Let $\mathrm{\Delta }OAB$ be an equilateral triangle with side length unity ($O$ being the origin). Also, $M$ and $N$ are the points of trisection of $AB$, $M$ being closer to $A$ and $N$ being closer to $B$. Position vectors of $A, B, M$ and $N$ are $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{m}$ and $\overrightarrow{n}$ respectively, then the value of $\overrightarrow{m}\cdot \overrightarrow{n}$ is equal to