Generalized elastic model yields a fractional Langevin equation description

A resetting particle embedded in a viscoelastic bath

We examine the behavior of a colloidal particle immersed in a viscoelastic bath undergoing stochastic resetting at a rate r. Microscopic probes suspended in a viscoelastic environment do not follow the classical theory of Brownian motion. This is primarily because the memory from successive collisions between the medium particles and the probes does not necessarily decay instantly as opposed to the classical Langevin equation. To treat such a system, one needs to incorporate the memory effects into the Langevin equation. The resulting equation formulated by Kubo, known as the generalized Langevin equation (GLE), has been instrumental to describing the transport of particles in inhomogeneous or viscoelastic environments. The purpose of this work, henceforth, is to study the behavior of such a colloidal particle governed by the GLE under resetting dynamics. To this end, we extend the renewal formalism to compute the general expression for the position variance and the correlation function of the resetting particle driven by the environmental memory. These generic results are then illustrated for the prototypical example of the Jeffreys viscoelastic fluid model. In particular, we identify various timescales and intermittent plateaus in the transient phase before the system relaxes to the steady state; and further discuss the effect of resetting pertaining to these behaviors. Our results are supported by numerical simulations showing an excellent agreement.

Generalized Langevin equation for a tagged monomer in a Gaussian semiflexible polymer

In this study, we present a comprehensive analysis of the motion of a tagged monomer within a Gaussian semiflexible polymer model. We carefully derived the generalized Langevin equation (GLE) that governs the motion of a tagged central monomer. This derivation involves integrating out all the other degrees of freedom within the polymer chain, thereby yielding an effective description of the viscoelastic motion of the tagged monomer. A critical component of our analysis is the memory kernel that appears in the GLE. By examining this kernel, we characterized the impact of bending rigidity on the non-Markovian diffusion dynamics of the tagged monomer. Furthermore, we calculated the mean-squared displacement of the tagged monomer using the derived GLE. Our theoretical findings were corroborated by the Langevin dynamics simulation and scaling theory. Our results not only show remarkable agreement with previously known results in certain limiting cases but also provide dynamic features over the entire timescale.

Decorrelation of a leader by an increasing number of followers

We compute the connected two-time correlator of the maximum M_{N}(t) of N independent Gaussian stochastic processes (GSPs) characterized by a common correlation coefficient ρ that depends on the two times t_{1} and t_{2}. We show analytically that this correlator, for fixed times t_{1} and t_{2}, decays for large N as a power law N^{-γ} (with logarithmic corrections) with a decorrelation exponent γ=(1-ρ)/(1+ρ) that depends only on ρ, but otherwise is universal for any GSP. We study several examples of physical processes including the fractional Brownian motion (fBm) with Hurst exponent H and the Ornstein-Uhlenbeck process (OUP). For the fBm, ρ is only a function of τ=sqrt[t_{1}/t_{2}] and we find an interesting freezing transition at a critical value τ=τ_{c}=(3-sqrt[5])/2. For τ<τ_{c}, there is an optimal H^{*}(τ)>0 that maximizes the exponent γ and this maximal value freezes to γ=1/3 for τ>τ_{c}. For the OUP, we show that γ=tanh(μ|t_{1}-t_{2}|/2), where μ is the stiffness of the harmonic trap. Numerical simulations confirm our analytical predictions.

Anomalous phase diagram of the elastic interface with nonlocal hydrodynamic interactions in the presence of quenched disorder

We investigate the influence of quenched disorder on the steady states of driven systems of the elastic interface with nonlocal hydrodynamic interactions. The generalized elastic model (GEM), which has been used to characterize numerous physical systems such as polymers, membranes, single-file systems, rough interfaces, and fluctuating surfaces, is a standard approach to studying the dynamics of elastic interfaces with nonlocal hydrodynamic interactions. The criticality and phase transition of the quenched generalized elastic model are investigated numerically and the results are presented in a phase diagram spanned by two tuning parameters. We demonstrate that in the one-dimensional disordered driven GEM, three qualitatively different behavior regimes are possible with a proper specification of the order parameter (mean velocity) for this system. In the vanishing order parameter regime, the steady-state order parameter approaches zero in the thermodynamic limit. A system with a nonzero mean velocity can be in either the continuous regime, which is characterized by a second-order phase transition, or the discontinuous regime, which is characterized by a first-order phase transition. The focus of this research is to investigate the critical scaling features near the pinning-depinning threshold. The behavior of the quenched generalized elastic model at the critical depinning force is explored. Near the depinning threshold, the critical exponent is obtained numerically.

