Certain intriguing consequences of the discreteness of time on the time evolution of dynamical systems are discussed. In the discrete-time classical mechanics proposed here, there is an {\\it arrow of time} that follows from the fact that the replacement of the time derivative by the backward difference operator alone can preserve the non-negativity of the phase space density. It is seen that, even for free particles, all the degrees of freedom are {\\it correlated} in principle. The forward evolution of functions of phase space variables by a finite number of time steps, in this discrete-time mechanics, depends on the entire continuous-time history in the interval $[0, \\infty]$ . In this sense, discrete time evolution is {\\it nonlocal} in time from a continuous-time point of view. A corresponding quantum mechanical treatment is possible {\\it via} the density matrix approach. The interference between non-degenerate quantum mechanical states decays exponentially. This {\\it decoherence} is present, in principle, for all systems; however, it is of practical importance only in macroscopic systems, or in processes involving large energy changes.Certain intriguing consequences of the discreteness of time on the time evolution of dynamical systems are discussed. In the discrete-time classical mechanics proposed here, there is an {\\it arrow of time} that follows from the fact that the replacement of the time derivative by the backward difference operator alone can preserve the non-negativity of the phase space density. It is seen that, even for free particles, all the degrees of freedom are {\\it correlated} in principle. The forward evolution of functions of phase space variables by a finite number of time steps, in this discrete-time mechanics, depends on the entire continuous-time history in the interval $[0, \\infty]$. In this sense, discrete time evolution is {\\it nonlocal} in time from a continuous-time point of view. A corresponding quantum mechanical treatment is possible {\\it via} the density matrix approach. The interference between non-degenerate quantum mechanical states decays exponentially. This {\\it decoherence} is present, in principle, for all systems; however, it is of practical importance only in macroscopic systems, or in processes involving large energy changes.

May 7, 20053 citations

Arrow of Time in Quantum and Cosmology: Energy Dissipation Driven by the Energy-Gradient Rule

Zhi Kai Zou

Abstract:Building upon the analytic quantum thermodynamic framework I previously established (Zou, 2025), this article presents a discrete ontological model in which the arrow of time emerges from irreversible energy transfers driven by local energy gradients on a topologically fixed spatial network, with the temporal sequence identified as the successive updates of spatial energy redistribution. A multiplicative entropy, defined as the product of all node energies, increases exactly with every transfer, providing a non-statistical, step-by-step measure of temporal progression. I argue that energy dissipation is fundamentally identical to decoherence: the former supplies the physical mechanism, the latter the observational signature. This framework synthesizes three intellectual traditions—computational time (Wolfram, 't Hooft), gradient-driven thermodynamics (Annila), and decoherence (Joos & Zeh, Kiefer)—into a unified picture where time is the counting of irreversible energy redistribution on discrete space.Abstract:Building upon the analytic quantum thermodynamic framework I previously established (Zou, 2025), this article presents a discrete ontological model in which the arrow of time emerges from irreversible energy transfers driven by local energy gradients on a topologically fixed spatial network, with the temporal sequence identified as the successive updates of spatial energy redistribution. A multiplicative entropy, defined as the product of all node energies, increases exactly with every transfer, providing a non-statistical, step-by-step measure of temporal progression. I argue that energy dissipation is fundamentally identical to decoherence: the former supplies the physical mechanism, the latter the observational signature. This framework synthesizes three intellectual traditions—computational time (Wolfram, 't Hooft), gradient-driven thermodynamics (Annila), and decoherence (Joos & Zeh, Kiefer)—into a unified picture where time is the counting of irreversible energy redistribution on discrete space.

Jul 6, 20260 citations

[0, \\infty]

. In this sense, discrete time evolution is {\\it nonlocal} in time from a continuous-time point of view. A corresponding quantum mechanical treatment is possible {\\it via} the density matrix approach. The interference between non-degenerate quantum mechanical states decays exponentially. This {\\it decoherence} is present, in principle, for all systems; however, it is of practical importance only in macroscopic systems, or in processes involving large energy changes.