We develop a perturbative framework to calculate the mean-squared displacement (MSD) of active Brownian particles with a finite moment of inertia. Starting from the corresponding Fokker–Planck equation, we employ a Fourier transform for the spatial coordinates and Hermite polynomials as eigenfunctions for the angular velocity, which enables a systematic perturbative expansion of the MSD order by order. By resumming the resulting series in Laplace space and performing the inverse transform, we obtain an explicit expression for the MSD as a function of the moment of inertia. The analytical results are further validated by comparison with numerical simulations.

In previous studies of relativistic thermodynamics, the temperature of a static system, as perceived by a moving observer, has traditionally been treated as a scalar. This assumption has also been extended to research on the cosmic microwave background. However, the validity of this assumption is a consequence of the massless nature of photons. More generally, when an observer is in relative motion to a system, the thermal equilibrium state is characterized by a inverse temperature four-vector. In this article, we study the non-interacting massive bosonic and fermionic field systems. We derive the Lorentz transformation of the energy spectral density in the equilibrium state of these fields. In the massless limit for bosonic field, our results recover the transformation of black body radiation [G. W. Ford and R. F. O'Connell, Phys. Rev. E, 88, 044101 (2013)], which corresponds to a scalar temperature with dipole anisotropy. For the massive fields, the moving equilibrium state cannot be characterized by a corresponding scalar temperature. This result shows the necessity of introducing inverse temperature four-vector in relativistic thermodynamics.

Reaction–diffusion infectious disease models are widely used to describe the spatial distribution of infected individuals. In this study, we construct network-based reaction–diffusion models that incorporate both higher-order interactions and advection mechanisms, formulated on a triangular lattice torus network. This idealized structure is adopted to facilitate explicit derivation and linear stability analysis of the theoretical conditions for Turing instability—analyses that would be considerably more challenging in complex heterogeneous geometries. To address the computational challenge of generating higher-order node Laplacian matrices in large-scale networks, we develop a dimensionality reduction strategy using graph-structured edge Laplacians. The theoretical analysis reveals how higher-order interactions and advection jointly influence the onset of Turing patterns. Furthermore, model fitting to real epidemic data—using a mobility network constructed from 2020 inter-prefectural commuting flows across Japan's 47 prefectures—shows that incorporating higher-order interactions results in better fitting performance compared to conventional models.

Cooperation, fairness, trust, and resource coordination are cornerstones of modern civilization, yet their emergence remains inadequately explained, largely due to persistent discrepancies between theoretical predictions and behavioral experiments. Part of this gap may arise from the imitation learning paradigm commonly used in prior theoretical models, which assumes individuals merely copy successful neighbors according to predetermined, fixed rules. This review examines recent advances in evolutionary game dynamics that employ reinforcement learning (RL) as an alternative paradigm. In RL, individuals learn through trial and error and introspectively refine their strategies based on environmental feedback. We begin by introducing key concepts in evolutionary game theory and the two learning paradigms, then synthesize progress in applying RL to elucidate cooperation, trust, fairness, optimal resource coordination, and ecological dynamics. Collectively, these studies indicate that RL offers a promising unified framework for understanding the diverse social and ecological phenomena observed in human and natural systems.

Omnivory, where species feed across multiple trophic levels, is a widespread feature of ecological networks. A key mechanism underlying such complexity is intraguild predation (IGP), in which a top predator consumes both an intermediate predator and a shared resource. Here, we show that Shilnikov homoclinic orbits emerge in a minimal intraguild predation model, triggering a cascade of homoclinic bifurcations near a saddle-focus equilibrium that culminates in chaos. Numerical simulations and Lyapunov spectrum analysis reveal multiple coexistence modes, ranging from regular oscillations to Shilnikov homoclinic orbits and chaos. Our model quantitatively reproduces patterns observed in natural omnivore networks, providing mechanistic insights into complex population fluctuations in ecological systems.

