Evolution of energy density distribution

The energy absorption in a stopping media and its transport can be described analogously in the classical and quantum-mechanical approaches. As an instructive case and for further reference we first consider the classical gas of N interacting electrons, the system described by the Hamiltonian 

To elarify the general procedure we will first consider a system consisting of only one electron (At = 1, W = 0). The general case can he treated analogously although this needs a more complicated procedure. By a simple transformation equation (3) can he rewritten as 

Taking the integral over p of both sides of this equation we arrive at the continuity equation in ordinary space 

multiplying both sides of equation (4) by Hq v/e obtain the equation for the evolution of the energy distribution in phase space (the change of Hq averaged over the distribution/(r, p, t) presents the energy deposited in the system)  

Note that, at V = 0, the divergence type of the right-hand side of this equation ensures the conservation of total energy. Again, the integration over p results in the equation 

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