The dissipation in the relaxation of an ensemble of conservative stochastic systems toward the equilibrium steady state is quantified by the free-energy difference. Functional inequalities within the framework of Otto and Villani [J. Funct. Anal. 173, 361 (2000)] are here revisited which connect the free-energy dynamics and optimal transport, offering a geometric perspective on the instantaneous speed of relaxation in the presence of potential barriers. These are illustrated with numerical relaxation experiments on Landau-Ginzburg potentials. The same inequalities hold for relaxation toward nonequilibrium steady states once the potential is replaced by the pseudopotential defined by the stationary density.

Investigating the dynamics of vibroimpact self-sustaining systems is of immense scientific interest, because systems that undergo vibroimpact interactions can be found in various engineering contexts. Repercussions and associated noise are prevalent in mechanical engineering. Conversely, in a self-sustaining system including viscoelastic material, the damping component is influenced not only by the current time and position, but also by previous states. The memory property of viscoelastic materials is determined by an exponential relaxation function and noise. Based on this viewpoint, this study investigates a self-sustained birhythmic vibroimpact oscillator with improved memory damping subjected to correlated noise. In the beginning, the nonsmooth transformation is implemented to eliminate the discontinuity in the vibroimpact oscillator. The detailed feature space analysis reveals that the correlation time and vibroimpact coefficient can effectively control the oscillator's rhythmic characteristics. To identify the maximal birhythmic zone in parameter space, select an approximate restitution coefficient and correlated noise intensity. In a stochastic environment, decreased noise intensity reduces the essential quantity of the vibroimpact coefficient and control parameters required for birhythmicity. The Rayleigh oscillator is a traditional nonlinear dynamics paradigm, and varying the nonlinear Rayleigh parameters together with vibroimpact coefficients enables control of the nonlinear dynamic multirhythmic system. The reported results demonstrate how the strength of correlated noise, correlation time, restitution coefficient, and system characteristics influence the transition from one potential well to another, and Monte Carlo simulation is used to confirm the stochastic generalized method's effectiveness. Furthermore, for this self-sustaining system, the method of recurrence plots is employed for numerical time series analysis, and the impacts of the restitution coefficient and noise correlation time on recurrence quantification have been demonstrated.

Continuous attractor neural networks traditionally rely on prewired recurrent connectivity to model spatial representations, path integration, and anticipatory dynamics. However, the biological mechanisms through which this structured connectivity emerges via learning remain relatively unexplored. This work presents a theoretical framework revealing how continuous attractor connectivity and its computational properties self-organize through Hebbian plasticity, firing-rate adaptation, and global inhibition. We show that translationally invariant inputs naturally drive the emergence of stable, Gaussian-profiled feedforward weights. Crucially, anticipatory dynamics arise spontaneously within these feedforward architectures, shifting the activity bump forward without requiring recurrent excitatory collaterals. This predictive shift can be linearly amplified across multilayer networks, consistent with anticipatory activity observed in the superficial layers of the entorhinal cortex. Furthermore, introducing recurrent interactions allows the network to learn connections capable of self-sustaining a moving bump of activity. Finally, by modulating the network with an external, time-varying baseline current that encodes speed, the system adjusts its intrinsic velocity to function as a precise unidirectional path integrator. Ultimately, this study suggests that prospective coding and path integration are not manually engineered features, but rather naturally coemergent properties of a single self-organizing competitive network.

Dewetting of a thin polymer film probes the stress and mobility frozen in during preparation. A companion theory showed that self-propulsion enhances the elastic modulus of flexible active films through grip forces at entanglements. Here I derive the semiflexible case relevant to biological filaments. Because a semiflexible chain reorients along its tube, grip forces from successive segments add in quadrature. At fixed persistence length ${\ell }_p$ and propulsion strength, this changes the activity parameter $\alpha$ (the ratio of active to passive plateau modulus) from $\alpha \propto L$ to $\alpha \propto {\left({\ell }_{p}L\right)}^{1/2}$ in the chain contour length $L$, and the active viscosity from $L^2$ to $L^{3/2}$, a scaling class lying between the passive $\left(L^3\right)$ and flexible active $\left(L^2\right)$ melts. The biexponential relaxation is retained; an exact relation between the fast-channel weight and $\alpha$ renders three observables stiffness-independent, while the kink in sequential nucleation and hole growth dynamics fixes the propulsion speed alone, while its amplitude ratio returns the activity parameter and hence the stiffness. A parameter-free wormlike-chain numerical simulation confirms the quadrature scaling and its crossover to the coherent limit, and published T. Tubifex worm rheology qualitatively supports the two-channel picture.

