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Consider the function $$\Omega(N,E)=\int dE_1 \int dE_2 \cdots \int dE_N \Omega_1(E_1)\Omega_2(E_2) \cdots \Omega_N(E_N)\delta(E-E_1-E_2\cdots -E_N)$$

Is there a necessary condition on the $\Omega_i$'s such that under the limit $N \rightarrow \infty$, $E \rightarrow \infty$, while $E/N = \epsilon$ remains finite, one has

$$\lim \frac{\partial^k}{\partial E^k}\log \Omega(N,E) = 0$$ $\forall k\ge 2$?

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  • $\begingroup$ Yes. For example, if $\Omega_1 \equiv 1$ and $\Omega_k$ is the characteristic function of $[0,1]$, for all $k$, then $ \Omega(N,E) \equiv 1$. You can certainly reformulate this into some sort of sufficiency condition. If this is not what you are asking for, please edit the question to clarify. $\endgroup$ Apr 20, 2016 at 19:41
  • $\begingroup$ Thank you for answering. Sorry! "Necessary" and "sufficient" always confuse me. I mean necessary. Corrected in the question. $\endgroup$
    – velut luna
    Apr 20, 2016 at 20:05

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