Entropy is a consequence of a discrete time

7th International Workshop DICE2014 Spacetime – Matter – Quantum Mechanics

物理学;力学

Riek, Roland^1

Laboratory of Physical Chemistry, ETH Zurich, Switzerland^1

关键词: Boltzmann constants;    Constant step sizes;    Microscopic levels;    Number of degrees of freedom;    Second Law of Thermodynamics;    Thermodynamic equilibria;    Time irreversibility;    Time reversibilities;   

Others  :  https://iopscience.iop.org/article/10.1088/1742-6596/626/1/012025/pdf
DOI  :  10.1088/1742-6596/626/1/012025

学科分类：力学，机械学

来源: IOP

【 摘 要 】

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 snleading to the Nosé-Hoover Lagrangian comprising a term NdfkBT In sn(kBthe Boltzmann constant, T the temperature, and Ndfthe number of degrees of freedom) which is defined as the microscopic entropy Snat 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.