Moving beyond simple associations, researchers need tools to quantify how variables influence each other in space and time. Correlation functions provide a mathematical framework for characterizing these essential dependencies, revealing insights into causality, structure, and hidden patterns within complex systems. In physical systems with many degrees of freedom, such as gases, liquids, and solids, a statistical analysis of these correlations is essential. For a field $\mathrm{\Psi }(\mathbf{x},t)$ that depends on spatial position $\mathbf{x}$ and time $t$, it is often necessary to understand the correlation with itself at another position and time $\mathrm{\Psi }({\mathbf{x}}_0,t_0)$. This specific function is called the autocorrelation function. In this context, the autocorrelation function for order-parameter fluctuations, introduced by Fisher [J. Math. Phys. 5, 944 (1964)], provides an important mathematical framework for understanding the second-order phase transition at equilibrium. However, his analysis is restricted to a Euclidean space of dimension $d$, and an exponent $\eta$ is introduced to correct the spatial behavior of the correlation function at $T=T_c$. In recent work, Lima et al. [Phys. Rev. E 110, L062107 (2024)] demonstrated that at $T_c$ a fractal analysis is necessary for a complete description of the correlation function. In this study, we investigate the fundamental physics and mathematics underlying phase transitions, emphasizing the deep interplay between scaling behavior, critical exponents, and fractal geometry. In particular, we show that the application of modern fractional differentials allows us to write an equation for the correlation function that recovers the correct exponents below the upper critical dimension. We obtain the exact expression for the Fisher exponent $\eta$. Furthermore, we examine the Rushbrooke scaling relation, which has been questioned in certain magnetic systems, and, drawing on results from the Ising model, we confirm that both our relations and the Rushbrooke scaling law hold even when $d$ is not an integer.

Brownian diffusion subject to stochastic resetting to a fixed position has been widely studied for applications to random search processes. In an unbounded domain, the mean first-passage time at a target site can be minimized for a convenient choice of the resetting rate. Here we study this optimization problem in one dimension when resetting occurs to random positions, chosen from a probability density function with compact support that does not include the target. Depending on the shape of this distribution, the optimal resetting rate either varies smoothly with the mean distance to the target, as in single-site resetting, or exhibits a discontinuity caused by the presence of a second local minimum in the mean first-passage time. These two regimes are separated by a critical line containing a singular point that we characterize through a Ginzburg-Landau theory. To quantify how useful is a given resetting point for the search, we calculate the probability density function of the last resetting position before absorption. The discontinuous transition separates two markedly different optimal strategies: one with a small resetting rate where the last path before absorption starts from a rather distant but likely position, while the other strategy has a large resetting rate, favoring last paths starting from not-so-likely points but which are closer to the target.

Krylov complexity (K-complexity) is a measure of quantum state complexity that minimizes wave-function spreading across all the possible bases. It serves as a key indicator of operator growth and quantum chaos. In this work, we use K-complexity and Arnoldi coefficients to investigate diverse localization phenomena in the quantum kicked rotor (QKR). We analyze four distinct localization scenarios—ranging from strong localization effect arising from quantum antiresonance to a weaker form of power-law localization—each one exhibiting distinct K-complexity signatures and Arnoldi coefficient variations. The long-time behavior of K-complexity and the wave-function evolution on Krylov chain can distinguish various types of observed localization in QKR. We show that K-complexity not only captures the degree of localization but also the nature of localization. In particular, the time-averaged K-complexity and scaling of the variance of Arnoldi coefficients with effective Planck's constant can distinguish the localization effects induced by the classical regular phase structures and the dynamical localization arising from quantum interferences. Further, we also show that the Arnoldi coefficients effectively reveal the onset of chaos, even with the quantum dynamics being localized.

Many examples from quantum and classical physics are known where topological protection is responsible for the robustness of the dynamics. Less explored is the role of topological protection in the context of classical oscillatory systems. As on-site dynamics, we consider prototypical oscillator models with possible applications in biochemical systems. However, our choice of coupling geometry is inspired by models from condensed matter physics which—in isolation—guarantee nontrivial topology in momentum space. We choose directed couplings between units on a two-dimensional grid, alternating between weak and strong values, such that oscillations become localized at the edges of the grid while bulk units transition to oscillation-death states, resulting in a frequency-chimera-like state. These patterns are resilient to parameter mismatches, additive noise, and structural defects. To explain the robustness of these edge oscillations we use topological characteristics and calculate Zak phases. As it turns out, the edge-localized oscillations result from a bulk-boundary correspondence that applies to our system even though our derived effective Hamiltonian is non-Hermitian. By appropriately tuning the system parameters, it is possible to control which regions of the two-dimensional grid are in an oscillatory state and which settle to oscillation-death states.

