A ubiquitous observation in cell biology is that the diffusive motion of macromolecules and organelles is anomalous, and a description simply based on the conventional diffusion equation with diffusion constants measured in dilute solution fails. This is commonly attributed to macromolecular crowding in the interior of cells and in cellular membranes, summarising their densely packed and heterogeneous structures. The most familiar phenomenon is a sublinear, power-law increase of the mean-square displacement as function of the lag time, but there are other manifestations like strongly reduced and time-dependent diffusion coefficients, persistent correlations in time, non-gaussian distributions of spatial displacements, heterogeneous diffusion, and a fraction of immobile particles. After a general introduction to the statistical description of slow, anomalous transport, we summarise some widely used theoretical models: gaussian models like fractional Brownian motion and Langevin equations for visco-elastic media, the continuous-time random walk (CTRW) model, and the Lorentz model describing obstructed transport in a heterogeneous environment. Particular emphasis is put on the spatio-temporal properties of the transport in terms of twopoint correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even if the mean-square displacements are identical. Then, we review the theory underlying commonly applied experimental techniques in the presence of anomalous transport like single-particle tracking, fluorescence correlation spectroscopy (FCS), and fluorescence recovery after photobleaching (FRAP). We report on the large body of recent experimental evidence for anomalous transport in crowded biological media: in cyto-and nucleoplasm as well as in cellular membranes, complemented by in vitro experiments where a variety of model systems mimic physiological crowding conditions. Finally, computer simulations are discussed which play an important role in testing the theoretical models and corroborating the experimental findings. The review is completed by a synthesis of the theoretical and experimental progress identifying open questions for future investigation.

The localization transition and the critical properties of the Lorentz model in three dimensions are investigated by computer simulations. We give a coherent and quantitative explanation of the dynamics in terms of continuum percolation theory and obtain an excellent matching of the critical density and exponents. Within a dynamic scaling ansatz incorporating two divergent length scales we achieve data collapse for the mean-square displacements and identify the leading corrections to scaling. We provide evidence for a divergent non-Gaussian parameter close to the transition. DOI: 10.1103/PhysRevLett.96.165901 PACS numbers: 66.30.Hs, 05.40.ÿa, 61.43.ÿj, 64.60.Ht Transport in heterogeneous and disordered media has important applications in many fields of science including composite materials, rheology, polymer and colloidal science, and biophysics. Recently, dynamic heterogeneities and growing cooperative length scales in structural glasses have attracted considerable interest [1,2]. The physics of gelation, in particular, of colloidal particles with short range attraction [3][4][5][6], is often accompanied by the presence of a fractal cluster generating subdiffusive dynamics. It is of fundamental interest to demonstrate the relevance of such heterogeneous environments on slow anomalous transport.The minimal model for transport of particles through a random medium of fixed obstacles is known as the Lorentz model, and already incorporates the generic ingredients for slow anomalous transport. Earlier, the Lorentz model played a significant role as a testing ground for elaborate kinetic theories, shortly after the discovery of long-time tails in autocorrelation functions for simple liquids in the late 1960s [7], since the nonanalytic dependence of transport coefficients on frequency, wave number, and density predicted for simple liquids [8][9][10][11][12] has a close analog in the Lorentz model [13,14].The simplest variant of the Lorentz model consists of a structureless test particle moving according to Newton's laws in a d-dimensional array of identical obstacles. The latter are distributed randomly and independently in space and interact with the test particle via a hard-sphere repulsion. Consequently, the test particle explores a disordered environment of possibly overlapping regions of excluded volume; see Fig. 1. Because of the hard-core repulsion, the magnitude of the particle velocity, v jvj, is conserved. Then, the only control parameter is the dimensionless obstacle density, n : n d , where denotes the radius of the hard-core potential. At high densities, the model exhibits a localization transition, i.e., above a critical density, the particle is always trapped by the obstacles.Significant insight into the dynamic properties of the Lorentz model has been achieved by a low-density expansion for the diffusion coefficient by Weijland and van Leeuwen [13] rigorously demonstrating the nonanalytic dependence on n . As expected, for low densities the theoretical results compare well with molecular dynamics (...

A second order phase transition for the three dimensional Gross-Neveu model is established for one fermion species N = 1. This transition breaks a parity-like discrete symmetry. It constitutes its peculiar universality class with critical exponent ν = 0.63 and scalar and fermionic anomalous dimension ησ = 0.31 and η ψ = 0.11, respectively. We also compute critical exponents for other N . Our results are based on exact renormalization group equations.An understanding of systems with many fermionic degrees of freedom is one of the big challenges in statistical physics. Due to the anticommuting nature of the variables numerical simulations are not straightforward-analytical methods are crucially needed. One typically has to solve a functional integral for a d dimensional system with Grassmann variables. Approximate solutions for "test models" would be of great value. The Gross-Neveu (GN) model [1] is one of the simplest fermionic models. In three dimensions a discrete symmetry forbids a mass term unless it is spontaneously broken. In the symmetric phase the GN model is therefore a realization of a statistical system of gapless fermions. For a large number N of fermion species it is known [2-5] that a second order phase transition separates the symmetric phase from an ordered phase where the symmetry is spontaneously broken and the fermions become massive. Using methods based on an exact renormalization group equation [6] a second order transition for N ≥ 2 was confirmed. We know, however, of no previous work which clarifies the existence and nature of the phase transition in the simplest model with only one fermion species. The model with one fermion species is inaccessible to lattice simulations due to the fermion doubling problem and the 1/N expansion is not expected to give reasonable results for N = 1. The case N = 1 is also of special interest since an order parameter ψ j ψ j = 0 leads to a ground state which does not admit any discrete symmetry involving the reflection of all coordinates, in contrast to the models with N ≥ 2.In this letter we improve the exact renormalization group approach and establish a second order phase transition for N = 1. We also compute the critical exponents. This is important beyond a possible relevance for real physical systems: the GN model constitutes a peculiar universality class due to the presence of massless fermions at the critical point. Just as the O(N )-Heisenberg models for bosons, the GN model could in the future become a benchmark for our understanding of critical systems in presence of fermions.The GN model describes N fermionic fields with local quartic interaction. Here ψj, j = 1...N , are irreducible representations of the group O(d) including parity reflections, i. e. 2 d/2 component Dirac spinors for d even and 2 (d−1)/2 for d odd.The classical Euclidean actionis symmetric under a coordinate reflection ψ(x) → −ψ(−x) ,ψ(x) →ψ(−x). (We note that ψ andψ are independent variables in an Eucidean formulation.) A nonvanishing expectation value ofψj ψ j spontaneously bre...

The dynamic properties of a classical tracer particle in a random, disordered medium are investigated close to the localization transition. For Lorentz models obeying Newtonian and diffusive motion at the microscale, we have performed large-scale computer simulations, demonstrating that universality holds at long times in the immediate vicinity of the transition. The scaling function describing the crossover from anomalous transport to diffusive motion is found to vary extremely slowly and spans at least five decades in time. To extract the scaling function, one has to allow for the leading universal corrections to scaling. Our findings suggest that apparent power laws with varying exponents generically occur and dominate experimentally accessible time windows as soon as the heterogeneities cover a decade in length scale. We extract the divergent length scales, quantify the spatial heterogeneities in terms of the non-Gaussian parameter, and corroborate our results by a thorough finite-size analysis.