SciPost Submission Page
Multi species asymmetric simple exclusion process with impurity activated flips
by Amit Kumar Chatterjee, Hisao Hayakawa
|As Contributors:||Amit Chatterjee|
|Arxiv Link:||https://arxiv.org/abs/2205.03082v2 (pdf)|
|Date submitted:||2022-05-17 08:30|
|Submitted by:||Chatterjee, Amit|
|Submitted to:||SciPost Physics|
We obtain an exact matrix product steady state for a class of multi species asymmetric simple exclusion process with impurities, under periodic boundary condition. Alongside the usual hopping dynamics, an additional flip dynamics is activated only in the presence of impurities. Although the microscopic dynamics renders the system to be non-ergodic, exact analytical results for observables are obtained in steady states for a specific class of initial configurations. Interesting physical features including negative differential mobility and transition of correlations from negative to positive with changing vacancy density, have been observed. We discuss plausible connections of this exactly solvable model with multi lane asymmetric simple exclusion processes as well as enzymatic chemical reactions.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 2022-7-8 (Invited Report)
1- Defines a new stochastic particle model with an exact matrix product stationary state
2-Calculations of physical observables carried out.
3- Negative differential mobility elucidated
I would say the paper satisfies SciPost expectation 3.
1- Perhaps a technical advance rather than a discovery of new physics
The paper considers a multi-species variant of the well-studied totally asymmetric exclusion process, on a periodic lattice. The model contains "defect" particles which activate switches of particle species when particles become adjacent to them. The key finding is that the model admits a matrix product solution of the stationary state. This adds to the range of models that are known to have matrix product solutions.
Although the model is of intrinsic interest, due to its matrix product stationary state, various further motivations for the model are given in the introduction. I find the mapping to a multi-lane TASEP of most interest.
The proof of the matrix product solution follows the usual construction outlined in equations (4)-(6). The auxiliary `tilde' matrices turn out to be scalars which is a crucial simplification.
It is noted that the model is non-ergodic, meaning that the configuration space breaks up into different sectors corresponding to different initial conditions and for each sector of initial conditions the partition function has to be computed separately. The partition function is computed for a special class of initial conditions and observables such as density profiles and currents are computed using the grand canonical ensemble in the large system limit.
I find the results interesting and worthy of eventual publication in SciPost.
First, the authors should consider the following points.
1) The first matrix product solution for a two species ASEP was
Derrida, B., Janowsky, S.A., Lebowitz, J.L. , Speer E.R.. Exact solution of the totally asymmetric simple exclusion process: Shock profiles. J Stat Phys 73, 813–842 (1993). https://doi.org/10.1007/BF01052811. (I will refer to this paper as [DJLS] below.)
The work [DJLS] should certainly be cited. It also showed how the stationary state factorises about the defect (a second-class particle in that case) which implies a projector form for the matrix $A$, the same as in equations (10) and (13) of the present work. This factorisation property due to the projector form of $A$ should be acknowledged in the current paper.
2) I am not sure I understand the last sentence of Section 2
`However, we should mention that.. this is not the general expression for $\alpha$ ..'.
Does this mean that generally $\alpha$ would appear as $\alpha_I$ in equation (15)? Perhaps this point can be clarified.
3) Second sentence of Section 3. I would put transfer matrix in inverted commas as this is not the same as a usual equilibrium transfer matrix.
i.e. Here the "transfer matrix'' $T$ refers to ..
4) In section 3 the grand canonical ensemble is used , which results in a fugacity $z_0$ which is fixed by the density.
It would be helpful to have the solution for $z_0$ appear somewhere, perhaps in an appendix if it is really very complicated. Does the solution for $z_0$ simplify in some limits?
5) Section 5.4 considers Negative Differential Mobility (NDM).
Some references and discussion of specific, related models, exhibiting NDM would be appropriate e.g.
Cividini J, Mukamel D and Posch H A
``Driven tracer with absolute negative mobility''
(2018) J. Phys. A: Math. Theor. 51 085001
6) Some typos:
p.3 paragraph 2 "with variety" $\to$ "with a variety"
p.4. paragraph 2 "For specific choice of" $\to$ "For a specific choice of"
p.5 last line "itlaics" $\to$ "italics"
p.7 after equation (10) "resembles to that of the defect or second class" $\to$ "resembles that of the defect of second class". Here reference [DJLS] (see point 1. above) should be cited.
p.36 reference  author is J. Szavits-Nossan
-List of 6 suggested points for improvement in the above report