Non-Markovian Modeling of Nonequilibrium Fluctuations and Dissipation in Active Viscoelastic Biomatter

Based on a Hamiltonian that incorporates the elastic coupling between a tracer particle and the embedding active viscoelastic biomatter, we derive a generalized non-Markovian Langevin model for the nonequilibrium mechanical tracer response. Our analytical expressions for the frequency-dependent tracer response function and the tracer positional autocorrelation function agree quantitatively with experimental data for red blood cells and actomyosin networks with and without adenosine triphosphate over the entire frequency range and in particular reproduce the low-frequency violation of the fluctuation-dissipation theorem. The viscoelastic power laws, the elastic constants and effective friction coefficients extracted from the experimental data allow straightforward physical interpretation.

Heterogeneous anomalous transport in cellular and molecular biology

It is well established that a wide variety of phenomena in cellular and molecular biology involve anomalous transport e.g. the statistics for the motility of cells and molecules are fractional and do not conform to the archetypes of simple diffusion or ballistic transport. Recent research demonstrates that anomalous transport is in many cases heterogeneous in both time and space. Thus single anomalous exponents and single generalised diffusion coefficients are unable to satisfactorily describe many crucial phenomena in cellular and molecular biology. We consider advances in the field ofheterogeneous anomalous transport(HAT) highlighting: experimental techniques (single molecule methods, microscopy, image analysis, fluorescence correlation spectroscopy, inelastic neutron scattering, and nuclear magnetic resonance), theoretical tools for data analysis (robust statistical methods such as first passage probabilities, survival analysis, different varieties of mean square displacements, etc), analytic theory and generative theoretical models based on simulations. Special emphasis is made on high throughput analysis techniques based on machine learning and neural networks. Furthermore, we consider anomalous transport in the context of microrheology and the heterogeneous viscoelasticity of complex fluids. HAT in the wavefronts of reaction-diffusion systems is also considered since it plays an important role in morphogenesis and signalling. In addition, we present specific examples from cellular biology including embryonic cells, leucocytes, cancer cells, bacterial cells, bacterial biofilms, and eukaryotic microorganisms. Case studies from molecular biology include DNA, membranes, endosomal transport, endoplasmic reticula, mucins, globular proteins, and amyloids.

Nuclear magnetic resonance signal decay in the presence of a background gradient: Normal and anomalous diffusion

A novel way for calculating the diffusion-weighted nuclear magnetic resonance (NMR) attenuation signal expression in the presence of a background gradient is developed. This method is easily applicable to NMR-attenuated signals arising from any pulse field gradient sequence experiments. Here, we provide detailed calculations for the classical pulsed gradient stimulated echo and the pulsed gradient spin echo, as the particular cases. Within this general theoretical framework, devised for Gaussian processes with stationary increments, we recover and extend the previous Stejskal-Tanner results in the case of normal diffusion and we furnish a new expression in the case of anomalous diffusion.

A new perspective of molecular diffusion by nuclear magnetic resonance

The diffusion-weighted NMR signal acquired using Pulse Field Gradient (PFG) techniques, allows for extrapolating microstructural information from porous materials and biological tissues. In recent years there has been a multiplication of diffusion models expressed by parametric functions to fit the experimental data. However, clear-cut criteria for the model selection are lacking. In this paper, we develop a theoretical framework for the interpretation of NMR attenuation signals in the case of Gaussian systems with stationary increments. The full expression of the Stejskal-Tanner formula for normal diffusing systems is devised, together with its extension to the domain of anomalous diffusion. The range of applicability of the relevant parametric functions to fit the PFG data can be fully determined by means of appropriate checks to ascertain the correctness of the fit. Furthermore, the exact expression for diffusion weighted NMR signals pertaining to Brownian yet non-Gaussian processes is also derived, accompanied by the proper check to establish its contextual relevance. The analysis provided is particularly useful in the context of medical MRI and clinical practise where the hardware limitations do not allow the use of narrow pulse gradients.

Driven transport of soft Brownian particles through pore-like structures: Effective size method

Single-file transport in pore-like structures constitutes an important topic for both theory and experiment. For hardcore interacting particles, a good understanding of the collective dynamics has been achieved recently. Here, we study how softness in the particle interaction affects the emergent transport behavior. To this end, we investigate the driven Brownian motion of particles in a periodic potential. The particles interact via a repulsive softcore potential with a shape corresponding to a smoothed rectangular barrier. This shape allows us to elucidate effects of mutual particle penetration and particle crossing in a controlled manner. We find that even weak deviations from the hardcore case can have a strong impact on the particle current. Despite this fact, knowledge about the transport in a corresponding hardcore system is shown to be useful to describe and interpret our findings for the softcore case. This is achieved by assigning a thermodynamic effective size to the particles based on the equilibrium density functional of hard spheres.

