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

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 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 s n leading to the Nosé-Hoover Lagrangian. The corresponding Nosé-Hoover Hamiltonian comprises a term N df k B T ln s n (k B the Boltzmann constant, T the temperature, and N df the number of degrees of freedom) which is defined as the microscopic entropyS n at time point n multiplied by 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 n 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.

On the Einstein-Podolsky-Rosen paradox using discrete time physics

“…While not popular upon energy quantization in quantum mechanics the introduction of a discrete time appears to be obvious [11][12][13][14][15][16][17]. With the recent success in the derivation of a microscopic entropy under the assumption of a discrete time the hypothesis that time in classical physics may be discrete [10] is strengthend and worth to be disputed further.…”

“…This assumption is nourished on the finding that the arrow of time and entropy can be derived by the introduction of a discrete time [10] and on the request of a dual relationship between energy and time from the corresponding uncertainty principle ΔE Δt > h/2 (i.e. since in quantum mechanics energy is quantized time is (allowed to be) continuous and vice versa, since in classical physics energy is continuous time is requested to be discrete).…”

“…since in quantum mechanics energy is quantized time is (allowed to be) continuous and vice versa, since in classical physics energy is continuous time is requested to be discrete). Thus, by introducing a discrete time, any time-dependent observable A(t) is represented by a sequence of discrete values [10], [11]:…”

On the time continuous evolution of the universe if time is discrete and irreversible in nature

“…It is this friction term, which enables discrete time physics to be time reversible with the following request (see ref. [11]) that holds for the symmetric case selected.…”

“…In discrete mechanics there are many possible definitions of the velocityṙ n . The following definition at time point n follows ref [11]:ṙ n = r n − r n−1 ∆t…”