conclusion $$\label{eq:q30}
\lim_{x\to 0} q^{(3)}(x) > 0.$$ In Figure \[fig:q2-3x.T0.6-1.000\] we show the second and third derivative of $q(x)$ obtained from numerical differentiation of $q(x)$. The agreement with the perturbative result is sufficiently good, moreover from the right panel of Figure \[fig:q2-3x.T0.6-1.000\] we clearly see that the prediction (\[eq:q30\]) is verified.

We conclude this Section with a short discussion on the entropy which, using the stationarity of the free energy functional (\[eqfrev\]), can be written as: $$s = -\frac{\beta^2}{4}\bigl[1-q(1)\bigr]^2
     + \int_{-\infty}^{\infty} dy \ P(1,y)
\bigl[\log 2 \cosh \beta y - y \ \tanh(\beta y)\bigr].
\label{s}$$ For other equivalent forms see, e.g., Ref. [@Luca]. The entropy as function of temperature is shown in the left panel of Figure \[fig:s-q1\]. The entropy must vanish quadratically with the temperature as $T\to 0$ [@SDJPC84]. From our numerical data we find $$\lim_{T\to 0} \frac{s(T)}{T^2} = a \simeq 0.72$$ to be compared with $0.718\pm 0.004$ of the analytic expansions.

In the limit $T\to 0$ the quantity $1-q(1)$ must also vanish as $T^2$ [@SDJPC84]. The behaviour of $q(1)$ as function of $T$ is shown in the right panel of Figure \[fig:s-q1\]. Using this data we obtain $$\lim_{T\to 0} \frac{1 - q(1)}{T^2} \simeq 1.60$$ in very good agreement 