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

Laboratory of Physical Chemistry, ETH Zuerich, Wolfgang-Pauli-Strasse 10, HCI F 225,Zurich CH-8093, Switzerland

Received: 2 April 2014 / Revised: 4 May 2014 / Accepted: 21 May 2014 / Published: 6 June 2014

# 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 Δ*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) ≠ 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

*N*(

_{df}k_{B}T ln s_{n}*k*the Boltzmann constant,

_{B}*T*the temperature, and Ndf the number of degrees of freedom) which is defined as the microscopic entropy

*S*at time point n multiplied by

_{n}*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*which is in equilibrium close to 1, but that the entropy

_{n}
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