fermionic string is then driven by a classical chaotic system.\
The classical chaos in the dynamics of $X(t)$ is established, but we want to know if the quantum Hamiltonian $H^{eff}$ presents quantum chaos. A commonly used criterion of quantum chaos is the level spacing distribution (LSD) of the spectum [@Haake]. A regular system presents a LSD as Dirac peaks, a (pseudo)-random system presents a LSD as a Poisson distribution and a chaotic system presents a LSD as a Wigner-Dyson distribution. The LSD for $H^{eff}$ for a fuzzy sphere at a time after thermalization is represented fig. \[LSD\].

![\[LSD\] Level spacing distribution of $H^{eff} = \sigma^i \otimes (X_i-x_i)$ at a time after the thermalization for a fuzzy sphere with the initial fluctuations choosen following a gaussian law with $\sigma=0.01$, compared with the Poisson distribution (green) and with the Wigner-Dyson distribution (red). The number of D0-branes is $N=201$.](LSDsphere.eps){width="6.4cm"}

We see clearly that the LSD of $H^{eff}$ (after thermalization) follows a Wigner-Dyson distribution characteristic of the quantum chaos (we have the same results with the other models). It is interesting to note that the thermalization does not involve a pseudo-random behaviour but well a quantum chaotic behaviour.\
We have then both classical and 