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Register a new userSteady states in a non-conserving zero-range process with extensive rates as a model for the balance of selection and mutation
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Authors
Pascal Grange
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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 removed from the system with a uniform annihilation rate. On a fully-connected lattice with a large number of sites, the mean-field 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 of particles is mapped to a model of population dynamics: the site label is interpreted as a level of fitness, the site-dependent creation rate is interpreted as a selection function, and the hopping process is interpreted as the introduction of mutants. In the limit of large density, the fraction of the total population occupying each site approaches the limiting distribution in the house-of-cards model of selection-mutation, introduced by Kingman. A single site can be occupied by a macroscopic fraction of the particles if the mutation rate is below a critical value (which matches the critical value worked out in the house-of-cards model). This feature generalises to classes of selection functions that increase sufficiently fast at high fitness. The process can be mapped to a model of evolving networks, inspired by the Bianconi--Barabasi model, but involving a large and fixed set of nodes. Each node forms links at a rate biased by its fitness, moreover links are destroyed at a uniform rate, and redirected at a certain rate. If this redirection rate matches the mutation rate, the number of links pointing to nodes of a given fitness level is distributed as the numbers of particles in the non-conserving zero-range process. There is a finite critical redirection rate if the density of quenched fitnesses goes to zero sufficiently fast at high fitness.
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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 mean-field 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--Barabasi 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 Kingmans 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.
Explicit expressions for arrival times of particles moving in a one-dimensional Zero-Range Process (ZRP) are computed. Particles are fed into the ZRP from an injection site and can also evaporate from anywhere in the interior of the ZRP. Two dynamics are considered; bulk dynamics, where particle hopping and decay is proportional to the numqber of particles at each site, and surface dynamics, where only the top particle at each site can hop or evaporate. We find exact solutions in the bulk dynamics case and for a single-site ZRP obeying surface dynamics. For a multisite ZRP obeying surface dynamics, we compare simulations with approximations obtained from the steady-state limit, where mean interarrival times for both models are equivalent. Our results highlight the competition between injection and evaporation on the arrival times of particles to an absorbing site.
A generalized zero-range process with a limited number of long-range interactions is studied as an example of a transport process in which particles at a T-junction make a choice of which branch to take based on traffic levels on each branch. The system is analysed with a self-consistent mean-field approximation which allows phase diagrams to be constructed. Agreement between the analysis and simulations is found to be very good.
We calculate the exact stationary distribution of the one-dimensional zero-range process with open boundaries for arbitrary bulk and boundary hopping rates. When such a distribution exists, the steady state has no correlations between sites and is uniquely characterized by a space-dependent fugacity which is a function of the boundary rates and the hopping asymmetry. For strong boundary drive the system has no stationary distribution. In systems which on a ring geometry allow for a condensation transition, a condensate develops at one or both boundary sites. On all other sites the particle distribution approaches a product measure with the finite critical density rho_c. In systems which do not support condensation on a ring, strong boundary drive leads to a condensate at the boundary. However, in this case the local particle density in the interior exhibits a complex algebraic growth in time. We calculate the bulk and boundary growth exponents as a function of the system parameters.
We study asymmetric exclusion processes (TASEP) on a nonuniform one-dimensional ring consisting of two segments having unequal hopping rates, or {em defects}. We allow weak particle nonconservation via Langmuir kinetics (LK), that are parameterised by generic unequal attachment and detachment rates. For an extended defect, in the thermodynamic limit the system generically displays inhomogeneous density profiles in the steady state - the faster segment is either in a phase with spatially varying density having no density discontinuity, or a phase with a discontinuous density changes. Nonequilibrium phase transitions between them are controlled by the inhomogeneity and LK. The slower segment displays only macroscopically uniform bulk density profiles in the steady states, reminiscent of the maximal current phase of TASEP but with a bulk density generally different from half. With a point defect, there are low and high density spatially uniform phases as well, in addition to the inhomogeneous density profiles observed for an extended defect. In all the cases, it is argued that the the mean particle density in the steady state is controlled only by the ratio of the LK attachment and detachment rates.
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