The convergence of cycle expansions in non-hyperbolic systems has long been an outstanding challenge, primarily due to the presence of singularities in natural measures and the attributed weak shadowing properties of unstable periodic orbits. Here, we introduce a novel layered approximation scheme that effectively decomposes the natural measure into distinct singular and hyperbolic layers. For the singular components, we employ analytical approximations, whereas for the hyperbolic components, we utilize cycle expansions. A key innovation of our approach is the incorporation of dynamical information beyond periodic orbits, which is essential for achieving a more accurate approximation of the natural measure and for precisely calculating observable averages. By integrating additional dynamical details, the new scheme more effectively captures the subtleties of the system's behavior that are not fully represented by a limited set of periodic orbits alone. This invariance allows us to refine the approximation of the natural measure in both the singular and the hyperbolic layers, thereby yielding a more reliable estimation of observable averages.

Voting is an important social activity for expressing public opinions. By conceptually considering a group of voting agents to be intelligent matter, the impact of real-time information on voting results is quantitatively studied by an intelligent Ising model, which is formed by adding nonlinear instantaneous feedback of the overall magnetization to the conventional Ising model. In the new model, the interaction strength becomes a variable depending on the total magnetization rather than a constant, which mimics the scenario that the decision of an individual during vote influenced by the dynamically changing polling result during the election process. Our analytical derivations along with Monte Carlo simulations reveal that, with a positive feedback, the intelligent Ising model exhibits phase transitions at any finite temperatures, a feature lacked in the conventional one-dimensional Ising model. In all dimensions, by varying the feedback strength, the system changes from going through a second-order phase transition to going through a first-order phase transition with increasing temperature, and the two types of phase transitions are connected by a tricritical point. This study on the one hand demonstrates that the intelligent matter with a nonlinear adaptive interaction can exhibit qualitatively different phase behaviors from conventional matter, and on the other hand shows that, during voting, even unbiased feedback may possibly induce spontaneous symmetry breaking, leading to a biased outcome where one side of the vote becomes favored.

The Wiener path integral framework is proposed to model military combat dynamics by incorporating the neglected stochastic effects to the Lanchester’s square law. This framework is applied to evaluate the empirical 3:1 combat rule, which posits that an attacker requires a threefold force superiority to achieve victory. Specifically, the attacker’s winning probability is computed utilizing a semi-analytical Rayleigh–Ritz method. Numerical results demonstrate that the validity of the rule critically depends on specific parameter regimes, primarily contingent upon the relative combat effectiveness ratio between the opposing forces and the tolerance for attrition. This work establishes a physics-informed theoretical bridge between statistical mechanics and military operations research for analyzing uncertain combat systems.

This article investigates the pseudo phase transition behavior of the six-state clock model on two-dimensional finite-size lattices. By employing the Wang-Landau sampling method and microcanonical inflection-point analysis, we identified two phase transition points and classified both as continuous transitions within the microcanonical framework. Using Metropolis sampling and canonical ensemble analysis, we verified the accuracy of the transition points obtained from the microcanonical approach and further pinpointed the location of a fourth-order dependent transition with high accuracy. Moreover, the fourth-order dependent transition points extracted from two canonical order parameters——the average cluster perimeter and the ${M}_{{{\mathbb{Z}}}_{6}}$ symmetry——are in excellent agreement and consistently reinforce each other. Through a detailed analysis of the ${M}_{{{\mathbb{Z}}}_{6}}$ symmetry, we further demonstrate that the position of this fourth-order dependent transition may function as a critical transition warning indicator for both primary phase transitions in the system.

We developed a model of a quantum Otto engine using two coupled two-level atoms. Based on the platform, we show that frequency detuning and the coupling strength induced by dipole-dipole interactions can lead to decoherence by disrupting coherent energy exchange. We focus on fundamental thermodynamic quantities, including heat absorption, release to heat baths, work done and efficiency. It is noteworthy that the interatomic coupling strength and frequency detuning do not merely affect the shape of the work and the efficiency but ultimately govern its quantitative magnitude. In the field of quantum thermodynamics, we have established an upper bound efficiency that is stricter than the Carnot limit. Moreover, our analysis confirms that quantum coherence enables the system to exceed the efficiency threshold of a classical Otto heat engine. The second law of thermodynamics holds all the while. Our results constitute a step forward in the design of conceptually new quantum thermodynamic devices which take advantage of uniquely quantum resources of quantum coherence.

