Introduction to Statistical Mechanics
It is probably clear by now that the gas particles, of which there are several different varieties, have electronic excitation, vibrational, rotational, spin and translational energies. While it is impossible to keep track of the exact total quantum energ
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It is probably clear by now that the gas particles, of which there are several different varieties, have electronic excitation, vibrational, rotational, spin and translational energies. While it is impossible to keep track of the exact total quantum energy an individual particle may possess at any given time, it is possible to determine for a large number of particles the approximate percentage distribution of the energy with suitable auxiliary conditions like the total energy being kept constant. Methods by which the statistics of the possible energy distributions are made do vary no doubt, but the most probable distribution for a large number of particles to be accommodated in a still larger number of energy levels seems to give similar results, as will be shown later. While trying to obtain the statistics of the energy distributions of the particles, we denote the particles, considered to be balls, with letters like a, b, c, d, . . . , so that we can exactly distinguish these from each other, and place these in boxes each having a definite energy. Thus, merely placing one ball in one particular box allows the ball to attain the particular energy. Now let the number of balls (particles) be N and the number of boxes (energy levels or energy states) be g. If the balls are distinguishable from each other (since they are labelled) and they are put into the boxes without any restriction on the number of balls in each box, the distribution is called Boltzmann statistic. In case the balls are not distinguishable, then it is Bose statistic developed by Satyendra Nath Bose oflndia in the twenties of the twentieth century, and subsequently used by Einstein for the statistics of electromagnetic radiating particles (photons). However, if the particles are not distinguishable from each other, and if the number of particles in each box is restricted to a maximum of one for each box, since, according to Pauli principle no two particles may have exactly the same energy, then it is the Fermi statistic. Obviously in this last case, N is less or equal to g. While the consequence of these different statistical procedures are examined later, in the sections that follow we discuss without proof some possible arrangements, without any auxiliary restrictions on the total energy.
(a) N balls are all distinguishable from each other, and they are placed maximum one in g boxes. Obviously, N is less or equal to g. While the first ball can be placed in any of the g boxes, the second ball can be placed only in T. K. Bose, High Temperature Gas Dynamics © Springer-Verlag Berlin Heidelberg 2004
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3 Introduction to Statistical Mechanics
(g- 1) boxes. Thus for the two balls there are g(g- 1) possibilities, and in a similar fashion for N balls placed maximum one in each box, the number of possibilities is
W1 = g(g- 1)(g- 2) .. .(g- N
+ 1) =
g! (g _ N)!
(3.1)
If g = N, obviously, W1 = g! since 0! = 1. Example: N = g = 3, the balls are numbered a, b, c. Possible arrangements
in the three boxes are a b c and
b c a
b a c
a c b
c a b
c b
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