on the particle density.](dfases2.eps "fig:"){width="0.49\linewidth"}

We can also examine the transition to the condensate through the behavior of specific heat. In particular we consider the specific heat at constant volume, defined by

$$\label{specificheat}
C_v = \frac{\partial E}{\partial T},$$

where $E = T S+\Omega+\mu N$ is the internal energy of the system. The entropy of the vector bosons gas is

$$\label{entropy}
S=-\frac{\partial \Omega}{\partial T}= -\beta \left(\mu^{\prime} N +\frac{5}{2} \Omega_{st} + \beta \frac{\partial \mu^{\prime}}{\partial \beta} N\right ),$$

with

$$\mu^{\prime} \cong -\frac{\zeta(3/2)T}{4 \pi} \left ( 1-\left(\frac{T_{cond}}{T}\right)^{3/2} \right )\Theta(T-T_{cond})$$

in the low temperature limit. Here $\Theta(x)$ is the Heaviside theta function.

With the use of Eq. (\[entropy\]) the internal energy can be written as

$$\label{energy}
E = \varepsilon(0,B) N + \Omega_{vac}- \frac{3}{2} \Omega_{st} -\beta \frac{\partial \mu^{\prime}}{\partial \beta} N.$$

Eq. (\[energy\]) allows us to obtain for the specific heat the following expression

$$\label{specificheat-1D}
C_v = -\beta \left(\frac{15}{4} \Omega_{st} + \frac{3}{2} \mu^{\prime} N +\frac{1}{2} \beta \frac{\partial \mu^{\prime}}{\partial \beta} N- \beta^2 \frac{\partial^2 \mu^{\prime}}{\partial \beta^2}N\right ).$$

The specific heat has been plotted in Fig. 2 as a function of the temperature for a fixed value of the density $N=10^{32} cm^{-3}$ and three values of the magnetic field. As it is apparent, it has a maximum that is a 