Phototaxis of many species of green algae relies upon directional sensitivity of their membrane-bound photoreceptors, which arises from the presence of a pigmented “eyespot” behind them that blocks light passing through the cell body from reaching the photoreceptor. A decade ago it was discovered that the spherical cell body of the alga Chlamydomonas reinhardtii acts as a lens to concentrate incoming light, and that in “eyeless” mutants of Chlamydomonas the consequence of that focused light reaching the photoreceptor from behind is a reversal in the sign of phototaxis relative to the wild type behavior. We present a quantitative theory of this sign reversal by completing a recent simplified analysis of lensing [Yang et al., Phys. Rev. E 113, 024401 (2026)] and incorporating it into an adaptive model for Chlamydomonas phototaxis. This model shows that phototactic dynamics in the presence of lensing is subtle because of the existence of internal light caustics when the cellular index of refraction exceeds that of water. During each period of cellular rotation about its body-fixed axis, the photoreceptor receives two competing signals: a relatively long, slowly varying signal from the direct illumination, and a stronger, shorter, rapidly varying lensed signal. The reversal of the sign of phototaxis is then a consequence of the dominance of the flagellar photoresponse to the signal with the higher time derivative. These features lead to a quantitative understanding of phototaxis sign reversal, including bistability in the direction choice, a prediction that can be tested in single-cell tracking studies of mutant phototaxis.

We present a systematic study of isolated hopfions in confined layers of chiral magnets and cholesteric liquid crystals with strong surface anchoring and a film thickness fixed to one spiral pitch. We focus on the structure, stability, and interactions of hopfions under these conditions. Using numerical simulations combined with scaling-energy analysis, we demonstrate that hopfions possess well-defined equilibrium radii and energy minima in several background phases, including homogeneous states, undulated conical phases, and cholesteric fingers of the first type. Strong easy-plane or easy-axis anisotropy reduces the hopfion size and can ultimately drive a collapse into torons, whereas moderate anisotropy stabilizes hopfions against both collapse and expansion. Interhopfion interactions are quantified and found to depend strongly on the surrounding phase. Hopfions repel each other in the homogeneous state, whereas the “soft” undulated conical phase mediates attraction. In the latter case, a shell of positive energy density formed between a hopfion and the surrounding background promotes the formation of hopfion clusters, whose energetics is governed by the number of pairwise overlaps, providing a simple geometric framework for describing their self-assembly. By analyzing hexagonally ordered clusters with increasing numbers of hopfions, we extract the interaction energy. It saturates in the thermodynamic limit and amounts to several percent ($\approx 6\%$) of the single-hopfion energy. Finally, we examine the possibility of periodic hopfion crystals. Within the considered one-pitch confinement, competing modulated phases of lower energy, together with the inability of isolated hopfions to attain a negative eigenenergy relative to the background, preclude the formation of genuine hopfion lattices. These findings provide a rigorous framework for reassessing previous claims of stable hopfion lattices and indicate that experimentally observed hopfion assemblies are better interpreted as finite clusters rather than true periodic crystals. Our study elucidates the mechanisms controlling hopfion stability and assembly in confined chiral systems, offering guidance for future experimental and theoretical investigations of topological solitons.

Occupation times quantify how long a stochastic process remains in a region, and their single-time statistics are famously given by the arcsine law for Brownian and Lévy processes. By contrast, two-time occupation statistics—which directly probe temporal correlations and aging—have resisted exact characterization beyond renewal processes. In this paper we derive exact results for generic one-dimensional jump processes, a central framework for intermittent and discretely sampled dynamics. Using generalized Wiener-Hopf methods, we obtain the joint distribution of occupation time and position, the aged occupation-time law, and the autocorrelation function. In the continuous-time scaling limit, universal features emerge that depend only on the tail of the jump distribution, providing a starting point for exploring aging transport in complex environments.

Fluctuation-response relations must be modified to describe nonequilibrium systems with non-Markovian dynamics. Here, we experimentally demonstrate that such relation is quantitatively recovered when the appropriate Markovian embedding of the dynamics is explicitly resolved. Using a colloidal particle optically trapped in a harmonic potential and driven out of equilibrium by a controlled colored noise, we study the response to a perturbation of the stiffness of the confining potential. While the reduced dynamics violates equilibrium fluctuation-response relations, we show that the dynamical response to the stiffness perturbation is fully determined by steady-state correlations involving the exact conjugate observable in the Markovian embedding.

Multifractal detrended fluctuation analysis has emerged as a standard tool for characterizing scale invariance in complex systems, yet its application to discrete spin models is frequently marred by reports of “spurious multifractality” that contradict established theory. In this work, we resolve this controversy by establishing a rigorous protocol for the analysis of discrete lattice snapshots. Using the two-dimensional Ising model as a benchmark, we demonstrate that the previously reported broad singularity spectra are finite-size artifacts dominated by lattice discreteness effects in the negative moment regime $(q<0)$. By restricting the analysis to positive moments and performing a systematic finite-size scaling (FSS) analysis, we show that the spectral width collapses to zero $(\mathrm{\Delta }\alpha \to 0)$ in the thermodynamic limit. The method accurately recovers the monofractal exponent of the Ising universality class $(\alpha \approx H\approx 0.875)$, consistent with conformal field theory. Applying this FSS protocol to the random bond Ising model, we reveal that the broad multifractal dispersion often attributed to quenched disorder also flattens systematically with increasing system size. These results demonstrate that apparent multifractality in finite discrete lattices is predominantly driven by preasymptotic discreteness artifacts, underscoring the universal necessity of FSS to avoid false positives in statistical mechanics.

