Abstract
All of 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 S larger than 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 (i.e. not 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 Nose-Hoover Lagrangian. The corresponding Nose-Hoover Hamiltonian comprises a term Ndf *kB*T*ln(sn) (with 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 term itself has its root not in statistical mechanics but rather in the discreteness of time.
Highlights
While the basic physical laws including the fundamental differential equations of mathematical physics Hamilton’s, Lagrange’s, Maxwell’s, Newton’s, Einstein’s, and Schroedinger’s are time reversible, only the second law of thermodynamics describing macroscopic systems brings the arrow of time into play by requesting that the entropy increases in an isolated macroscopic system [1,2,3,4]
That complex processes with many particles under a complex force are time irreversible is in line with our daily experiences, and sound, albeit most of the physical laws are time reversible including the Newton’s mechanics
It was demonstrated that under the hypothesis that time is discrete with constant time steps the microscopic physical laws in presence of a complex force (F (r) 6= const) are time irreversible
Summary
While the basic (microscopic) physical laws including the fundamental differential equations of mathematical physics Hamilton’s, Lagrange’s, Maxwell’s, Newton’s, Einstein’s, and Schroedinger’s are time reversible, only the second law of thermodynamics describing macroscopic systems brings the arrow of time into play by requesting that the entropy increases in an isolated macroscopic system [1,2,3,4]. In attempts to unify the general relativity theory with quantum mechanics including in particular quantum loop theory [15,29] it is assumed at the most fundamental level that time has a granular structure with the Planck time as the smallest time step It appears that the Dirac equation which describes the free electron is more sound in presence of a discrete time than its continuous analog without loosing. We follow here the approach by Lee [22] introducing time as a dynamical discrete variable yielding a scaling of time that depends on the potential present This Ansatz is only applied to classical physics because of simplicity. In (Section 2.10) the microscopic and macroscopic entropies of the volume expanding gas are calculated, followed by a discussion on time-irreversibility within time reversible descriptions of physical laws (Section 2.11).
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