Computational Complexity as a Source of Thermodynamic Irreversibility

Abstract

In spite of the fact that the Newtonian dynsmics of the underlying molecules do not seem to favor one direction of time over its opposite, the non-decreasing entropy of macroscopic physical systems provides a unique direction to 'the arrow of time'. This paper proposes that the directionality of time arises from the intrinsic computational complexity of molecular dynamics. In particular, the following question about a hard sphere gas is proven to be Np-complete: Given a set of input signals, in which TRUE is represented by an actual sphere trajectory and FALSE is represented by a trajectory that is not followed because an input signal is not TRUE; a set of 'routing' spheres that follow the same trajectory for all choices of input signals and that collide with the signal trajectories at pre-specified points; does there exist a set of inputs that would produce a specified final configuration? The time asymmetry arises because the computational effort, and thus the actual time required to determine the final configuration from the starting configuration grows only linearly with the number of particles, N, but the computational effort required to determine the answer to the above Np-complete problem probably grows faster than any polynomial in N. We conclude by arguing that, as N approaches infinity, the resulting temporal asymmetry is sufficient to justify the molecular chaos assumption (Stosszahlansatz) used in deriving the Boltzmann equation.