Thin sheets have been predicted to transition from a flat to a crumpled state at a high enough temperature driven by thermal fluctuations. However, such a thermal crumpling transition has been elusive in experiments. We report the observation of a capillary crumpling transition in thin solid sheets driven by the sheet's surface energy. Using a sensitive experimental technique developed by us, we gently compress a thin polymer sheet weakly adhered to a hydrogel substrate. Initially, the sheet resists compression; however, beyond a well-defined critical compression, it transitions to a self-crumpling state. The transition is closely related to the percolation of a fold network and a power-law increase in fold density, bearing signatures of a critical phenomenon. Most remarkably, unlike the common notion, the crumpled state is not necessarily disordered but can consist of an ordered pattern of folds which can be conveniently tuned through the substrate geometry. We demonstrate the potential of this technique as a powerful and scalable method to do origami with extremely thin sheets, which can be used to sculpt them into desired three-dimensional (3D) shapes and design nontrivial emergent mechanical responses and can have important applications in 2D materials based nanotechnology.

How cross-sectional heterogeneity evolves in nonequilibrium populations of competing, aging units is a collective question that cannot be resolved from a representative trajectory or the ensemble mean alone. Analysis of more than two million age-aligned English Wikipedia pageview trajectories from six yearly cohorts spanning 2015–2020 shows that the variance of logarithmic attention grows linearly with $lnt$, while the ensemble mean relaxes on the same logarithmic clock. This dispersion law recurs across all six cohorts and is systematically favored over the tested power-law alternative; temporal increments also exhibit consistent antipersistent scaling. A complementary trajectory-level analysis reveals broad, non-Gaussian variation in effective relaxation rates, showing that this collective regularity coexists with heterogeneous observed trajectories. A minimal scale-free stochastic model combines mean relaxation with self-affine temporal fluctuations of age-dependent amplitude. Within this model class, logarithmic spreading emerges when fluctuation accumulation balances aging attenuation, defining the marginal boundary between algebraic broadening and saturation. Together, age-resolved cross-sectional dispersion and temporal-increment scaling provide a transferable ensemble-level criterion for distinguishing amplification, marginal spreading, and saturation in other populations of competing, aging units.

We study coupled FitzHugh-Nagumo oscillators on a small-world network in which the activator and the inhibitor diffuse through independent channels, with an added three-body coupling whose structure is varied among an all-to-all form, a simplicial complex built on the network triangles, and a random hypergraph. At fixed coupling strength and a fixed number of three-body interactions, the structure decides whether multistability and coexisting oscillatory-quiescent states survive. With matched triple count on the same backbone, the simplicial complex retains most of its partial-quenching basin whereas the random hypergraph removes nearly all of it, a separation far larger than the variation across network realizations, while total death falls comparably under both. The difference is not set by the spectrum of the normalized higher-order Laplacian, whose leading eigenvalue the normalization forces to be the same for both structures, but by the alignment of that operator with the pairwise eigenbasis: simplicial triples join adjacent nodes and leave long-wavelength perturbations almost undamped, whereas random triples damp them as strongly as short ones. Randomizing this alignment at fixed spectrum moves the transverse stability of synchrony under the simplicial complex onto that under the hypergraph. Sampling the attraction basins shows that the total-death basin shrinks under both structures, but only under the simplicial complex does the partial-quenching basin grow, and the quiescent domains of these mixed states are long-wavelength patterns. An all-to-all three-body control acts approximately as a pairwise mean field whose strength is reduced by an amplitude gate, set by the cycle-averaged squared activator of the collective oscillation.

We consider systems characterized by the presence of a rapidly oscillating force. A general method is presented for the construction of the effective action governing the large-scale nonlinear dynamics of such systems order by order in inverse powers of the oscillation frequency $\omega$. The explicit expression for the effective Lagrangian is derived up to $\mathcal{O}(1/{\omega }^6)$ next-to-next-to-leading approximation in the strong-drive regime. The general structure of the high-frequency expansion reveals a broad class of nonlinear systems whose transition curves are identical to those of the linear Mathieu equation, which enables a fully nonperturbative stability analysis in the case of strong driving and nonlinearity. The method is generalized to velocity-dependent forces and configuration space with curvature, characteristic to systems with constraints. Several applications are discussed in detail, including the dynamical magnetic trapping of electric charges.

