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Maximum entropy principle, evolution equations, and physics education

 

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

 

J-H. Schonfeldt^a, G.B. Roston^a, A.R. Plastino^a, and A. Plastino^a,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

 

Abstract

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.

 

Resumen

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

 

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