| MadSci Network: Physics |
Boltzmann, Gibbs and Maxwell are credited with formulating the main ideas of statistical mechanics, and they used classical mechanics to do it. Their basic assumption was the existence of atoms, but they did not assumed discrete energy levels. To them atoms simply were microscopic balls of some sort that moved smoothly in space with definite momentum. Each arrangement of the atoms in position and momentum defined a different state, and the thermodynamic state was the most probable state. With this idea Boltzmann was able to prove the second law of thermodynamics without assuming anything else about atoms. For his belief in atoms he received a lot of criticism, which partly led to his suicide in 1906. It was actually Gibbs the one who formulated more generally the original ideas of Boltzmann about entropy. Neither Boltzmann nor Gibbs lived to see their ideas formulated in terms of quantum states and many inconsistencies of their work solved by quantum mechanics. An interesting discussion of these ideas are found in chapters 17 to 19 of "Chance and Chaos" by David Ruelle.
Today Boltzmann's and Gibbs' ideas are more conveniently presented in terms of quantum mechanics, but it is possible to take the limit to classical physics, and go back to the math that Boltzmann and Gibbs did. The classical limit applies when the number of states that are available to molecules or atoms are much greater than the number of molecules or atoms in the system. This happens generally at high temperature and for large mass particles. In that limit the number of quantum states becomes a volume of phase space defined by the coordinates and momenta of the particles, and the sum over quantum states becomes integrals over phase space. The molecular partition function q and the canonical partition function Q for a system of N bodies described by s pairs (q,p) in the classical limit are:
Planck's constant h appears in the classical partition functions in order to obtain the values that today we know that are correct from quantum mechanics. Look up the chapter on classical statistical mechanics in "Statistical Thermodynamics" by Donald A. McQuarrie or any other book on statistical mechanics.
Vladimir Escalante Ramírez
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