. Unifying mass-action kinetics and Newtonian mechanics by means of Nambu brackts. J. Biol. Phys.
**2011**, 37, 375–385. [Google Scholar] [CrossRef] [PubMed] - Frank, T.D. Nambu brackt formulation of nonlinear biochemical reactions beyond elementary mass action kinetics. J. Nonlinear Math. Phys.
**2012**, 19, 81–97. [Google Scholar] [CrossRef] - Mongkolsakulvong, S.; Chaikhan, P.; Frank, T.D. Oscillatory nonequilibrium Nambu systems: The canonical-dissipative Yamaleev oscillator. Eur. Phys. J. B
**2012**, 85. [Google Scholar] [CrossRef] - Chaikhan, P.; Frank, T.D.; Mongkolsakulvong, S. In-phase and anti-phase synchronization in an active Nambu mechanics system. Acta Mech.
**2016**, 10, 2703–2717. [Google Scholar] [CrossRef] - Gordon, J.M.; Kim, S.; Frank, T.D. Linear non-equilibrium thermodynamics of human voluntary behavior: A canonical-dissipative Fokker-Planck equation approach involving potentials beyond the harmonic case. Condens. Matter Phys.
**2016**, 19, 1–6. [Google Scholar] [CrossRef] - Schweitzer, F. Brownian Agents and Active Particles; Springer: Berlin/Heidelberg, Germany, 2003. [Google Scholar]
- Romanczuk, P.; Bär, M.; Ebeling, W.; Lindner, B.; Schimansky-Geier, L. Active Brownian particles: From individual to collective stochastic dynamics. Eur. Phys. J. Spec. Top.
**2012**, 202, 1–162. [Google Scholar] [CrossRef] - Haken, H. Synergetics: An Introduction; Springer: Berlin/Heidelberg, Germany, 1977. [Google Scholar]
- Frank, T.D. Nonlinear Fokker-Planck Equations: Fundamentals and Applications; Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar]
- Tsallis, C. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys.
**1988**, 52, 479–487. [Google Scholar] [CrossRef] - Tsallis, C. Non-extensive thermostatistics: Brief review and comment. Physica A
**1995**, 221, 227–290. [Google Scholar] [CrossRef] - Abe, S.; Okamoto, Y. Nonextensive Statistical Mechanics and Its Applications; Springer: Berlin/Heidelberg, Germany, 2001. [Google Scholar]
- Haken, H. Distribution function for classical and quantum systems far from thermal equilibrium. Z. Phys.
**1973**, 263, 267–282. [Google Scholar] [CrossRef] - Graham, R. Statistical Theory of Instabilities in Stationary Nonequilibrium Systems with Applications to Lasers and Nonlinear Optics. In Springer Tracts in Modern Physics; Höhler, G., Ed.; Springer: Berlin/Heidelberg, Germany, 1973; Volume 66, pp. 1–97. [Google Scholar]
- Ebeling, W.; Sokolov, I.M. Statistical Thermodynamics and Stochastic Theory of Nonequilibrium Systems; World Scientific: Singapore, 2004. [Google Scholar]
- Plastino, A.R.; Plastino, A. Non-extensive statistical mechanics and generalized Fokker-Planck equation. Physica A
**1995**, 222, 347–354. [Google Scholar] [CrossRef] - Crank, J. The Mathematics of Diffusion; Clarendon Press: Oxford, UK, 1975. [Google Scholar]
- Peletier, L.A. The Porous Media Equation; Applications of Nonlinear Analysis in the Physical Science; Amann, H., Bazley, N., Kirchgässner, K., Eds.; Pitman Advanced Publishing Program: Boston, MA, USA, 1981; pp. 229–241. [Google Scholar]
- Barenblatt, G.I.; Entov, V.M.; Ryzhik, V.M. Theory of Fluid Flows through Natural Rocks; Kluwer Academic Publisher: Dordrecht, The Netherlands, 1990. [Google Scholar]
- Risken, H. The Fokker-Planck Equation. Methods of Solution and Applications; Springer: Berlin/Heidelberg, Germany, 1989. [Google Scholar]
- Kaniadakis, G. H-theorem and generalized entropies within the framework of nonlinear kinetics. Phys. Lett. A
**2001**, 288, 283–291. [Google Scholar] [CrossRef] - Shiino, M. 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.; Daffertshofer, A. H-theorem for nonlinear Fokker-Planck equations related to generalized thermostatistics. Physica A
**2001**, 295, 455–474. [Google Scholar] [CrossRef] - Chavanis, P.H. Generalized Fokker-Planck equations and effective thermodynamics. Physica A
**2004**, 340, 57–65. [Google Scholar] [CrossRef] - Schwämmle, V.; Nobre, F.D.; Curado, E. Consequences of the H-theorem for nonlinear Fokker-Planck equations. Phys. Rev. E
**2007**, 76, 041123. [Google Scholar] [CrossRef] [PubMed] - Frank, T.D. On the boundedness of free energy functionals. Nonlinear Phenom. Complex Syst.
**2003**, 6, 696–704. [Google Scholar] - Dotov, D.G.; Frank, T.D. From the W-method to the canonical-dissipative method for studying uni-manual rhythmic behavior. Motor Control
**2011**, 15, 550–567. [Google Scholar] [CrossRef] [PubMed] - Dotov, D.G.; Kim, S.; Frank, T.D. Non-equilibrium thermodynamical description of rhythmic motion patterns of active systems: A canonical-dissipative approach. BioSystems
**2015**, 128, 26–36. [Google Scholar] [CrossRef] [PubMed] - Kim, S.; Gordon, J.M.; Frank, T.D. Nonequilibrium thermodynamic state variables of human self-paced rhythmic motions: Canonical-dissipative approach, augmented Langevin equation, and entropy maximization. Open Syst. Inf. Dyn.
**2015**, 22. [Google Scholar] [CrossRef] - Mongkolsakulvong, S.; Frank, T.D. Canonical-dissipative limit cycle oscillators with a short-range interaction in phase space. Condens. Matter Phys.
**2010**, 13, 13001. [Google Scholar] [CrossRef] - Frank, T.D. On a moment-based data analysis method for canonical-dissipative oscillator systems. Fluct. Noise Lett.
**2010**, 9, 69–87. [Google Scholar] [CrossRef] - Frank, T.D.; Kim, S.; Dotov, D.G. Canonical-dissipative nonequilibrium energy distributions: Parameter estimation via implicit moment method, implementation and application. Int. J. Mod. Phys. B
**2013**, 27. [Google Scholar] [CrossRef] - Frank, T.D. Virial theorem and non-equilibrium canonical-dissipative distributions characterizing Parkinson tremor. Int. J. Mod. Phys. B
**2011**, 25, 1465–1469. [Google Scholar] [CrossRef] - Bödeker, H.U.; Beta, C.; Frank, T.D.; Bodenschatz, E. Quantitative analysis of random ameboid motion. Europhys. Lett.
**2010**, 90, 28005. [Google Scholar] [CrossRef] - Frank, T.D. Nonextensive cutoff distributions of postural sway for the old and the young. Physica A
**2009**, 388, 2503–2510. [Google Scholar] [CrossRef]

