Using computer simulations in two dimensions, we investigate the dynamics and structure of a passive polymeric tracer with different topologies immersed in a low-density active particle bath. One of the key observations is that polymers exhibit faster dynamics compared to passive colloidal particles at high activity, for the same particle density, in both linear and star polymer topologies. This enhanced motion is attributed to the accumulation of active particles, which induces prolonged and persistent movement of the polymer. Further analysis reveals that star polymers exhibit more complex and intriguing behavior than their linear counterparts. Notably, the accumulation of active particles promotes the pairing of arms in star polymers. For instance, a three-armed star polymer adopts a conformation similar to a linear polymer with two arms due to this pairing; as a result, at high activity, the dynamics of both the polymers converge. Finally, we explore the dynamics of a linear polymer with the same total number of beads as the star polymer. Interestingly, at high activity—where arm pairing in the star polymer is significant—the star polymer demonstrates faster dynamics than the linear polymer, despite having the identical number of beads. These findings contribute to a broader understanding of the interactions between active and passive components of varying topologies in dilute systems and highlight their potential for innovative applications ranging from materials science to biomedicine.

Cohesin-driven loop extrusion shapes chromosome organization, yet what determines loop lengths and coordination between the extruding arms in vivo remains unclear. Broad extrusion "fountains" in Hi-C maps point to arm desynchronization, whose physical origin is unknown. Here we develop a kinetic theory of extrusion through transient chromatin roadblocks and neighbouring cohesins. Roadblock abundance, lifetime and partial permeability, together with cohesin crowding, combine into an effective obstacle density that renormalizes processivity and sets the mean loop length. Full loop-length distributions reveal extrusion symmetry: one-sided extrusion remains exponential, whereas effectively two-sided extrusion can generate a finite-length peak. ChIA-PET and MNase HiChIP data match the two-sided predictions, disfavouring purely one-sided extrusion. Cohesin crowding further desynchronizes the arms while leaving a residual correlation approaching ρ=1/4. Fountain anisotropy across three vertebrates yields correlations near this value, consistent with local cohesin crowding as a sufficient mechanism for fountain formation. Fits further reveal 30-100-kb cohesin-loading regions and, in Danio rerio , a loading width comparable to the extent of enhancer enrichment, motivating a model in which an enhancer-rich loading platform acts as a collective barrier sustaining outward extrusion toward surrounding promoters. Our framework shows how cohesin and roadblock kinetics jointly determine loop statistics, shape chromosome-contact patterns, and may facilitate enhancer-promoter search.

Chiral fluids are defined by broken mirror or time-reversal symmetry, giving rise to tensorial transport coefficients with antisymmetric components. A key example is the odd mobility tensor, which governs the response of a chiral tracer to an applied force and induces a characteristic transverse drift. While this response is well understood in the infinite dilution limit, the impact of interparticle interactions on the tracer dynamics remains largely unexplored. Here, we conduct an analytical and computational study of a chiral fluid with interparticle interactions and show that, under an external driving force, a chiral tracer can undergo a complete reversal of both its transverse Hall drift and its advection along the force. This reversal emerges from the interplay between odd mobility and interaction-mediated forces, resulting in a phenomenon reminiscent of absolute negative mobility.

Myopic self-avoiding fractional Brownian motion (FBM) is a stochastic process in which an ensemble of particles is driven by fractional Gaussian noise while being repelled by the gradient of the time-integrated ensemble density [J. House, R. Bakhshizada, S. Janušonis, R. Metzler, and T. Vojta, Phys. Rev. E 112, 034119 (2025)]. Depending on the anomalous diffusion exponent α characterizing the noise, the process features two dynamical regimes: an interaction-dominated regime (α< α_c=4/(d+2)) where the mean-density interaction governs long-time dynamics, and a noise-dominated regime (α> α_c) where FBM correlations prevail. In the interaction-dominated regime, the mean-squared displacement grows as ⟨ r^2(t) ⟩∼ t^4/(d+2) regardless of α, while for α> α_c the standard FBM scaling ⟨ r^2(t) ⟩∼ t^α is recovered. Here, we develop an analytical mean-field theory of myopic self-avoiding FBM, based on a Fokker-Planck approach to the interaction-dominated regime. This allows us to derive closed-form polynomial solutions for the probability density. To compare with computer simulations, we develop an efficient radial binning algorithm that significantly reduces the computational complexity, making large-scale three-dimensional simulations feasible. Extensive simulations in one, two, and three dimensions confirm the analytical predictions. We also discuss the application of the process to the self-organization of serotonergic axons (fibers) in vertebrate brains, where FBM paths with self-avoidance provide a natural framework for understanding spatial heterogeneities of fiber densities.

