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Duality for Stochastic Models of Transport

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Abstract

We study three classes of continuous time Markov processes (inclusion process, exclusion process, independent walkers) and a family of interacting diffusions (Brownian energy process). For each model we define a boundary driven process which is obtained by placing the system in contact with proper reservoirs, working at different particle densities or different temperatures. We show that all the models are exactly solvable by duality, using a dual process with absorbing boundaries. The solution does also apply to the so-called thermalization limit in which particles or energy is instantaneously redistributed among sites.

The results shows that duality is a versatile tool for analyzing stochastic models of transport, while the analysis in the literature has been so far limited to particular instances. Long-range correlations naturally emerge as a result of the interaction of dual particles at the microscopic level and the explicit computations of covariances match, in the scaling limit, the predictions of the macroscopic fluctuation theory.

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Acknowledgements

We are extremely grateful to Bernard Derrida, with whom we discussed some of the topics in this work. In particular we own to him the results of Sect. 7 for the comparison to macroscopic fluctuation theory.

We acknowledge financial support from the by the Italian Research Funding Agency (MIUR) through FIRB project “Stochastic processes in interacting particle systems: duality, metastability and their applications”, grant n. RBFR10N90W and the Fondazione Cassa di Risparmio Modena through the International Research 2010 project.

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Authors and Affiliations

  1. University of Modena and Reggio Emilia, via G. Campi 213/b, 41125, Modena, Italy

    Gioia Carinci & Cristian Giardinà

  2. University of Modena and Reggio Emilia, via Amendola 2, Pad. Morselli, 42122, Reggio Emilia, Italy

    Claudio Giberti

  3. University of Delft, Mekelweg 4 2628 CD, Delft, The Netherlands

    Frank Redig

Authors
  1. Gioia Carinci
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  2. Cristian Giardinà
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  3. Claudio Giberti
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  4. Frank Redig
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Corresponding author

Correspondence to Cristian Giardinà.

Appendix: Equations for the two points correlations

Appendix: Equations for the two points correlations

We provide the linear systems that must be satisfied by the two points correlation functions in the steady state, i.e. X i,ℓ =〈η i η ℓ 〉 with 1≤i≤ℓ≤L. In the following, equations (1), (2), (3) are obtained by letting act the generator on a couple of sites at distance larger or equal than two, equations (4), (5), (6) are derived from nearest-neighbouring sites, equations (7), (8), (9) correspond to the diagonal, equation (10) is obtained from the couple (1,L).

Inclusion/Exclusion Walkers: the equations for the inclusion walkers \(\operatorname{SIP}(2k)\) and for the exclusion walkers \(\operatorname{SEP}(2j)\) are similar, with some relevant change of signs in the two cases; therefore we write them together. With the convention to use upper symbol for inclusion and lower symbol for exclusion in ± and ∓ and with the further convention that h=k for \(\operatorname{SIP}(2k)\) and h=j for \(\operatorname{SEP}(2j)\), the equations read

(8.1)

Brownian Energy Process \(\operatorname{BEP}(2k)\): the equations for the \(\operatorname{BEP}(2k)\) read

(8.2)

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Carinci, G., Giardinà, C., Giberti, C. et al. Duality for Stochastic Models of Transport. J Stat Phys 152, 657–697 (2013). https://doi.org/10.1007/s10955-013-0786-9

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