Spiral waves, self-organized patterns prevalent in diverse nonequilibrium systems from cardiac tissue to chemical reactions, exhibit complex dynamics influenced by medium properties such as bistability. A type of structure, the bistable Yin-Yang (BYY) spiral waves, characterized by a dynamic line defect separating two regions oscillating with distinct wave crests, emerges uniquely in oscillatory bistable media. Controlling such intricate spiral structures—particularly modifying their internal features like line defects—remains challenging and underexplored. Here we establish a methodology for the structural control of the BYY spiral waves. Through targeted pulse application, we demonstrate three fundamental operations: elimination of the line defect in the BYY spiral wave, driving the system to a homogeneous wave crest state; programmable addition of line defects whose evolution depends critically on their spatial placement; and induction of persistent spatiotemporal oscillations, where wave crests rhythmically alternate during wave propagation. Phase space analysis reveals that these controlled states emerge from symmetric limit cycles with intertwined basins of attraction, with pulse thresholds dictated by geometric relationships between these basins. This work provides a method for manipulating complex spiral wave architectures and reveals elimination, addition, and modulation of line defects.

We investigate the role of inertia in the asynchronous state of a disordered Kuramoto model. We extend an iterative simulation scheme to the case of the Kuramoto model with inertia in order to determine the self-consistent fluctuation statistics, specifically, the power spectra of network noise and single oscillators. Comparison with network simulations demonstrates that this works well whenever the system is in an asynchronous state. We also find an unexpected effect when varying the degree of inertia: the correlation time of the oscillators becomes minimal at an intermediate mass of the oscillators; correspondingly, the power spectra appear flatter and thus more similar to white noise around the same value of mass. We also find a similar effect for the Lyapunov spectra of the oscillators when the mass is varied.

We study the effects of nonreciprocity and network structure on percolation. To this end, we investigate nonreciprocal random networks—directed networks for which the probability of a link occurring from node $i$ to node $j$ differs from the probability of the reverse link occurring from node $j$ to node $i$. We analytically determine the degree and percolation properties of such networks with exactly two types of link probability, demonstrating that whether the networks are structured such that the nodes are not statistically indistinguishable has profound effects on these measures, both quantitatively and in how such networks need to be approached. In particular, we develop a technique for solving the percolation problem which can be applied to both structured and unstructured networks. The method entails writing self-consistent integral and differential equations for the probability that each node will belong to the network's giant component. Exact solutions to these equations are obtained and simulations which confirm our analytic predictions are presented.

Here, we propose a valveless nanopump model, featuring a two-dimensional graphene slit that can only accommodate a single layer of water molecules. Using molecular dynamics simulations, we demonstrate the feasibility of inducing water transport via mechanical vibrations at the slit entrance, even in the absence of a pressure difference. Simulation results indicate that the vibrational energy of the graphene plane can be converted into kinetic energy for water molecules within the nano slit, resulting in directed water flow. The water flow is dependent sensitively on the vibration frequency. Unlike previous one-dimensional systems, we observe that nanoscale vortices form within the two-dimensional slit when the frequency exceeds a specific threshold. The emergence of the vortices impedes the directed transport of water, consequently leading to a reduction in the efficiency of the water pump. Our findings provide valuable insights into the development of two-dimensional valveless nanopump.

Projections of hypercubes have been applied to visualize high-dimensional binary state spaces in various scientific fields. Conventional methods for projecting hypercubes, however, face practical difficulties. Manual methods require nontrivial adjustments of the projection basis, while optimization-based algorithms limit the interpretability and reproducibility of the resulting plots. These limitations motivate us to explore theoretically analyzable projection algorithms such as principal component analysis (PCA). Here, we investigate the mathematical properties of PCA-projected hypercubes. Our numerical and analytical results show that PCA effectively captures polarized distributions within the hypercubic state space. This property enables the assessment of the asymptotic distribution of projected vertices and error bounds, which characterize the performance of PCA in the projected space. We demonstrate the application of PCA to visualize the hypercubic energy landscapes of Ising spin systems, specifically finite artificial spin-ice systems, including those with geometric frustration. By adding projected hypercubic edges, these visualizations reveal pathways of correlated spin flips. We confirm that the time-integrated probability flux exhibits patterns consistent with the pathways identified in the projected hypercubic energy landscapes. Using the mean-field model, we show that dominant state transition pathways tend to emerge around the periphery of the projected hypercubes. Our work provides a better understanding of how PCA discovers hidden patterns in high-dimensional binary data.

