BOLTZMAN'S DERIVATION OF THE MOST PROBABLE DISTRIBUTION OF MOLECULES IN A SYSTEM (according to Gerhard A. Blass, Theoretical Physics, pp.236-43)
In general, a system of N molecules with certain properties (total enery content, volume occupied, etc.) can exist in variously diverse states, some many in type, others few. The type of state most assumed is "the most probable one", from which deviations are possible, but -- because of the multitude of molecules involved (1 mole = 6 x 1029 molecules) -- the law of large numbers of proability theory implies that such deviations are of very low probaility.

The tendency of a system to move toward the most probable state is equivalent to the second law of thermodynamics: entropy does not macroscopically decrease. So, in calculating this most probable state, Boltzmann was dealing with the foundations of thermodynamics.

  1. Start with a system of N molecules whose representative points move through phase space, subdivided into numbered cells of unequal size, i, each containing unequal numbers of representative points , Ni.
  2. The probability of this arrangement, for equal size cells, is: P = N!/Ni! .
  3. For unequal size cells, modify the numerator by a factor, (i)Ni
  4. This yields the following probability: P = N!(i)Ni/iNi!.
  5. In maximizing such a probability, we conveniently substitute a continuous density function:
             Ni f(i) i D.
  6. And replace the factorial by the appropriate Sterling approximation: N! = (N/e)N, for exponential , e, which, upon substitution of numerator and denominator of probability fraction, divides out.

  7. Also the i to power f(i) i divides out of numerator and denominator, yielding:
    P = NN/f(i)D.
  8. We can then apply Boltzmann's form for entropy, with Boltzmann's constant, k:
          S = k ln P - k N ln N - k ( D) ln f(i).
  9. Now we vary the density factor. f(i), to maximize entropy, S, or, more conveniently, to minimize - S/k. To do so, we keep total number of molecules , so that N = 0; and total energy constant : U = 0. Minimization then leads to - S/k + N + U = 0.
  10. This leads to a term, = exp(- u(i), with S = k U + k N ln .
  11. Then we arrive at Boltzmann's most probable distribution of molecules in a system:
    S = U/T + k ln , where = exp{{u(i)/kT) i).
  12. However, a DIMENSIONAL PROBLEM ARISES IN THIS LAST EQUATION, OVERLOOKED BY BOLTZMANN, OVERLOOKED BY PHYSICISTS OF HIS TIME AND LATER, DOWN TO OUR TIME, EXCEPT FOR BLASS! The single term on the left-side of the equation, namely, entropy, has the dimension of energy divided by temperature, which is exactly the dimension of the first term on the right-hand side. And also part of the second (additive) term on the right-hand side, namely, k N. But this requires that the rest of the part, namely, ln must be dimensionless! which it is not! Rather, since it denotes i, it has dimensions of the phase-space volume, i.e., the dimension of action, the dimension of Planck's constant. How can this part be rendered dimensionless within the equation?
  13. When dS= dQ/T is integrated, it can have a constant of integration. Let it have the dimension of action, which is that of Planck's constant: h. Then, we can find: S = U/T + k ln /hd, where d denotes the number of dimensions on which the molecules move.
  14. Thus, BOLTZMANN'S MOST PROBABLE DISTRIBUTION OF MOLECULES IN A SYSTEM DEMANDS SOMETHING REPRESENTING PLANCK'S CONSTANT! Attention to dimension analysis and constants of integration -- matters often ignored by physicists -- would have exposed this many years before Planck's work! And in statistical mechanics, instead of the crossblend of thermodynamics and optics that Planck dealt with.