[QUE/SM-02013] SM-PROBLEM

Consider a one dimensional damped motion of a particle, given by the equations
$$ \frac{dq}{dt}\,=\,\frac{p}{m}\,,\qquad \frac{dp}{dt}\,=\,mg\,-\,\gamma \frac{p}{m}\,$$
where $p$ and $q$ are the momentum and the position of the oscillator.

(a) Calculate the change in volume in phase space $\Omega(t)$ as a function of $t$. In particular, start with rectangular region $ABCD$ with coordinates $A(Q_1\,,\,P_1);\,B(Q_2\,,\,P_1)\,;\,C(Q_1\,,\,P_2)$ and $D(Q_2\,,\,P_2)$ and use its development in time to show that
$$ \Omega(t)\,=\,\Omega(0)e^{-\gamma t/m} $$
(b) What does it imply for the entropy of the system ? ( assume the damping is such that the system can be treated to be in equilibrium at all times)

(c) Does this violate the second law of thermodynamics? Give arguments to support your answer.