![\[fig1\] Density of bosons as a function of $p_3$ for several values of magnetic field(left plot) and density of boson as a function of $p_3$ for different values of temperatures (right plot). In both graphics $N=10^{34} cm^{-3}$.](bosonsfixedT.eps "fig:"){width="0.49\linewidth"} ![\[fig1\] Density of bosons as a function of $p_3$ for several values of magnetic field(left plot) and density of boson as a function of $p_3$ for different values of temperatures (right plot). In both graphics $N=10^{34} cm^{-3}$.](bosonsfixedb.eps "fig:"){width="0.49\linewidth"}

The region of temperatures around which the diffuse phase transition occurs may be estimated from the specific heat. To compute the specific heat, we need the energy and the entropy of the one dimensional gas. They read as

$$\label{entropy-1D}
S^{1D}=-\frac{\partial \Omega^{1D}}{\partial T}= -\frac{1}{T}\left(\mu^{\prime} N +\frac{3}{2} \Omega^{1D}_{st}+ \frac{2 \varepsilon(0,B)T^2}{N}\right),$$

$$\label{energy-1D}
E^{1D} = T S^{1D}+\Omega^{1D}+\mu N = \varepsilon(0,B) \left ( N - \frac{2 T^2}{N}\right ) + \Omega^{1D}_{vac}- \frac{1}{2} \Omega^{1D}_{st}.$$

From Eq.(\[energy-1D\]) the specific heat is

$$\label{specificheat-1D}
C^{1D}= \frac{\partial E^{1D}}{\partial T} = -\frac{1}{2 T} \left ( \mu^{\prime} N +\frac{3}{2} \Omega_{st} + 10 \frac{\varepsilon(0,B) T^2}{N}\right ).$$

Eq.(\[specificheat-1D\]) has been plotted as a function of temperature in the right panel of Fig. 4. From this figure it can be seen that the specific heat has a maximum. Is position 