We propose a non-integer-dimensional spatial model for anisotropic solids by incorporating a q-deformed derivative operator, inspired by the Tsallis nonadditive entropy framework. This generalization provides an analytical framework to explore anisotropic thermal properties, within a unified and flexible mathematical formalism. We derive explicit expressions for the phonon density of states and specific heat capacity, highlighting the impact of the deformation parameter q on the thermodynamic behavior. We apply the model to various solid-state materials, achieving excellent agreement with experimental data across a wide temperature range, and demonstrating its effectiveness in capturing anisotropic and subextensive effects in real systems. Beyond providing accurate fits, we anchor the q-deformation in a microscopic disorder/kinetics exponent μemerging from conformable dynamics, thereby linking nonextensive statistics to measurable heterogeneity and memory effects.

We study analytically and numerically the mean fastest first-passage time (fFPT) to an immobile target for an ensemble of N independent finite-speed random searchers driven by dichotomous noise and described by the telegrapher's equation. In stark contrast to the well-studied case of Brownian particles – for which the mean fFPT vanishes logarithmically with N – we uncover that the mean fFPT is bounded from below by the minimal ballistic travel time, with an exponentially fast convergence to this bound as N →∞. This behavior reveals a dramatic efficiency advantage of physically realistic, finite-speed searchers over Brownian ones and illustrates how diffusive macroscopic models may be conceptually misleading in predicting the short-time behavior of a physical system. We extend our analysis to anomalous diffusion generated by Riemann-Liouville-type dichotomous noises and find that target detection is more efficient in the superdiffusive regime, followed by normal and then subdiffusive regimes, in agreement with physical intuition and contrary to earlier predictions.

In single-particle tracking experiments measuring anomalous diffusion dynamics, understanding ergodicity is crucial, as it ensures that the time average of an observable matches the ensemble average, and can thus be fitted with known ensemble-averaged observables. A commonly used criterion for assessing the ergodicity of a stochastic process is based on the comparison of the mean-squared displacement (MSD) with the time-averaged MSD (TAMSD). This approach has been widely applied and proves effective in cases of weak ergodicity breaking across various systems in both theoretical and experimental studies. However, there is relatively little discussion regarding the theoretical justification and limitations of this definition. Here, we demonstrate that this widely accepted criterion to some extent contradicts the classical definition of ergodicity as well as physical intuition, leading to spurious (non-)ergodicity results when applied to several well-known stochastic models. To address this limitation, we propose using the mean-squared increment (MSI) instead of the MSD for comparison of ensemble- and time-averaged observables. Several well-established examples demonstrate that our MSI-TAMSD criterion not only effectively reveals weak ergodicity breaking, equivalent to the MSD-TAMSD approach, but also provides a more accurate characterization of the genuine (non-)ergodicity of systems where the MSD-TAMSD method fails. Additionally, for systems exhibiting "ultraweak" ergodicity breaking, the MSI can reveal the asymptotic stationarity and ergodic nature of the process' increments. Our findings emphasize the important role of the MSI observable for SPT experiments and anomalous diffusion studies.

Chiral fluids, for which the mobility tensor has antisymmetric, off-diagonal components, exhibit transport phenomena absent in conventional systems, including interaction-enhanced diffusion and negative mobility. While these effects have been predicted theoretically and observed in simulations, their microscopic origin has remained unclear. Here, we address this question using a mobility-based nonequilibrium approach, analysing the steady-state drift of a tracer driven through an interacting chiral fluid. We show that, under strong chirality, the tracer generates a reversed density wake, in which regions of particle accumulation and depletion are inverted compared to the achiral case. This structural inversion of the wake provides a unified physical mechanism underlying both enhanced diffusion and negative mobility. Furthermore, we demonstrate that these phenomena are robust to changes in the interaction potential, highlighting their generality as a consequence of odd mobility.

The non-Markovian Fokker-Planck equation (FPE) provides an effective description of anomalous diffusion within the classical continuous time random walk framework. However, its nonlocal memory kernel leads to strong non-Markovian behaviors, making the non-Markovian FPE difficult to solve. Conventional numerical approaches must rely on intricate discretized implementations of fractional-calculus differentiation to approximate the fractional operators, leading to substantial computational and storage costs. To overcome these challenges and obtain the probability density function (PDF) across the entire temporal regime, we propose a subordination-based neural network (SNN) that leverages an equivalent subordinated integral representation of non-Markovian dynamics. This representation transforms the original non-Markovian FPE into a Markovian FPE formulated in an operational time where all memory effects are encapsulated by the subordination mechanism, enabling the solution via a deep neural network and direct transfer to FPEs with different memory kernels that share the same auxiliary Markovian FPE, without retraining. We validate the method for non-Markovian FPEs under nonlinear external forces, general boundary conditions, and with arbitrary memory kernels. With four representative examples, the SNN maintains high accuracy in both the central and tail regions of the PDFs, as well as in additional statistical quantities, e.g., survival probability and mean velocity, that are relevant to practical applications. The SNN approach provides a promising route for future extensions to higher-dimensional systems.

