A Derivation of a Microscopic Entropy and Time Irreversibility From the Discreteness of Time

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 Delta S > 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 (F (r) not equal 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 s(n) leading to the Nose-Hoover Lagrangian. The corresponding Nose-Hoover Hamiltonian comprises a term N(df)k(B)Tln s(n) (k(B) the Boltzmann constant, T the temperature, and N-df the number of degrees of freedom) which is defined as the microscopic entropy S-n at time point n multiplied by T. Upon ensemble averaging this microscopic entropy S-n 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 s(n) 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.