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### Abstract

A two-dimensional model of distinguishable particles that obey reversible deterministic laws of motion is used to illustrate the role played by thermal noise in the emergence of irreversible macroscopic behavior. Particles move in an array of square cells with four possible directions of motion. Particles that coincide with one or more particles in the same cell rotate by 90°. We study the evolution of the system from a fine-grained perspective, which allows us to follow the trajectory of each particle, and a coarse-grained perspective. The loss of information due to coarse-graining is compared with the loss of information resulting from noise. It is shown how particle and system trajectories can be used to determine if a system is quasi-ergodic. A modification of the interaction rule ensures that particles undergo momentum-conserving collisions equivalent to those of a simple lattice gas.

The author, who is indebted to two anonymous reviewers for their insightful commentaries and suggestions, devised the clockwise-interaction rule as a student in a course on statistical mechanics given by Dr. Jorge Flores Valdés at the Instituto de Física of the Universidad Nacional Autónoma de México (UNAM).

I. INTRODUCTION

II. THE MODEL

III. FINE-GRAINED AND COARSE-GRAINED PERSPECTIVES

IV. TIME INVERSION

V. EMERGENCE OF THE ARROW OF TIME

VI. QUANTIFYING DISORDER AND LACK OF INFORMATION

VII. PARTICLE AND SYSTEM TRAJECTORIES

VIII. SPECIAL INITIAL CONFIGURATIONS

IX. MORE REALISTIC LATTICE GAS MODELS

### Key Topics

- Entropy
- 6.0
- Hydrological modeling
- 5.0
- Statistical mechanics models
- 5.0
- Collision theories
- 4.0
- Thermal models
- 4.0

## Figures

A -cell system with seven particles. (a) Initial configuration at ; (b) the configuration after translation; (c) configuration at after translation and interaction; and (d) configuration at .

A -cell system with seven particles. (a) Initial configuration at ; (b) the configuration after translation; (c) configuration at after translation and interaction; and (d) configuration at .

A -cell system with 200 particles. (a) Initial configuration at ; (b) configuration at ; (c) configuration at ; and (d) configuration at .

A -cell system with 200 particles. (a) Initial configuration at ; (b) configuration at ; (c) configuration at ; and (d) configuration at .

as a function of time. The solid line represents the case with no inversion. The broken line represents the case with inversion at .

as a function of time. The solid line represents the case with no inversion. The broken line represents the case with inversion at .

Configuration at with the initial condition shown in Fig. 2(a) after inversion at . The particle distribution is the same as in Fig. 2(b), although the directions correspond to the system running backward.

Configuration at with the initial condition shown in Fig. 2(a) after inversion at . The particle distribution is the same as in Fig. 2(b), although the directions correspond to the system running backward.

Breaking of reversibility by noise. The inversion of directions and rules of motion occurs at for a system with the initial configuration in Fig. 2(a). The solid line represents the case when noise is introduced at . The broken line shows reversibility for noise-free inversion.

Breaking of reversibility by noise. The inversion of directions and rules of motion occurs at for a system with the initial configuration in Fig. 2(a). The solid line represents the case when noise is introduced at . The broken line shows reversibility for noise-free inversion.

Breaking of reversibility by noise. Inversion of directions and rules of motion occurs at for a system with the initial configuration in Fig. 2(a). The solid line represents the case when noise is introduced at . The broken line shows reversibility for the noise-free inversion.

Breaking of reversibility by noise. Inversion of directions and rules of motion occurs at for a system with the initial configuration in Fig. 2(a). The solid line represents the case when noise is introduced at . The broken line shows reversibility for the noise-free inversion.

Partition of a -cell system into 25 -cell regions.

Partition of a -cell system into 25 -cell regions.

The entropy as a function of time for the system depicted in Fig. 2 (compare with the solid line of Fig. 3).

The entropy as a function of time for the system depicted in Fig. 2 (compare with the solid line of Fig. 3).

Path traveled by the particle initially at the lower-left corner of a -cell system with one particle pointing up in each of its 2500 cells at . (a) ; (b) ; and (c) .

Path traveled by the particle initially at the lower-left corner of a -cell system with one particle pointing up in each of its 2500 cells at . (a) ; (b) ; and (c) .

Distance (in cell-length units) between two particles adjacent at as a function of time. Note the logarithmic scale of time.

Distance (in cell-length units) between two particles adjacent at as a function of time. Note the logarithmic scale of time.

Distance (in cell-length units) as a function of time between trajectories whose initial states are almost identical (see Sec. VII). Note the logarithmic scale of time.

Distance (in cell-length units) as a function of time between trajectories whose initial states are almost identical (see Sec. VII). Note the logarithmic scale of time.

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