Confinement often influences the properties of fluids. Using molecular dynamics simulations with explicit solvents, we show that strong slit confinement induces a first-order transition between solvation states, where a nanoparticle either resides at the slit center or binds to the confining surfaces. We find that the stability transition of this bistable state follows a subcritical pitchfork bifurcation. Furthermore, along with the transition, the nanoparticle's lateral diffusion, which is Brownian in bulk, exhibits a discontinuous change. Although the transition depends on various parameters, such as interparticle interactions and confinement interface properties, the bifurcation framework consistently captures the observed stability transitions across all conditions studied. These results underscore the need to develop molecular-level hydrodynamic theories capable of accurately describing strongly confined systems.

We study the porous medium equation (PME) in one space dimension in the presence of additive nonconservative white noise and interpreted as a stochastic growth equation for the height field of an interface. We predict the values of the two growth exponents $\alpha$ and $\beta$ using the functional renormalization group. Extensive numerical simulations show agreement with the predicted values for these exponents; however, they also show anomalous scaling with an additional local exponent ${\alpha }_{\mathrm{loc}}$ as well as multiscaling originating from broad distributions of local height differences. The stationary measure of the stochastic PME is found to be well described by a random walk model related to a Bessel process. This model allows for several predictions about the multiscaling properties.

Can we deduce the total length of a random trajectory by observing only its local path segments within a confined domain? Surprisingly, the answer is yes—for curves randomly placed and oriented in space, whether stochastic or deterministic; generated by ballistic or diffusive dynamics; possibly interrupted by stopping or branching; and in two or more dimensions. More precisely, the mean total length $\langle L\rangle$ relates to the mean in-domain path length $\langle \ell \rangle$ and the mean chord length of the domain $\langle \sigma \rangle$ via the following simple and universal relation: $$\frac{1}{\langle \ell \rangle }=\frac{1}{\langle L\rangle }+\frac{1}{\langle \sigma \rangle }.$$Here, $\langle \sigma \rangle$ is a purely geometric quantity, dependent only on the volume-to-surface ratio of the domain. Derived solely from the kinematic formula of integral geometry, the result is independent of step-length statistics, memory, absorption, and branching, making it equally relevant to photons in turbid tissue, active bacteria in microchannels, cosmic rays in molecular clouds, or neutron chains in nuclear reactors. Monte Carlo simulations spanning straight needles, Y shapes, and isotropic random walks in two and and three dimensions confirm the universality and demonstrate how a local measurement of $\langle \ell \rangle$ yields $\langle L\rangle$ without ever tracking the full trajectory.

The time evolution of Markovian open quantum systems is governed by Lindblad master equations, whose solution can be formally written as the Lindbladian exponential acting on the initial density matrix. By expanding this Lindbladian exponential into the Taylor series, we propose a generic method for integrating Lindblad master equations. In this method, the series is truncated, retaining a finite number of terms, and the iterative actions of Lindbladian on the density matrix follow the corresponding master equation. Our method offers significant improvements in numerical efficiencies both in memory cost and computation time, especially for systems with many degrees of freedom. Moreover, our proposed method can be integrated seamlessly with tensor networks. Two illustrative examples, a two-level system exhibiting damped Rabi oscillations and a driven dissipative Heisenberg chain, are used to demonstrate the validity of our method. The superiority of our method is benchmarked with detailed performance tests.

This work explores the thermal-siphon phenomenon in networks and its regulation via on-site potential. By locally reconstructing a two-dimensional lattice, we induce the thermal-siphon effect and investigate its behavior through heat conduction simulations. Our results demonstrate that increasing the strength of the on-site potential gradually suppresses and ultimately eliminates this phenomenon. We further confirm the modulation mechanism in networks with more complex topologies. Utilizing phonon spectral theory, we elucidate the fundamental physical mechanisms underlying the emergence and control of the thermal-siphon effect. These findings could be realized in nanoscale experiments, deepening the understanding of microscopic thermal transport mechanisms and providing new perspectives for efficient thermal management.

This work investigates the unidirectional transport of charged particles across magnetic field lines driven by magnetosonic solitons in a collisionless plasma. Within the framework of a magnetohydrodynamic model, numerical solutions as well as analytical expressions for small-amplitude solitons are obtained, and it is shown that particles experience a unidirectional displacement proportional to the square root of the wave amplitude, indicating that solitons generate substantial matter transport across the magnetic field.

We analytically solve the critical dynamics of the Kob-Andersen kinetically constrained model of supercooled liquids on the Bethe lattice, employing a combinatorial argument based on the cavity method. For arbitrary values of graph connectivity $z$ and facilitation parameter $m$, we demonstrate that the critical behavior of the order parameter is governed by equations of motion equivalent to those found in mode-coupling theory. The resulting predictions for the dynamical exponents are validated through direct comparisons with numerical simulations that include both continuous and discontinuous transition scenarios.

