We present a class of stochastic processes in which the large deviation functions of time-integrated observables exhibit singularities that relate to dynamical phase transitions of trajectories. These illustrative examples include Brownian motion with a death rate or in the presence of an absorbing wall, for which we consider a set of empirical observables such as the net displacement, local time, residence time, and area under the trajectory. Using a backward Fokker-Planck approach, we derive the large deviation functions of these observables and demonstrate how singularities emerge from a competition between survival and diffusion. Furthermore, we analyze this scenario using an alternative approach with tilted operators, showing that at the singular point, the effective dynamics undergoes an abrupt transition. Extending this approach, we show that similar transitions may generically arise in Markov chains with transient states. This scenario is robust and generalizable for non-Markovian dynamics and for many-body systems, potentially leading to multiple dynamical phase transitions. We have confirmed most of our findings on the singular large-deviation function using rare-event simulation techniques.

Wahl et al. [Phys. Rev. E 93, 022213 (2016)] introduced Granger causality (GC) maps for Langevin systems: Dynamics are localized linearly at each point in phase space as vector Ornstein-Uhlenbeck (VOU) processes, for which GCs may in principle be calculated, thus constructing a GC map on phase space. These maps may, furthermore, be averaged over phase space to yield a systemwide global GC value. The implementation, however, suffered some significant drawbacks: GCs were approximated from models based on discrete-time stroboscopic sampling of local VOU processes, which is not only computationally inefficient but, more seriously, infeasible on regions of phase space where local dynamics are unstable, leaving “holes” in the GC maps. We solve these problems by deriving an analytical expression for GC rates associated with a VOU process which, under quite general conditions, yields a meaningful solution even in the unstable case. Applied to GC maps, this not only “fills in the holes” but also furnishes a computationally efficient method of calculation devolving to solution of algebraic Riccati equations which, in the case of a univariate source, become simple quadratic equations. We show, furthermore, that the GC rate for VOU processes is invariant under rescaling of the overall fluctuations intensity, so that GC maps may effectively be calculated for deterministic nonlinear dynamical systems, with a residual “ghost of noise” represented by a variance-covariance map.

We consider a quasi-one-dimensional (quasi-1D) system that is an optical hard-wall nanotube with axial harmonic potential. This quasi-1D system possesses a quasi-1D harmonic potential that consists of a 1D harmonic trap along the $z$ axis and a two-dimensional box trap in the $x\text{−}y$ plane. Within the framework of quantum statistical mechanics, we find that in a quasi-1D harmonic potential, the generalized Bose-Einstein condensation (GBEC) can be classified into the two types. The first type of GBEC $\left({\mathrm{GBEC}}_1\right)$ corresponds to $T_m<T_c$. The second type of GBEC $\left({\mathrm{GBEC}}_2\right)$ corresponds to $T_c<T_m$. Here, $T_c$ is the critical temperature of the first-step condensation and $T_m$ is the critical temperature of the second-step condensation. There is a competitive mechanism between normal Bose-Einstein condensation (NBEC) and GBEC. We shall show that the standard transition temperature $T_c$ is much larger than the transition temperature $T_n$ of NBEC. Therefore, the state of NBEC does not occur in a quasi-1D harmonic potential. We have proposed an analytical solution to the problem of GBEC of ideal atoms in the quasi-1D harmonic potential. The number of noncondensed atoms in the first-step condensation is characterized by a series of elliptic theta functions and the number of noncondensed atoms in the second-step condensation is characterized by a single $q$-digamma function. If $r$ denotes the tube radius, then this quasi-1D system can show the quantum size effect. There is a critical tube radius $r_c$. (1) If $r>r_c$, then the mediate condensate fraction $N_m/N=0$; (2) if $r>r_c$, then the standard transition temperature $T_c=0$; (3) if $r>r_c$, then $T_m>T_c$ and this is the state of ${\mathrm{GBEC}}_2$. The results of numerical calculation of the analytical solution predict many new experimental phenomena in the GBEC of ideal atoms in the quasi-1D harmonic potential. In the thermodynamic limit, the analytical expressions of the three critical temperatures and the three condensate fractions are derived.