This study investigates the thermal and statistical properties of the Dirac oscillator within the framework of two prominent formulations of doubly special relativity (DSR): the Amelino-Camelia and Magueijo-Smolin models. DSR extends Einstein’s special relativity by introducing an additional invariant scale—the Planck energy—leading to modified energy-momentum relations that encode potential quantum-gravitational effects at ultra-high energies. In this context, we derive the modified Dirac equations for both DSR scenarios and analytically determine the corresponding energy spectra. These spectra are subsequently used to compute the partition function and key thermodynamic quantities, including specific heat, by employing the Euler–Maclaurin formula to facilitate an efficient approximation of the partition function. The analysis is restricted to the positive-energy sector, enabled by the exact Foldy–Wouthuysen transformation, which effectively decouples positive and negative energy states. The findings reveal that Planck-scale deformation parameters induce significant modifications in the energy spectrum and thermodynamic behavior of the Dirac oscillator in each DSR framework, thereby offering valuable insights into possible observable imprints of quantum gravitational phenomena in relativistic quantum systems.

We present a novel nonlinear state transition model for inositol 1,4,5-trisphosphate receptors (IP₃Rs) that incorporates a pre-activated state, as suggested by electron microscopy observations. Our model provides a theoretical framework for the biphasic Ca²⁺ dependence of IP₃Rs and accurately reproduces their experimentally observed state distribution under saturating IP₃ conditions. By integrating receptor dynamics with cytoplasmic and endoplasmic reticulum (ER) calcium exchange, we simulate IP₃R-mediated Ca²⁺ oscillations governed by six key conformational states. A pivotal finding is that IP₃ regulates these oscillations in a switch-like manner: once a critical IP₃ concentration is reached, the system abruptly transitions to sustained, constant-amplitude oscillations that quickly terminate when the concentration exceeds a secondary threshold. These results underscore the crucial role of the pre-activated state in modulating calcium signaling.

We introduce a minimal model consisting of a two-body system with stochastically broken reciprocity (i.e. random violation of Newton’s third law) and then investigate its statistical behaviors, including fluctuations of velocity and position, time evolution of probability distribution functions, energy gain, and entropy production. The effective temperature of this two-body system immersed in a thermal bath is also derived. Furthermore, we heuristically present an extremely minimal model where the relative motion adheres to the same rules as in classical mechanics, while the effect of stochastically broken reciprocity only manifests in the fluctuating motion of the center of mass.

We study Onsager vortex clustered states in a shell-shaped superfluid containing a large number of quantum vortices. In the incompressible limit and at low temperatures, the relevant problem can be boiled down to the statistical mechanics of neutral point vortices confined on a sphere. We analyze rotation-free vortex-clustered states within the mean-field theory in the microcanonical ensemble. We find that the sandwich state, which involves the separating of vortices with opposite circulation and the clustering of vortices with the same circulation around the poles and the equator, is the maximum entropy vortex distribution, subject to a zero angular momentum constraint. The dipole moment vanishes for the sandwich state and the quadrupole tensor serves as an order parameter to characterize the vortex cluster structure. For a given finite angular momentum, the equilibrium vortex distribution forms a dipole structure, i.e., vortices with opposite sign are separated and accumulate around the south and north poles, respectively. The conditions for the onset of clustering and the exponents associated with the quadrupole moment and the dipole moment as functions of energy are obtained within the mean field theory. At large energies, we obtain asymptotically exact vortex density distributions using the stereographic projection method, yielding the parameter bounds for the vortex clustered states. The analytical predictions are in excellent agreement with microcanonical Monte Carlo simulations.

