energy change $\Delta E_{\rm rand}$ after interaction with a spiral arm. Here, the random energy is calculated as the difference between the total energy and the circular energy, i.e., $E_{\rm rand} = E - E_{\rm cir}(R_{\rm gc})$. It is noteworthy that the random energy change is [*not*]{} always positive; a significant fraction of stars [*lose*]{} their random energy. This is because the perturbation from the spiral arm shifts the guiding center of the stars’ epicycle motion without increase in orbital eccentricity. It is to be noted that this argument is rigorously correct; in fact, Figure \[fig:L-t12\] (bottom left panel) shows the changes in the random energy of the stars as a function of change in the angular momentum. We can see a weak trend: the outer migrators (i.e., $L_{\rm fin}-L_{\rm ini}>0$) undergo a decrease in their random energy ($E_{\rm rand,fin}-E_{\rm rand,ini}<0$), and vice versa. A similar effect of the radial migration of stars around spiral arms upon disk heating has been noticed in recent numerical simulations [@Grand+2012; @Roskar+2011; @Grand+2012b; @Minchev+2012].

In summary, the gravitational interaction between the stars in the spiral arm and the spiral density enhancement changes the angular momentum and random energy of the stars, and this process 