initial conditions (in the absence of a magnetic field) $$\begin{aligned}
m(1,y) & = & \tanh (y/T)\label{condm} \\
P(0,y) & = & \delta (y)\label{condP}\end{aligned}$$ These equations are the starting point of both the expansion discussed in this Section and the numerical solution.

A time scale $\tau_x$ can be associated to the order parameter $q(x)$ such that for times of order $\tau_x$ states with an overlap equal to $q(x)$ or greater can be reached by the system. In this picture the $P(x,y)$ and $m(x,y)$ become respectively the probability distribution of frozen local fields $y$ and the local magnetization in a local field $y$ at the time scale labeled by $x$ [@SDJPC84; @MPV].

The derivatives of $q(x)$ can be obtained by successive $x$-derivation of eq. (\[SP1\]). The procedure is simplified by the use of the following identity [@S85]: $$\label{int1}
\frac{d}{dx}\int dy\,P(x,y)\,f(x,y)=\int dy\,P(x,y)\,\Omega(x,y)\, f(x,y)$$ where $$\Omega(x,y) =\frac{\partial }{\partial x}+\frac{\dot{q}}{2}
       \left( \frac{\partial ^{2}}{\partial y^{2}} + 2\,\beta\,x\,m(x,y)
              \frac{\partial }{\partial y}\right)$$ The application of the operator $\Omega(x,y)$ generates derivatives of the function $m(x,y)$ with respect to $x$ and $y$. Mixed derivatives such as $m^{(1,j)}(x,y)$ can be eliminated in favor of derivatives of $m(x,y)$ respect only to $y$ by deriving equation (\[SP2\]) $j$ times with respect to $y$.