Among thermoresponsive hydrogels, poly-N-isopropylacrylamide gels are well known for their marked deswelling behavior near $T_c\approx {32}^{\circ }\mathrm{C}$; under slow, quasistatic heating, the gel shrinks affinely. In contrast, a rapid temperature quench above $T_c$ produces an effectively impermeable outer skin that fixes the total volume of the gel, inducing the phase coexistence of solvent-rich and solvent-poor regions. For cylindrical gels, or gels characterized by a centerline with circular cross section more generically, the simplest phase-coexistent scenario consists in having an axisymmetrically arranged, collapsed outer region and an interior swollen region, all centered along the centerline. Introducing a polarization vector to describe the solvent distribution in the phase separated state reveals that breaking axisymmetry lowers the system's free energy. If the gel's centerline is initially curved, this polarization couples to the local curvature and tends to point opposite to the normal vector of the centerline, thereby biasing the phase separation process. This coupling generates a feedback loop: curvature influences polarization, and polarization in turn drives further shape change. Interestingly, in the cylindrical case, a spontaneous polarization lowers the free energy, which then induces the bending of the cylindrical gel by the coupling between polarization and deformation. Finally, we show that cylindrical gels of different diameters quenched above $T_c$ also exhibit other size- and temperature-dependent instabilities.

Particle number fluctuations $N\left(t\right)$, measured in virtual observation boxes of an image or a simulation, offer a way to quantify particle dynamics when particle tracking is impractical, such as in high-density systems. While traditionally limited to equilibrium diffusive systems, we extend this approach—named “Countoscope”—to out-of-equilibrium self-propelled particles: Active Brownian (ABPs), run and tumble (RTPs), and active Ornstein-Uhlenbeck particles (AOUPs). For AOUPs, we leverage their Gaussian statistics to derive a general formula applicable to any Gaussian system. For ABPs and RTPs, we derive the self-intermediate scattering function (ISF)—and thus the correlations of $N\left(t\right)$—using an exact perturbative expansion over the probability density fields, revealing key physical features of the ISF and of the number correlations. Our theoretical predictions for the mean-squared number difference $\langle \mathrm{\Delta }{N}^2\left(t\right)\rangle =\langle {(N\left(t\right)-N\left(0\right))}^2\rangle$ match stochastic simulations and exhibit three time-dependent scaling regimes: diffusive, ballistic, and long-time enhanced diffusive, reflecting the regimes of the mean-squared particle displacement. We further uncover limiting laws in each of these regimes that are useful to quantify self-propulsion properties.

Family-Vicsek scaling of island-size distribution (ISD) is well known for homogeneous epitaxial islands with infinitely large ratios of surface diffusion coefficient over deposition rate, where the capture numbers are asymptotically linear in size. Linear rows of surface adatoms, droplets in the cavitation regime, and vertical nanowires present the examples of growth systems with size-linear capture rates for any size. Theory of Bartelt and Evans does not provide any solution in this case, because the ISD is established in the short-range nucleation step rather than in a much longer asymptotic growth stage. Here, we present the exact solution of discrete rate equations for the monomer concentration and ISD in homogeneous irreversible growth with size-linear capture rates for any size, and obtain the corresponding scaling function. The monomer concentration is a unimodal function of time, whereas the ISD is monotonically decreasing with size at any time. The Family-Viscek scaling function is also monotonically decreasing and close to exponential decay. These parameter-free solutions of the well-defined problem should be useful for general understanding of the ISD scaling properties and the analysis of concrete systems.

We develop a mean-field description including spatial structure for a simplified version of the three-state active matter model studied by Venzel et al. [Phys. Rev. E 110, 014109 (2024)]. The resulting triangular lattice of coupled nonlinear differential equations are integrated numerically using a fourth-order Runge-Kutta scheme. Starting from various ordered initial configurations, we probe the stability of the corresponding stationary states, revealing the presence of various high-density ordered structures in the density-noise plane. The results are compared with Monte Carlo simulations of the simplified model, yielding, in certain cases, unexpected transitions between ordered configuration types.

Time-reversal symmetry plays an essential role in the thermodynamic uncertainty relations, which bound the fluctuations of observables in terms of the associated dissipation. In fact, thermodynamic uncertainty relations are typically derived under the assumption that the observable of interest is antisymmetric under time reversal. This also suggests that existing thermodynamic uncertainty relations are restricted to a limited class of observables. In this paper, we mitigate this restriction by introducing a new class of observables that do not exhibit exact antisymmetry but change sign under time reversal. We call it generalized time reversal and derive a broadly applicable thermodynamic uncertainty relation for observables with this condition. The generalization is achieved by direct statistical arguments on the observable distributions and holds for both deterministic and stochastic dynamics. We demonstrate the derived thermodynamic uncertainty relation with observables subjected to rectifications, showing that the precision of generalized observables remains expressible in terms of the dissipative cost. The result extends the scope of thermodynamic uncertainty relations beyond the reach of previous frameworks relying on the exact antisymmetry.

