  All of the basic microsopic physical laws are time reversible. In contrast,
the second law of thermodynamics, which is a macroscopic physical
representation of the world, is able to describe irreversible processes in an
isolated system through the change of entropy S larger than 0. It is the
attempt of the present manuscript to bridge the microscopic physical world with
its macrosocpic one with an alternative approach than the statistical mechanics
theory of Gibbs and Boltzmann. It is proposed that time is discrete with
constant step size. Its consequence is the presence of time irreversibility at
the microscopic level if the present force is of complex nature (i.e. not
const). In order to compare this discrete time irreversible mechamics (for
simplicity a classical, single particle in a one dimensional space is selected)
with its classical Newton analog, time reversibility is reintroduced by scaling
the time steps for any given time step n by the variable sn leading to the
Nose-Hoover Lagrangian. The corresponding Nose-Hoover Hamiltonian comprises a
term Ndf *kB*T*ln(sn) (with kB the Boltzmann constant, T the temperature, and
Ndf the number of degrees of freedom) which is defined as the microscopic
entropy Sn at time point n multiplied by T. Upon ensemble averaging this
microscopic entropy Sn in equilibrium for a system which does not have fast
changing forces approximates its macroscopic counterpart known from
thermodynamics. The presented derivation with the resulting analogy between the
ensemble averaged microscopic entropy and its thermodynamic analog suggests
that the original description of the entropy by Boltzmann and Gibbs is just an
ensemble averaging of the time scaling variable sn which is in equilibrium
close to 1, but that the entropy term itself has its root not in statistical
mechanics but rather in the discreteness of time.
