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Entropy 2014, 16(6), 3149-3172; doi:10.3390/e16063149
Article

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

Received: 2 April 2014; in revised form: 4 May 2014 / Accepted: 21 May 2014 / Published: 6 June 2014
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Abstract: 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 ΔS > 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) ≠ 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 Nosé-Hoover Lagrangian. The corresponding Nos´e-Hoover Hamiltonian comprises a term Ndf kB T ln 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 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
Keywords: Entropy; discrete time; inflation of the universe; scaling of time; time reversibility Entropy; discrete time; inflation of the universe; scaling of time; time reversibility
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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MDPI and ACS Style

Riek, R. A Derivation of a Microscopic Entropy and Time Irreversibility From the Discreteness of Time. Entropy 2014, 16, 3149-3172.

AMA Style

Riek R. A Derivation of a Microscopic Entropy and Time Irreversibility From the Discreteness of Time. Entropy. 2014; 16(6):3149-3172.

Chicago/Turabian Style

Riek, Roland. 2014. "A Derivation of a Microscopic Entropy and Time Irreversibility From the Discreteness of Time." Entropy 16, no. 6: 3149-3172.


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