A mechanism of occurrence of irreversibility is found. It is based on allowance for the relativistic factor in a mechanical system consisting of a finite number of particles interacting with one another in accordance with the law of elastic collision and moving in a container of fixed volume with a natural law of reflection from the container walls. A new concept of thermodynamic entropy of the system is introduced, and it is shown that for fairly general conditions both the new entropy and the Gibbs entropy for this system of particles increases unboundedly with the time when the relativistic factor is taken into account, whereas when it is not the new entropy is constant, not dependent on the time, while the Gibbs entropy fluctuates in a bounded range as the time increases without limit. It is also shown that in the relativistic case the energy of all the particles tends to infinity exponentially with the time.