corresponds to the set of coupled equations: $$\begin{aligned}
  \ddot X_i & = & [X_j,[X_i,X^j]] \nonumber \\
  & & + [X_j,|\delta x_i\rangle \langle \delta x^j|-|\delta x^j\rangle \langle \delta x_i|] - \{|\delta x_j\rangle \langle \delta x^j|,X_i-x_i\} \nonumber \\
  & & + |\delta x_j\rangle \langle \delta x_i|(X^j-x^j) - (X^j-x^j)|\delta x_i\rangle \langle \delta x_j|\end{aligned}$$ $$\ddot x_i = 2\langle \delta x_j|(X_i-x_i)|\delta x^j\rangle -2\Re\mathrm{e}(\langle \delta x_j|(X^j-x^j)|\delta x_i\rangle)$$ $$\begin{aligned}
 \label{equfluct}
  |\delta \ddot x_i \rangle & = & - |X-x|^2|\delta x_i\rangle -([X_i,X^j]-(X^j-x^j)(X_i-x_i))|\delta x_j \rangle \nonumber \\
  & & -\langle \delta x^j|\delta x_j\rangle |\delta x_i\rangle +(2\langle \delta x_i|\delta x^j\rangle - \langle \delta x^j|\delta x_i\rangle)|\delta x_j\rangle\end{aligned}$$ where $|A|^2 = A_i A^i$ for a set of matrices $\{A_i\}_{i=1,2,3}$ and $\{\cdot,\cdot\}$ denoting the anticommutator. We are interested by the last equation which governs the evolution of the fluctuations. Some examples are drawn figure \[fluctuations\].

![\[fluctuations\] Evolution of the amplitudes of the fluctuations $|\delta x_i(t) \rangle$, for the fuzzy sphere, ellipsoid, torus and the noncommutative plane. Initial conditions for $|\delta x_i(t=0) \rangle$ are choosen following a gaussian law with $\sigma=0.01$. The number of D0-branes is $N=201$ (for the noncommutative plane the cutoff is $N=11$) (simulations show that the thermalization duration grows with $N$).](fluctSphere201.eps "fig:"){width="6.4cm"} ![\[fluctuations\] Evolution of the amplitudes of the fluctuations $|\delta 