# Why all energies have the same contribution in statistics of a system?

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?

• 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. Apr 3, 2017 at 1:41

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 $$\beta = \frac{1}{k_b T}$$