We formulate a scaling theory for the long-time diffusive motion in a space occluded by a high density of moving obstacles in dimensions 1, 2 and 3. Our tracers diffuse anomalously over many decades in time, before reaching a diffusive steady state with an effective diffusion constant $D_\mathrm{eff}$, which depends on the obstacle diffusivity and density. The scaling of $D_\mathrm{eff}$, above and below a critical regime, is characterized by two independent critical parameters: the conductivity exponent $μ$, also found in models with frozen obstacles, and an exponent $ψ$, which quantifies the effect of obstacle diffusivity.
We summarise different results on the diffusion of a tracer particle in lattice gases of hard-core particles with stochastic dynamics, which are confined to narrow channels -- single-files, comb-like structures and quasi-one-dimensional channels with the width equal to several particle diameters. We show that in such geometries a surprisingly rich, sometimes even counter-intuitive, behaviour emerges, which is absent in unbounded systems. We also present a survey of different results obtained for a tracer particle diffusion in unbounded systems, which will permit a reader to have an exhaustivel…
We calculate the diffusion coefficient of an active tracer in a schematic crowded environment, represented as a lattice gas of passive particles with hardcore interactions. Starting from the master equation of the problem, we put forward a closure approximation that goes beyond trivial mean-field and provides the diffusion coefficient for an arbitrary density of crowders in the system. We show that our approximation is accurate for a very wide range of parameters, and that it correctly captures numerous nonequilibrium effects, which are the signature of the activity in the system. In addition …
We analyze a pair of diffusion equations which are derived in the infinite system--size limit from a microscopic, individual--based, stochastic model. Deviations from the conventional Fickian picture are found which ultimately relate to the depletion of resources on which the particles rely. The macroscopic equations are studied both analytically and numerically, and are shown to yield anomalous diffusion which does not follow a power law with time, as is frequently assumed when fitting data for such phenomena. These anomalies are here understood within a consistent dynamical picture which app…
We study the transport of self-propelled particles in dynamic complex environments. To obtain exact results, we introduce a model of run-and-tumble particles (RTPs) moving in discrete time on a $d$-dimensional cubic lattice in the presence of diffusing hard core obstacles. We derive an explicit expression for the diffusivity of the RTP, which is exact in the limit of low density of fixed obstacles. To do so, we introduce a generalization of Kac's theorem on the mean return times of Markov processes, which we expect to be relevant for a large class of lattice gas problems. Our results show the …
The influence of crowding on the diffusion of tagged particles in a dense medium is investigated in the framework of a mean-field model, derived in the continuum limit from a microscopic stochastic process with exclusion. The probability distribution function of the tagged particles obeys to a nonlinear Fokker-Planck equation, where the drift and diffusion terms are determined self-consistently by the concentration of crowders in the medium. Transient sub-diffusive or super-diffusive behaviours are observed, depending on the selected initial conditions, that bridge normal diffusion regimes cha…
We study memory based random walk models to understand diffusive motion in crowded heterogeneous environment. The models considered are non-Markovian as the current move of the random walk models is determined by randomly selecting a move from history. At each step, particle can take right, left or stay moves which is correlated with the randomly selected past step. There is a perfect stay-stay correlation which ensures that the particle does not move if the randomly selected past step is a stay move. The probability of traversing the same direction as the chosen history or reversing it depend…
Diffusion-limited reactions (DLR) are usually described within the Smoluchowski theory, which neglects interactions between the diffusing components. We propose a first extension of such frame- work that incorporates excluded-volume effects, considering hard diffusing agents in the presence of an absorbing sink. For large values of the absorber-to-diffuser size ratio Rs, the encounter rate obtained from the simulations is in very good agreement with a simple generalization of the Smolu- chowski equation at high densities. Reducing Rs, the rate gets substantially depressed at interme- diate pac…
A topic of intense current investigation pursues the question how the highly crowded environment of biological cells affects the dynamic properties of passively diffusing particles. Motivated by recent experiments we report results of extensive simulations of the motion of a finite sized tracer particle in a heterogeneously crowded environment. For given spatial distributions of monodisperse crowders we demonstrate how anomalous diffusion with strongly non-Gaussian features arises in this model system. We investigate both biologically relevant situations of particles released either at the sur…
Diffusion of a two component fluid is studied in the framework of differential equations, but where these equations are systematically derived from a well-defined microscopic model. The model has a finite carrying capacity imposed upon it at the mesoscopic level and this is shown to lead to non-linear cross diffusion terms that modify the conventional Fickean picture. After reviewing the derivation of the model, the experiments carried out to test the model are described. It is found that it can adequately explain the dynamics of two dense ink drops simultaneously evolving in a container fille…
We develop a model to compute the first-passage time of a random walker in a crowded environment. Hard-core particles with the same size and diffusion coefficient than the tracer diffuse, and the model allows to compute the first passage time of the tracer on euclidian lattices. The result is compared to classical Nakazato-Kitahara model, and extends previous results obtained for persistent random walker. The crowding in a confined media acts as a memory effect, and thus lead to a persistent-like behavior.