Understanding biochemical processes in the presence of sub-diffusive behavior of biomolecules in solution and living cells

In order to maintain cellular function, biomolecules like protein, DNA, and RNAs have to diffuse to the target spaces within the cell. Changes in the cytosolic microenvironment or in the nucleus during the fulfillment of these cellular processes affect their mobility, folding, and stability thereby impacting the transient or stable interactions with their adjacent neighbors in the organized and dynamic cellular interior. Using classical Brownian motion to elucidate the diffusion behavior of these biomolecules is hard considering their complex nature. The understanding of biomolecular diffusion inside cells still remains elusive due to the lack of a proper model that can be extrapolated to these cases. In this review, we have comprehensively addressed the progresses in this field, laying emphasis on the different aspects of anomalous diffusion in the different biochemical reactions in cell interior. These experiment-based models help to explain the diffusion behavior of biomolecules in the cytosolic and nuclear microenvironment. Moreover, since understanding of biochemical reactions within living cellular system is our main focus, we coupled the experimental observations with the concept of sub-diffusion from in vitro to in vivo condition. We believe that the pairing between the understanding of complex behavior and structure-function paradigm of biological molecules would take us forward by one step in order to solve the puzzle around diseases caused by cellular dysfunction.

Single-file transport in periodic potentials: The Brownian asymmetric simple exclusion process

Single-file Brownian motion in periodic structures is an important process in nature and technology, which becomes increasingly amenable for experimental investigation under controlled conditions. To explore and understand generic features of this motion, the Brownian asymmetric simple exclusion process (BASEP) was recently introduced. The BASEP refers to diffusion models where hard spheres are driven by a constant drag force through a periodic potential. Here we derive general properties of the rich collective dynamics in the BASEP. Average currents in the steady state change dramatically with the particle size and density. For an open system coupled to particle reservoirs, extremal current principles predict various nonequilibrium phases, which we verify by Brownian dynamics simulations. For general pair interactions we discuss connections to single-file transport by traveling-wave potentials and prove the impossibility of current reversals in systems driven by a constant drag and by traveling waves.

Protein-driven lipid domain nucleation in biological membranes

Lipid rafts are heterogeneous dynamic lipid domains of the cell membranes that are involved in several biological processes, such as protein and lipid specific transport and signaling. Our understanding of lipid raft formation is still limited due to the transient and elusive nature of these domains in vivo, in contrast with the stable phase-separated domains observed in artificial membranes. Inspired by experimental findings highlighting the relevance of transmembrane proteins for lipid rafts, we investigate lipid domain nucleation by coarse-grained molecular dynamics and Ising-model simulations. We find that the presence of a transmembrane protein can trigger lipid domain nucleation in a flat membrane from an otherwise mixed lipid phase. Furthermore, we study the role of the lipid domain in the diffusion of the protein showing that its mobility is hindered by the presence of the raft. The results of our coarse-grained molecular-dynamics and Ising-model simulations thus coherently support the important role played by transmembrane proteins in lipid domain formation and stability.

Fractional Langevin equation from damped bath dynamics

We consider the stochastic dynamics of a system linearly coupled to a hierarchical thermal bath with two well-separated inherent timescales: one slow, and one fast. The slow part of the bath is modeled as a set of harmonic oscillators and taken into account explicitly, while the effects of the fast part of the bath are simulated by dissipative and stochastic Langevin forces, uncorrelated in space and time, acting on oscillators of the slow part of the bath. We demonstrate for this model the robust emergence of a fractional Langevin equation with a power-law decaying memory kernel. The conditions of such an emergence and the specific value of the fractional exponent depend only on the asymptotic low-frequency spectral properties of the slow part of the bath.