This study investigates the influence of material density, disorder in onsite energies, and localization length on relaxation dynamics within a two-dimensional random site Coulomb glass model at half-filling. To explore relaxation laws, we calculate the eigenvalue distribution of the linear dynamical matrix using mean-field approximations. Our findings indicate that the system initially undergoes rapid relaxation through energy-lowering transitions. The depletion of the single-particle density of states (DOS) near the Fermi level leads to slow relaxation, with fluctuations diminishing according to a power law. Subsequently, the system adheres to an exponential decay law after a specific period, defined as the relaxation time, which is inversely related to the minimum eigenvalue of the dynamical matrix. As the density of the system decreases, the relaxation rate slows down, resulting in an increase in the relaxation time. For a constant density and localization length, an increase in the disorder of onsite energies results in a longer relaxation time. A significant portion of the eigenvalue spectrum remains unaffected, suggesting that a reduction in localization length concurrent with increased disorder may play an equally vital role in the slow dynamics observed.

Simulations of the dynamics of two consumers $C_1, C_2$ and two resources $R_1, R_2$ are presented within the Rosenzweig-MacArthur model (RMA). Consumers $i=1,2$ share their time/effort devoted to particular resources; for each $i$, the consumption rates of the two resources are ${\beta }_i$ and $1-{\beta }_i$. Here, the model dynamics are enriched by the evolution of these rates over time. Namely, at discrete time instants, the consumer with a smaller density $C_i$ updates her rates to imitate the behavior of the more successful opponent, while the rates of the latter remain unchanged. These rules can be written as ‘‘lose-shift’’ and ‘‘win-stay,’’ respectively, where ‘‘shift’’ is directed to imitate the temporal winner. The phase diagram in the plane (${\beta }_1,{\beta }_2$) shows periodic orbits and chaos. The latter behavior is more prominent when resources are acquired unevenly. Compared with three alternative strategies, the ‘‘win-stay, lose-shift’’ strategy is an equilibrium in both the periodic and chaotic regimes.

People often express opinions that differ from their privately held views, a phenomenon known in economy as preference falsification. Expressed-private opinion (EPO) models capture this by assigning each agent two dynamical variables: a private (internal) and an expressed (external) opinion. Within the nonlinear $q$-voter model, two EPO variants have been studied so far: with and without self-anticonformity. In both formulations, agents update private and expressed binary opinions, one after another and at the same rate, which has led to two update schemes studied previously: act then think, in which an agent first updates its expressed and then its private opinion, and think then act, in which the order is reversed. To eliminate this ad hoc distinction and quantify the interplay between private and expressed opinions, we introduce the $\alpha$-EPO $q$-voter model with asynchronous updating—in each elementary step, an agent updates its private opinion with probability $\alpha$ or its expressed opinion with complementary probability $1-\alpha$. We derive mean-field theory and a pair approximation, and validate them with Monte Carlo simulations on artificial and real organizational networks. Comparing the two model variants, we show that the collective outcome controlled by $\alpha$ strongly depends on self-anticonformity: With self-anticonformity, the results are robust to $\alpha$, whereas without it $\alpha$ shifts the agreement-disagreement threshold and can change the type of phase transition. In the mean-field limit, this change occurs only for $q=3$, but the pair approximation reveals an additional low-connectivity regime in which both $\alpha$ and the average degree $k$ also control the emergence and width of hysteresis for larger influence groups.

Due to cell destruction in sequencing, time-series data yield unpaired snapshots, obscuring lineages and gene dynamics in individual cells. Current data-driven methods rely on single-variate expression, ignoring cell types and rotational effects in nonsteady states, limiting interpretability and accuracy. Each cell's fate diverges to different endpoints and can also flow backward to the starting point. In summary, we propose a bidirectional causal neural network of stochastic differential equation. It jointly trains the forward and reverse processes of the diffusion model to ensure reversible consistency between them, enhancing the stability of the learning process. In the forward process, it models cell differentiation as a stochastic process, whereby the drift term of the stochastic differential equation is learned by discovering causal relationships between principal components. Through multivariate dynamic modeling of unstructured data, the model has better expressiveness and interpretability. Then, we introduce the consistency of cell-type distribution between the predictions and the real data. In the reverse process, we account for the influence of the rotational term on the stochastic differential equation under nonstationary conditions. Instead of simply defining the drift term as the negative gradient of a potential function, we learn it through reverse modeling, thereby constructing an interpretable Waddington potential landscape.