We investigate numerically the effects of long-range temporal and spatial correlations based on the rescaled distributions of the squared interface width W²(L, t) and the interface height h(x, t) in the (1+1)-dimensional Kardar–Parisi–Zhang (KPZ) growth system within the early growth regime. Through extensive numerical simulations, we find that long-range temporally correlated noise does not significantly impact the distribution form of the interface width. Generally, W²(L, t) approximately obeys a lognormal distribution when the temporal correlation exponent θ ≥ 0. On the other hand, the effects of long-range spatially correlated noise are evidently different from the temporally correlated case. Our results show that, when the spatial correlation exponent ρ ≤ 0.20, the distribution forms of W²(L, t) approach the lognormal distribution, and when ρ > 0.20, the distribution becomes more asymmetric, steep, and fat-tailed, and tends to an unknown distribution form. As a comparison, probability distributions of the interface height are also provided in the temporally and spatially correlated KPZ system, exhibiting quite different characteristics from each other within the whole correlated strengths. For the temporal correlation, the height distributions follow Tracy–Widom Gaussian orthogonal ensemble (TW-GOE) when θ → 0, and with increasing θ, the height distributions crossover continuously to an unknown distribution. However, for the spatial correlation, the height distributions gradually transition from the TW-GOE distribution to the standard Gaussian form.

Signal transduction in a cell is mostly mediated with biochemical reactions which are noisy and often modeled with chemical master equations or Langevin type of dynamics. Thus stochastic simulation constitutes a major part of computation in cell signaling. Nevertheless, the presence of a wide span of time scales or molecular numbers in various pathways may lead to trouble in computation efficiency or accuracy. To avoid this problem, the commonly employed variational method evolves the whole probability distribution and reduces the stochastic equations to deterministic ones of only a few parameters. However, the design of the left variational basis is essential for its successful application, especially to large networks. In this paper, we extend the conventional polynomial basis to the Fourier and further the Gaussian basis, much facilitating description of multi-peaked or localized non-Gaussian distributions and at the same time avoiding numerical instability and computational complexity frequently encountered with conventional basis. The extension here is demonstrated in several typical biochemical signaling networks and achieves similar accuracy as the benchmark Gillespie algorithm, but with much less running time, which seems to open new opportunities in the variational approach to efficient analysis of noisy dynamics.

Systems with quenched disorder possess complex energy landscapes that are challenging to explore under conventional Monte Carlo methods. In this work, we implement an efficient entropy sampling scheme for accurate computation of the entropy function in low-energy regions. The method is applied to the two-dimensional ±J random-bond Ising model, where frustration is controlled by the fraction p of ferromagnetic bonds. We investigate the low-temperature paramagnetic–ferromagnetic phase boundary below the multicritical point at T_N = 0.9530(4), P_N = 0.89078(8), as well as the zero-temperature ferromagnetic–spin-glass transition. Finite-size scaling analysis reveals that the phase boundary for T < T_N exhibits reentrant behavior. By analyzing the evolution of the magnetization-resolved density of states g(E, M) and ground-state spin configurations against increasing frustration, we provide strong evidence that the zero-temperature transition is a mixed-order. Finite-size scaling conducted on the spin-glass side supports the validity of β = 0, where β is the magnetization exponent, with a correlation length exponent ν = 1.50(8). Our results provide new insights into the nature of the ferromagnetic-to-spin-glass phase transition in an extensively degenerate ground state.

Combinatorial optimization problems and ground state problems of spin glasses are crucial in various fields of science and technology. However, they often belong to the computational class of NP-hard, presenting significant computational challenges. Traditional algorithms inspired by statistical physics like simulated annealing have been widely adopted. Recently, advancements in Ising machines, such as quantum annealers and coherent Ising machines, offer new paradigms for solving these problems efficiently by embedding them into the analog evolution of nonlinear dynamical systems. However, existing dynamics-based algorithms often suffer from low convergence rates and local minima traps. In this work, we introduce the dual mean-field dynamics into Ising machines. The approach integrates the gradient force and the transverse force into the dynamics of Ising machines in solving combinatorial optimization problems, making it easier for the system to jump out of the local minimums and allowing the dynamics to explore wider in configuration space. We conduct extensive numerical experiments using the Sherrington–Kirkpatrick spin glass up to 10 000 spins and the maximum cut problems with the standard G-set benchmarks. The numerical results demonstrate that our dual mean-field dynamics approach enhances the performance of base Ising machines, providing a more effective solution for large-scale combinatorial optimization problems.

