In the Callen's paper pointed out two approaches for evolution of systems out of equilibrium. In one part he says

The statistical approach consists of generalizing the methods of statistical mechanics, and a theorem proved in this framework must then, of course, be projected into the macroscopic thermodynamic domain by a reasoning similar to that which projects the statistical mechanical H theorem into the macroscopic second law of thermodynamics.

what does he mean by "by a reasoning similar to that which projects the statistical mechanical H theorem into the macroscopic second law of thermodynamics."? What is this reasoning?

In the following he says

The alternative, or thermodynamic, approach consists in generalizing the methods of thermodynamics itself, which is already on a macroscopic level. Whereas the results of the thermodynamic approach are thus in more immediate contact with macroscopic irreversibility than are the results of the statistical approach, the postulates of the latter are closer to fundamental theory.

Why "the postulates of the latter are closer to fundamental theory."?

  • $\begingroup$ The linkage of thermodynamics and statistical mechanics is covered in most physics textbooks on thermodynamics (not chemical engineering for example). Have you had a statistical mechanics course? $\endgroup$
    – Jon Custer
    Commented Apr 2 at 14:17
  • $\begingroup$ Yes. But I've not seen these before. For example what is fundamental theory in this sense?@JonCuster $\endgroup$
    – Testina
    Commented Apr 2 at 14:48
  • 2
    $\begingroup$ If you have access to a copy of Reif, appendix A-12 is "The H theorem and the approach to equilibrium". The basis is in statistical mechanics, and then you need to understand the connection of stat mech to thermodynamics. $\endgroup$
    – Jon Custer
    Commented Apr 2 at 15:06

1 Answer 1


By "fundamental theory," Callan means that when you do statistical mechanics, you appeal to the "underlying physics" in a direct sense. If you're studying the statmech of a gas, you'll think about what happens when two gas particles collide with one another—they'll conserve total momentum and maybe energy, but there'll be some dependence of the outgoing trajectories on the particulars of the interaction between them and their ingoing trajectories. You then use this to say something about measurable properties: here, the two-body collision dynamics will tell you something about the relationship between pressure and volume (more specifically, it'll tell you about the virial coefficients, which parametrize that relationship).

The thermodynamic perspective, in this case, starts with the virial coefficients, and you never think about collisions explictly. You're doing thermodynamics if you are making statements about pressure, temperature, volume, entropy, etc., but you never stop to think about individual, microscopic degrees of freedom. This is a very powerful and general perspective, because you don't have to care (or calculate) about microscopic degrees of freedom and still can get quite a lot of mileage. Historically, thermodynamics was invented first, and we were able to use it to build pretty good steam engines without having to think about molecules.

Callan speaks of "projecting" into the macroscopic thermodynamic framework because it is less expressive than the statistical mechanics—for one thermodynamic macrostate, which is defined by enumerating the thermodynamic variables, there are many statistical microstates, which are defined by enumerating the positions and momenta and whatever other microscopic degrees of freedom there are.

The H-theorem is a classic way of justifying why you can get irreversible behavior on the macroscopic, thermodynamic level from seemingly reversible dynamics on the microscopic level (after all, particle collisions (and their quantum versions) are very usually reversible.) It makes a key assumption that the ingoing momenta of a collision are not correlated: this is similar in spirit to saying that if two particles collide, those two won't see each other ever again.


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