  In the thermodynamics of nanoscopic systems the relation between classical
and quantum mechanical description is of particular importance. To scrutinize
this correspondence we have picked out two 2-dim billiard systems. Both systems
are studied in the classical and the quantum mechanical setting. The classical
conditional probability density $p(E,L|E_0,L_0)$ as well as the quantum
mechanical transition probability $P(n,l|n_0,l_0)$ are calculated which build
the basis for statistical analysis. We calculate the work distribution for a
particle. Especially the results in the quantum case are of special interest
since already a suitable definition of mechanical work in small quantum systems
is controversial. Furthermore we analysed the probability of both zero work and
zero angular momentum difference. Using connections to an exact solvable system
analytical formulas are given in both systems. In the quantum case we get
numerical results with some interesting relations to the classical case.