Following significant advances in microscopic and macroscopic single-particle tracking and supercomputing, the theoretical investigation of fluctuations and anomalous dynamics in complex systems is currently of high interest. Stochastic processes and their generalizations represent an important tool for the statistical description of such systems. Modeling random walks and stochastic processes in complex systems, including complex networks and graphs, requires an interdisciplinary approach due to the different applications in various fields, such as physics, biology, chemistry, engineering, computer science, and economy. Various studies of active and passive tracer diffusion, for instance, in biological cells and in heterogeneous and porous media showed that the underlying structure of the environment has a strong effect on the particle movement, leading to anomalous dynamics due to the constrained particle motion or the variation of the local diffusion coefficient and the potential energy function. Moreover, determining optimal search strategies is central in diverse fields, from physics to computer science, from biology to robotics. In particular, random search strategies have been widely observed for animal foraging, in reaction pathways in DNA-binding proteins, in intracellular transport, etc. Furthermore, it has been shown that the resetting of the searcher to its initial position can improve the search strategy by appropriate optimal resetting rate, which results in minimizing the mean first-passage time. This Editorial is meant to serve as an Introduction to this Focus Issue in the form of a mini-review of the field.

Mathematical models of diffusion based on stochastic processes of the type of continuous-time random walks (CTRWs) are widely used for the description of various transport characteristics in complex media. An external or internal bias in a given CTRW model is often attributed to an asymmetric distribution of jump lengths, while the waiting times are assumed to possess identical distributions in all directions. However, for a number of real-world transport phenomena the latter assumption is violated. To account for such situations, we examine the properties of such biased CTRWs in one dimension. We propose a CTRW-based model of diffusion with the waiting-time distributions (WTDs) of the tracers chosen unequal for right- and left-directed jumps. Below we name CTRWs with such asymmetric WTDs (AWTDs) as A_{t}-CTRWs. We derive the corresponding generalized Montroll-Weiss and master equations in Laplace and Z space. We consider both exponential and power-law WTDs as the standard and most useful functional choices for the purposes of data analysis. For these two WTD classes, we analyze the average displacement (generally nonzero) and the mean-squared displacement of the diffusing particles and analytically compute the variance as effective statistical measures. Our analytical predictions favorably agree with the results of the performed computer simulations. Specifically, for a CTRW with power-law WTDs we find that at long times the emerging anomalous diffusion depends not only on the smallest exponent of the left- and right-directed WTDs. Such diffusion models can, e.g., be used to describe the anomalous diffusion in various anisotropic media in the presence of a drift. One hydrological example is a gravity-affected spreading of contaminants in porous rocks, a system where the CTRW-based models are widely known to be applicable. A biological example of a biased-CTRW-based transport is a regulated driven bidirectional transport of cargoes between the center of a living biological cell and its periphery taking place along the microtubule filaments.

We investigate the application of conformable derivatives to model critical phenomena near continuous phase transition points. By incorporating a deformation parameter into the differential structure, we derive unified expressions for thermodynamic observables such as heat capacity, magnetisation, susceptibility, and coherence length, each exhibiting a power-law behaviour near the critical temperature. The conformable derivative framework naturally embeds scale invariance and critical slowing down into the dynamics without resorting to fully nonlocal fractional calculus. Modified Ginzburg-Landau equations are constructed to model superconducting transitions, leading to analytical expressions for the order parameter and the London penetration depth. Experimental data from niobium confirm the model's applicability, showing excellent fits and capturing asymmetric scaling behaviour around the critical point. This work offers a bridge between classical mean-field theory and generalised scaling frameworks, with implications for both theoretical modelling and experimental analysis.