Allosteric regulation in proteins arises from collective dynamics distributed over networks of residue contacts, but how multiple communication pathways contribute to signal transmission and noise suppression remains unclear. Here we develop a spanning-tree-based framework to quantify allosteric communication as an ensemble of pathways in protein contact networks. We introduce a dynamic distance measure linking local perturbations of residue interactions to global changes in network entropy, establishing a local-to-global scaling between local dynamics and global sensitivity. Using spanning-tree calculus, we derive exact probabilities for all simple paths connecting prespecified functional residue pairs. This enables a comparison between an approximate description based on uniform path usage and a topology-aware description in which path probabilities are determined by the Burton-Pemantle theorem and reflect network dependencies. From these path ensembles, we define corresponding signal-to-noise ratios and quantify how pathway multiplicity and statistical weighting shape noise suppression. Applied to KRAS and to 20 additional allosteric proteins spanning diverse functional classes, the analysis shows large variability in path usage, entropy reduction, and signal-to-noise enhancement, while consistently demonstrating that topology-aware weighting concentrates signal transmission onto dominant short pathways. This suggests that the robustness of allosteric signaling is a fundamental emergent property of protein contact topology. The spanning tree ensemble constitutes a distinct physical model whose partition function is the matrix tree theorem; the Gaussian network model is recovered as its high temperature limit. These results provide a quantitative framework linking protein structure, dynamics, and information flow, and show that robustness in allosteric communication, manifested as noise suppression through pathway redundancy, can be interpreted as an intrinsic noise averaging mechanism arising from network topology.

An ensemble model of turbulence based on states with constant flux in wavenumber space was proposed in [K. Yoshida, Phys. Rev. E 106, 045106 (2022)]. The justification of this ensemble model relies on the conjecture that almost all states with constant flux correspond to turbulence states. To verify this conjecture, a numerical search for states with constant enstrophy flux in wavenumber space over finite time intervals in two-dimension turbulence is conducted using a Monte Carlo method. Properties of the obtained states, such as energy spectrum and spectra related to fourth-order moments, are examined and compared with those of turbulence states obtained from direct numerical simulations. The dependence of the obtained states on the time interval and the initial conditions used in the numerical search is also discussed.

Identifying and classifying quantum phases from measurable time series in many-body dynamics have significant values, yet lack details and face formidable challenges, requiring the profound knowledge of physicists. Here, to achieve a purely data-driven machine intelligent classification, we introduce a temporal fluctuation-amplified distance measure that captures the inherent temporal fluctuation complexity of dynamic evolution series in different quantum many-body phases. Significantly, the introduction of complexity-powered distance leads to remarkable improvements of unsupervised manifold learning of quantum many-body dynamics, as exemplified in models such as the discrete time crystal (DTC) and Aubry-André (AA) models. Our method does not require any prior knowledge and exhibits effectiveness even in imperfect, disordered, and noisy situations that are challenging for human scientists. Successful classification of dynamic phases in many-body systems holds the potential to enable crucial applications, including identification of tsunamis, earthquakes, catastrophes, and future trends in finance.

Boltzmann sampling is a central component of many computational frameworks, including numerous algorithms in machine learning. Although quantum annealers have been investigated as potential fast Boltzmann samplers, their dependence on environmental noise makes precise control of the effective temperature difficult, introducing uncertainty into the sampling process. As an alternative, we propose diabatic quantum annealing—a faster, purely unitary process—as a controllable Boltzmann sampler in which the effective temperature is determined by the annealing rate. Using the ferromagnetic Ising model and the Sherrington-Kirkpatrick model as test cases, we demonstrate that this method achieves rapid and accurate sampling in the high-temperature regime.

Nonreciprocal interactions, where the influence of agent $i$ on $j$ differs from that of $j$ on $i$, are fundamental in active and living matter. Yet, most models implement such asymmetry phenomenologically. Here, we show that nonreciprocity can emerge from internal topology alone. Agents evolve on an internal “context manifold” coupled to a Chern–Simons gauge field. Because the gauge field is first order in time, it relaxes rapidly; eliminating it yields an effective transverse, antisymmetric interaction kernel that generically produces chiral waves, persistent vorticity, and irreversible state transitions. Numerical simulations reveal clear signatures of broken reciprocity characterized by long-lived vortex cores, nonzero total rotational energy of the flow, and signed circulation dominance. The dynamics further exhibit pronounced hysteresis under parameter sweeps, demonstrating memory effects that cannot occur in reciprocal or potential-driven systems. These results identify Chern–Simons gauge fields as a minimal and universal source of directional influence and robust nonreciprocal collective behavior.