Scale-invariant Green-Kubo relation for time-averaged diffusivity

In recent years it was shown both theoretically and experimentally that in certain systems exhibiting anomalous diffusion the time- and ensemble-averaged mean-squared displacement are remarkably different. The ensemble-averaged diffusivity is obtained from a scaling Green-Kubo relation, which connects the scale-invariant nonstationary velocity correlation function with the transport coefficient. Here we obtain the relation between time-averaged diffusivity, usually recorded in single-particle tracking experiments, and the underlying scale-invariant velocity correlation function. The time-averaged mean-squared displacement is given by 〈δ^{2}[over ¯]〉∼2D_{ν}t^{β}Δ^{ν-β}, where t is the total measurement time and Δ is the lag time. Here ν is the anomalous diffusion exponent obtained from ensemble-averaged measurements 〈x^{2}〉∼t^{ν}, while β≥-1 marks the growth or decline of the kinetic energy 〈v^{2}〉∼t^{β}. Thus, we establish a connection between exponents that can be read off the asymptotic properties of the velocity correlation function and similarly for the transport constant D_{ν}. We demonstrate our results with nonstationary scale-invariant stochastic and deterministic models, thereby highlighting that systems with equivalent behavior in the ensemble average can differ strongly in their time average. If the averaged kinetic energy is finite, β=0, the time scaling of 〈δ^{2}[over ¯]〉 and 〈x^{2}〉 are identical; however, the time-averaged transport coefficient D_{ν} is not identical to the corresponding ensemble-averaged diffusion constant.

Model of chromosomal loci dynamics in bacteria as fractional diffusion with intermittent transport

The short-time dynamics of bacterial chromosomal loci is a mixture of subdiffusive and active motion, in the form of rapid relocations with near-ballistic dynamics. While previous work has shown that such rapid motions are ubiquitous, we still have little grasp on their physical nature, and no positive model is available that describes them. Here, we propose a minimal theoretical model for loci movements as a fractional Brownian motion subject to a constant but intermittent driving force, and compare simulations and analytical calculations to data from high-resolution dynamic tracking in E. coli. This analysis yields the characteristic time scales for intermittency. Finally, we discuss the possible shortcomings of this model, and show that an increase in the effective local noise felt by the chromosome associates to the active relocations.

Scaling characteristics of one-dimensional fractional diffusion processes in the presence of power-law distributed random noise

Here, we present results of 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. We study the scaling properties of the global width and two-point correlation functions and then compare the analytical and numerical results for the growth exponent β and the roughness exponent α. We also investigate the fractional Fokker-Planck equation for heavy-tailed random fluctuations. We show that the fractional diffusion processes in the presence of μ-stable Lévy noise display special scaling properties in the probability distribution function (PDF). Finally, we numerically study the scaling properties of the heavy-tailed random fluctuations by using the diffusion entropy analysis. This method is based on the evaluation of the Shannon entropy of the PDF generated by the random fluctuations, rather than on the measurement of the global width of the process. We apply the diffusion entropy analysis to extract the growth exponent β and to confirm the validity of our numerical analysis.

Procedures for Model-Guided Data Analysis of Chromosomal Loci Dynamics at Short Time Scales

This chapter provides theoretical background and practical procedures for model-guided analysis of mobility of chromosomal loci from movies of many single trajectories. We guide the reader through existing physical models and measurable quantities, illustrating how this knowledge is useful for the interpretation of the measurements.

Single file dynamics in soft materials

The term single file (SF) dynamics refers to the motion of an assembly of particles through a channel with cross-sections comparable to the particles' diameter. Single file diffusion (SFD) is then the diffusion of a tagged particle in a single file, i.e., under the condition that particle passing is not allowed. SFD accounts for a large variety of processes in nature, including diffusion of colloids in synthetic and natural channels, biological motors along molecular chains, electrons in proteins and liquid helium, ions through membranes, just to mention a few examples. Albeit introduced in 1965s, over the last decade the classical notion of SF dynamics has been generalised to account for a more realistic modelling of the particle properties, file geometry, particle-particle and channel-particle interactions, which paves the way to remarkable applications of the SF model, for instance, in the technology of bio-integrated nanodevices. We provide here a comprehensive review of the recent advances in the theory of SF dynamics with the purpose of spurring further experimental work.

Aging Wiener-Khinchin theorem and critical exponents of 1/f^{β} noise

The power spectrum of a stationary process may be calculated in terms of the autocorrelation function using the Wiener-Khinchin theorem. We here generalize the Wiener-Khinchin theorem for nonstationary processes and introduce a time-dependent power spectrum 〈S_{t_{m}}(ω)〉 where t_{m} is the measurement time. For processes with an aging autocorrelation function of the form 〈I(t)I(t+τ)〉=t^{Υ}ϕ_{EA}(τ/t), where ϕ_{EA}(x) is a nonanalytic function when x is small, we find aging 1/f^{β} noise. Aging 1/f^{β} noise is characterized by five critical exponents. We derive the relations between the scaled autocorrelation function and these exponents. We show that our definition of the time-dependent spectrum retains its interpretation as a density of Fourier modes and discuss the relation to the apparent infrared divergence of 1/f^{β} noise. We illustrate our results for blinking-quantum-dot models, single-file diffusion, and Brownian motion in a logarithmic potential.

