particle needs to change its velocity by a factor of 2 through weak Coulomb scattering events.

The soft core radius $x_0=r_0/R=5\times 10^{-3}$ is chosen to be well below the critical value $x_{gr}\approx 0.021$, above which the collapse-explosion transition is replaced by a normal first-order phase transition [@chi].

The MF theory of the system is described in detail in Ref.[@usg]. The equilibrium velocity distribution is Maxwellian and isothermal, while the equilibrium (saddle point) density profile $\r({ x})$ corresponding a stable or a metastable state, is a spherically symmetric solution of the integral equation (\[extr\]). This equation replaces the Poisson-Boltzmann differential equation for the self-consistent potential (see, for example, Ref.[@pr]) since the interparticle interaction considered here is not pure Coulombic.

![\[fig\_rho\] MF density profiles $\r_u({x})$ of a uniform state (dashed line) and $\r_{c-h}({x})$ of a core-halo state (solid line) for $\e=\e_{coll}$ ](fig2.eps){width=".45\textwidth"}

$$\begin{aligned}
\label{extr}
\nonumber
\r({\bf x})=\r_0 F[\r(.), {\bf x}]\\
\nonumber
F[\r(.),{\bf x}]=\exp\left [\b \int {\r({\bf x}')\over
\sqrt{({\bf x}-{\bf x}')^2+x_0^2}}d^3{\bf x}'\right ]\\
%\nonumber
\b={3\over 2} \left[\e + {1\over 2}\int \int{\r({\bf x}_1) \r({\bf x}_2)
\over \sqrt{({\bf x}_1-{\bf x}_2)^2+x_0^2} }d^3{\bf x}_1 d^3{\bf x}_2
\right ]^{-1}\\
\nonumber
\r_0=\left\{\int F[\r_s(.), {\bf x}]d^3{\bf x}\right\}^{-1}\end{aligned}$$

The equilibrium density profile $\r({x})$ obtained from this equation is then used to 