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On convergence of dynamics of hopping particles to a birth-and-death process in continuum  [PDF]
Dmitri Finkelshtein,Yuri Kondratiev,Eugene Lytvynov
Mathematics , 2008,
Abstract: We show that some classes of birth-and-death processes in continuum (Glauber dynamics) may be derived as a scaling limit of a dynamics of interacting hopping particles (Kawasaki dynamics)
Diffusion of interacting particles in discrete geometries  [PDF]
T. Becker,K. Nelissen,B. Cleuren,B. Partoens,C. Van den Broeck
Physics , 2013, DOI: 10.1103/PhysRevLett.111.110601
Abstract: We evaluate the self-diffusion and transport diffusion of interacting particles in a discrete geometry consisting of a linear chain of cavities, with interactions within a cavity described by a free-energy function. Exact analytical expressions are obtained in the absence of correlations, showing that the self-diffusion can exceed the transport diffusion if the free-energy function is concave. The effect of correlations is elucidated by comparison with numerical results. Quantitative agreement is obtained with recent experimental data for diffusion in a nanoporous zeolitic imidazolate framework material, ZIF-8.
Long term behaviour of locally interacting birth-and-death processes  [PDF]
Vadim Shcherbakov,Stanislav Volkov
Mathematics , 2014, DOI: 10.1007/s10955-014-1122-8
Abstract: In this paper we study long-term evolution of a finite system of locally interacting birth-and-death processes labelled by vertices of a finite connected graph. A detailed description of the asymptotic behaviour is obtained in the case of both constant vertex degree graphs and star graphs. The model is motivated by modelling interactions between populations and is related to interacting particle systems, Gibbs models with unbounded spins, as well as urn models with interaction.
Self diffusion in a system of interacting Langevin particles  [PDF]
D. S. Dean,A. Lefèvre
Physics , 2004, DOI: 10.1103/PhysRevE.69.061111
Abstract: The behavior of the self diffusion constant of Langevin particles interacting via a pairwise interaction is considered. The diffusion constant is calculated approximately within a perturbation theory in the potential strength about the bare diffusion constant. It is shown how this expansion leads to a systematic double expansion in the inverse temperature $\beta$ and the particle density $\rho$. The one-loop diagrams in this expansion can be summed exactly and we show that this result is exact in the limit of small $\beta$ and $\rho\beta$ constant. The one-loop result can also be re-summed using a semi-phenomenological renormalization group method which has proved useful in the study of diffusion in random media. In certain cases the renormalization group calculation predicts the existence of a diverging relaxation time signalled by the vanishing of the diffusion constant -- possible forms of divergence coming from this approximation are discussed. Finally, at a more quantitative level, the results are compared with numerical simulations, in two-dimensions, of particles interacting via a soft potential recently used to model the interaction between coiled polymers.
Mean field approach for diffusion of interacting particles  [PDF]
G. Suárez,M. Hoyuelos,H. Mártin
Physics , 2015,
Abstract: A nonlinear Fokker-Planck equation is obtained in the continuous limit of a one-dimensional lattice with an energy landscape of wells and barriers. Interaction is possible among particles in the same energy well. A parameter $\gamma$, related to the barrier's heights, is introduced. Its value is determinant for the functional dependence of the mobility and diffusion coefficient on particle concentration, but has no influence on the equilibrium solution. A relation between the mean field potential and the microscopic interaction energy is derived. The results are illustrated with classical particles with interactions that reproduce fermion and boson statistics.
Diffusion of interacting particles in discrete geometries: equilibrium and dynamical properties  [PDF]
T. Becker,K. Nelissen,B. Cleuren,B. Partoens,C. Van den Broeck
Physics , 2014, DOI: 10.1103/PhysRevE.90.052139
Abstract: We expand on a recent study of a lattice model of interacting particles [Phys. Rev. Lett. 111, 110601 (2013)]. The adsorption isotherm and equilibrium fluctuations in particle number are discussed as a function of the interaction. Their behavior is similar to that of interacting particles in porous materials. Different expressions for the particle jump rates are derived from transition state theory. Which expression should be used depends on the strength of the inter-particle interactions. Analytical expressions for the self- and transport diffusion are derived when correlations, caused by memory effects in the environment, are neglected. The diffusive behavior is studied numerically with kinetic Monte Carlo (kMC) simulations, which reproduces the diffusion including correlations. The effect of correlations is studied by comparing the analytical expressions with the kMC simulations. It is found that the Maxwell-Stefan diffusion can exceed the self-diffusion. To our knowledge, this is the first time this is observed. The diffusive behavior in one-dimensional and higher dimensional systems is qualitatively the same, with the effect of correlations decreasing for increasing dimension. The length dependence of both the self- and transport diffusion is studied for one-dimensional systems. For long lengths the self-diffusion shows a one over length dependence. Finally, we discuss when agreement with experiments and simulations can be expected. The assumption that particles in different cavities do not interact is expected to hold quantitatively at low and medium particle concentrations, if the particles are not strongly interacting.