Aging Wiener-Khinchin Theorem

The Wiener-Khinchin theorem shows how the power spectrum of a stationary random signal I(t) is related to its correlation function ⟨I(t)I(t+τ)⟩. We consider nonstationary processes with the widely observed aging correlation function ⟨I(t)I(t+τ)⟩∼t(γ)ϕ(EA)(τ/t) and relate it to the sample spectrum. We formulate two aging Wiener-Khinchin theorems relating the power spectrum to the time- and ensemble-averaged correlation functions, discussing briefly the advantages of each. When the scaling function ϕ(EA)(x) exhibits a nonanalytical behavior in the vicinity of its small argument we obtain the aging 1/f-type of spectrum. We demonstrate our results with three examples: blinking quantum dots, single-file diffusion, and Brownian motion in a logarithmic potential, showing that our approach is valid for a wide range of physical mechanisms.

A simple non-chaotic map generating subdiffusive, diffusive, and superdiffusive dynamics

Analytically tractable dynamical systems exhibiting a whole range of normal and anomalous deterministic diffusion are rare. Here, we introduce a simple non-chaotic model in terms of an interval exchange transformation suitably lifted onto the whole real line which preserves distances except at a countable set of points. This property, which leads to vanishing Lyapunov exponents, is designed to mimic diffusion in non-chaotic polygonal billiards that give rise to normal and anomalous diffusion in a fully deterministic setting. As these billiards are typically too complicated to be analyzed from first principles, simplified models are needed to identify the minimal ingredients generating the different transport regimes. For our model, which we call the slicer map, we calculate all its moments in position analytically under variation of a single control parameter. We show that the slicer map exhibits a transition from subdiffusion over normal diffusion to superdiffusion under parameter variation. Our results may help to understand the delicate parameter dependence of the type of diffusion generated by polygonal billiards. We argue that in different parameter regions the transport properties of our simple model match to different classes of known stochastic processes. This may shed light on difficulties to match diffusion in polygonal billiards to a single anomalous stochastic process.

Numerical approach to unbiased and driven generalized elastic model

From scaling arguments and numerical simulations, we investigate the properties of the generalized elastic model (GEM) that is used to describe various physical systems such as polymers, membranes, single-file systems, or rough interfaces. We compare analytical and numerical results for the subdiffusion exponent β characterizing the growth of the mean squared displacement 〈(δh)(2)〉 of the field h described by the GEM dynamic equation. We study the scaling properties of the qth order moments 〈∣δh∣(q)〉 with time, finding that the interface fluctuations show no intermittent behavior. We also investigate the ergodic properties of the process h in terms of the ergodicity breaking parameter and the distribution of the time averaged mean squared displacement. Finally, we study numerically the driven GEM with a constant, localized perturbation and extract the characteristics of the average drift for a tagged probe.

Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking

This Perspective summarises the properties of a variety of anomalous diffusion processes and provides the necessary tools to analyse and interpret recorded anomalous diffusion data.

Renewal and memory origin of anomalous diffusion: a discussion of their joint action

The adoption of the formalism of fractional calculus is an elegant way to simulate either subdiffusion or superdiffusion from within a renewal perspective where the occurrence of an event at a given time t does not have any memory of the events occurring at earlier times. We illustrate a physical model to assign infinite memory to renewal anomalous diffusion and we find (i) a condition where the simultaneous action of a renewal and a memory source of subdiffusion generates localization and (ii) a condition where they make subdiffusion weaker and superdiffusion emerge. We argue that our approach may provide important contributions to the current search to distinguish the renewal from the memory source of subdiffusion.

Dynamics of a tagged monomer: effects of elastic pinning and harmonic absorption

We study the dynamics of a tagged monomer of a Rouse polymer for different initial configurations. In the case of free evolution, the monomer displays subdiffusive behavior with strong memory of the initial state. In the presence of either elastic pinning or harmonic absorption, we show that the steady state is independent of the initial condition that, however, strongly affects the transient regime, resulting in nonmonotonic behavior and power-law relaxation with varying exponents.