We study a two-dimensional diffusive motion of a tracer particle in restricted, crowded anisotropic geometries. The underlying medium is the same as in our previous work [J. Chem. Phys. 140, 044706 (2014)] in which standard, gaussian diffusion was studied. Here, a tracer is allowed to perform Cauchy random walk with uncorrelated steps. Our analysis shows that presence of obstacles significantly influences motion, which in an obstacle-free space would be of a superdiffusive type. At the same time, the selfdiffusive process reveals different anomalous properties, both at the level of a single tr…
We study a strongly interacting crowded system of self-propelled stiff filaments by event-driven Brownian dynamics simulations and an analytical theory to elucidate the intricate interplay of crowding and self-propulsion. We find a remarkable increase of the effective diffusivity upon increasing the filament number density by more than one order of magnitude. This counter-intuitive 'crowded is faster' behavior can be rationalized by extending the concept of a confining tube pioneered by Doi and Edwards for highly entangled crowded, passive to active systems. We predict a scaling theory for the…
We study the behavior of the stationary velocity of a driven particle in an environment of mobile hard-core obstacles. Based on a lattice gas model, we demonstrate analytically that the drift velocity can exhibit a nonmonotonic dependence on the applied force, and show quantitatively that such negative differential mobility (NDM), observed in various physical contexts, is controlled by both the density and diffusion time scale of obstacles. Our study unifies recent numerical and analytical results obtained in specific regimes, and makes it possible to determine analytically the region of the f…
We study dynamically crowded solutions of stiff fibers deep in the semidilute regime, where the motion of a single constituent becomes increasingly confined to a narrow tube. The spatiotemporal dynamics for wave numbers resolving the motion in the confining tube becomes accessible in Brownian dynamics simulations upon employing a geometry-adapted neighbor list. We demonstrate that in such crowded environments the intermediate scattering function, characterizing the motion in space and time, can be predicted quantitatively by simulating a single freely diffusing phantom needle only, yet with ve…
Diffusion in the crowded environments of the biological membranes or materials interfaces often involves intermittent binding to surface proteins or defects. To account for this situation we study a 2-dimensional lattice gas in a field of immobilized traps. Using kinetic Monte Carlo simulations, we calculate the effective diffusion coefficient in the long-time limit as a function of the traps and particle densities. We find a remarkable result - an increase of the diffusion coefficient with particle density, an effect that we coin as crowding-enhanced diffusion. We rationalize this result usin…
30% of the DNA in E. coli bacteria is covered by proteins. Such high degree of crowding affect the dynamics of generic biological processes (e.g. gene regulation, DNA repair, protein diffusion etc.) in ways that are not yet fully understood. In this paper, we theoretically address the diffusion constant of a tracer particle in a one dimensional system surrounded by impenetrable crowder particles. While the tracer particle always stays on the lattice, crowder particles may unbind to a surrounding bulk and rebind at another or the same location. In this scenario we determine how the long time di…
Under dilute in vitro conditions transcription factors rapidly locate their target sequence on DNA by using the facilitated diffusion mechanism. However, whether this strategy of alternating between three-dimensional bulk diffusion and one-dimensional sliding along the DNA contour is still beneficial in the crowded interior of cells is highly disputed. Here we use a simple model for the bacterial genome inside the cell and present a semi-analytical model for the in vivo target search of transcription factors within the facilitated diffusion framework. Without having to resort to extensive simu…
We are discussing long-time, scaling limit for the anomalous diffusion composed of the subordinated Lévy-Wiener process. The limiting anomalous diffusion is in general non-Markov, even in the regime, where ensemble averages of a mean-square displacement or quantiles representing the group spread of the distribution follow the scaling characteristic for an ordinary stochastic diffusion. To discriminate between truly memory-less process and the non-Markov one, we are analyzing deviation of the survival probability from the (standard) Sparre-Andersen scaling.
Diffusion-limited aggregation is consistent with simple scaling. However, strong subdominant terms are present, and these can account for various earlier claims of anomalous scaling. We show this in detail for the case of multiscaling.
