# Are temperature and chemical potential of a black hole independent quantities?

I am a bit confused about the independent parameters in a charged black hole in AdS spaces. From equation (63) of this lecture notes we see that the temperature (T) of the black hole has chemical potential ($\mu$) dependence.

1. I was wondering, if one wishes to work in this background (using holography) what are the free parameters one can play with?

2. What are the basic differences between canonical and grand canonical ensemble description of this system? Also which ensemble should one choose to work with depending on the problem?

EDIT :

1. What does $\mu \to 0$ mean? Here it seems like charge-free limit. But this limit appears also in Bose-Einstein condensate. Isn't this a bit subtle?
• In principle you have $T$ and $\mu$ as parameters, but only the $T/\mu$ dependence is nontrivial since this is dimensionless. E.g. if $X$ is some quantity with (scaling) dimension $\delta$, you can write $X=X(T,\mu)$ but also $X=X(\frac{T}{\mu},\mu)$. Dimensional analysis then makes the $\mu$ dependence trivial, i.e. $X=f(T/\mu)\mu^\delta$, and you're only left with a function depending on $T/\mu$. – ScroogeMcDuck Jul 15 '15 at 10:10
• In the canonical ensemble you would use the charge density $\rho$ rather than $\mu$ as a parameter and consider the $T/\sqrt{\rho}$ dependence (for 3+1 bulk dimensions) of $X$. I guess the choice of ensemble depends on what you want to describe in the boundary theory – ScroogeMcDuck Jul 15 '15 at 10:15
• @ScroogeMcDuck Thanks a lot for your response! But why do you think they are two independent parameters? – Physics Moron Jul 15 '15 at 16:12
• @ScroogeMcDuck Can you also elaborate on the second comment, I mean, some example of boundary theories in this context.. – Physics Moron Jul 15 '15 at 16:20
• Simply because if I fix the chemical potential, I can still change the temperature. There's still $r_+$. Yes, you can scale to $r_+=1$, but that basically just implies you're considering quantities $\tilde{T}=Tr_+$ and $\tilde{\mu}=\mu r_+$ instead of the physical $T$ and $\mu$. Indeed, then $\tilde{\mu}$ fixes $\tilde{T}$. But I can still fix the physical $\mu$ and change $\tilde{\mu}$ (by changing $r_+$), thereby changing $\tilde{T}$ through (63). – ScroogeMcDuck Jul 15 '15 at 20:59