Stationary uphill currents in locally perturbed zero-range processes

Cirillo, Emilio N. M.; Colangeli, Matteo

doi:10.1103/physreve.96.052137

Cited by 24 publications

(28 citation statements)

References 36 publications

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“…The steady state is unique and does not depend on the initial conditions ( ( )) ] ] Î n 0 l l 0,1 . In the rest of the paper we will assume that the rates δ and β satisfy the conditions of equation (16) leading to a steady state. We are going to study the master equation at finite time in order to work out how fast each fitness level forgets initial conditions, in the absence of resetting events.…”

Section: Master Equationmentioning

confidence: 99%

“…with ζ>0 (this assumption, equation (16) ensures that the sum of the vanishing and hopping rates is high enough to prevent an exponential divergence of the occupation number at maximum fitness). The non-linearity can be addressed using the functional method (see [40] for an application of this method to the backgammon model).…”

Section: From the Master Equation To The Generating Functionmentioning

confidence: 99%

“…The zero-range process is well-known to give rise to condensation, whose emergence and dynamics are well studied in the case of decreasing hopping rates [7][8][9][10][11][12][13][14]. Non-conserving ZRPs coupled to reservoirs through open boundaries are studied in [15], and stationary currents can be induced by asymmetry of the hopping rate at a defect [16].…”

Section: Introductionmentioning

confidence: 99%

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Non-conserving zero-range processes with extensive rates under resetting

Grange

2020

J. Phys. Commun.

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We consider a non-conserving zero-range process with hopping rate proportional to the number of particles at each site. Particles are added to the system with a site-dependent creation rate, and vanish with a uniform annihilation rate. On a fully-connected lattice with a large number of sites, the meanfield geometry leads to a negative binomial law for the number of particles at each site, with parameters depending on the hopping, creation and annihilation rates. This model can be mapped to population dynamics (if the creation rates are reproductive fitnesses in a haploid population, and the hopping rate is the mutation rate). It can also be mapped to a Bianconi-Barabási model of a growing network with random rewiring of links (if creation rates are the rates of acquisition of links by nodes, and the hopping rate is the rewiring rate). The steady state has recently been worked out and gives rise to occupation numbers that reproduce Kingman's house-of-cards model of selection and mutation. In this paper we solve the master equation using a functional method, which yields integral equations satisfied by the occupation numbers. The occupation numbers are shown to forget initial conditions at an exponential rate that decreases linearly with the fitness level. Moreover, they can be computed exactly in the Laplace domain, which allows to obtain the steady state of the system under resetting. The result modifies the house-of-cards result by simply adding a skewed version of the initial conditions, and by adding the resetting rate to the hopping rate.

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Section: Master Equationmentioning

confidence: 99%

Section: From the Master Equation To The Generating Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Non-conserving zero-range processes with extensive rates under resetting

Grange

2020

J. Phys. Commun.

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“…In this paper we focus on the behavior of the profiles ρ i and the mass displacement χ as a function of the model parameters. For given parameters, we obtain the exact value by first solving equations (8) to compute the π i 's, and then equation (9) and (10). In the simulations, the total number of particles is N = 200 and L = 50.…”

Section: Resultsmentioning

confidence: 99%

Uphill migration in coupled driven particle systems

Cirillo¹,

Colangeli²,

Dickman³

2019

J. Stat. Mech.

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In particle systems subject to a nonuniform drive, particle migration is observed from the driven to the non-driven region and vice-versa, depending on details of the hopping dynamics, leading to apparent violations of Fick's law and of steady-state thermodynamics. We propose and discuss a very basic model in the framework of independent random walkers on a pair of rings, one of which features biased hopping rates, in which this phenomenon is observed and fully explained. arXiv:1904.04325v2 [cond-mat.stat-mech]

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“…We show that the wave function time independent profile of the Kronig-Penney model can be associated with the stationary particle profile of independent random walkers with suitable hopping rates. For convenience, we deal with this model, using the language of the Zero Range Process (ZRP) [4][5][6]15,19,20]. The regular quantum system considered here is the same as the one studied in Ref.…”

Section: Introductionmentioning

confidence: 99%

Transport in Quantum Multi-barrier Systems as Random Walks on a Lattice

2019

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A quantum finite multi-barrier system, with a periodic potential, is considered and exact expressions for its plane wave amplitudes are obtained using the Transfer Matrix method [10]. This quantum model is then associated with a stochastic process of independent random walks on a lattice, by properly relating the wave amplitudes with the hopping probabilities of the particles moving on the lattice and with the injection rates from external particle reservoirs. Analytical and numerical results prove that the stationary density profile of the particle system overlaps with the quantum mass density profile of the stationary Schrödinger equation, when the parameters of the two models are suitably matched. The equivalence between the quantum model and a stochastic particle system would mainly be fruitful in a disordered setup. Indeed, we also show, here, that this connection, analytically proven to hold for periodic barriers, holds even when the width of the barriers and the distance between barriers are randomly chosen.

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Stationary uphill currents in locally perturbed zero-range processes

Cited by 24 publications

References 36 publications

Non-conserving zero-range processes with extensive rates under resetting

Non-conserving zero-range processes with extensive rates under resetting

Uphill migration in coupled driven particle systems

Transport in Quantum Multi-barrier Systems as Random Walks on a Lattice

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