The long wavelength diffusion coefficient of a critical fluid confined between two parallel plates separated by a distance L is strongly affected by the finite size. Finite size scaling leads us to expect that the vanishing of the diffusion coefficient as ξ^{-1} for ξ<<L, ξbeing the correlation length, would crossover to Ł^{-1} for ξ>>L. We show that this is not strictlytrue. There is a logarthmic scaling violation. We construct a Kawasaki like scaling function that connects the thermodynamic regime to the extreme critical (ξ>>L) regime.
We study the dynamics of solutions of infinitely thin needles up to densities deep in the semidilute regime by Brownian dynamics simulations. For high densities, these solutions become strongly entangled and the motion of a needle is essentially restricted to a one-dimensional sliding in a confining tube composed of neighboring needles. From the density-dependent behavior of the orientational and translational diffusion, we extract the long-time transport coefficients and the geometry of the confining tube. The sliding motion within the tube becomes visible in the non-Gaussian parameter of the…
What happens when a continuously evolving stochastic process is interrupted with large changes at random intervals $τ$ distributed as a power-law $\sim τ^{-(1+α)};α>0$? Modeling the stochastic process by diffusion and the large changes as abrupt resets to the initial condition, we obtain {\em exact} closed-form expressions for both static and dynamic quantities, while accounting for strong correlations implied by a power-law. Our results show that the resulting dynamics exhibits a spectrum of rich long-time behavior, from an ever-spreading spatial distribution for $α< 1$, to one that is time i…
A dynamic scaling of a diffusion process involving the Langmuir type adsorption is studied. We find dynamic scaling functions in one and two dimensions and compare them with direct numerical simulations, and we further study the dynamic scaling law on fractal surfaces. The adsorption-diffusion process obeys the fracton dynamics on the fractal surfaces.
We introduce a heterogeneous continuous time random walk (HCTRW) model as a versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as porous or disordered media, multiscale or crowded environments, weighted graphs or networks. We derive the exact form of the propagator and investigate the effects of spatio-temporal heterogeneities onto the diffusive dynamics via the spectral properties of the generalized transition matrix. In particular, we show how the distribution of first passage times changes due to local and global heterogeneities of…
Combining experiments and theory, we address the dynamics of self-propelled particles in crowded environments. We first demonstrate that motile colloids cruising at constant speed through random lattices undergo a smooth transition from diffusive, to subdiffusive, to localized dynamics upon increasing the obstacle density. We then elucidate the nature of these transitions by performing extensive simulations constructed from a detailed analysis of the colloid-obstacle interactions. We evidence that repulsion at a distance and hard-core interactions both contribute to slowing down the long-time …
We investigate the diffusion limited aggregation of particles executing persistent random walks. The scaling properties of both random walks and large aggregates are presented. The aggregates exhibit a crossover between ballistic and diffusion limited aggregation models. A non-trivial scaling relation $ξ\sim\ell^{1.25}$ between the characteristic size $ξ$, in which the cluster undergoes a morphological transition, and the persistence length $\ell$, between ballistic and diffusive regimes of the random walk, is observed.
We study diffusion-controlled single-species annihilation with a finite number of particles. In this reaction-diffusion process, each particle undergoes ordinary diffusion, and when two particles meet, they annihilate. We focus on spatial dimensions $d>2$ where a finite number of particles typically survive the annihilation process. Using the rate equation approach and scaling techniques we investigate the average number of surviving particles, $M$, as a function of the initial number of particles, $N$. In three dimensions, for instance, we find the scaling law $M\sim N^{1/3}$ in the asymptoti…
Random deposition model with surface diffusion over several next nearest neighbours is studied. The results agree with the results obtained by Family for the case of nearest neighbour diffusion [F. Family, J. Phys. A 19(8), L441, 1986]. However for larger diffusion steps, the growth exponent and the roughness exponent show interesting dependence on diffusion length.
