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Time arrow without past hypothesis: a toy model explanation

Pablo Arrighi, Gilles Dowek, Amélia Durbec

TL;DR

The paper tackles the arrow-of-time problem by proving, within a rigorously defined toy model, that an entropic arrow can emerge from time-reversible local dynamics featuring expansion, without relying on the past hypothesis. The authors construct reversible causal graph dynamics on unbounded finite graphs, showing that almost all initial configurations lead to graph expansion and concomitant entropy growth, with entropy quantified as $S'=\,\log\binom{2|V|}{p}$ based on graph size and particle count. They further explore time-symmetric variants, inflation-like exponential growth, and mechanisms to counter thermal death via inelastic shocks that couple matter with radiation, demonstrating qualitatively robust arrow-of-time behavior. The results provide a conceptually local explanation for the thermodynamic arrow that could have cosmological relevance, while acknowledging the toy nature and dimensional limitations of the model. Overall, the work offers a rigorous, CS-informed framework for understanding how expansion-driven entropy can establish a time direction without presupposing a special low-entropy past.

Abstract

The laws of Physics are time-reversible, making no qualitative distinction between the past and the future -- yet we can only go towards the future. This apparent contradiction is known as the "arrow of time problem". Its current resolution states that the future is the direction of increasing entropy. But entropy can only increase towards the future if it was low in the past, and past low entropy is a very strong assumption to make, because low entropy states are rather improbable, non-generic. Recent works from the Physics literature suggest, however, we may do away with this so-called "past hypothesis", in the presence of reversible dynamical laws featuring expansion. We prove that this can be the case in principle, within a toy model. It consists in graphs upon which particles circulate and interact according to local reversible rules. Some rules locally shrink or expand the graph. We prove that almost all states expand; entropy always increases as a consequence of expansion -- thereby providing a local explanation for the rise of an entropic arrow of time without the need for a past hypothesis. The discrete setting of this toy model allows us to deploy the full rigour of theoretical Computer Science proof techniques. It also allows for the numerical exploration of several physically-motivated variants: a time-symmetric variant; two inflationary variants; and a damping variant -- which slows down thermal death. The fact that all of these models exhibit similar behaviours suggests that local reversible expansion mechanisms constitute a robust recipe for a time arrow without past hypothesis. In this qualitative sense, the explanation may therefore also be relevant at the cosmological level.

Time arrow without past hypothesis: a toy model explanation

TL;DR

The paper tackles the arrow-of-time problem by proving, within a rigorously defined toy model, that an entropic arrow can emerge from time-reversible local dynamics featuring expansion, without relying on the past hypothesis. The authors construct reversible causal graph dynamics on unbounded finite graphs, showing that almost all initial configurations lead to graph expansion and concomitant entropy growth, with entropy quantified as based on graph size and particle count. They further explore time-symmetric variants, inflation-like exponential growth, and mechanisms to counter thermal death via inelastic shocks that couple matter with radiation, demonstrating qualitatively robust arrow-of-time behavior. The results provide a conceptually local explanation for the thermodynamic arrow that could have cosmological relevance, while acknowledging the toy nature and dimensional limitations of the model. Overall, the work offers a rigorous, CS-informed framework for understanding how expansion-driven entropy can establish a time direction without presupposing a special low-entropy past.

Abstract

The laws of Physics are time-reversible, making no qualitative distinction between the past and the future -- yet we can only go towards the future. This apparent contradiction is known as the "arrow of time problem". Its current resolution states that the future is the direction of increasing entropy. But entropy can only increase towards the future if it was low in the past, and past low entropy is a very strong assumption to make, because low entropy states are rather improbable, non-generic. Recent works from the Physics literature suggest, however, we may do away with this so-called "past hypothesis", in the presence of reversible dynamical laws featuring expansion. We prove that this can be the case in principle, within a toy model. It consists in graphs upon which particles circulate and interact according to local reversible rules. Some rules locally shrink or expand the graph. We prove that almost all states expand; entropy always increases as a consequence of expansion -- thereby providing a local explanation for the rise of an entropic arrow of time without the need for a past hypothesis. The discrete setting of this toy model allows us to deploy the full rigour of theoretical Computer Science proof techniques. It also allows for the numerical exploration of several physically-motivated variants: a time-symmetric variant; two inflationary variants; and a damping variant -- which slows down thermal death. The fact that all of these models exhibit similar behaviours suggests that local reversible expansion mechanisms constitute a robust recipe for a time arrow without past hypothesis. In this qualitative sense, the explanation may therefore also be relevant at the cosmological level.
Paper Structure (27 sections, 11 theorems, 35 equations, 19 figures)

This paper contains 27 sections, 11 theorems, 35 equations, 19 figures.

Table of Contents

  1. Introduction
  2. In short.
  3. The problem.
  4. Comparison with related models.
  5. The conventional argument
  6. Time arrow without past hypothesis proof
  7. State space
  8. The toy model
  9. Size increases
  10. Entropy increases with size
  11. Recovering an arrow of time
  12. Time-symmetric model
  13. Exponential growth models
  14. Fighting thermal death
  15. Prognosis
  16. ...and 12 more sections

Key Result

Lemma 1

Let $X \in \mathcal{C}_1$ be a circular graph. Given a pair $u$ and $v$ of adjacent vertices of $X$, these are said to belong to a merger pattern if and only if $a_1(u)=b_1(v)=1$ and $b_1(u)=a_1(v)=0$. For any $u,v$ forming a merger pattern in $X$, there are two vertices $u.r, v.l \in V((\sqrt{\tau}

Figures (19)

  • Figure 1: A configuration. All configurations are cyclic graphs. The nodes are connected via ports $a$ or $b$. At the nodes, each full half-disk represents the presence of a particle that is about to hop along the port it is facing. The names identifying each node are omitted in this picture.
  • Figure 2: Rules of the toy model.
  • Figure 3: Spacetime diagram of dynamics $\sqrt{\tau}I$. Dynamical clock time flows towards the bottom. Particles moving along port $a$ (resp. $b$) are represented green (resp. blue). When they cross, $I$ gets applied, leading to shrinking or expansion of the graph. Expansion wins: cells are smaller at the bottom, their packing leading to this "curtain folds" appearance.
  • Figure 4: Typical size and entropy curves for dynamics $\sqrt{\tau}I$. The horizontal axis represents the number of steps of the dynamics, aka dynamical clock time. The initial configuration is drawn uniformly at random amongst all graphs of size $100$.
  • Figure 5: Instability of merger patterns under $\sqrt{\tau}I$.
  • ...and 14 more figures

Theorems & Definitions (32)

  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Definition 4: $\sqrt{\tau}$
  • Lemma 1: Merger Instability
  • proof
  • Definition 5
  • Theorem 1: Expansion
  • proof
  • ...and 22 more