Continuous-time random walks (CTRWs) are highly appreciated as a model of anomalous diffusion (AD) in complex media such as structured and heterogeneous environments. These walks are used to describe AD phenomena characterized by a nonlinear functional dependence of the mean-squared displacement (MSD) of particles with time. The introduction of asymmetric waiting-time distributions (WTDs) depending on the diffusion directions leads to biased CTRWs and to AD phenomena. Here, we examine the general formulation of such CTRWs with asymmetric WTDs in continuous space. From the balance equations, we derive the corresponding generalized Montroll-Weiss equation in Fourier-Laplace space as well as the generalized master equations with asymmetric waiting times. For random jump lengths obeying an asymmetric modified Gaussian distribution (AMGD), we analytically obtain the probability-density function of the particles as well as the first two moments in Laplace space. We unveil that CTRWs with exponential WTDs and jump lengths following the AMGD exhibit a ballistic diffusion, whereas in a symmetric case for the jump lengths both the MSD and variance at long time have a linear scaling with time. The AD properties of the MSD and of the variance at long times depend on a smaller exponent of the power-law WTDs, while the AMGD-distributed jump lengths double the AD scaling exponents for such CTRWs. In this continuous-space model, the possibility of length-distributed jumps-in contrast to the case of only neighboring-site jumps allowed in the discrete version of a similar model we proposed recently [G. Li et al., Phys. Rev. E 113, 064116 (2026)2470-004510.1103/6c55-c9tp]-leads to different resulting scaling properties of the MSD. The analytical results are in close agreement with the results of computer simulations, both for the case of asymmetric exponential and of power-law WTDs. The current theoretical results are extendable to more elaborate CTRW models. The latter are applicable for the description of more complicated AD-related phenomena taking place in the presence of external fields.

The mean squared displacement (MSD) is a cornerstone in the analysis of diffusion processes in complex media. When the system is heterogeneous and, in particular, when single-particle trajectories are short, it is essential to extract maximal information from each measured trajectory. This is typically done by time-averaging squared increments and examining the scaling of the time-averaged MSD in log-log space. However, classical regression methods perform poorly in this setting because time-averaging introduces correlations aggravated by those inherent to anomalous diffusion. We tackle these limitations by applying a generalized least-squares framework, which substantially reduces variance and bias in diffusion parameter estimates, especially for short (≈100 points) and ultra-short (≈10 points) trajectories. The method is fully automated and requires no supervision. Furthermore, it enables the prediction of estimation error probability density, which is asymptotically Gaussian, for both classical and enhanced approaches. Leveraging this prediction, we introduce a specialized deconvolution algorithm to reconstruct the underlying particle ensemble structure from experimental data.

Diffusive motion is a fundamental transport mechanism in physical and biological systems, governing dynamics across a wide range of scales-from molecular transport to animal foraging. In many complex systems, however, diffusion deviates from classical Brownian behaviour, exhibiting striking phenomena such as Brownian yet non-Gaussian diffusion (BYNGD) and anomalous diffusion. BYNGD describes a frequently observed statistical feature characterised by the coexistence of linear mean-square displacement (MSD) and non-Gaussian displacement distributions. Anomalous diffusion, in contrast, involves a nonlinear time dependence of the MSD and often reflects mechanisms such as trapping, viscoelasticity, heterogeneity, or active processes. Both phenomena challenge the conventional framework based on constant diffusivity and Gaussian statistics. This review focuses on the theoretical modelling of such behaviour via the Langevin equation with fluctuating diffusivity (LEFD)-a flexible stochastic framework that captures essential features of diffusion in heterogeneous media. LEFD not only accounts for BYNGD but also naturally encompasses a wide range of anomalous transport phenomena, including subdiffusion, ageing, and weak ergodicity breaking. Ergodicity is discussed in terms of the correspondence between time and ensemble averages, as well as the trajectory-to-trajectory variability of time-averaged observables. The review further highlights the empirical relevance of LEFD and related models in explaining diverse experimental observations and underscores their value to uncovering the physical mechanisms governing transport in complex systems.

Fractional diffusion equations (FDEs) are widely used to model anomalous diffusion in a variety of complex systems. In this work, we investigate a distributed-order FDE that incorporates the Hilfer-Prabhakar (HP) time-fractional derivative, which introduces a generalized memory kernel characterized by composite power-law scaling. This formulation extends the classical Riemann-Liouville (RL) and Caputo derivatives. We identify an admissible parameter regime under which the solution of the FDE behaves as a probability density function (PDF), and the HP derivative reduces to the Caputo-type under integer-order initial conditions. For the single-order case, we show that the PDF defined on an infinite domain can be expressed in terms of Fox H-functions in both short-and long-time limits, revealing distinctive subdiffusive behavior. Additionally, we derive asymptotic expressions for the mean squared displacement (MSD) corresponding to three representative order distributions: two-point, uniform, and beta. Notably, the uniform and beta distributions give rise to ultraslow diffusion. This study offers practical guidance for experimentalists seeking to apply FDEs with HP derivatives, emphasizing the importance of working within a physically consistent parameter regime. Our findings highlight the HP fractional framework as a powerful extension of classical FDEs, enabling a more versatile description of anomalous diffusion in systems with complex memory effects or trapping dynamics.

