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Forums
Physics
Quantum Physics
Minimize grand potential functional for density matrix
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[QUOTE="dRic2, post: 6396823, member: 638830"] [B]TL;DR Summary:[/B] ##\rho## is the density matrix ##\Omega## is the grand potential ##\text{Tr}## stands for 'trace' I'd like to show that, by minimizing this functional $$\Omega[\hat \rho] = \text{Tr} \hat \rho \left[ \hat H - \mu \hat N + \frac 1 {\beta} \log \hat \rho \right]$$ I get the well known expression $$\Omega[\hat \rho_0] = - \frac 1 {\beta} \log \text{Tr} e^{-\beta (\hat H - \mu \hat N )}$$ I'm familiar with minimizing a functional of the form ##F[g] = \int dx f(g(x))##, but this notations for operators eludes me. Thanks, Ric [/QUOTE]
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Physics
Quantum Physics
Minimize grand potential functional for density matrix
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