Continuum limits of particles interacting via diffusion  [PDF]
Nicholas D. Alikakos,Giorgio Fusco,Georgia Karali
Abstract and Applied Analysis , 2004, DOI: 10.1155/s1085337504310080
Abstract: We consider a two-phase system mainly in three dimensions and weexamine the coarsening of the spatial distribution, driven by thereduction of interface energy and limited by diffusion asdescribed by the quasistatic Stefan free boundary problem. Underthe appropriate scaling we pass rigorously to the limit by takinginto account the motion of the centers and the deformation of thespherical shape. We distinguish between two different cases and wederive the classical mean-field model and another continuum limitcorresponding to critical density which can be related to acontinuity equation obtained recently by Niethammer andOtto. So, the theory of Lifshitz, Slyozov, and Wagner is improved by takinginto account the geometry of the spatial distribution.
Infinite interacting diffusion particles I: Equilibrium process and its scaling limit  [PDF]
Yuri Kondratiev,Eugene Lytvynov,Michael R?ckner
Mathematics , 2003,
Abstract: A stochastic dynamics $({\bf X}(t))_{t\ge0}$ of a classical continuous system is a stochastic process which takes values in the space $\Gamma$ of all locally finite subsets (configurations) in $\Bbb R$ and which has a Gibbs measure $\mu$ as an invariant measure. We assume that $\mu$ corresponds to a symmetric pair potential $\phi(x-y)$. An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics--the so-called gradient stochastic dynamics, or interacting Brownian particles--has been investigated. By using the theory of Dirichlet forms, we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form ${\cal E}_\mu^\Gamma$ on $L^2(\Gamma;\mu)$, and under general conditions on the potential $\phi$, prove its closability. For a potential $\phi$ having a ``weak'' singularity at zero, we also write down an explicit form of the generator of ${\cal E}_\mu^\Gamma$ on the set of smooth cylinder functions. We then show that, for any Dirichlet form ${\cal E}_\mu^\Gamma$, there exists a diffusion process that is properly associated with it. Finally, we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in $C([0,\infty),{\cal D}')$, where ${\cal D}'$ is the dual space of ${\cal D}{:=}C_0^\infty({\Bbb R})$.
A hierarchical kinetic theory of birth, death, and fission in age-structured interacting populations  [PDF]
Tom Chou,Chris D Greenman
Quantitative Biology , 2015,
Abstract: We study mathematical models describing the evolution of stochastic age-structured populations. After reviewing existing approaches, we present a full kinetic framework for age-structured interacting populations undergoing birth, death and fission processes, in spatially dependent environments. We define the complete probability density for the population-size-age-chart and find results under specific conditions. Connections with more classical models are also explicitly derived. In particular, we show that factorial moments for non-interacting processes are described by a natural generalization of the McKendrick-von Foerster equation, which describes mean-field deterministic behaviour. Our approach utilizes mixed type, multi-dimensional probability distributions similar to those employed in the study of gas kinetics, with terms that satisfy BBGKY-like equation hierarchies.
A linear birth-death process on a star graph and its diffusion approximation  [PDF]
Antonio Di Crescenzo,Barbara Martinucci,Abdelaziz Rhandi
Mathematics , 2014,
Abstract: Consider a continuous-time random walk on a lattice formed by the integers of $d$ semiaxes joined at the origin, i.e. a star graph. The motion on each ray behaves as a one-dimensional linear birth-death process with immigration. When the walk reaches the origin, then it may jumps toward any semiaxis. We investigate transient and asymptotic behaviours of the resulting stochastic process, as well as its diffusion approximation. As a byproduct, we obtain a closed form of the number of permutations with a fixed number of components, and a new series form of the polylogarithm function involving the Gauss hypergeometric function.
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