SciPost Submission Page
Ergodic behaviors in reversible 3-state cellular automata
by Rustem Sharipov, Matija Koterle, Sašo Grozdanov, Tomaž Prosen
Submission summary
| Authors (as registered SciPost users): | Rustem Sharipov |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2503.16593v2 (pdf) |
| Data repository: | https://zenodo.org/records/17117722 |
| Date submitted: | Oct. 31, 2025, 2:55 p.m. |
| Submitted by: | Rustem Sharipov |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
The author(s) disclose that the following generative AI tools have been used in the preparation of this submission:
AI tools were used for grammar editing.
Abstract
Classical cellular automata represent a class of explicit discrete spacetime lattice models in which complex large-scale phenomena emerge from simple deterministic rules. With the goal to uncover different physically distinct classes of ergodic behavior, we perform a systematic study of three-state cellular automata (with a stable `vacuum' state and `particles' with $\pm$ charges). The classification is aided by the automata's different transformation properties under discrete symmetries: charge conjugation, spatial parity and time reversal. In particular, we propose a simple classification that distinguishes between types and levels of ergodic behavior in such system as quantified by the following observables: the mean return time, the number of conserved quantities, and the scaling of correlation functions. In each of the physically distinct classes, we present examples and discuss some of their phenomenology. This includes chaotic or ergodic dynamics, phase-space fragmentation, Ruelle-Pollicott resonances, existence of quasilocal charges, and anomalous transport with a variety of dynamical exponents.
Author indications on fulfilling journal expectations
- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
2-Introduces a new classification of these types of models
Weaknesses
Report
The authors perform a highly extensive and impressive numerical study of a large number of three state cellular automata. They identify different dynamical behaviours and classify the models accordingly. They highlight the fact that multiple quantities are required to distinguish different behaviours. The work is thorough and the results convincing. It meets the acceptance criteria of the journal, in particular "Provide a novel and synergetic link between different research areas".
I have some small comments that I would ask the authors to address before publication.
1-It is not clear to me how some of the results are produced and it would be worthwhile to add a comment on this. For example, is table 1 calculated by brute force or is there a systematic way to exhaustively generate rules which obey a certain symmetry. Likewise, how are the average return times calculated, is this also by brute force?
2- It is stated that the lower bound in equation 13 is achieved by the trivial system. I would have thought that the more ergodic the system the smaller the number of orbits while the trivial rule has the most number of distinct orbits.
3-In figure 10 (a) , there appear to be eigenvalues at 1 in spite of there being no conserved quantities. Perhaps this just is the commutation with the transfer matrix itself but maybe the authors comment on this .
4-Is the table on page 30 supposed to show an exponential growth of the number of conserved quantities? If so, it is not that convincing, perhaps the authors could provide more evidence for this.
5-The authors restrict a certain class of models 3 states which possess C, P, T symmetries and combinations there of. What is the expectation of the absence of these, will the classification I-IV still hold or could other dynamical behaviours be present. Similarly, what is the expectation for a higher number of internal states?
Requested changes
1-Please respond to the comments above. 2- add more details on how some of the figures are produced. 3- could the authors add more explanation on the significance of the bi-norms. It was not clear to me what they were meant to show. 4- Please correct several typos e.g BCA->RCA, operat ->operator, eigenovector ->eigenvector. There are also some inconsistencies in notation e.g below equation 14 T(L)->T_U(L)
Recommendation
Publish (easily meets expectations and criteria for this Journal; among top 50%)
Strengths
1- Comprehensive classification of a broad class of reversible cellular automata 2- Systematic analysis (chaotic/integrable behavior, number of local/quasilocal charges, transport etc) 3- Uncovers unexpected transport behavior in some models, including subdiffusion with dynamical exponent z=3 (which to my knowledge is a first, I do not know of any other many-body system exhibiting z=3), or superdiffusion with z=3/2 but non-KPZ scaling functions.
Weaknesses
1- Due to the classification-like nature of this work, it might be hard for the reader to jump to the most "interesting" examples. 2- Most of the analysis is numerical, although I do not believe this is a real "weakness" given the scope of the number of models studied, and the surprising nature of some of the results.
Report
Within class II, they uncover a number of interesting chaotic and integrable models with surprising and unexpected transport properties. The most intriguing result is the existence of models with dynamical exponent z=3, which do not fit any previously studied examples of subdiffusion, and models with z=3/2 but non-KPZ scaling functions. In my opinion, these results alone clearly "Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work". I expect a number of interesting follow-up works motivated by these results to understand the underlying hydrodynamics of these models.
The classification work itself is also very impressive and systematic, and I expect that a many extensions could follow in the future. This is a very impressive work, and I strongly recommend publication.
Requested changes
1- Minor typos: page 6: posses --> possesses 2- page 7: "the exhibit" --> "they exhibit" 3- Eq 41: very minor, but maybe clarify that the limit is x and t \to \infty with x^2/t fixed. 4- I would suggest extending section 2 on the summary of results to include the most interesting transport results. In the current version some of the most interesting results from the perspective of transport and hydrodynamics are buried deep in the text, and could be highlighted here.
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)