occasional meanderings in physics' brave new world

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Location: New Zealand

Marni D. Sheppeard

## Friday, June 06, 2008

### M Theory Lesson 194

The denominator of the Fermi function is derived from the partition function

$Z = 1 + \textrm{exp}(- \frac{E - \mu}{kT})$

for the 2 possible occupancies of a fermion state, namely 0 or 1. A ternary analogue resulting in tripled Pauli statistics would require

$Z = 1 + 3 \textrm{exp} (- \frac{E}{kT})$

where we arbitrarily shift the energy scale, momentarily. Presumably this corresponds to the three possible ways of occupying the state with one particle, whereas for ordinary fermions there is only one way of occupying a state. Another interpretation is to write

$3 \textrm{exp} (- \frac{E}{kT}) = \textrm{exp} (- \frac{E}{kT} + \textrm{log} 3)$

where $\textrm{log} 3$ is an energy level (for one prime object) in the Riemann gas system, whose complete partition function is the Riemann zeta function $\zeta (s)$ for $s = (kT)^{-1}$.