Everlasting effect of initial conditions on single-file diffusion

We study the dynamics of a tagged particle in an environment of point Brownian particles with hard-core interactions in an infinite one-dimensional channel (a single-file model). In particular, we examine the influence of initial conditions on the dynamics of the tagged particle. We compare two initial conditions: equal distances between particles and uniform density distribution. The effect is shown by the differences of mean-square-displacement and correlation function for the two ensembles of initial conditions. We discuss the violation of Einstein relation, and its dependence on the initial condition, and the difference between time and ensemble averaging. More specifically, using the Jepsen line, we will discuss how transport coefficients, like diffusivity, depend on the initial state. Our work shows that initial conditions determine the long time limit of the dynamics, and in this sense the system never forgets its initial state in complete contrast with thermal systems (i.e., a closed system that attains equilibrium independent of the initial state).

Evolution equation for tagged-particle density and correlations in single-file diffusion

We derive and study a theoretical description for single-file diffusion, i.e., diffusion in a one-dimensional lattice of particles with hard core interaction. It is well known that for this system a tagged particle has anomalous diffusion for long times. The novelty of the present approach is that it allows for the derivation of correlations between a tagged particle and other particles of the system at a given distance with empty sites in between. The behavior of the correlation gives deeper insights into the processes involved. The numerical integration of differential equations are in good agreement with Monte Carlo simulations.

Dynamics of discrete semiflexible chains under dihedral constraints: analytic results

Here we consider the dynamics of semiflexible polymers subject both to angular and to dihedral constraints. We succeed in obtaining analytically the dynamical matrix of such systems by extending the formalism developed by Dolgushev and Blumen [J. Chem. Phys. 131, 044905 (2009)]. This leads to a set of Langevin equations whose eigenvalues determine many dynamical properties. Exemplarily, we display the mechanical relaxation loss moduli [G"(ω)] as a function of several, distinct sets of microscopic stiffness parameters; it turns out that such differences lead to macroscopically distinct patterns.

Anomalous transport in the crowded world of biological cells

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, summarizing their densely packed and heterogeneous structures. The most familiar phenomenon is a sublinear, power-law increase of the mean-square displacement (MSD) as a 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 summarize some widely used theoretical models: Gaussian models like fractional Brownian motion and Langevin equations for visco-elastic media, the continuous-time random walk 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 two-point correlation functions, dynamic scaling behaviour, and how the models are distinguished by their propagators even if the MSDs 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.

Fractional Brownian motors and stochastic resonance

We study fluctuating tilt Brownian ratchets based on fractional subdiffusion in sticky viscoelastic media characterized by a power law memory kernel. Unlike the normal diffusion case, the rectification effect vanishes in the adiabatically slow modulation limit and optimizes in a driving frequency range. It is shown also that the anomalous rectification effect is maximal (stochastic resonance effect) at optimal temperature and can be of surprisingly good quality. Moreover, subdiffusive current can flow in the counterintuitive direction upon a change of temperature or driving frequency. The dependence of anomalous transport on load exhibits a remarkably simple universality.

Inequivalence of time and ensemble averages in ergodic systems: exponential versus power-law relaxation in confinement

Single-particle tracking has become a standard tool for the investigation of diffusive properties, especially in small systems such as biological cells. Usually the resulting time series are analyzed in terms of time averages over individual trajectories. Here we study confined normal as well as anomalous diffusion, modeled by fractional Brownian motion and the fractional Langevin equation, and show that even for such ergodic systems time-averaged quantities behave differently from their ensemble-averaged counterparts, irrespective of how long the measurement time becomes. Knowledge of the exact behavior of time averages is therefore fundamental for the proper physical interpretation of measured time series, in particular, for extraction of the relaxation time scale from data.

Correlations in a generalized elastic model: fractional Langevin equation approach

The generalized elastic model (GEM) provides the evolution equation which governs the stochastic motion of several many-body systems in nature, such as polymers, membranes, and growing interfaces. On the other hand a probe (tracer) particle in these systems performs a fractional Brownian motion due to the spatial interactions with the other system's components. The tracer's anomalous dynamics can be described by a fractional Langevin equation (FLE) with a space-time correlated noise. We demonstrate that the description given in terms of GEM coincides with that furnished by the relative FLE, by showing that the correlation functions of the stochastic field obtained within the FLE framework agree with the corresponding quantities calculated from the GEM. Furthermore we show that the Fox H -function formalism appears to be very convenient to describe the correlation properties within the FLE approach.