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While playing around with a variant of the one-dimensional Ising model with periodic boundary conditions I came up with a formula, let's call it $F$, whose form looks suspiciously like an entropy formula, although as far as I can tell it shouldn't have anything to do with entropy.

The form is:

$F = -\sum \frac{1}{b_i}ln(1 - \alpha^{b_i})$

where $\alpha$ is a fixed real number $0 < \alpha < 1$, and $b_i > 0$. Without getting into the gritty details, my question is a simple one: Has anyone encountered such a formula before, possibly related to entropy?

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It looks like a free energy of a system of bosons, you're summing over their states $i$ where you may have $N_i$ bosons in each. The term $1/b_i$ is somewhat strange, the closest reason why it's inverse is that $b_i=\beta$, a kind of inverse temperature, which explains why it's in the exponent and why $1/b_i=T_i$ is in front of the log. But not sure why the temperature would depend on $i$. With $b_i$ interpreted as the energy, I get a problem with the prefactor. – LuboŇ° Motl Oct 12 '11 at 16:44
Hi LuboŇ°. Thank you for your great answer! If you could post this as an answer I will most definitely accept it. – Unreasonable Sin Oct 13 '11 at 17:14

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