Abstract
The n-species totally asymmetric zero range process (n-TAZRP) on a one-dimensional periodic chain studied recently by the authors is a continuous time Markov process where arbitrary number of particles can occupy the same sites and hop to the adjacent sites only in one direction with a priority constraint according to their species. In this paper we introduce an n-parameter generalization of the n-TAZRP having inhomogeneous transition rate. The steady state probability is obtained in a matrix product form and also by an algorithm related to combinatorial R.
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Notes
The enumeration of particles is reversed from [16] where the smaller species ones had the priority.
Here and in what follows, a multiset (set accounting for multiplicity of elements), say \(\{1,1,3,5,6\}\), is abbreviated to 11356, which does not cause a confusion since all the examples in this paper shall be concerned with the case \(n\le 9\).
We write \(A_{(\mu ),(\alpha ^1,\alpha ^2)}\) for example simply as \(A_{\mu ,\alpha ^1\alpha ^2}\).
The convention of labeling the particle species here is opposite from [16] causing many changes.
The notation \(\Phi _{\mathbf{x}^a}\) is slightly incomplete in that the a-dependence becomes invisible when \(\mathbf{x}^a\) is written generally as \(\mathbf{y}\) for example. However we prefer it for simplicity. A proper alternative is to formulate it as a map \(S(m_1,\ldots , m_{a-1})\times B_{\ell _a}\; \longrightarrow S(m_1,\ldots , m_a)\). A similar caution applies to \(\varpi _{\mathbf{x}^a}\).
This is an abbreviation of \((\emptyset , \{1,3\}, \{2\}, \{3\}, \emptyset , \{1,2\}, \{1,1\})\) as in Example 2.1.
The arbitrariness of the choice does not spoil the well-definedness. See Remark 4.2 (i).
Hence every line starts from a particle with the shape \(\lceil \) rather than \(\rfloor \).
Possible incoming H-lines originating from the NE neighbor box have not been drawn here for simplicity but are included in the argument.
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Acknowledgments
This work is supported by Grants-in-Aid for Scientific Research No. 15K04892, No. 15K13429 and No. 23340007 from JSPS.
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Appendix
Appendix
Let us show the equality of the coefficients of \(w_n^3\) in (3.7). In the left hand side, it is given by
where we used \({\bar{\alpha }}={\bar{\gamma }} \iff {\bar{\beta }}={\bar{\delta }}\) due to the conservation law. In the right hand side of (3.7), the coefficient of \(w_n^3\) is
coinciding with the above.
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Kuniba, A., Maruyama, S. & Okado, M. Inhomogeneous Generalization of a Multispecies Totally Asymmetric Zero Range Process. J Stat Phys 164, 952–968 (2016). https://doi.org/10.1007/s10955-016-1555-3
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DOI: https://doi.org/10.1007/s10955-016-1555-3