We study Lévy walks in quenched disordered one-dimensional media, with scatterers spaced according to a long-tailed distribution. By analyzing the scaling relations for the random-walk probability and for the resistivity in the equivalent electric problem, we obtain the asymptotic behavior of the mean square displacement as a function of the exponent characterizing the scatterers distribution. We demonstrate that in quenched media different average procedures can display different asymptotic behavior. In particular, we estimate the moments of the displacement averaged over processes starting from scattering sites. Our results are compared with numerical simulations, with excellent agreement. Random walks in quenched random environments occur in many fields of statistical and condensed matter physics [1], as they represent the simplest model of diffusion phenomena and non-deterministic motion. Disorder and geometrical confinement are known to strongly influence transport properties. In particular, in highly spatial inhomogeneous media, the diffusion process is often characterized by large distance diffusion events, which play a crucial role in transport phenomena and can strongly enhance them [2]. Molecular diffusion at low pressure in porous media is dominated by collision with pore walls, with ballistic motion inside the large pores [3], diffusion in chemical space over polymer chains can be described by a distribution of step length with power law behavior [4]. In addition, recent experiments on new disordered optical materials paved the way to a tuned engineering of Lévy-like distributed step lengths [5]. These and many other processes can often be successfully analyzed using the Lévy walks formalism [6]: The random walker can perform long steps, whose distribution is characterized by a power law behavior λ(r) ∼ 1/r α+1 , with α > 0, for large distance displacements r.An important feature of these experimental settings is that the random walk is in general correlated, and the correlation is induced by the topology of the quenched medium. Diffusing agents moving in highly inhomogeneous regions, where they just experienced a long distance jump without being scattered, have a high probability of being backscattered at the subsequent step undergoing a jump of similar size, and this leads to a correlation in step lengths. While the effect of annealed disorder on transport properties in Lévy walks is quite well understood [7], the role of correlations in Lévy-like motion is still an open problem.If the motion occurs in low dimensional samples, spatial correlations in jump probabilities can deeply influence the diffusion properties. This was first evidenced in models of Lévy flights, [8] and more recently discussed in one-dimensional models for Lévy walks on quenched and correlated random environments. The recent studies focused, respectively, on the mean square displacement in a Lévy-Lorentz gas [9], and on the conductivity and transmission through a chain of barriers with Lévy-distributed spacings [10]. Both studi...

Lévy-type walks with correlated jumps, induced by the topology of the medium, are studied on a class of one-dimensional deterministic graphs built from generalized Cantor and Smith-Volterra-Cantor sets. The particle performs a standard random walk on the sets but is also allowed to move ballistically throughout the empty regions. Using scaling relations and the mapping onto the electric network problem, we obtain the exact values of the scaling exponents for the asymptotic return probability, the resistivity, and the mean-square displacement as a function of the topological parameters of the sets. Interestingly, the system undergoes a transition from superdiffusive to diffusive behavior as a function of the filling of the fractal. The deterministic topology also allows us to discuss the importance of the choice of the initial condition. In particular, we demonstrate that local and average measurements can display different asymptotic behavior. The analytic results are compared to the numerical solution of the master equation of the process.

In a cell, the folding of a protein molecule into tertiary structure can begin while it is synthesized by the ribosome. The rate at which individual amino acids are incorporated into the elongating nascent chain has been shown to affect the likelihood that proteins will populate their folded state, indicating that co-translational protein folding is a far from equilibrium process. Developing a theoretical framework to accurately describe this process is, therefore, crucial for advancing our understanding of how proteins acquire their functional conformation in living cells. Current state-of-the-art computational approaches, such as molecular dynamics simulations, are very demanding in terms of the required computer resources, making the simulation of co-translational protein folding difficult. Here, we overcome this limitation by introducing an efficient approach that predicts the effects that variable codon translation rates have on co-translational folding pathways. Our approach is based on Markov chains. By using as an input a relatively small number of molecular dynamics simulations, it allows for the computation of the probability that a nascent protein is in any state as a function of the translation rate of individual codons along a mRNA's open reading frame. Due to its computational efficiency and favorable scalability with the complexity of the folding mechanism, this approach could enable proteome-wide computational studies of the influence of translation dynamics on co-translational folding.

We consider a random walk on one-dimensional inhomogeneous graphs built from Cantor fractals. Our study is motivated by recent experiments that demonstrated superdiffusion of light in complex disordered materials, thereby termed Lévy glasses. We introduce a geometric parameter α which plays a role analogous to the exponent characterizing the step length distribution in random systems. We study the large-time behavior of both local and average observables; for the latter case, we distinguish two different types of averages, respectively over the set of all initial sites and over the scattering sites only. The "single long jump approximation" is applied to analytically determine the different asymptotic behaviours as a function of α and to understand their origin. We also discuss the possibility that the root of the mean square displacement and the characteristic length of the walker distribution may grow according to different power laws; this anomalous behaviour is typical of processes characterized by Lévy statistics and here, in particular, it is shown to influence average quantities.PACS numbers:

Protein translation is one of the most important processes in cell life, but despite being well-understood biochemically, the implications of its intrinsic stochastic nature have not been fully elucidated. In this paper we develop a microscopic and stochastic model which describes a crucial step in protein translation, namely the binding of the tRNA to the ribosome. Our model explicitly takes into consideration tRNA recharging dynamics, spatial inhomogeneity, and stochastic fluctuations in the number of charged tRNAs around the ribosome. By analyzing this nonequilibrium system we are able to derive the statistical distribution of the times needed by the tRNAs to bind to the ribosome, and to show that it deviates from an exponential due to the coupling between the fluctuations of charged and uncharged populations of tRNA.