**Figure 1.**Trajectories of ${L}_{1}$ (top) and ${H}_{1}$ (bottom) of the active spinning top model without noise. Equation (66) for $D=0$ was solved numerically using a Euler forward scheme with a single time step of 0.01 time units. Model parameters: ${I}_{1}=1.1$, ${I}_{2}=1.3$, ${I}_{3}=1.5$, ${\gamma}_{1}=3/1000$, ${\gamma}_{2}=0$, ${B}_{1}=3$. Initial values: ${L}_{1}\left(0\right)=0.1$, ${L}_{2}\left(0\right)=0.2$, ${L}_{3}\left(0\right)=0.3$.

**Figure 3.**Stationary probability density of the stochastic active spinning top model. Analytical (solid line) and numerical results (symbols) are shown. The diffusion constant D was relatively small. The full stochastic model defined by Equation (66) was solved numerically using a stochastic Euler forward scheme with a single time step of 0.005 time units in the time interval $[0,1000]$. From $\mathbf{L}\left(t\right)$ thus obtained, the invariant ${H}_{1}\left(t\right)$ was calculated. The numerical results show the probability density estimated from ${H}_{1}\left(t\right)$ in $[500,1000]$ (neglecting the transient period) using kernel density estimation with positive support. The analytical results were drawn from Equation (73). The effective integration factor ${\mathrm{Z}}^{\prime}\mathrm{Z}$ was determined numerically. Model parameters: ${\gamma}_{1}=0.3$, $D=0.1$. Other parameters as in Figure 2: ${I}_{1}=1.1$, ${I}_{2}=1.3$, ${I}_{3}=1.5$, ${\gamma}_{2}=0$, ${B}_{1}=3$.

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Frank, T.D.
Active and Purely Dissipative Nambu Systems in General Thermostatistical Settings Described by Nonlinear Partial Differential Equations Involving Generalized Entropy Measures. *Entropy* **2017**, *19*, 8.
https://doi.org/10.3390/e19010008

**AMA Style**

Frank TD.
Active and Purely Dissipative Nambu Systems in General Thermostatistical Settings Described by Nonlinear Partial Differential Equations Involving Generalized Entropy Measures. *Entropy*. 2017; 19(1):8.
https://doi.org/10.3390/e19010008

**Chicago/Turabian Style**

Frank, T. D.
2017. "Active and Purely Dissipative Nambu Systems in General Thermostatistical Settings Described by Nonlinear Partial Differential Equations Involving Generalized Entropy Measures" *Entropy* 19, no. 1: 8.
https://doi.org/10.3390/e19010008