Newtonian Dynamics from the Principle of Maximum Caliber
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.
KeywordsNewtonian mechanics Maximum caliber
- 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.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
- 11.Feynman, R.P., Hibbs, A.R.: Quantum Mechanics and Path Integrals, Dover Books, (2005)Google Scholar
- 12.Gelfand, I.M., Fomin, S.V.: Calculus of Variations. Dover Publications, (2000)Google Scholar