Irreversible random sequential deposition of interacting particles is widely used to model aggregation phenomena in physical, chemical, and biophysical systems. We show that in one dimension the exact time dependent solution of such processes can be found for arbitrary interaction potentials with finite range. The exact solution allows to rigorously prove characteristic features of the deposition kinetics, which have previously only been accessible by simulations. We show in particular that a unique interaction potential exists that leads to a maximally dense line coverage for a given interaction range. Remarkably, this distribution is singular and can only be expressed as a mathematical limit. The relevance of these results for models of nucleosome packing on DNA is discussed. The results highlight how the generation of an optimally dense packing requires a highly coordinated packing dynamics, which can be effectively tuned by the interaction potential even in the presence of intrinsic randomness.
A Generalized Model for Disordered Random Sequential Adsorption with Charge-Dependent Deposition
Centro de Ciências Exatas e Sociais Aplicadas, Universidade Estadual da Paraíba（帕拉伊巴州立大学精确科学与应用科学学院）；Departamento de Física, Universidade Federal de Pernambuco（伯南布哥联邦大学物理系）
We revisit a known model in which (conducting) blocks are hierarchically and randomly deposited on a $d$-dimensional substrate according to a hyperbolic size law with the block size decreasing by a factor $λ\, > 1$ in each subsequent generation. In the first part of the paper the number of coastal points (in $D=1)$ or coastlines (in $D=2)$ is calculated, which are points or lines that separate a region at "sea level" and an elevated region. We find that this number possesses a non-universal character, implying a Euclidean geometry below a threshold value $P_c$ of the deposition probability $P$…
Dynamical phase diagram of synchronization in one dimension: universal behavior from Edwards-Wilkinson to random deposition through Kardar-Parisi-Zhang
A recent breakthrough in glass science has been the synthesis of ultrastable glasses via physical vapor deposition techniques. These samples display enhanced thermodynamic, kinetic and mechanical stability, with important implications for fundamental science and technological applications. However, the vapor deposition technique is limited to atomic, polymer and organic glass-formers and is only able to produce thin film samples. Here, we propose a novel approach to generate ultrastable glassy configurations in the bulk, via random particle bonding, and using computer simulations we show that …
Active walker models have proved to be extremely effective in understanding the evolution of a large class of systems in biology like ant trail formation and pedestrian trails. We propose a simple model of a random walker which modifies its local environment that in turn influences the motion of the walker at a {\em later} time. We perform direct numerical simulations of the walker in a discrete lattice with the walker actively depositing a chemical which attracts the walker trajectory and also evaporates in time. We propose a method to look at the structural transitions of the trajectory usin…
The elephant random walk is a history-dependent random walk. We study a class of interacting elephant random walks. Our model includes the exclusion process as a special case. By means of Monte Carlo simulations and mean-field arguments, we find that it exhibits a condensation phenomenon with a phase transition that is manifestly of first order. The transition point depends on the initial configuration.
Self-interacting random walks are endowed with long range memory effects that emerge from the interaction of the random walker at time $t$ with the territory that it has visited at earlier times $t'<t$. This class of non Markovian random walks has applications in a broad range of examples, ranging from insects to living cells, where a random walker modifies locally its environment -- leaving behind footprints along its path, and in turn responds to its own footprints. Because of their inherent non Markovian nature, the exploration properties of self-interacting random walks have remained elusi…
We study the fluctuations of time-additive random observables in the stochastic dynamics of a system of $N$ non-interacting Ising spins. We mainly consider the case of all-to-all dynamics where transitions are possible between any two spin configurations with uniform rates. We show that the cumulant generating function of the time-integral of a normally distributed quenched random function of configurations, i.e., the energy function of the random energy model (REM), has a phase transition in the large $N$ limit for trajectories of any time extent. We prove this by determining the exact limit …
We study approach to the large-time jammed state of the deposited particles in the model of random sequential adsorption. The convergence laws are usually derived from the argument of Pomeau which includes the assumption of the dominance, at large enough times, of small landing regions into each of which only a single particle can be deposited without overlapping earlier deposited particles and which, after a certain time are no longer created by depositions in larger gaps. The second assumption has been that the size distribution of gaps open for particle-center landing in this large-time sma…
Exercises in simplest dynamical random walk, or Quantum path integral approach to true diffusion law and 1/f noise of classical particle interacting with ideal gas
In ergodic quantum systems, physical observables have a non-relaxing component if they "overlap" with a conserved quantity. In interacting microscopic models, how to isolate the non-relaxing component is unclear. We compute exact dynamical correlators governed by a Hamiltonian composed of two large interacting random matrices, $H=A+B$. We analytically obtain the late-time value of $\langle A(t) A(0) \rangle$; this quantifies the non-relaxing part of the observable $A$. The relaxation to this value is governed by a power-law determined by the spectrum of the Hamiltonian $H$, independent of the …