In this paper, we investigate the distinctions between dynamical quantum chaotic systems and random models from the perspective of observable properties, particularly focusing on the eigenstate thermalization hypothesis (ETH). Through numerical simulations, we find that for dynamical systems, the envelope function of off-diagonal elements of observables exhibits an exponential decay at large ΔE, while for randomized models, it tends to be flat. We demonstrate that the correlations of chaotic eigenstates, originating from the delicate structures of Hamiltonians, play a crucial role in the non-trivial structure of the envelope function. Furthermore, we analyze the numerical results from the perspective of the dynamical group elements in Hamiltonians. Our findings highlight the importance of correlations in physical chaotic systems and provide insights into the deviations from random matrix theory (RMT) predictions. These understandings offer valuable directions for future research.

Ribonucleic Acid (RNA) contact prediction holds great significance for modeling RNA 3D structures and further understanding RNA biological functions. The rapid growth of RNA sequencing data has driven the development of diverse computational methods for RNA contact prediction, and a benchmark evaluation of these methods remains essential. In this work, we first classified RNA contact prediction methods into statistical inference-based and neural network-based ones. We then evaluated eight state-of-the-art methods on three test sets: a sequence-diverse set, a structurally non-redundant set and a CASP RNA targets set. Our evaluation shows that for identifying non-local and long-range contacts, neural network-based methods outperform statistical inference-based ones, with SPOT-RNA-2D achieving the best performance, followed by CoCoNet and RNAcontact. However, for identifying the long-range tertiary contacts, which are vital for stabilizing RNA tertiary structure, statistical inference-based methods exhibit superior performance with GREMLIN emerging as the top performer. This work provides a comprehensive benchmarking of RNA contact prediction methods, highlighting their strengths and limitations to guide further methodological improvements and applications in RNA structure modeling.

The theory of statistical physics relies on ergodicity, whereby in large or interacting systems lacking integrability, trajectories eventually explore nearly all points in the phase space. It has been believed that chaotic dynamics provide a possible pathway to ergodicity. Here, we examine the phase space density distributions and their recurrence in the harmonic oscillator, the linear and nonlinear Mathieu equations, the Lorenz attractor, and the Nosé–Hoover model. We show that in models with periodic or quasiperiodic dynamics, sharp peaks can be found in the phase space density distributions. However, for the chaotic dynamics, their distributions display totally different behaviors. We understand these differences using recurrence plots. Our results show that while chaotic dynamics provide an efficient way for the trajectory to explore a large portion of the phase space, which is necessary for ergodicity, the chaotic dynamics are not sufficient for this goal. For instance, despite the Nosé–Hoover model being chaotic, it is not sufficiently large for ergodicity. Therefore, our results may lead to an important conclusion, which is that ergodicity may be realized from large chaotic systems. These findings in these simple models can be explored in experiments in the future, which may provide some key insights into ergodic dynamics.

Computing free energy is a fundamental problem in statistical physics. Recently, two distinct methods have been developed and have demonstrated remarkable success: the tensor-network-based contraction method and the neural-network-based variational method. Tensor networks are accurate, but their application is often limited to low-dimensional systems due to the high computational complexity in high-dimensional systems. The neural network method applies to systems with general topology. However, as a variational method, it is not as accurate as tensor networks. In this work, we propose an integrated approach, tensor-network-based variational autoregressive networks (TNVAN), that leverages the strengths of both tensor networks and neural networks: combining the variational autoregressive neural network's ability to compute an upper bound on free energy and perform unbiased sampling from the variational distribution with the tensor network's power to accurately compute the partition function for small sub-systems, resulting in a robust method for precisely estimating free energy. To evaluate the proposed approach, we conducted numerical experiments on spin glass systems with various topologies, including two-dimensional lattices, fully connected graphs, and random graphs. Our numerical results demonstrate the superior accuracy of our method compared to existing approaches. In particular, it effectively handles systems with long-range interactions and leverages GPU efficiency without requiring singular value decomposition, indicating great potential in tackling statistical mechanics problems and simulating high-dimensional complex systems through both tensor networks and neural networks.

