Abstract:
The diffusion of finite-size hard-core interacting particles in two- or three-dimensional confined domains is considered in the limit that the confinement dimensions become comparable to the particle's dimensions. The result is a nonlinear diffusion equation for the one-particle probability density function, with an overall collective diffusion which depends on both the excluded-volume and the narrow confinement. By including both these effects the equation is able to interpolate between severe confinement (for example, single-file diffusion) and unconfined diffusion. Numerical solutions of both the effective nonlinear diffusion equation and the stochastic particle system are presented and compared. As an application, the case of diffusion under a ratchet potential is considered, and the change in transport properties due to excluded-volume and confinement effects is examined.

Abstract:
Individual-based models of chemical or biological dynamics usually consider individual entities diffusing in space and performing a birth-death type dynamics. In this work we study the properties of a model in this class where the birth dynamics is mediated by the local, within a given distance, density of particles. Groups of individuals are formed in the system and in this paper we concentrate on the study of the properties of these clusters (lifetime, size, and collective diffusion). In particular, in the limit of the interaction distance approaching the system size, a unique cluster appears which helps to understand and characterize the clustering dynamics of the model.

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.

Abstract:
We prove the existence of solutions of a cross-diffusion parabolic population problem. The system of partial differential equations is deduced as the limit equations satisfied by the densities corresponding to an interacting particles system modeled by stochastic differential equations. According to the values of the diffusion parameters related to the intra and inter-population repulsion intensities, the system may be classified in terms of an associated matrix. For proving the existence of solutions when the matrix is positive definite, we use a fully discrete finite element approximation in a general functional setting. If the matrix is only positive semi-definite, we use a regularization technique based on a related cross-diffusion model under more restrictive functional assumptions. We provide some numerical experiments demonstrating the weak and strong segregation effects corresponding to both types of matrices.

Abstract:
In this paper we study the motion of two particles diffusing on low-dimensional discrete structures in presence of a hard-core repulsive interaction. We show that the problem can be mapped in two decoupled problems of single particles diffusing on different graphs by a transformation we call 'diffusion graph transform'. This technique is applied to study two specific cases: the narrow comb and the ladder lattice. We focus on the determination of the long time probabilities for the contact between particles and their reciprocal crossing. We also obtain the mean square dispersion of the particles in the case of the narrow comb lattice. The case of a sticking potential and of 'vicious' particles are discussed.

Abstract:
Describing the diffusion of particles through crowded, confined environments with which they can interact is of considerable biological and technological interest. Under conditions where the confinement dimensions become comparable to the particle dimensions, steric interactions between particles, as well as particle-wall interactions, will play a crucial role in determining transport properties. To elucidate the effects of these interactions on particle transport, we consider the diffusion and binding of finite-size particles within a channel whose diameter is comparable to the size of the particles. Using a simple lattice model of this process, we calculate the steady-state current and density profiles of both bound and free particles in the channel. We show that the system can exhibit qualitatively different behavior depending on the ratio of the channel width to the particle size. We also perform simulations of this system, and find excellent agreement with our analytic results.

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.

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.

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 ${\cal D}{:=}C_0^\infty({\Bbb R})$.

Abstract:
We study the dynamics of a tracer particle (TP) on a comb lattice populated by randomly moving hard-core particles in the dense limit. We first consider the case where the TP is constrained to move on the backbone of the comb only, and, in the limit of high density of particles, we present exact analytical results for the cumulants of the TP position, showing a subdiffusive behavior $\sim t^{3/4}$. At longer times, a second regime is observed, where standard diffusion is recovered, with a surprising non analytical dependence of the diffusion coefficient on the particle density. When the TP is allowed to visit the teeth of the comb, based on a mean-field-like Continuous Time Random Walk description, we unveil a rich and complex scenario, with several successive subdiffusive regimes, resulting from the coupling between the inhomogeneous comb geometry and particle interactions. Remarkably, the presence of hard-core interactions speeds up the TP motion along the backbone of the structure in all regimes.