SciELO - Scientific Electronic Library Online

vol.52 issue2Análisis del grado de conocimiento declarativo y procedural de estudiantes en cursos de física universitariaNewton's missing experiment author indexsubject indexsearch form
Home Pagealphabetic serial listing  

Services on Demand




Related links

  • Have no similar articlesSimilars in SciELO


Revista mexicana de física E

Print version ISSN 1870-3542

Rev. mex. fís. E vol.52 n.2 México Dec. 2006




Maximum entropy principle, evolution equations, and physics education


Principio de máxima entropía como herramienta didáctica para discutir ecuaciones de evolución temporal


J-H. Schonfeldta, G.B. Rostona, A.R. Plastinoa, and A. Plastinoa,b


a Department of Physics, University of Pretoria, Pretoria 0002, South Africa.

b Facultad de Ciencias Exactas, Universidad Nacional de La Plata, IFLP-CONICET, C.C. 727, 1900 La Plata, Argentina.


Recibido el 27 de 09 de 2005;
aceptado el 22 de 03 de 2006



The landscape of Physics is in a constant state of change and the structure of the University level Physics Curriculum needs to be adapted to this state of affairs. One of the most interesting current features of physics is the increasing importance of multidisciplinary studies. Methods and ideas from physics are being applied to diverse areas of science ranging from biology and economics to sociology and linguistics. Statistical Physics (SP) provides the most fertile set of methods for these kind of applications. The aim of the present contribution is to show how a powerful idea from SP that is widely applied in many fields, the maximum entropy principle (MaxEnt), can be integrated into the physics curriculum. First of all, the constrained maximization of an entropic measure provides an important illustration of the Lagrange multipliers technique, which is part of the standard calculus course for physics students. Secondly, MaxEnt provides the basis for an alternative foundation for statistical mechanics, which is nowadays being considered in some modern textbooks on SP. In point of fact, the main role usually assigned to MaxEnt (as a tool for teaching theoretical physics) is in connection with the Gibbs canonical and grand canonical ensembles. However, as we shall here explain, MaxEnt also constitutes a useful tool in the teaching of other aspects of theoretical physics: it provides an elegant and simple method for obtaining analytical solutions for several evolution equations, like the Liouville equation, the diffusion equation, and the Fokker-Planck equation. Last but certainly not least, MaxEnt belongs to the tool-kit that physicist use to solve concrete "real-world" problems.

Keywords: Maximum entropy principle; continuity equations; Liouville equation.



El panorama de la física contemporánea se encuentra en un estado de continuo cambio y por ende la estructura de los planes de estudio del área necesita adaptarse a tal situación. La creciente importancia de la multidisciplinariedad es hoy faceta típica de la actividad en física. Técnicas e ideas de origen físico están siendo aplicados con éxito en áreas diversas. Biología y economía constituyen ejemplos importantes. La física estadística (FE) es la principal proveedora de métodos para este tipo de aplicaciones. Este trabajo se refiere a una idea muy fecunda (y de amplia aplicación) de la FE, el llamado "principio de máxima entropía" (PME). Pretendemos aquí mostrar como puede el PME ser integrado con provecho en la currícula de la física. Se verá que los cursos de mecánica estadística no son los únicos donde este principio puede ser exitosamente incorporado. En particular, ilustraremos como el PME puede ser empleado para construir soluciones analíticas relativamente sencillas para ecuaciones de evolución muy importantes, tales como las de Liouville y Fokker-Planck.

Descriptores: Principio de máxima entropía; ecuaciones de continuidad; ecuación de Liouville.


PACS: 05.40.-a; 05.20.Gg





1. J. Willard Gibbs, Elementary Principles in Statistical Mechanics (New Haven, Yale University Press, 1902).         [ Links ]

2. E.T. Jaynes, Phys. Rev. 106 (1957) 620; 108 171; Papers on probability, statistics and statistical physics, edited by R.D. Rosenkrantz (Dordrecht, Reidel, 1987).         [ Links ]

3. M. Tribus, Thermostatics and Thermodynamics (D. Van Nos-trand Company, New York, 1961).         [ Links ]

4. A. Katz, Principles of Statistical Mechanics (W.H. Freeman and Company, San Francisco and London, 1967).         [ Links ]