Synchrony patterns describe network states in which nodes of a coupled dynamical system are grouped into clusters based on synchronization between nodes. Beyond simple synchrony, synchronized clusters may also exhibit active or inactive states, and the collection of all such clusters constitutes an activity pattern. Although these patterns may arise naturally in networks with permutation symmetries, the requirement of symmetries imposes a restrictive and often unrealistic assumption, as many real-world networks lack such symmetries. In this work we present synchrony patterns of coexisting active-inactive clusters that cannot be identified through symmetries. Considering dynamical systems in which intrinsic dynamics and coupling functions are odd functions in phase space, we identify all possible patterns a network can exhibit through symmetry breaking of identically synchronized clusters in the amplitude death state. The symmetry breaking of invariant clusters generates antisynchronized clusters, allowing active-inactive clusters to coexist. We show that while active clusters are external equitable partitions, inactive clusters can be purely dynamics-induced. Starting with symmetry-broken states, we show that the existence of different invariant patterns is a function of coupling strength and intercluster weights. Finally, by combining synchronization manifolds with the Laplacian eigenvectors, we identify transversal perturbations for these patterns and present a stability analysis.

The phenomenological renormalization group (PRG) has been applied to the study of scale-invariant phenomena in neuronal data, providing evidence for critical phenomena in the brain. However, it remains unclear how reliably these observed signatures indicate genuine critical behavior, as it is not well established how close to criticality a system must be for them to emerge. Here, we rely on neuronal models with known critical points to investigate under which conditions the PRG procedure yields consistent results. We show that the PRG method detects scaling behavior in neuronal models only within a narrow vicinity of the critical point, reinforcing the interpretations drawn from PRG results in experimental data. We also demonstrate that time-binning choices can substantially affect the results and introduce a data-driven adaptive binning procedure to circumvent this issue.

The solute transport due to mixed electro-osmotic and pressure-driven flows of viscoelastic fluids in microchannels is studied here. The Reynolds decomposition technique, in combination with the assumptions underlying the Taylor-Aris theory, is used to derive a reduced-order model that yields the dispersion coefficient, which is then solved using the general Duhamel theorem to obtain the cross-sectional average concentration. The dispersion is evaluated for the general case of mixed electro-osmotic and pressure-driven flows of viscoelastic fluids, and for three special cases: combined electro-osmotic and pressure-driven flows of a Newtonian fluid, electro-osmotic flow of a viscoelastic fluid, and pressure-driven flow of a viscoelastic fluid. The dispersion coefficient is characterized by four main nondimensional parameters: the ratio of advection to diffusion (Pe), the inverse of the electric double-layer thickness (κ), the ratio of pressure-driven to electro-osmotic forces (λ), and the fluid viscoelasticity $(\varepsilon {\mathrm{De}}^2)$. As expected, the larger the Peclet number, the greater the dispersion for fixed $\lambda$ and $\varepsilon \mathrm{D}{\mathrm{e}}^2$ values. It is generally observed that the dispersion coefficient increases as the fluid viscoelasticity increases, for constant values of the Peclet number and the ratio of pressure-driven to electro-osmotic forces. The results also indicate that the dispersion exhibits nonmonotonic behavior as $\lambda$ increases from negative to positive values, for fixed Peclet number and fluid viscoelasticity. The dispersion coefficient and the cross-sectional average concentration are examined for special cases and compared with the general case. The developed mathematical model and findings have implications for the design of solute transport in microfluidic systems.

The phase transition in the Vicsek model is widely believed to be associated with spontaneous symmetry breaking of the two-dimensional rotational symmetry $\text{O}\left(2\right)$. In this paper, we revisit the original model introduced by Vicsek et al. [Phys. Rev. Lett. 75, 1226 (1995)] and demonstrate that the original angle-based update rule is not invariant under global phase shifts at the level of angular increments when it is implemented using the principal-value arctangent, as in the original definition. As a consequence, we numerically demonstrate that the phase transition reported in the original paper vanishes when the global phase is adaptively chosen.

We introduce a thermodynamic formalism for nonequilibrium transport networks based on generalized transport potentials (GTPs). When endowed with Legendre duality and the associated contact structure, this framework quantifies the distance from nonequilibrium operating points in analogy with thermodynamic availability, and clarifies how different optimality principles emerge under distinct constraints. Classical extremizations of dissipation and entropy production are recovered as special cases, requiring power-law resistances and, for entropy production, isothermal conditions. Stability analysis shows that changes between multiple operating points can be interpreted as nonequilibrium phase transitions governed by generalized equations of state. Finally, cost-based optimization of transport properties reveals new connections to generalized dissipation, particularly for power-law costs and linear resistance laws, though typically associated with unstable states that give rise to branched optimal transport.

