consider such distributions since on average one does not expect an effect that would lead systematically to values grater or smaller than the mean value), we see that one would have obtained the traditional evolution to leading order plus a corrective term, $${\partial \rho(T)\over \partial T} =i [\rho(T),H] +\sigma(T) [H,[H,\rho(T)]].$$ and the extra term is dominated by the rate of change of the width of the distribution $\sigma(T)=\partial b(T)/\partial T$.

An equation of this form has been considered in the context of decoherence due to environmental effects, it is called the Lindblad equation [@lindblad], $$\frac{d}{dt}\rho=-i[H,\rho]-{\cal D}(\rho)\label{rho},$$ with $${\cal D}(\rho)=\sum_n[D_n,[D_n,\rho]], \;\;\;
D_n=D_n^{\dagger},\;\;\; [D_n,H]=0,$$ and in our case there is only one $D_n$ that is non-vanishing and it coincides with $H$. This is a desirable thing, since it implies that conserved quantities are automatically preserved by the modified evolution. Other mechanisms of decoherence coming from a different set of effects of quantum gravity have been criticized in the past because they fail to conserve energy [@hag]. It should be noted that Milburn arrived at a similar equation as ours from different assumptions [@milburn]. Egusquiza, Garay and Raya derived a similar expression from considering imperfections in the clock due to thermal fluctuations [@egusquiza]. 