In earlier works we have shown how it is possible to modify a largely arbitrary Hamiltonian H to a different Hamiltonian depending on an arbitrarily assigned (positive) parameter Ω, so that all orbits of have period but the dynamics of resembles closely that yielded by H over times much shorter than the period . In this paper, we apply our approach to a standard Hamiltonian model describing the interaction of a particle with a large, but finite, set of harmonic oscillators, which can approximate well the behaviour of a thermal bath. We thereby show that the second law of thermodynamics can be strongly violated for times of order , which can be chosen independently of N, in a system whose dynamics resembles closely that of an 'ordinary' N-particle system over times of the order of the relaxation times. Since the recurrence period is independent of the initial condition, no averaging over initial conditions allows us to recover the usual irreversible behaviour. The consequences concerning the hypotheses necessary for the validity of the second law are briefly discussed.