A central problem in statistical physics is to extend the Boltzmann distribution to nonequilibrium steady states (NESS). We prove that NESS on large dense disordered networks have Boltzmann-like occupation, despite extensive entropy production. One consequence is that such NESS can conceal their nonequilibrium nature from measurements of static response to energy perturbations. We further prove that the active-matter heuristic of “low rattling” is asymptotically exact for these NESS. Intuitively, they spend a greater fraction of their time in states they leave more slowly. We show that rattling effectively extends the Boltzmann distribution to such NESS, as well as the broader class of “equiaccessible” steady states, which play a role in our analysis akin to that of equilibrium in linear response.

We investigate integrability-breaking transitions in two classes of stadiumlike billiards with parabolic boundaries. While focusing boundaries generate a mixed phase space in which regular islands coexist with a chaotic sea, dispersing boundaries produce a fully chaotic phase space for any finite boundary deformation. By analyzing the scaling behavior of the roughness $\omega$, we identify two qualitatively distinct transitions: a continuous transition for the focusing geometry and a first-order transition for the dispersing one. We determine the corresponding critical exponents and establish the associated scaling laws. For the continuous transition, we further provide a complete characterization within the framework of critical phenomena by identifying the broken symmetry, the order parameter and its diverging susceptibility, the elementary excitations responsible for chaotic diffusion, and the topological defects governing transport. These results establish a statistical-mechanics framework for describing integrability-breaking transitions in Hamiltonian billiards and suggest that the concepts of critical phenomena naturally extend to deterministic nonlinear dynamical systems.

Adaptivity and learning are ubiquitous in living systems, from cytoplasm and biofilms to animal flocks. However, the problem of designing nonliving mechanical systems capable of self-learning behavior continues to present theoretical and experimental challenges. Here we introduce self-learning mechanical circuits: passive systems which take stochastic input forces from changing environments and constantly update their internal state in response, thus representing a mechanical unsupervised learning unit. Motivated by simplified models of synapses, our circuits are composed of adaptive directed springs, which change stiffness in a directional manner. We provide an experimental realization of these elastic learning units and demonstrate how stability arises from adaptivity in certain network architectures. In particular, we theoretically demonstrate the ability of a network to autonomously uncover patterns in external inputs. Our results have applications towards the design of adaptive materials that can self-optimize in different environments.

For a network of $n$ nodes under random gossip dynamics, we show that by introducing the additional resetting mechanism (with rate $\beta$) to a designated subset of $r$ nodes, the expected sum of all node values is asymptotically equal to $\frac{n}{r}$ times the sum of the initial values of the $r$ designated nodes, in the small-$\beta$ and long-time limit. We also analyze the convergence speed to this asymptotic state and provide an approximate high-probability confidence interval for the expected sum at finite time $t$ and finite $\beta$. Operationally, we propose a protocol that distributes the initial node values over the designated subset evenly (up to a prescribed precision), by appropriately turning on and off the resetting dynamics. Our work not only offers insights into classical problems such as distributed computing and network control, but also serves as a potential foundation for designing quantum algorithms that prepare desired states in quantum networks using only local operations.

In Escherichia coli (E. coli), entropic repulsion between the two daughter DNA ring polymers under cylindrical confinement is believed to be an important factor governing chromosomal segregation. The repulsion can be enhanced by topological modifications, i.e., by the introduction of internal loops at certain locations along the contour of the circular DNA. However, the effect of topological modifications on the rate of segregation of ring polymers remains unclear. Therefore, we systematically varied the number and the contour length of loops introduced at selected locations by crosslinking monomers. The appropriate crosslinking was motivated by observations that extruded loops are located mainly near the origin of replication (ori-proximal) region of the E. coli chromosome. This resulted in the chains becoming intrinsically anisotropic. Using Langevin dynamics simulations of these topologically modified bead-spring polymers, we calculated the time required for segregation under cylinder confinement. With certain caveats, we found that increasing the number of loops resulted in a decrease in the time of segregation. In line with past work, we propose that this is due to the increase in the entropic repulsion between the polymers upon increasing the number of loops. In addition to the number of loops, the contour length of the loops and the mutual orientation of the (anisotropic) chains in the initial configurations played a role in determining the time of segregation.