 spatial dimension of
the D-brane bound state. The only change is the appearance of
extra charges and  chemical potentials, but the temperature $T$,
entropy $S$, angular velocity $\Omega_i$ and angular momentum
$J_i$ were unchanged. We now consider the thermodynamics of the
near-horizon solution given in Section \ref{secnearhorsol}, which
can be obtained from \eqref{pbranethermo} using the rescalings
\eqref{rescal1}, \eqref{rescal2}. We also note that the energy is
computed from the energy above extremality $E=M- (\sum Q^2)^{1/2}$
in the near-horizon limit, where the sum is over all charges in
the bound state. Here, it is used that at extremality the bound
state is a 1/2-BPS state, and we recall that the expressions of
the charges and chemical potentials are given in Appendix
\ref{apprrpot}.

One then finds that  the charges as well as the chemical
potentials are constant in the near-horizon limit, and hence do
not appear in the thermodynamics. For the remaining thermodynamic
quantities we find in the near-horizon limit
\begin{subequations}
\label{nhthermdyn}
\begin{equation}
\label{nhtands}
T = \frac{w_p}{4\pi} \lambda^{-1/2} (7-p-2\kappa)
\frac{r_0^{\frac{7-p}{2}}}{r_H}
\sp
S = 4\pi \hat{w}_p V_p N^2 \lambda^{-3/2} r_0^{\frac{7-p}{2}} r_H
\end{equation}
\begin{equation}
\label{nhoandj}
\Omega_i = w_p \lambda^{-1/2} r_0^{\frac{7-p}{2}} \frac{l_i}{l_i^2+r_H^2}
\sp
J_i = 2 \hat{w}_p V_p N^2 \lambda^{-3/2} r_0^{\frac{7-p}{2}} l_i
\end{equation}
\begin{equation}
\label{nhenergy}
 E = \frac{9-p}{2} \tilde{w}_p V_p N^2 \lambda^{-2}
r_0^{7-p}
\end{equation}
\end{subequations}
where we have defined the NC 't Hooft coupling,
\begin{equation}
\label{nchooft}
 \lambda = g_{\mathrm{YM}}^2 N \prod_{k=1}^m b_k
\end{equation}
and where
\begin{subequations}
\begin{equation}
w_p = \sqrt{ \frac{(7-p)V(S^{8-p})}{(2\pi)^{9-2p}}}
\sp
\hat{w}_p = (2\pi)^{2p-11} \sqrt{\frac{ (2\pi)^{9-2p} V(S^{8-p})}{7-p} }
\end{equation}
\begin{equation}
\tilde{w}_p = w_p \hat{w}_p = (2\pi)^{2p-11} V(S^{8-p})
\end{equation}
\end{subequations}
It is not difficult to verify that the energy
satisfies the first law of thermodynamics
\begin{equation}
\label{nhfirstlaw}
 dE = T dS + \sum_{i=1}^n \Omega_i dJ_i \sp
E=E(S,\{ J_i \})
\end{equation}
and the integrated Smarr formula \cite{Harmark:1999xt}
\begin{equation}
(7-p) E = \frac{9-p}{2} TS + \frac{9-p}{2} \sum_{i=1}^n \Omega_i
J_i
\end{equation}

Comparing the thermodynamics \eqref{nhthermdyn} with the one obtained
for the near-horizon spinning D$p$-brane in Ref.
\cite{Harmark:1999xt} we observe that they are identical up to the
replacement
\begin{equation}
g_{\mathrm{YM}}^2 \rightarrow g_{\mathrm{YM}}^2 \prod_{k=1}^m b_k
\end{equation}
where $g_{\mathrm{YM}}$ on the left-hand side is the YM coupling constant of
the commutative theory and $g_{\mathrm{YM}}$ on the right-hand side of the
non-commutative theory. For the non-rotating case this was found
in
\cite{Maldacena:1999mh,Alishahiha:1999ci,Barbon:1999mx,Cai:1999aw}
and argued at weak coupling from the field theory point of view in
\cite{Bigatti:1999iz}.

For various applications it is also useful to compute the Gibbs
free energy $F$ from Eqs. \eqref{nhthermdyn}, with the result
\begin{equation}
\label{nhgibbs}
F = E-TS -\sum_{i=1}^n \Omega_i J_i = - \frac{5-p}{9-p} E
= - \frac{5-p}{2} \tilde{w}_p V_p N^2 \lambda^{-2} r_0^{7-p}
\end{equation}
satisfying
\begin{equation}
dF = - S dT - \sum_{i=1}^n J_i d\Omega_i
\sp
F = F(T,\{ \Omega_i \})
\end{equation}
For $p=2,3,4$ and one non-zero angular momentum the exact
form of the free energy in terms of the intensive thermodynamic
quantities is \cite{Harmark:1999xt}
\begin{subequations}\label{strongdpfe}
\begin{equation}
 F (T,\Omega) = - c_p V_p N^2
\lambda^{-\frac{p-3}{p-5}} T^{\frac{2(7-p)}{5-p}}
(1+x)^{2(6-p)/(5-p)} \left( 1 + \frac{x }{x_c}
\right)^{-2\frac{7-p}{5-p}}
\end{equation}
\begin{equation}
\label{xsol} x \equiv \frac{l^2}{r_H^2}
= 8 \left( \frac{7-p}{4 \pi} \right)^2
 \frac{\omega_c^4}{ \omega^2}
\left( 1 - \frac{1}{2} \left( \frac{ \omega}{
\omega_c} \right)^2 - \sqrt{ 1- \left( \frac{ \omega}{
\omega_c} \right)^2 } \right)
\sp \omega \equiv \frac{\Omega}{T}
\end{equation}
\end{subequations}
where the constants $c_p$ can be found in Ref. \cite{Harmark:1999xt}
and
\begin{equation}
\label{xc}
 x_c = \frac{7-p}{5-p} \sp \omega_c = \frac{2\pi}{\sqrt{(7-p)(5-p)}}
\end{equation}
The boundaries of stability and critical exponents for these cases
can also be found in Ref. \cite{Harmark:1999xt}, and we note in
particular the stability bound $2 \pi J/S \leq \sqrt{x_c}$, with
the boundary of   stability at $\omega = \omega_c$. For more than
one angular momentum the free energy has the same prefactor as in
\eqref{strongdpfe} and the expansion in powers of $\omega$
including the first two subleading terms can be found in
\cite{Harmark:1999xt}. For the case $p=5$ the free energy is zero
since $T$ and $\Omega_i$ are not independent variables anymore.
For the corresponding results in the D6-brane case we refer to
Section \ref{secd6}, which includes a more detailed account of the
thermodynamics as it shows some exotic behavior and will be
relevant for  the 7-dimensional non-commutative field theory that
is dual to this supergravity solution.




\subsection{Ranges of validity
\label{secrangeval} }

In this section we discuss the range of validity in which the
thermodynamics of the near-horizon  limit (see  Section
\ref{secthermquan}) describes the thermodynamics of