Entropy is a consequence of a discrete time

Abstract

While the basic microscopic physical laws are time reversible, the arrow of time and time irreversibility appears only at the macroscopic physical laws by the second law of thermodynamics with its entropy term S. It is the attempt of the present work to bridge the microscopic physical world with its macroscopic one with an alternative approach than the statistical mechanics theory of Gibbs and Boltzmann. For simplicity a “classical”, single particle in a one dimensional space is selected. In addition, it is assumed that time is discrete with constant step size. As a consequence time irreversibility at the microscopic level is obtained if the present force is of complex nature (F(r) ≠ const). In order to compare this discrete time irreversible mechanics 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 Nosé-Hoover Lagrangian comprising a term NdfkB T In sn (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 of the microscopic entropy in a many particles system in thermodynamic equilibrium it approximates its macroscopic counterpart known from statistical mechanics. The presented derivation with the resulting analogy between the ensemble averaged microscopic entropy and its statistical mechanics analog suggests that the entropy term itself has its root not in statistical mechanics but rather in the discreteness of time.