We extend the random walk framework to include compounded steps, providing first-passage time (FPT) properties for a class of superdiffusive processes which are governed by the space-fractional spectral Fokker-Planck equation. This first-passage process introduces FPT properties, different from Lévy flights, that account for space-dependent forces and hitting boundaries throughout the path of a jump. The FPT distribution can be derived for different types of barriers and potentials, for which we also provide specific examples. For the one-sided absorbing boundary with no potential on the semi-infinite line, we find that the FPT density scales asymptotically as $t^{-1/\left(2\alpha \right)-1}$ for large times, where the parameter $\alpha \in (0,1]$ relates to the power-law behavior for the distribution of the number of compounded steps. This is in agreement with the method of images but different from the Sparre-Andersen scaling $t^{-3/2}$ for corresponding Lévy flights of order $2\alpha$. In this case, there exists an optimal space-fractional exponent $\alpha$ to minimize the mean FPT.

Zermelo's navigation problem seeks the trajectory of minimal travel time between two points in a fluid flow. We address this problem for an agent—such as a floating drone or active particle—that is advected by a two-dimensional flow, self-propels at a fixed speed smaller than or comparable to the characteristic flow velocity, and can steer its direction. The flows considered span increasing levels of complexity, from steady solid-body rotation and time-dependent sink-vortex to the Taylor-Green flow and turbulence in the inverse energy cascade regime. Although optimal-control theory provides time-minimizing trajectories, these solutions become unstable in chaotic regimes characterized by positive finite-time Lyapunov exponents. To design robust navigation strategies, we apply reinforcement learning and compare $Q$-learning with a one-step actor-critic algorithm. Both methods achieve successful navigation, yielding mean travel times within 3–10% of optimal-control solutions in regular flows, while the discrepancy increases to 35–75% in time-dependent turbulent flows. Finally, we show that agents trained on coarse-grained turbulent flows generalize to the full velocity field. This robustness to incomplete flow information is essential for practical navigation in real-world oceanic and atmospheric environments.

The addition of a small fraction of flexible fibers to a granular medium is known to enhance the bulk mechanical strength of the material. However, their effect on the free-surface flow of grains remains largely unexplored. To address this gap, we investigate the influence of flexible fibers on dry granular avalanches. Using a drum partially filled with a grain–fiber mixture and rotated at low speeds, we observed successive avalanches and analyzed their dynamics. Specifically, we examined the effects of fiber volume fraction and fiber length on both the statistical properties and the individual dynamics of avalanches. Our results show that the average starting and stopping angles increase with fiber volume fraction, demonstrating that the addition of fibers stabilizes the granular slope. Fiber length is found to have only a minor influence on slope stabilization, i.e., on the increase in avalanche angles induced by fibers. While the increase in avalanche angles aligns with the bulk rheology of grain–fiber mixtures, the role of fiber length appears different in the case of free-surface flows. Increasing the fiber volume fraction also alters the distribution of avalanche angles, leading to a broader distribution and a higher probability of small-amplitude avalanches. Analysis of individual avalanche dynamics further reveals that fibers affect avalanche duration, suggesting enhanced energy dissipation within the system. To capture these effects, we propose a theoretical model that links avalanche dynamics to the rheological properties of the material. This framework provides an estimate of how fibers influence the inertial term in the rheology of grain–fiber mixtures, thereby extending existing empirical formulations for their behavior.

We investigate biased random walks with stochastic resetting to previously visited sites and demonstrate how directional drift fundamentally alters the emergence of localized stationary states. We show that bias acts as a competing mechanism and can weaken or suppress localization. This competition produces an anisotropic phase transition: while the walker remains localized transverse to the bias, the localization length diverges only along the direction of the drift, independently of spatial dimension. We derive the stationary distribution, identify the conditions under which localization survives in the presence of bias, and compute the critical behavior of the localization length near the transition. These results reveal how the interplay of bias and resetting reshapes the phase structure of nonequilibrium random walks and establishes directional divergence as a hallmark of biased resetting dynamics.

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)]—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.

Stochastic out-of-equilibrium processes often involve asymmetric contributions that break detailed balance and lead to nonmonotonic entropy production, limiting thermodynamic interpretations and inference techniques. Here we use Dyson maps to restore monotonic entropy growth in those processes, allowing the use of standard tools from statistical physics, providing a general and computationally tractable method applicable to a broad class of Markovian systems.

We investigate the Stirling-cycle performance of a Heisenberg–Kitaev magnonic medium with Dzyaloshinskii–Moriya (DM) interactions. Using linear spin-wave theory, we show the DM interaction preserves spectral symmetry, yielding even caloric responses and symmetric Stirling engine efficiency. In contrast, bond-dependent Kitaev exchange asymmetrically distorts the magnonic density of states, enabling distinct direct and inverse caloric effects. Consequently, Kitaev-driven cycles achieve significantly higher efficiencies than DM-driven protocols, approaching a high-performance saturation regime for negative couplings. This establishes exchange-anisotropic magnets as highly tunable platforms for nanoscale solid-state energy conversion.