Strong Hydrodynamic Limit for Attractive Particle Systems on $\mathbb{Z}$

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

doi:10.1214/ejp.v15-728

Cited by 18 publications

(72 citation statements)

References 41 publications

(95 reference statements)

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“…We recall that a function f is said to be monotonous if f (η) ≤ f (η ) whenever η ≤ η (in the sense of coordinate-wise order) and a Markov process with semigroup S(t) is said to be monotonous if, for every time t ≥ 0, S(t) f is monotonous function if f is a monotonous function. In this paper we do not investigate the consequence of monotonicity which is for instance very useful for the hydrodynamic limit (see [1]). …”

Section: Basic Properties Of the Asep(q J)mentioning

confidence: 99%

A generalized asymmetric exclusion process with $$U_q(\mathfrak {sl}_2)$$ stochastic duality

Carinci

Giardinà

Redig

et al. 2015

Probab. Theory Relat. Fields

134

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We study a new process, which we call ASEP(q, j), where particles move asymmetrically on a one-dimensional integer lattice with a bias determined by q ∈ (0, 1) and where at most 2 j ∈ N particles per site are allowed. The process is constructed from a (2 j + 1)-dimensional representation of a quantum Hamiltonian with U q (sl 2 ) invariance by applying a suitable ground-state transformation. After showing basic properties of the process ASEP(q, j), we prove self-duality with several selfduality functions constructed from the symmetries of the quantum Hamiltonian. By making use of the self-duality property we compute the first q-exponential moment of the current for step initial conditions (both a shock or a rarefaction fan) as well as when the process is started from a homogeneous product measure. Mathematics Subject Classification

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Section: Basic Properties Of the Asep(q J)mentioning

confidence: 99%

A generalized asymmetric exclusion process with $$U_q(\mathfrak {sl}_2)$$ stochastic duality

Carinci

Giardinà

Redig

et al. 2015

Probab. Theory Relat. Fields

134

View full text Add to dashboard Cite

show abstract

“…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: 70%

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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“…Then for any t > 0, the sequence (η α,N N t ) N ∈N\{0} has limiting density profile ρ(., t). for the constant c in (7). The additional requirement (33) sets a restriction on the sparsity of slow sites (where by "slow sites" we mean sites where the disorder variable becomes arbitrarily close or equal to the infimum value c).…”

Section: Assumption 22mentioning

confidence: 99%

“…The measure (6) is always supported on X if β ∈ (0, c) ∪ {0} (11) When β = c > 0, conventions (9)-(10) yield a measure supported on configurations with infinitely many particles at all sites x ∈ Z that achieve the infimum in (7), and finitely many particles at other sites. In particular, this measure is supported on X when the infimum in (7) is not achieved.…”

Section: Introductionmentioning

confidence: 99%

Hydrodynamics in a condensation regime: The disordered asymmetric zero-range process

Bahadoran¹,

Mountford²,

Ravishankar³

et al. 2020

Ann. Probab.

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We study asymmetric zero-range processes on Z with nearestneighbour jumps and site disorder. The jump rate of particles is an arbitrary but bounded nondecreasing function of the number of particles. For any given environment satisfying suitable averaging properties, we establish a hydrodynamic limit given by a scalar conservation law including the domain above critical density, where the flux is shown to be constant.MSC 2010 subject classification: 60K35, 82C22.

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Strong Hydrodynamic Limit for Attractive Particle Systems on $\mathbb{Z}$

Cited by 18 publications

References 41 publications

A generalized asymmetric exclusion process with $$U_q(\mathfrak {sl}_2)$$ stochastic duality

A generalized asymmetric exclusion process with $$U_q(\mathfrak {sl}_2)$$ stochastic duality

How to initialize a second class particle?

Hydrodynamics in a condensation regime: The disordered asymmetric zero-range process

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