We present results of the numerical simulations and the scaling characteristics of one-dimensional random fluctuations with heavy tailed probability distribution functions. Assuming that the distribution function of the random fluctuations obeys Lévy statistics with a power-law scaling exponent, we investigate the fractional diffusion equation in the presence of $μ$-stable Lévy noise. e study the scaling properties of the global width and two point correlation functions, we then compare the analytical and numerical results for the growth exponent $β$ and the roughness exponent $α$. We also inv…
The passive and active motion of micron-sized tracer particles in crowded liquids and inside living biological cells is ubiquitously characterised by "viscoelastic" anomalous diffusion, in which the increments of the motion feature long-ranged negative and positive correlations. While viscoelastic anomalous diffusion is typically modelled by a Gaussian process with correlated increments, so-called fractional Gaussian noise, an increasing number of systems are reported, in which viscoelastic anomalous diffusion is paired with non-Gaussian displacement distributions. Following recent advances in…
The methods currently used to determine the scaling exponent of a complex dynamic process described by a time series are based on the numerical evaluation of variance. This means that all of them can be safely applied only to the case where ordinary statistical properties hold true even if strange kinetics are involved. We illustrate a method of statistical analysis based on the Shannon entropy of the diffusion process generated by the time series, called Diffusion Entropy Analysis (DEA). We adopt artificial Gauss and Lévy time series, as prototypes of ordinary and anomalus statistics, respect…
This paper presents a simple, one-dimensional model of a randomly advected passive scalar. The model exhibits anomalous inertial range scaling for the structure functions constructed from scalar differences. The model provides a simple computational test for recent ideas regarding closure and scaling for randomly advected passive scalars. Results suggest that high order structure function scaling depends on the largest velocity eddy size, and hence scaling exponents may be geometry-dependent and non-universal.
The equation which describes a particle diffusing in a logarithmic potential arises in diverse physical problems such as momentum diffusion of atoms in optical traps, condensation processes, and denaturation of DNA molecules. A detailed study of the approach of such systems to equilibrium via a scaling analysis is carried out, revealing three surprising features: (i) the solution is given by two distinct scaling forms, corresponding to a diffusive (x ~ \sqrt{t}) and a subdiffusive (x >> \sqrt{t}) length scales, respectively; (ii) the scaling exponents and scaling functions corresponding to bot…
We analyze the role of spectral diffusion in the problem of many-body delocalization in quantum dots and in extended systems. The spectral diffusion parametrically enhances delocalization, modifying the scaling of the delocalization threshold with the interaction coupling constant.
To solve the obscureness in measurement brought about from the weak ergodicity breaking appeared in anomalous diffusions we have suggested the time-averaged mean squared displacement (MSD) $\bar{δ^2 (τ)}_τ$ with a integral interval depending linearly on the lag time $τ$. For the continuous time random walk describing a subdiffusive behavior, we have found that $\bar{δ^2 (τ)}_τ\sim τ^γ$ like that of the ensemble-averaged MSD, which makes it be possible to measure the proper exponent values through time-average in experiments like a single molecule tracking. Also we have found that it is origina…
Brownian particles suspended in disordered crowded environments often exhibit non-Gaussian normal diffusion (NGND), whereby their displacements grow with mean square proportional to the observation time and non-Gaussian statistics. Their distributions appear to decay almost exponentially according to "universal" laws largely insensitive to the observation time. This effect is generically attributed to slow environmental fluctuations, which perturb the local configuration of the suspension medium. To investigate the microscopic mechanisms responsible for the NGND phenomenon, we study Brownian d…
We consider a continuous-space and continuous-time diffusion process under resetting with memory. A particle resets to a position chosen from its trajectory in the past according to a memory kernel. Depending on the form of the memory kernel, we show analytically how different asymptotic behaviours of the variance of the particle position emerge at long times. These range from standard diffusive ($σ^2 \sim t$) all the way to anomalous ultraslow growth $σ^2 \sim \ln \ln t$.
Fundamental thermodynamics and an earlier elastic solid-state point defect model [P. Varotsos and K. Alexopoulos, Phys. Rev B 15, 4111 (1977); 18, 2683 (1978)] are employed to formulate an analytical second-order polynomial function describing the density scaling of the diffusion coefficient in viscous liquids. The function parameters are merely determined by the scaling exponent, which is directly connected with the Gruneisen constant. Density scaling diffusion coefficient isotherms obtained at different pressures collapse on a unique master curve, in agreement with recent computer simulation…
The standard setup for single-file diffusion is diffusing particles in one dimension which cannot overtake each other, where the dynamics of a tracer (tagged) particle is of main interest. In this article we generalise this system and investigate first-passage properties of a tracer particle when flanked by crowder particles which may, besides diffuse, unbind (rebind) from (to) the one-dimensional lattice with rates $k_{\rm off}$ ($k_{\rm on}$). The tracer particle is restricted to diffuse with rate $k_D$ on the lattice. Such a model is relevant for the understanding of gene regulation where r…
The question of universality class of pair contact process with diffusion (PCPD) is revisited with an alternative approach. We study persistence in Generalized Pair-Contact Process with diffusion (GPCPD) introduced by Noh and Park, (Phys. Rev. E 69,016122(2004)). This model allows us to interpolate between directed percolation (DP) and PCPD universality classes. We find that transition to nonzero persistence is at same parameter value as transition to zero number density. We obtain finite size scaling and off-critical scaling collapse for persistence and find critical exponents by fitting phen…
We prove a law of large numbers and a central limit theorem for a tagged particle in a symmetric simple exclusion process in the one-dimensional lattice with variable diffusion coefficient. The scaling limits are obtained from a similar result for the current through -1/2 for a zero-range process with bond disorder. For the CLT, we prove convergence to a fractional Brownian motion of Hurst exponent 1/4.
