On the Arrow of Time

We believe the following three ingredients are enough to explain the mystery of the arrow of time: (1). equations of dynamics of gas molecules, (2). chaotic instabilities of the equations of dynamics, (3). unavoidable perturbations to the gas molecules. The level of physical rigor or mathematical rigor that can be reached for such a theory is unclear. 1. The theory. There have been a lot of writings on the arrow of time; nevertheless, the problem still remains a mystery. In the equilibrium thermodynamics context, the arrow of time refers to the second law of thermodynamics. Consider the example of gas in a box, each gas molecule moves according to the equation of dynamics (free flying and collision) which is reversible in the coordinate time; on the other hand, the whole body of gas can only move in an entropy increasing direction among thermodynamic equilibria, e.g. moving from half box to whole box. The entropy increasing direction can be viewed as an intrinsic time direction — the arrow of time. In fact, there are many similar irreversible features in nature; for instances, living things can only grow older, sugar and water do not spontaneously un-mix, broken glass does not spontaneously fix itself. Explaining all these arrows of time is a challenging task. According to L. Boltzmann [1]:“The second law (of thermodynamics) can never be proved mathematically by means of the equations of dynamics alone.” Here we shall approach the problem with the following three ingredients: 1. equations of dynamics of gas molecules, 2. chaotic instabilities of the equations of dynamics, 3. unavoidable perturbations to the gas molecules. Trying to use chaos to explain irreversibility is not a new attempt, see for example I. Prigogine’s writings [7] [9] [10] [8]. Of course, there are many other attempts from different perspectives. The most recent one is that of Maccone [4] from quantum perspective, see also its critiques [2] [3] [6] [5]. As Boltzmann said, the equations of dynamics alone will not be able to explain irreversibility. Unavoidable perturbations to the gas molecules have to be taken into account. These perturbations are inside the isolated system, and they can 2010 Mathematics Subject Classification. Primary: 82; Secondary: 37.