By Giovanni Gallavotti
This transparent booklet offers a severe and glossy research of the conceptual foundations of statistical mechanics as laid down in Boltzmann's works. the writer emphasises the relation among microscopic reversibility and macroscopic irreversibility, explaining basic techniques in detail.
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A Historical Note. 8. A1. 9 (1) The etymology of the word \ergodic" and the heat theorems. Trying to nd the meaning of the word ergodic one is led to a paper by Boltzmann, Bo84]: see the footnote of S. Brush in his edition, Bo64], of the Lectures on Gas Theory, on p. 10): here Boltzmann's paper is quoted as the rst place where the word is introduced. Brush acutely warns the reader that Ehrenfests' paper misrepresents the opinions and even the terminology of Boltzmann and Maxwell and dates (in agreement with Gibbs, 9 Readers might be interested in the referee report to one of my papers, Ga95a], as it shows, in my opinion, how blind to evidence an historian of Science can be at times.
7. 7), hence for an a priori justi cation of the connection between the microcanonical ensemble and the set of states of macroscopic thermodynamic equilibrium, one meets three main di culties. 3, as a mathematical problem. e. with energy xed within microscopic uncertainty E ) one has to solve the di culty that, in spite of the ergodicity, the elements of the microcanonical ensemble are not ergodic because the (trivial) non ergodicity is due to the fact that in the microcanonical ensemble the energy varies by a small but macroscopic quantity DE E.
Dp d! )=R2 are independent of R (recall that we are considering the case of hard sphere systems). Hence the Boltzmann's equation should describe correctly the evolution of a rare ed gas for arbitrary times t: in fact we expect that in the Grad limit the recurrence times grow in nitely large while the collisions make the system evolve on a time scale xed by the ight time: (( R2 ) v);1 . 17) in which the time scale is xed by R2 jp0 ; p00 j=m which, in the average, is just R2 v. 17) has been proved with complete mathematical rigor only recently and for times t 1= R2 v for systems of hard spheres and for interesting classes of initial data f0 : this is the content of Lanford's theorem on the Grad conjecture, La74].
