- #1

dRic2

Gold Member

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- TL;DR Summary
- ##\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

$$\Omega[\hat \rho] = \text{Tr} \hat \rho \left[ \hat H - \mu \hat N + \frac 1 {\beta} \log \hat \rho \right]$$

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