We present the results of a study of random sequential adsorption of linear segments (needles) on sites of a square lattice. We show that the percolation threshold is a nonmonotonic function of the length of the adsorbed needle, showing a minimum for a certain length of the needles, while the jamming threshold decreases to a constant with a power law. The ratio of the two thresholds is also nonmonotonic and it remains constant only in a restricted range of the needles length. We determine the values of the correlation length exponent for percolation, jamming, and their ratio.
Porous media are often modelled as systems of overlapping obstacles, which leads to the problem of two percolation thresholds in such systems, one for the porous matrix and the other one for the void space. Here we investigate these percolation thresholds in the model of overlapping squares or cubes of linear size k > 1 randomly distributed on a regular lattice. We find that the percolation threshold of obstacles is a nonmonotonic function of k, whereas the percolation threshold of the void space is well approximated by a function linear in 1/k. We propose a generalization of the excluded volume approximation to discrete systems and use it to investigate the transition between continuous and discrete percolation, finding a remarkable agreement between the theory and numerical results. We argue that the continuous percolation threshold of aligned squares on a plane is the same for the solid and void phases and estimate the continuous percolation threshold of the void space around aligned cubes in a 3D space as 0.036(1). We also discuss the connection of the model to the standard site percolation with complex neighborhood.
Random sequential adsorption (RSA) is a standard method of modeling adsorption of large molecules at the liquid-solid interface. Several studies have recently conjectured that in the RSA of rectangular needles, or k-mers, on a square lattice, percolation is impossible if the needles are sufficiently long (k of order of several thousand). We refute these claims and present rigorous proof that in any jammed configuration of nonoverlapping, fixed-length, horizontal, or vertical needles on a square lattice, all clusters are percolating clusters.
We study random sequential adsorption of flexible chains onto a two-dimensional lattice by computer Monte Carlo simulations. The flexibility of chains is controlled by the temperature of the solution via the Boltzmann factor. We investigate the percolation threshold in the system as a function of chain length and temperature. Several temperature regimes are identified, and respective characteristic types of behavior of the system are discussed. Especially, nonmonotonicity of percolation threshold is observed—there appears a characteristic temperature unique for all chain lengths for which the percolation threshold attains its minimum.
We analyze the unforced and deterministically forced Burgers equation in the framework of the (diffusive) interpolating dynamics that solves the socalled Schrödinger boundary data problem for the random matter transport. This entails an exploration of the consistency conditions that allow to interpret dispersion of passive contaminants in the Burgers flow as a Markovian diffusion process. In general, the usage of a continuity equation ∂ t ρ = −∇( vρ), where v = v( x, t) stands for the Burgers field and ρ is the density of transported matter, is at variance with the explicit diffusion scenario. Under these circumstances, we give a complete characterisation of the diffusive transport that is governed by Burgers velocity fields. The result extends both to the approximate description of the transport driven by an incompressible fluid and to motions in an infinitely compressible medium. Also, in conjunction with the Born statistical postulate in quantum theory, it pertains to the probabilistic (diffusive) counterpart of the Schrödinger picture quantum dynamics. We give a generalisation of this dynamical problem to cases governed by nonconservative force fields when it appears indispensable to relax the gradient velocity field assumption. The Hopf-Cole procedure has been appropriately generalised to yield solutions in that case.
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