A scaling theory is developed for diffusion-limited cluster aggregation in a porous medium, where the primary particles and clusters stick irreversibly to the walls of the pore space as well as to each other. Three scaling regimes are predicted, connected by smooth crossovers. The first regime is at low primary particle concentrations where the primary particles stick individually to the walls. The second regime is at intermediate concentrations where clusters grow to a certain size, smaller than the pore size, then stick individually to the walls. The third regime is at high concentrations wh…
We present a (heuristic) theoretical derivation for the scaling of the diffusion coefficient $D_f$ for fluctuating ``pulled'' fronts. In agreement with earlier numerical simulations, we find that as $N\to\infty$, $D_f$ approaches zero as $1/\ln^3N$, where $N$ is the average number of particles per correlation volume in the stable phase of the front. This behaviour of $D_f$ stems from the shape fluctuations at the very tip of the front, and is independent of the microscopic model.
We discuss the scaling of characteristic lengths in diffusion limited aggregation (DLA) clusters in light of recent developments using conformal maps. We are led to the conjecture that the apparently anomalous scaling of lengths is due to one slow crossover. This is supported by an analytical argument for the scaling of the penetration depth of newly arrived random walkers, and by numerical evidence on the Laurent coefficients which uniquely determine each cluster. We find a single crossover exponent of -0.3 for all the characteristic lengths in DLA. This gives a hint about the structure of th…
The scaling invariance for chaotic orbits near a transition from unlimited to limited diffusion in a dissipative standard mapping is explained via the analytical solution of the diffusion equation. It gives the probability of observing a particle with a specific action at a given time. We show the diffusion coefficient varies slowly with the time and is responsible to suppress the unlimited diffusion. The momenta of the probability are determined and the behavior of the average squared action is obtained. The limits of small and large time recover the results known in the literature from the p…
We address the problem of the statistical analysis of a time series generated by complex dynamics with a new method: the Diffusion Entropy Analysis (DEA) (Fractals, {\bf 9}, 193 (2001)). This method is based on the evaluation of the Shannon entropy of the diffusion process generated by the time series imagined as a physical source of fluctuations, rather than on the measurement of the variance of this diffusion process, as done with the traditional methods. We compare the DEA to the traditional methods of scaling detection and we prove that the DEA is the only method that always yields the cor…
The manner in which the intermolecular potential u(r) governs structural relaxation in liquids is a long standing problem in condensed matter physics. Herein we show that diffusion coefficients for simulated Lennard-Jones m-6 liquids (8<m<36) in normal and moderately supercooled states are a unique function of the variable rho^g/T, where rho is density and T is temperature. The scaling exponent g is a material specific constant whose magnitude is related to the steepness of the repulsive part of u(r), evaluated around the distance of closest approach between particles probed in the supercooled…
Diffusion Limited Aggregation (DLA) is a model of fractal growth that had attained a paradigmatic status due to its simplicity and its underlying role for a variety of pattern forming processes. We present a convergent calculation of the fractal dimension D of DLA based on a renormalization scheme for the first Laurent coefficient of the conformal map from the unit circle to the expanding boundary of the fractal cluster. The theory is applicable from very small (2-3 particles) to asymptotically large (n \to \infty) clusters. The computed dimension is D=1.713\pm 0.003.
An early (and influential) scaling relation in the multifractal theory of Diffusion Limited Aggregation(DLA) is the Turkevich-Scher conjecture that relates the exponent α_{min} that characterizes the ``hottest'' region of the harmonic measure and the fractal dimension D of the cluster, i.e. D=1+α_{min}. Due to lack of accurate direct measurements of both D and α_{min} this conjecture could never be put to serious test. Using the method of iterated conformal maps D was recently determined as D=1.713+-0.003. In this Letter we determine α_{min} accurately, with the result α_{min}=0.665+-0.004. We…
Crystallization kinetics has features that are universal and independent of the type of crystallized system. The possibility of using scaling relations to describe the temperature dependences of the surface self-diffusion coefficient $D_s$, which is one of the key characteristics of crystallization kinetics, has been demonstrated in application to various crystallized molecular glasses. It has been shown that the surface self-diffusion coefficient $D_s$ as a function of the dimensionless temperature is reproduced by a power law and is universally scaled for all considered systems. The analysis…
