The final session of the workshop on Teaching and learning statistical physics was devoted to a plenary discussion about the main points emerged from the contributions of the invited speakers and of the students. All the participants have recognized the increasing interest for the applications of nonequilibrium statistical physics (NESP) and its remarkable interdisciplinary potential. Rather than summing up the discussion we want here to make some general considerations and proposals.
We think that the first and most important distinction to make is between physicists (and possibly mathematicians) and non physicists:
i) the former are expected to have a solid mathematical bases and to have the possibility to attend more than one class, therefore allowing the student to learn statistical physics in a progressive and rational way;
ii) the latter have weaker mathematical bases, are not even interested in learning statistical physics at large and typically approach statistical physics as an ensemble of tools rather than a discipline in its own.
In fact, the risk that NESP is seen as a mere list of recipes for cooking different dishes at will exists for physicists as well, because there isn’t a standard way to teach it.
For all of these reasons we will make distinct considerations for teaching NESP to physicists/mathematicians and to other students. We will conclude with an interdisciplinary proposal.
Teaching NESP to physicists/mathematicians
We assume that students have attended a course on equilibrium statistical physics and we think that such basic knowledge should be used at best to teach NESP. We confine here to some freewheeling considerations.
1) From equilibrium to nonequilibrium phase transitions.
Control parameter and order parameter, spontaneous symmetry breaking, upper and lower critical dimensions, the role of noise and its different relevance in different spatial dimensions: all these concepts, if they are clear to students from their knowledge of equilibrium phase transitions should also allow to introduce and understand more easily not only nonequilibrium phase transitions but also kinetic roughening phenomena, which are a critical phenomenon indeed. Here are some examples of possible, useful connections.
2) Stochastic differential equations
It is unlikely that a physicist has attended a course on stochastic PDEs, while it might be the case for mathematicians approaching statistical physics. However it is more likely that physics students have already heard about the Einstein's theory of Brownian motion and its formulation by Langevin.
The possibility to apply either the Itô or the Stratonovich integral is perhaps obscure to students (and researchers…) because there is no general recipe to choose between the two. In this sense it is extremely useful to present at least a case where the Stratonovich prescription should be used.
3) Hydrodynamic approximation
This continuum approach to study large scale properties of a system starting from conserved quantities might be emphasized and also related to the interpretation of the Fokker-Planck equation as a continuity equation.
Teaching statistical physics to non physicists
A provocative question has raised during the workshop: is it possible to teach NESP to students who have not a detailed knowledge of equilibrium statistical physics? While this question has not much sense for physicists it must be raised for non physicists, who have not the opportunity to attend a standard course of equilibrium statistical physics and who are generally interested to know a specific branch of statistical physics, typically an interdisciplinary one (in most cases in connection to biophysics or to sociophysics or to econophysics).
An opinion (that we share) is that the answer should be based on the role of temperature: if it plays an important role, then equilibrium statistical physics is hardly avoidable. At risk of being self-referential we refer to the Table of Contents of our book and we pretend to make the following statement: equilibrium statistical physics is unaivodable to teach most of Chap. 1 (Brownian motion, Langevin and Fokker-Planck equations), Chap. 2 (Linear response theory and transport phenomena) and Chap. 6 (Phase-ordering kinetics). Equilibrium statistical physics is not unavoidable but it is useful to teach Chaps. 3 and 4 (Out of Equilibrium phase transitions and critical phenomena). Finally, equilibrium statistical physics can be avoided to teach a part of Chap. 1 (Generalized Random Walk), Chap. 5 (Stochastic dynamics of surfaces and interfaces) and Chap. 7 (Pattern formation). As you can see the chapters where equilibrium theory is avoidable concern stochastic models and stochastic PDEs.
One general proposal
Teaching an advanced subject depends on so many variables and constraints that we do not hazard suggestions for institutional courses. Instead, we think that Summer Schools might be the appropriate place to test new ideas and syllabi.
Any serious course on NESP topics should last no less than 16 hours and it could be preceded by a few introductory lessons for those students who miss some prerequisites. This would be specially useful for interdisciplinary courses, e.g., a course on “Statistical methods in biophysics/genetics/economy”.
Roberto Livi (University of Florence)
Paolo Politi (Institute for Complex Systems, CNR, Florence)