Euler hydrodynamics for attractive particle systems in random environment

Bahadoran, Christophe; Guiol, Hervé; Ravishankar, K.; Saada, Ellen

doi:10.1214/12-aihp510

Cited by 14 publications

(34 citation statements)

References 35 publications

(65 reference statements)

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“…Indeed, the previous assertion holds in even more general context as well as with sharper conclusions, for details consult [3] and [4].…”

Section: Remarkmentioning

confidence: 69%

“…It is also known that for each t ≥ 0, this weak solution is continuous apart from a finite set of jump discontinuities (shocks), where we define u( · , t) to be left-continuous. For concepts and results in hyperbolic conservation laws, which were omitted here, we refer to [3] and further references therein (see also [24]). …”

Section: Definition 2 (Hydrodynamic Limit) a Sequence Of Processes (ωmentioning

confidence: 99%

“…al. in [3] for systems of bounded particle numbers per site. In particular, this result does not require the special algebraic structure of Section 5.…”

Section: Theorem 7 (F Rezakhanlou) Take Any Process From the Misantmentioning

confidence: 99%

“…We refer to [3] for the detailed definition of G in the general case. Indeed, the previous assertion holds in even more general context as well as with sharper conclusions, for details consult [3] and [4].…”

Section: Remarkmentioning

confidence: 99%

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How to initialize a second class particle?

Balázs¹,

Nagy²

2017

Ann. Probab.

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We identify the ballistically and diffusively rescaled limit distribution of the second class particle position in a wide range of asymmetric and symmetric interacting particle systems with established hydrodynamic behavior, respectively (including zero-range, misanthrope and many other models). The initial condition is a step profile which, in some classical cases of asymmetric models, gives rise to a rarefaction fan scenario. We also point out a model with non-concave, non-convex hydrodynamics, where the rescaled second class particle distribution has both continuous and discrete counterparts. The results follow from a substantial generalization of P. A. Ferrari and C. Kipnis' arguments ("Second class particles in the rarefaction fan", Ann. Inst. H. Poincaré, 31, 1995) for the totally asymmetric simple exclusion process. The main novelty is the introduction of a signed coupling measure as initial data, which nevertheless results in a proper probability initial distribution for the site of the second class particle and makes the extension possible. We also reveal in full generality a very interesting invariance property of the one-site marginal distribution of the process underneath the second class particle which in particular proves the intrinsicality of our choice for the initial distribution. Finally, we give a lower estimate on the probability of survival of a second class particle-antiparticle pair. Keywords. second class particle, limit distribution, rarefaction fan, shock, hydrodynamic limit, collision probability. Acknowledgement.The authors thank valuable discussions with Pablo A. Ferrari on the problem, with Ellen Saada on the hydrodynamic limit of asymmetric processes and with Cédric Bernardin on symmetric processes. We also thank the anonymous referee for his/her comments. M. Balázs acknowledges support from the Hungarian National Research, Development and Innovation Office, NKFIH grants K100473 and K109684.

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“…Indeed, the previous assertion holds in even more general context as well as with sharper conclusions, for details consult [3] and [4].…”

Section: Remarkmentioning

confidence: 69%

Section: Definition 2 (Hydrodynamic Limit) a Sequence Of Processes (ωmentioning

confidence: 99%

“…al. in [3] for systems of bounded particle numbers per site. In particular, this result does not require the special algebraic structure of Section 5.…”

Section: Theorem 7 (F Rezakhanlou) Take Any Process From the Misantmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

See 2 more Smart Citations

How to initialize a second class particle?

Balázs¹,

Nagy²

2017

Ann. Probab.

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show abstract

“…In this paper, we review successive stages ( [6,7,8,38,9]) of a constructive approach to hydrodynamic limits given by equations of the type (1), which ultimately led us in [9] to a very general hydrodynamic limit result for attractive particle systems in one dimension in ergodic random environment.…”

Section: Introductionmentioning

confidence: 99%