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Two-species hardcore reversible cellular automaton: matrix ansatz for dynamics and nonequilibrium stationary state

by Marko Medenjak, Vladislav Popkov, Tomaž Prosen, Eric Ragoucy, Matthieu Vanicat

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Submission summary

Authors (as registered SciPost users): Matthieu Vanicat
Submission information
Preprint Link:  (pdf)
Date accepted: 2019-06-06
Date submitted: 2019-05-22 02:00
Submitted by: Vanicat, Matthieu
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Mathematical Physics
  • Statistical and Soft Matter Physics
Approach: Theoretical


In this paper we study the statistical properties of a reversible cellular automaton in two out-of-equilibrium settings. In the first part we consider two instances of the initial value problem, corresponding to the inhomogeneous quench and the local quench. Our main result is an exact matrix product expression of the time evolution of the probability distribution, which we use to determine the time evolution of the density profiles analytically. In the second part we study the model on a finite lattice coupled with stochastic boundaries. Once again we derive an exact matrix product expression of the stationary distribution, as well as the particle current and density profiles in the stationary state. The exact expressions reveal the existence of different phases with either ballistic or diffusive transport depending on the boundary parameters.

List of changes


We thank the referee for her/his careful review of our manuscript. We have revised our paper according to his/her
remarks, and the remarks of the second referee.


Referee: Some criticism can still be addressed. It may be argued that models such as the one considered here may
not catch in full generality the richness of interacting quantum many-body system: the dynamics here is indeed
purely deterministic, and it would be interesting if the authors could discuss a little what it might miss compared
to non-deterministic models. For the latter, it seems reasonable to expect that no such simple matrix product
solution could exist.

Answer: This is an excellent point. And indeed, one can think of a stochastic deformation of our deterministic
model which still keeps part of its ballistic character, and still probably can be exactly solved. At least we
have empirical evidence that matrix product solutions of similar complexity should exist there, but that should
require a non-negligible amount of extra work.
This stochastic extension of the model and empirical evidence on its solvability are now discussed in a new
paragraph at the end of Discussion and conclusion section.

1. Referee: Many syntax or spelling errors, which need to be corrected. For instance, 2nd paragraph of first
page, misspell of "analytically". Later, "in the context of ["the" missing] Totally Asymmetric Simple
Exclusion Process". 3rd paragraph, "as a propagation of ["the" to be replaced by "a"] local observable".
3rd paragraph of section 2.1, "usefull" should be replaced by "useful", etc... In many other places the writing
seems a bit careless, and should be improved before publication.

Answer: We have polished the spelling and grammar.

2. Referee: In the introduction, since the TASEP is mentioned, it would be useful if the authors could recall in
a few sentences the state of the art for the latter, as far as the computation of out-of-equilibrium density
profiles is concerned.

Answer: We do not wish to specifically emphasise one stochastic model, even though TASEP is perhaps the most important
one, but rather put the emphasis on the matrix product ansatz method, as our context (that of deterministic lattice
gasses) is a very different one than TASEP. Therefore we feel that the mention in the introduction should be

3. Referee: The authors refer to a different approach to cellular automata, based on soliton counting. The
deterministic dynamics considered here looks like it could allowed such methods for the present problems. Have
these been considered by the authors? Do they fail where the matrix product approach works ? If so, this should
probably be explained in the introduction.

Answer: Our matrix product ansatz could be equivalently interpreted through "soliton counting" mechanism, but we
feel that such a solution would look more convoluted and less clear than the one we discuss here, which is obtained
through algebraic cancellation mechanism and its explicit representation in terms of Fock spaces. We quote in this
context our recent independent work on Rule 54 cellular automaton, where we could only obtain matrix product solution
for dynamics in terms of soliton counting and algebraic cancellation mechanism is still unknown.

4. Referee: After equation (6) the author mention a connection with the Yang-Baxter equation, and "the fact that
R(0)=U implies the integrability of the model with periodic boundary conditions.". This Last sentence is in fact
not completely clear to me : while integrable R matrices can indeed be used to build commuting sets of transfer
matrices in different geometries (row-to-row, or diagonal-to-diagonal), it is not clear whether the generator
U for the dynamics considered here really belongs to this setup. It would correspond to a diagonal-to-diagonal
array of R matrices all carrying an infinite spectral parameter, which seems difficult to relate to the usual
construction. Could the authors clarify whether the Yang-Baxter integrability plays any role here ?

