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The diffusion process of N hard rods in a 1D interval of length L (--> inf) is studied using scaling arguments and an asymptotic analysis of the exact N-particle probability density function (PDF). In the class of such systems, the universal scaling law of the tagged particles mean absolute displacement reads, <|r|>~ <|r|>_{free}/n^mu, where <|r|>_{free} is the result for a free particle in the studied system and n is the number of particles in the covered length. The exponent mu is given by, mu=1/(1+a), where a is associated with the particles density law of the system, rho~rho_0*L^(-a), 0<= a <=1. The scaling law for <|r|> leads to, <|r|>~rho_0^((a-1)/2) (<|r| >_{free})^((1+a)/2), an equation that predicts a smooth interpolation between single file diffusion and free particle diffusion depending on the particles density law, and holds for any underlying dynamics. In particular, <|r|>~t^((1+a)/2) for normal diffusion, with a Gaussian PDF in space for any value of a (deduced by a complementary analysis), and, <|r|>~t^((beta(1+a))/2), for anomalous diffusion in which the systems particles all have the same power-law waiting time PDF for individual events, psi~t^(-1-beta), 0<beta<1. Our analysis shows that the scaling <|r|>~t^(1/2) in a standard single file is a direct result of the fixed particles density condition imposed on the system, a=0.
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Single-file diffusion is a ubiquitous physical process exploited by living and synthetic systems to exchange molecules with their environment. It is paramount quantifying the escape time needed for single files of particles to exit from constraining synthetic channels and biological pores. This quantity depends on complex cooperative effects, whose predominance can only be established through a strict comparison between theory and experiments. By using colloidal particles, optical manipulation, microfluidics, digital microscopy and theoretical analysis we uncover the self-similar character of the escape process and provide closed-formula evaluations of the escape time. We find that the escape time scales inversely with the diffusion coefficient of the last particle to leave the channel. Importantly, we find that at the investigated {bf microscale}, bias forces as tiny as $10^{-15};{rm N}$ determine the magnitude of the escape time by drastically reducing interparticle collisions. Our findings provide crucial guidelines to optimize the design of micro- and nano-devices for a variety of applications including drug delivery, particle filtering and transport in geometrical constrictions.
We study propagation dynamics of a particle phase in a single-file pore connected to a reservoir of particles (bulk liquid phase). We show that the total mass $M(t)$ of particles entering the pore up to time $t$ grows as $M(t) = 2 m(J,rho_F) sqrt{D_0 t}$, where $D_0$ is the bare diffusion coefficient and the prefactor $m(J,rho_F)$ is a non-trivial function of the reservoir density $rho_F$ and the amplitude $J$ of attractive particle-particle interactions. Behavior of the dynamic density profiles is also discussed.
We study the statistics of a tagged particle in single-file diffusion, a one-dimensional interacting infinite-particle system in which the order of particles never changes. We compute the two-time correlation function for the displacement of the tagged particle for an arbitrary single-file system. We also discuss single-file analogs of the arcsine law and the law of the iterated logarithm characterizing the behavior of Brownian motion. Using a macroscopic fluctuation theory we devise a formalism giving the cumulant generating functional. In principle, this functional contains the full statistics of the tagged particle trajectory---the full single-time statistics, all multiple-time correlation functions, etc. are merely special cases.
Normal dynamics in a quasi-one-dimensional channel of length L (toinfty) of N hard spheres are analyzed. The spheres are heterogeneous: each has a diffusion coefficient D that is drawn from a probability density function (PDF), W D^(-{gamma}), for small D, where 0leq{gamma}<1. The initial spheres density {rho} is non-uniform and scales with the distance (from the origin) l as, {rho} l^(-a), 0leqaleq1. An approximation for the N-particle PDF for this problem is derived. From this solution, scaling law analysis and numerical simulations, we show here that the mean square displacement for a particle in such a system obeys, <r^2>~t^(1-{gamma})/(2c-{gamma}), where c=1/(1+a). The PDF of the tagged particle is Gaussian in position. Generalizations of these results are considered.
In this work, we present an effective discrete Edwards-Wilkinson equation aimed to describe the single-file diffusion process. The key physical properties of the system are captured defining an effective elasticity, which is proportional to the single particle diffusion coefficient and to the inverse squared mean separation between particles. The effective equation gives a description of single-file diffusion using the global roughness of the system of particles, which presents three characteristic regimes, namely normal diffusion, subdiffusion and saturation, separated by two crossover times. We show how these regimes scale with the parameters of the original system. Additional repulsive interaction terms are also considered and we analyze how the crossover times depend on the intensity of the additional terms. Finally, we show that the roughness distribution can be well characterized by the Edwards-Wilkinson universal form for the different single-file diffusion processes studied here.
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