the energy density $\mathcal{H}$ of the electro-magnetic field, $$\mathcal{H} = \frac 1 2 F_{0A} F_{0A} + \frac 1 4 F_{AB} F_{AB}\;,$$ minus the energy density $\frac{1}{2} E^2$ of the constant electric field. Thus, we are effectively considering a classical particle with coordinate $h(t)$ moving in the potential $$U= -\frac 1 2 k(4\alpha E - k)h^2 + 2\alpha^2 k^2 h^4\;.
\label{potential}$$ This is a double well potential for $0<k<4\alpha E$ as in Figure \[fig:doublewell\]. The original homogeneous phase corresponds to the point $h=0$, which is unstable. If we add some perturbation, the amplitude $h(t)$ starts oscillating as in the figure.

![Double well potential $U$ for a classical particle with coordinate $h$. If the particle starts slightly outside of the origin, say at $A$, then the particle will oscillate between $A$ and another point $B$ with the same potential energy.[]{data-label="fig:doublewell"}](doublewell.png){width="5cm"}

The Final Configuration
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We have seen that the instability induces an oscillatory solution in the potential (\[potential\]). Suppose that our system is weakly coupled to a heat reservoir with a large heat capacity at very low temperature. Eventually the oscillation will fade away by transferring its energy to the heat reservoir and the system will land on its lowest energy state. 