The idea of the evolutionarily stable state (ESS) of a population is a cornerstone of evolutionary game theory; moreover, it coincides with the game-theoretic concept of Nash equilibrium. Such a state corresponds to a strategy adopted by the population such that a rare mutant strategy cannot invade the population. In parallel, the dynamical formulation of evolutionary game theory—particularly through replicator dynamics embodying the tenet of survival of the fittest—provides a framework for modeling frequency-dependent selection over time. While it is well known that an ESS corresponds to the stable fixed point in replicator dynamics, the evolutionary game-theoretic characterization of limit cycles is unknown. Here we fill this lacuna by defining an oscillatory ESS (OESS), which we prove to be a stable limit cycle. We also determine the conditions for the existence and the phase-space locations of OESSs.

Dry active matter in an anisotropic medium is of experimental relevance, and the interplay between anisotropy and the dynamics of the active matter remains under-explored. Here, we derive the hydrodynamic equations of a generic dry polar active fluid that preferentially flows along a particular axis induced by the anisotropy of the medium. We then study its critical behavior at the order-disorder transition in which the symmetry between “forward” and “back” along the special axis is spontaneously broken. We obtain the critical static and dynamic exponents, mean velocity, and two-point correlation functions exactly in three dimensions, and to two-loop level in two dimensions, by mapping our class of systems to the equilibrium Ising model with dipolar interactions.

In the network model, the interactions between nodes represent the transmission of information, energy, or substance. Strengthening interactions to enhance collective dynamical performance has garnered extensive interest. However, the specific degree to which collective performance is enhanced is a critical issue that has yet to be clarified. This work reveals that, although interactions between individuals are gradually strengthening, the enhancement of collective dynamical performance has a ceiling, referred to as dynamical potential, which coincidentally relies on the network Laplacian eigen-spectrum. Different interaction groupings correspond to different dynamical potentials, which determine diverse enhancement degrees. Additionally, two fascinating phenomena—potential preservation and potential jump—are observed in a network. It inspires a further step in network control and optimization, for both conventional node-pair interactions and higher-order interactions.

We present an analytical 1D axisymmetric model describing the evolution of the dynamic $Z$ pinch. This model is capable of predicting the trajectories of the imploding sheath's magnetic piston and preceding shock front, along with the velocity, pressure, density, and magnetic field profiles, for any time-dependent current, spatially varying initial density profile, and weak initial axial field. The implosion is divided into stages, with each stage described by a set of coupled ordinary differential equations derived from the ideal MHD equations. Comparisons with experimental data from the COBRA pulsed-power facility are quite promising and imply this model could prove useful in designing and analyzing future pulsed-power experiments.

This study uses three-dimensional particle-in-cell simulations to investigate the nonlinear evolution of electromagnetic waves propagating perpendicular to an external magnetic field in a magnetized plasma, with an emphasis on wave breaking and electron energization. In the weakly nonlinear regime (small nonlinearity parameter), the response exhibits two prominent frequency components that produce a beating signal, whose relative spectral weights vary with the magnetization parameter. As the nonlinearity parameter increases, nonlinear steepening and waveform distortion develop, accompanied by spectral broadening in wave-number space, and wave breaking eventually occurs. During the breaking regime, energy is irreversibly transferred from the electromagnetic fields to electrons, leading to a rapid increase of electron kinetic energy. We characterize the process using the breaking time, the saturated field energy, and an electron heating measure defined from the kinetic-energy gain. Over the scanned parameter range, this electron heating measure follows power-law scalings with the nonlinearity parameter and the magnetization parameter, with fitted exponents of approximately 3.15 and 0.28, respectively. These results are consistent with wave breaking providing an efficient dissipation pathway in the strongly driven regime considered here and constrain the scaling of electron energization with wave amplitude and the magnetization parameter.

Energy injection and dissipation in nonequilibrium systems typically lock microscopic motion and collective flow into a coupled evolution. Our observations in electron-beam-driven dusty plasma reveal a distinct “kinetic decoupling” regime where this synchrony breaks down. While microscopic and collective entropies evolve in tandem at low energies, a sharp “entropic scissors” effect emerges at a critical threshold: microscopic velocity randomization reaches a maximum, while collective flow fluctuations are simultaneously suppressed. The resulting state behaves as an ergodic, thermal-like fluid where intense local mixing coexists with global transport stability. This phenomenology originates from a fundamental timescale separation between rapid energy injection and slower neutral-drag dissipation, sustaining microscopic randomization while quenching large-scale instabilities. The dual-entropy framework introduced here provides a model-independent diagnostic for emergent scale separation, applicable to diverse driven many-body platforms.