Foundations of Physics

, Volume 44, Issue 9, pp 923–931 | Cite as

Newtonian Dynamics from the Principle of Maximum Caliber

  • Diego González
  • Sergio Davis
  • Gonzalo Gutiérrez
Article

Abstract

The foundations of Statistical Mechanics can be recovered almost in their entirety from the principle of maximum entropy. In this work we show that its non-equilibrium generalization, the principle of maximum caliber (Jaynes, Phys Rev 106:620–630, 1957), when applied to the unknown trajectory followed by a particle, leads to Newton’s second law under two quite intuitive assumptions (both the expected square displacement in one step and the spatial probability distribution of the particle are known at all times). Our derivation explicitly highlights the role of mass as an emergent measure of the fluctuations in velocity (inertia) and the origin of potential energy as a manifestation of spatial correlations. According to our findings, the application of Newton’s equations is not limited to mechanical systems, and therefore could be used in modelling ecological, financial and biological systems, among others.

Keywords

Newtonian mechanics Maximum caliber 

References

  1. 1.
    Caticha, A., Cafaro, C.: From information geometry to newtonian dynamics. AIP Conf. Proc. 954, 165 (2007)ADSCrossRefGoogle Scholar
  2. 2.
    Jaynes, E.T.: Information theory and statistical mechanics. Phys. Rev. 106, 620–630 (1957)ADSCrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    J. E. Shore and R. W. Johnson. Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Trans. Info. Theory, IT. 26, 26–37 (1980)Google Scholar
  4. 4.
    Skilling, J.: The axioms of maximum entropy. In: Erickson, G.J., Smith, C.R., (eds.) Maximum Entropy and Bayesian Methods in Science and Engineering, pp. 173–187. Kluwer Academic Publishers, New York (1988)Google Scholar
  5. 5.
    Jaynes, E.T.: The minimum entropy production principle. Ann. Rev. Phys. Chem. 31, 579–601 (1980)ADSGoogle Scholar
  6. 6.
    Press, S., Ghosh, K., Lee, J., Dill, K.A.: Principles of maximum entropy and maximum caliber in statistical physics. Rev. Mod. Phys. 85, 1115 (2013)ADSCrossRefGoogle Scholar
  7. 7.
    Stock, G., Ghosh, K., Dill, K.A.: Maximum caliber: a variational approach applied to two-state dynamics. J. Chem. Phys. 128, 194102 (2008)ADSGoogle Scholar
  8. 8.
    Haken, H.: A new access to path integrals and fokker planck equations via the maximum calibre principle. Z. Phys. B. Cond. Matt. 63, 505–510 (1986)ADSCrossRefGoogle Scholar
  9. 9.
    Ge, H., Press, S., Ghosh, K., Dill, K.: Markov processes follow from the principle of maximum caliber. J. Chem. Phys. 136, 064108 (2012)ADSGoogle Scholar
  10. 10.
    Davis, S., Gutiérrez, G.: Conjugate variables in continuous maximum-entropy inference. Phys. Rev. E 86, 051136 (2012)ADSCrossRefGoogle Scholar
  11. 11.
    Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals, Dover Books, (2005)Google Scholar
  12. 12.
    Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Dover Publications, (2000)Google Scholar
  13. 13.
    Jaynes, E.T.: On the rationale of maximum-entropy methods. Proc. IEEE 10, 939–952 (1982)ADSCrossRefGoogle Scholar
  14. 14.
    Smolin, Lee: Quantum fluctuations and inertia. Phys. Lett. 113A, 408 (1986)ADSCrossRefMathSciNetGoogle Scholar
  15. 15.
    Nelson, E.: Derivation of the schrdinger equation from newtonian mechanics. Phys. Rev. 150, 1079–1085 (1966)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Diego González
    • 1
  • Sergio Davis
    • 1
  • Gonzalo Gutiérrez
    • 1
  1. 1.Grupo de Nanomateriales, Departamento de Física, Facultad de CienciasUniversidad de ChileSantiagoChile

Personalised recommendations