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Consider a quantum system with discrete states say $\left|E_n\right>$. If one wants to write the partition function for a bunch of similar systems, they would sum the terms $e^{-\beta E_n}$. The point is that they consider all states with the same weight, $\beta$, which is independent of the microscopic structures of the system. My question is that, is there any underlying physics law that dictates the same-weight assumption? Similarly, if one violates this assumption (or anything else you may call), would it indeed violate an important physics law?

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  • $\begingroup$ Well, not really the same weight. $E_n$ is part of the weight, the state probabilities are weighted by the values of their energies divided by $k\beta$T. A higher energy state contributes less to the partition function. $\endgroup$
    – Bob Bee
    Apr 3, 2017 at 1:41

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I don't think it so much "violates a law" as removes all the physical meaning from the expression. The partition function encodes the physics of the system in the structure of the energy levels. If you multiply each of those energies by an arbitrary constant, you are not going to have a lot of that structure left. Your partition function will be a fairly general function with no connection to reality.

To look at this from a different angle we arrive at the partition function in the canonical ensemble by maximising the entropy subject to the constraint that the system has a fixed average energy. To do this we introduce a Lagrange multiplier and find the stationary points of $$ \Lambda(\beta,p_1,p_1,\dots) = S(p_1,p_2\dots) + \beta\left(U -\sum_ip_iE_i\right) $$ where $p_i$ is the probability that the system is in state $i$ and $S$ is the Gibbs entropy. Notice that $\beta$ is already multiplying all all the $E_i$s in the first line where it enters the derivation. Introducing a multiple $\beta$s would mean you were individually constraining the average energy levels of different states separately, which does not strike me as a physically meaningful thing to do.

There is one situation, however, where you do encounter multiple weightings, and that is when two different systems, initially with different weightings $\beta_1$ and $\beta_2$ are brought into thermal contact. In this case the partition function is given by a slightly different expression from the one you were proposing $$ Z = \sum_i e^{-\beta_1E_{1i}-\beta_2E_{2i}} $$ where the sum is taken over states of the combined system. It is a standard exercise to show that if the total entropy is maximised $$\beta_1=\beta_2\,.$$ From the Zeroth Law we conclude that $\beta$ can be viewed as a direct function of the temperature and indeed a little further analysis shows that in fact $$ \beta = \frac{1}{k_b T} $$

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