Distributions and the Chemical Potential

The quantum mechanical world has a further complexity concerning distinguishability. In Chapter 10, we found that there are two kinds of indistinguishable particles, the difference related to the intrinsic spin of particles. A rigorous analysis of arrangemenfs requires that we first consider if or under which conditions these differences are important. 

If we considered the products in Equations 11.5 through 11.7 to go over only one level, we can show that 

Equation 11.9 uses the explicit expressions for the A numbers, but with the common n term removed. Consider the left inequality for the case s = 8 and n = 2 8(7) 8 9(8). This illustrates that the indistinguishable boltzon result may serve as an approximation to fermion or boson statistics. 

The three equations for arrangements, Equations 11.5 through 11.7, can be written as one by defining a quantity, 5, such that 

The differential of this expression, recognizing that the s/s are constants, is 

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