A few-degrees-of-freedom Hamiltonian model exhibiting one-directional long-term trends in energy exchange flows is introduced. The model includes a massive potential well - a container with one or few relatively light non-interacting particles \ - attached to a linearly elastic spring... No phenomenological dissipation is imposed, nevertheless, due to a similarity of the container shapes to typical stochastic soft-wall billiards, the energy is transferred from the container (donor) to the inner particles (acceptor) in almost irreversible way during physically reasonable time intervals. The potential well is introduced in such a way that, in the rigid-body limit, it resembles either Sinai billiards or the so-called Buminovich stadiums as the main geometrical parameter of the well switches its sign. In particular, using the nonlinear normal mode stability concept reveals conditions of stochasticity and determines the analogy with the dynamic properties of billiards. Possible applications to the design of macro-level energy absorbers, harvesters, and energy absorbing materials are discussed. (read more)