Molecular diffusion, or more specifically, diffusion-controlled transport, of tracers, contaminants, and chemical species --- in soil and rock formations, and in river, lake, and marine sediments --- plays a critical role in many dynamic processes that affect water chemistry and properties of the host domain. The spreading of dissolved ionic species via Brownian motion is generally described by a Gaussian law for the probability density function, with diffusion (embodying Fick’s second law) then being described by the classical diffusion equation. Solution of this equation shows that the spreading pattern of chemical species is characterized by a mean squared displacement that scales linearly with time. However, in other porous domains like biological tissues and cells, dense liquids, and gels, diffusion behavior often deviates from Fickian, instead exhibiting anomalous (or non-Fickian) diffusion. More specifically, tracer movements in these “crowded environments” exhibit a spreading pattern wherein the mean squared displacement scales as a power law. Somewhat surprisingly, in studies involving water-saturated porous rock, diffusion of chemical species is generally assumed to follow Fick’s second law, ignoring the possible occurrence of anomalous diffusion. To test this assumption, we measure molecular diffusion in five chalk and dolomite rock samples using a specially designed diffusion cell. The set-up enables high-resolution measurement of extended, long-time tailing at the measurement plane, which is required to distinguish between Fickian and anomalous diffusion behavior. In all of the rock samples, the diffusion behavior is demonstrated to be significantly different than Fickian, with extreme long-time tailing of tracer advance relative to conventional Fickian diffusion. The measured breakthrough curves are then analyzed using a continuous time random walk framework that describes anomalous diffusion in heterogeneous porous materials. The analysis (i) provides a framework to distinguish between Fickian and anomalous diffusion, and (ii) demonstrates that anomalous diffusion in geological formations is likely ubiquitous and implies that diffusion-controlled transport processes should be analyzed using tools that account for such behavior.

The diversity of diffusive systems exhibiting long-range correlations characterized by a stochastically varying Hurst exponent calls for a generic multifractional model. We present a simple, analytically tractable model which fills the gap between mathematical formulations of multifractional Brownian motion and empirical studies. In our model, called telegraphic multifractional Brownian motion, the Hurst exponent is modeled by a smoothed telegraph process which results in a stationary beta distribution of exponents as observed in biological experiments. We also provide a methodology to identify our model in experimental data and present concrete examples from biology, climate, and finance to demonstrate the efficacy of our approach.

We consider a general one-dimensional overdamped diffusion model described by the It & ocirc; stochastic differential equation (SDE) dX(t) = mu(X-t, t)dt + sigma(X-t, t)dW(t), where W-t is the standard Wiener process. We obtain a specific condition that mu and sigma must fulfil in order to be able to solve the SDE via mapping the generic process, using a suitable space-time transformation, onto the simpler Wiener process. By taking advantage of this transformation, we obtain the propagator in the case of open, reflecting, and absorbing time-dependent boundary conditions for a large class of diffusion processes. In particular, this allows us to derive the first-passage time statistics of such a large class of models, some of which were so far unknown. While our results are valid for a wide range of non-autonomous, non-linear and non-homogeneous processes, we illustrate applications in stochastic thermodynamics by focusing on the propagator and the first-passage-time statistics of isoentropic processes that were previously realised in the laboratory by Brownian particles trapped with optical tweezers.

Fractional Brownian motion is a Gaussian stochastic process with long-range correlations in time; it has been shown to be a useful model of anomalous diffusion. Here, we investigate the effects of mutual interactions in an ensemble of particles undergoing fractional Brownian motion. Specifically, we introduce a mean-density interaction in which each particle in the ensemble is coupled to the gradient of the total, time-integrated density produced by the entire ensemble. We report the results of extensive computer simulations for the mean-squared displacements and the probability densities of particles undergoing one-dimensional fractional Brownian motion with such a mean-density interaction. We find two qualitatively different regimes, depending on the anomalous diffusion exponent α characterizing the fractional Gaussian noise. The motion is governed by the interactions for α< 4/3 whereas it is dominated by the fractional Gaussian noise for α> 4/3. We develop a scaling theory explaining our findings. We also discuss generalizations to higher space dimensions and nonlinear interactions, the relation of our process to the “true” or myopic self-avoiding walk, as well as applications to the growth of strongly stochastic axons (e.g., serotonergic fibers) in vertebrate brains.