Quantum many-body systems present substantial technical challenges from both analytical and numerical perspectives. Despite these difficulties, some progress has been made, including studies of interacting atomic gases and interacting quantum spins. Furthermore, the potential for criticality to enhance engine performance has been demonstrated, suggesting a promising direction for future investigation. Here, we explore the performance of a quantum Otto cycle using a long-range Ising chain as the working substance. We consider an idealized cycle consisting of two adiabatic transformations and two perfect thermalizations, eliminating dissipation. Analyzing both engine and refrigerator modes, we investigate the influence of particle number, varied from 10 to 100, on efficiencies and behavior near the critical point of the phase transition, which we characterize using a scaling factor. We also examine how internal factors—specifically, the power-law exponent, the number of particles, and the hot and cold reservoir temperatures—affect the system's operation in different modes. Our results reveal that these factors have a different impact compared with their classical counterparts.

On-site potentials are ubiquitous in physical systems and strongly influence their heat transport and energy localization. These potentials will inevitably affect the dynamical properties of $q$ breathers (QBs), defined as periodic orbits exponentially localized in normal mode space. By integrating on-site terms into the Fermi-Pasta-Ulam-Tsingou-$\beta$ system, this work utilizes numerical simulations and Floquet analysis to systematically explore the influence of on-site potentials on QB stability. For most QBs, except those at the phonon band edges, the instability is primarily governed by parametric resonance and effectively described by coupled Mathieu equations. This approach provides a theoretical expression for the instability thresholds, which aligns well with numerical results. The instability threshold is highly sensitive to the seed mode, in stark contrast to systems without on-site potentials. In addition, a systematic stability analysis across three-dimensional parameter space shows that quartic intersite nonlinearity monotonically enhances QB instability, while quartic on-site nonlinearity yields nonmonotonic effects, in particular, an initial destabilization followed by restabilization at high amplitudes. The quadratic on-site potential uniformly promotes stability. Furthermore, these instability phase diagrams highlight both the individual contributions and combined effects of each possible component in regulating the QB dynamics. These findings offer valuable insights into QB stability and the manipulation of localized excitations in diverse physical systems with on-site potentials.

Understanding the interplay among processes that occur over different timescales is a challenging issue in the physics of systems regulation. In gene regulation, the timescales for changes in chromatin states can differ from those for changes in the concentration of product protein, raising questions about how to understand their coupled dynamics. In this study, we examine the effects of these different timescales on eukaryotic gene regulation using a stochastic model that describes the landscapes and probability currents of nonequilibrium fluctuations. This model shows that slow, nonadiabatic transitions of chromatin states significantly impact gene-regulation dynamics. The simulated circular flow of the probability currents indicates a maximum entropy production when the rates of chromatin-state transitions are low in the intensely nonadiabatic regime. In the mildly nonadiabatic regime, this circular flow fosters hysteresis, suggesting that changes in chromatin states precede changes in transcription activity. Furthermore, calculations using a model of a circuit involving three core genes in mouse embryonic stem cells illustrate how the timescale difference can tune fluctuations in individual genes. These findings highlight the rich effects of nonadiabatic chromatin-state transitions on gene regulation in eukaryotic cells.

${\mathrm{K}}_{2\mathrm{P}}$ channels are potassium leak channels known to play an essential role in modulating the membrane potential of neurons and glia. Their dysfunction has been associated with a wide range of diseases, including cancer, autoimmune, CNS, cardiovascular and urinary tract disorders, and dysfunction of ${\mathrm{K}}^{+}$ homeostasis. Despite voltage-dependent activity in most ${\mathrm{K}}_{2\mathrm{P}}$ channels, no voltage-sensing mechanism or complex similar to that present in voltage-gated potassium channels has been found. Due to the relatively small size of the ${\mathrm{K}}_{2\mathrm{P}}$ channel, and based on the data set, the only area that may be used for voltage sensing is the ion conduction pathway. Therefore, to determine whether the voltage-sensing area is located in this area, we checked the change in electric field in that area through physical analyses based on structural biological data. In this study, we showed that a voltage change occurs in the area of the ion conduction pathway in the ${\mathrm{K}}_{2\mathrm{P}}$ channel and therefore, also, the electric field changes in this area. We strengthened and created support for this claim through a physical model. The sensitivity to this voltage originates from the movement of three to four ions that cause a significant change in both the voltage and the electric field. Stimuli of different intensities and at different distances from the region of the ion conduction pathway show significant differences in the opening and closing of the channel and move the channel into a leaky state. The closer the stimulus is to the ion conduction path, the more significant the change in the field. This mechanism provides a clue to the location and possible existence of a voltage-sensing complex in the region of the ion conduction pathway of potassium leak channels.