5. G.A.P. Wyllie, Elementary Statistical Mechanics (Hutchinson, London, 1970).         [ Links ]

6. R. Baierlein, Atoms and Information Theory (W.H. Freeman and Company, San Francisco, 1971).         [ Links ]

7. L. Brillouin, Science and Information Theory (New York: Academic Press, New York, 1962).         [ Links ]

8. Deepak Agrawal, J.K. Singh, and Akhilesh Kumar, Biosystems Engineering 90 (2005) 103.         [ Links ]

9. A. Lukacs and E. Papp, Journal of Photochemistry and Photobiology B: Biology 77 (2004) 1.         [ Links ]

10. A.M. Blokhin, R.S. Bushmanov, and V. Romano, International Journal of Engineering Science 42 (2004) 915.         [ Links ]

11. Yihong Gong, Mei Han, Wei Hua, and Wei Xu, Computer Vision and Image Understanding 96 (2004) 181.         [ Links ]

12. Ali A. Shams, Sergio Picozzi, and F. Bary Malik, Physica B: Condensed Matter 352 (2004) 269.         [ Links ]

13. Kouichirou Amemiya, Shigenori Moriwaki, and Norikatsu Mio, Physics Letters A 315 (2003) 184.         [ Links ]

14. S. Israel, R. Saravanan, N. Srinivasan, and R. K. Rajaram, Journal of Physics and ChemistryofSolids 64 (2003) 43.         [ Links ]

15 . Sun Kyoung Kim and Woo Il Lee, International Journal of Heat and Mass Transfer 45 (2002) 381.         [ Links ]

16. Jonathan Clowsera and Costas Strouthosa, Nuclear Physics B -Proceedings Supplements 106-107 (2002) 489.         [ Links ]

17. A. Elgarayhi, Journal of Quantitative Spectroscopy and Radiative Transfer 75 (2002) 1.         [ Links ]

18. Soumya Raychaudhuri, Jeffrey T. Chang, Patrick D. Sutphin, and Russ B. Altman, Genome Research 12 (2002) 203.         [ Links ]

19. S.A. El-Wakil, A. Elhanbaly, and M. A. Abdou, Journal of Quantitative Spectroscopy and Radiative Transfer 69 (2001) 41.         [ Links ]

20. H.B. Callen, Thermodynamics (NY, J. Wiley, 1960); E.A. Desloge, Thermal Physics (NY, Holt, Rhinehart and Winston, 1968).         [ Links ]

21. R.K. Pathria, Statistical Mechanics (Exeter, Pergamon Press, 1993).         [ Links ]

22. F. Reif, Statistical and Thermal Physics (NY, McGraw-Hill, 1965).         [ Links ]

23. J.J. Sakurai, Modern Quantum Mechanics (Menlo Park, Ca., Benjamin, 1985).         [ Links ]

24. A. Katz, Principles of Statistical Mechanics: The Information Theory Approach (San Francisco, Freeman and Co., 1967).         [ Links ]

25. D.J. Scalapino, in Physics and Probability. Essays in Honor of Edwin T. Jaynes, edited by W.T. Grandy Jr. and P.W. Milonni (NY, Cambridge University Press, 1993).         [ Links ]

26. A.R. Plastino and A. Plastino, Phys. Lett. A 226 (1997) 257.         [ Links ]

27. E. Curado and A. Plastino, Phys. Rev. E (2005) (in press).         [ Links ]

28. A.R. Plastino, Tsallis Theory, the Maximum Entropy Principle, and Evolutions Equations, p. 157, in NonExtensive Statistical Mechanics and Its Applications, Lecture Notes in Physics, S. Abe and Y. Okamoto (Eds.) (Springer Verlag, Berlin, 2001).         [ Links ]

29. E.D. Malaza, H.G. Miller, A.R. Plastino, and F. Solms, Physica A 265 (1999) 224.         [ Links ]

30. A.R. Plastino and A. Plastino, Physica A 258 (1998)429.         [ Links ]

31. A.R. Plastino, H.G. Miller, and A. Plastino, Phys. Rev. E 56 (1997) 3927.         [ Links ]

32. T.D. Franck, Nonlinear Fokker-Planck Equations (Springer-Verlag, Berlin, 2005).         [ Links ]

Creative Commons License All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License