Statistical Physics, Second Revised and Enlarged Edition

52 Gases: the distributions

Grouped

distribution

True

distribution

ith level, energyi

containing gistates

gi states

Fig. 5. 1 Distribution ingroups. The true distribution, containingverymanystates indeed, can be replaced
bythegrouped distribution for the purpose of calculation.

out thedistribution. But as physicists we can rest assuredthat more sophisticated

approaches give the same mathematical results even if the groups are not assumed

large (orifthe groupingis not carriedout at all);however, why not accept a

short-cut if oneisoffered?

2. The grouping used in the distribution is arbitrary, but not capricious! The rule is
   that the number of states remains the same in both sides of Fig. 5.1. Or, more
   precisely, the densityof states (over a sufficientlycoarse energy graining) remains
   unchanged. If we choose the energy levelsεi, then the full microscopic details of
   the true physicalproblemdetermine thevaluesgi.


3. A test for whether thegroupingmakes physical sense is to consider the average
   number of particles per state, often called thedistribution functionor colloquially
   thefillingfactor.Thisisdefinedas

fffi=ni/gi

Effectively, this filling factorfffitells us the fractional occupation of a state of energy

εi.In other words, withinthedensityofstates approximation,it canbethought

of purelyas a functionf(ε)of the energyε.The testjust referred to is a simple

one. If the grouping has not violated the original problem, we expect thatfffiwill

beawell-definedfunction ofεiwhosevaluedoesnotdependon thedetailsofthe

grouping. In particular, it should not depend on the value ofgi.

4. Finally, if we return to the example of helium gas (section 4.3), we can see that this
   approachmakes some numericalsense. We notedpreviouslythat the energy level
   spacing between the true states (εεj)was about 1 0 −^19 timeskkkBT. Therefore,it is
   quite possible to use groups of, say, 10^10 states each(so thatgi 1 in anybody’s
   language), whilst maintainingan energy levelspacingεiwhichisstillminute
   (about 1 0 −^9 to 1) compared to the thermal energy scalekkkBT.