For a dynamical system governed by the equation.
$$\ddot{x}\left ( t \right ) + 2\zeta \omega _{n}\dot{x}\left ( t \right )+\omega _{n}^{2}x\left ( t \right ) = 0$$
the damping ratio $\zeta$ is equal to $\dfrac{1}{2\pi}\:\log_{e}2$. The displacement $x$ of this system is measured during a hammer test. A displacement peak in the positive displacement direction is measured to be $4$ $\text{mm}$. Neglecting higher powers $(>1)$ of the damping ratio, the displacement at the next peak in the positive direction will be __________________ $\text{mm}$ (in integer).