Answer: We have added a new paragraph after Eq. (7) (braid relation for U), explicitly defining the
spectral-parameter dependent R-matrix obeying the Yang-Baxter equation, and constructing a commuting set of
transfer matrices, through which we can derive the full evolution operator, and if needed, an infinite set of
conserved quantities of the dynamics. This clearly demonstrated the relevance of integrability for the present model.

5. Referee: in eqs (11) and (12), p' has not been defined.

Answer: We now explicitly define p and p' just after Eq. (15) (previous Eq. (12)).

6. Referee: eq (13) is slightly confusing, as the labels α0,±,β0,± have already been used in eq. (10), but are a priori
unrelated (even though in the particular case of boundary driving they turn out to be the same). Could this be
fixed ?

Answer: We believe the notation is fully consistent as it is.

7. Referee: The notation ⟨+0⟩12 (and other similar ones) should be explained.

Answer: A sentence has been added after Eq. (18) to explicitly explain this notation.

8. Referee: before equation (22) : "On [remove "the"] physical grounds"

Answer: This has been rewritten.

9. Referee: section 2.2 : Could the authors briefly discuss what happens when the condition (23) is not met ? Does the shock
smoothen and decay ?
Answer: see after question 10.

10. Referee: section 3.3, after (62) : maybe a few words to explain what a "marginally stable" shock means in practice

Answer to 10 and 11: A new paragraph is added before the section 2.3,
explaining in detail the notion of shock stability.

11. Referee: Section 4.1, after (64) : "This dependence is reminiscent [of] the defect..."

Answer: Done

12. Referee: Figure 4 : is there a specific reason why the authors chose ρ+=0.01, and not 0?
Would the plots look much different there?

Answer: For ρ+=0, we cannot observe the balistically left moving peak on the density of + particles. The height of the peak is
indeed proportional to ρ+, see eq. (89) (previously eq (87)). The fine-tuning of ρ+ to 0.01 was only done for graphical purpose.



We thank the referee for her/his insightful remarks concerning the hydrodynamic aspect of the paper. In fact, her/his remarks has
encouraged us to give more details and improve the presentation.


1. Referee: there is no problem in having many shocks at different velocities,
and the Riemann problem for two-component fluids will usually display a shock and a rarefaction wave,
or modifications of such two structures. I think what the authors are really doing here is establishing
the conditions for a single shock to separate two current-carrying state.I think what the authors are
really doing here is establishing the conditions for a single shock to separate two current-carrying state.

Answer: We agree with the referee: Indeed, several scenarios are possible hydrodynamically (formation of two
spatially separated distinct shocks, a combination of a shock and a rarefaction wave, etc.), if two current
carrying states are joined together. Our statements are now made more precise in the revised version, see the
paragraph following the Eq (19) and a paragraph following the Eq (21).

2. Referee: Overall, the fluid has three components. So don't we indeed have in general three normal modes,
and aren't these the three velocities that we see emerging?

Answer: With respect to the remark of the referee, that the fluid has three components, we would like to note
that only 2 quantities out of three are independent, since the sum of local densities of pluses, minuses and
vacancies is constant, i.e. $n_{k,+}+n_{k,-}+n_{k,0}=1$ for all sites $k$.

So generically, in absence of odd/even separation, we have two conservation laws and one would expect two
characteristic velocities. However, due to odd-even sublattice structure of the evolution operator, we find three
characteristic velocities: one associated with the diffusive mode, (which can be seen e.g. in Fig.4), and two
"free" modes, associated with a propagation of contact discontinuity Eq.(19),(20), which indeed result in
being normal modes of vacancy propagation. In Fig.4 one sees a diffusive mode and one of the free modes (another
free mode would appear, if one considers slightly more general initial condition with a local quench in two
neighbouring (even and odd) sites).

3. Referee: my understanding is that on both sides of the diffusive discontinuity, the authors expect the
state to be essentially one of the current-carrying states studied around eq 22. Can they see this analytically?

Answer: The point discontinuities in Fig.3, in accordance with the suggestion of the referee, do belong to the
contact singularities described in "hydrodynamic" section Eqs(19),(20). In the revised version, we have added
a paragraph between Eqs. (19) and (20), and three paragraphs after Eq. (64), following Fig. 3, where we now
discuss hydrodynamic interpretation of our microscopic results, summarized in Eqs. (60,61), in detail.

Published as SciPost Phys. 6, 074 (2019)

Reports on this Submission

Anonymous Report 2 on 2019-6-5 (Invited Report)


I thank the referees for clarifying the hydrodynamic properties of their results - this is very useful.

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Anonymous Report 1 on 2019-5-22 (Invited Report)


I thank the authors for their careful consideration of the points raised in my previous report, and am now fully satisfied with the present version.

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