The elastic properties of membranes are typically characterized by a few phenomenological parameters, including bending and Gaussian curvature moduli measuring the membrane rigidity against its deformation and topological change, as well as spontaneous curvature arising from the asymmetry between the two leaflets in the lipid bilayers. Though tether-based and fluctuation-based experiments are commonly utilized to measure the bending modulus, measuring the Gaussian curvature modulus and the spontaneous curvature of the membrane is considered to be much more difficult. In this paper, we study the buckling process of a circular membrane with nonzero spontaneous curvature under compressive stresses. It is found that when the stress exceeds a critical value, the circular membrane will transform from a spherical cap to a buckled shape, with its buckling degree enhanced with the increase of stress until its base is constricted to almost zero. As the stress-strain relationship of the buckled membrane strongly depends on the Gaussian curvature modulus and the spontaneous curvature,we therefore propose a method to determine the Gaussian curvature modulus and the spontaneous curvature simultaneously by measuring its stress-strain relationship during a buckling process.

Intracellular transports of cargoes are performed by biological molecular motors that move processively along their linear tracks. In some cases, the cargo can interact with the track. A typical example of these cases is the transport of a major mitotic signaling module, the chromosomal passenger complex (CPC), along the microtubule toward the equatorial cortex by a kinesin-6 motor during anaphase, where the CPC can interact with the microtubule. Here, an analytical theory is presented on the dynamics of the molecular motor transporting a track-interacted cargo. The theory is then applied to the transport of the track-interacted cargo by kinesin-6 and by kinesin-1 motors, with the theoretical results reproducing quantitatively the available experimental data. It is found that a diffusive cargo along the track, with the diffusion constant $\geqslant $ 0.1 μm² s^-1, can largely enhance the processivity relative to the non-diffusive cargo and relative to the cargo having no interaction with the track.

The theoretical challenges in understanding the nature of glass and glass transition raise significant questions in statistical and condensed matter physics. As a prototypical example of complex physical systems, glasses and the vitrification process have been central research topics, consistently attracting broad scientific interest. This focus has driven extensive studies on phenomena such as aging, non-exponential relaxation, dynamic anomalies, glass-forming ability, and the mechanical response of glasses under stress. Recent advances in computational and experimental techniques have enabled rigorous testing of theoretical models, shedding new light on glass behavior. However, the intrinsic complexity of glass and the glass transition that lies in their physics, which spans multiple length and time scales, makes the system challenging to characterize. In this review, we emphasize the need to move beyond conventional approaches and propose a topological perspective as a promising alternative to address these challenges. Specifically, our findings reveal that the diversity in particle relaxation behavior is statistically linked to a global topological feature of the transient network structures formed by the particles in a given liquid. This direction offers opportunities to uncover novel phenomena that could fundamentally reshape our understanding of glassy materials.

The Richtmyer–Meshkov (RM) instability plays an important role in various natural and engineering fields such as inertial confinement fusion. In this study, the effect of relaxation time on the RM instability under reshock impact is investigated using a two-component discrete Boltzmann method. The hydrodynamic and thermodynamic characteristics of the fluid system are comprehensively analyzed from the perspectives of the density gradient, vorticity, kinetic energy, mixing degree, mixing width and non-equilibrium intensity. Simulation results indicate that for longer relaxation time, the diffusion and dissipation are enhanced, the physical gradients decrease, and the growth of the interface is suppressed. Furthermore, the non-equilibrium manifestations show complex patterns, driven by the competitive physical mechanisms of the diffusion, dissipation, shock wave, rarefaction wave, transverse wave and fluid instabilities. These findings provide valuable insight into the fundamental mechanism of compressible fluid flows.

In this work, Langevin dynamics simulations were carried out to thoroughly investigate the swapping process of composite knots under tension in a cuboid nanochannel. From our analysis, the free energy profiles of knot swapping under different conditions were extracted from the overall probability distribution of the relative distance between the centers of composite knots. In addition, the impact of the stretching force, confinement size, and bending stiffness on the free energy profiles was directly identified. Especially, the influence of topology structure is for the first time reported. The increasing stretching force in a fixed confinement or the confinement size under a constant stretching force does not alter their respective equilibrium populations at the separate state and the entangled state. In contrast, a bending stiffness larger than 15 enhanced the formation of the entangled state. The topology structure of the 5₁ knot, which was different from the 5₂ knot, resulted in forming a metastable state in the free energy profiles. The increasing stretching forces yielded an enhancement of the following free energy barrier.

Active matter exhibits collective motions at various scales. Geometric confinement has been identified as an effective way to control and manipulate active fluids, with much attention given to external factors. However, the impact of the inherent properties of active particles on collective motion under confined conditions remains elusive. Here, we use a highly tunable active nematics model to study active systems under confinement, focusing on the effect of the self-driven speed of active particles. We identify three distinct states characterized by unique particle and flow fields within confined active nematic systems, among which circular rotation emerges as a collective motion involving rotational movement in both particle and flow fields. The theoretical phase diagram shows that increasing the self-driven speed of active particles significantly enhances the region of the circular rotation state and improves its stability. Our results provide insights into the formation of high quality vortices in confined active nematic systems.

Understanding neural dynamics is a central topic in machine learning, non-linear physics, and neuroscience. However, the dynamics are non-linear, stochastic and particularly non-gradient, i.e., the driving force cannot be written as the gradient of a potential. These features make analytic studies very challenging. The common tool is the path integral approach or dynamical mean-field theory. Still, the drawback is that one has to solve the integro-differential or dynamical mean-field equations, which is computationally expensive and has no closed-form solutions in general. From the associated Fokker–Planck equation, the steady-state solution is generally unknown. Here, we treat searching for the fixed points as an optimization problem, and construct an approximate potential related to the speed of the dynamics, and find that searching for the ground state of this potential is equivalent to running approximate stochastic gradient dynamics or Langevin dynamics. Only in the zero temperature limit, can the distribution of the original fixed points be achieved. The resultant stationary state of the dynamics exactly follows the canonical Boltzmann measure. Within this framework, the quenched disorder intrinsic in the neural networks can be averaged out by applying the replica method, which leads naturally to order parameters for the non-equilibrium steady states. Our theory reproduces the well-known result of edge-of-chaos. Furthermore, the order parameters characterizing the continuous transition are derived, and the order parameters are explained as fluctuations and responses of the steady states. Our method thus opens the door to analytically studying the fixed-point landscape of the deterministic or stochastic high dimensional dynamics.

We consider a relativistic two-fluid model of superfluidity, in which the superfluid is described by an order parameter that is a complex scalar field satisfying the nonlinear Klein–Gordon equation (NLKG). The coupling to the normal fluid is introduced via a covariant current–current interaction, which results in the addition of an effective potential, whose imaginary part describes particle transfer between superfluid and normal fluid. Quantized vorticity arises in a class of singular solutions and the related vortex dynamics is incorporated in the modified NLKG, facilitating numerical analysis which is usually very complicated in the phenomenology of vortex filaments. The dual transformation to a string theory description (Kalb–Ramond) of quantum vorticity, the Magnus force, and the mutual friction between quantized vortices and normal fluid are also studied.

This investigation focuses on the under-damped Brownian transport of a dimer characterized by two harmonically interacting components. The friction coefficients between the dimer components are different; thus the dynamic symmetry of the system is broken. In addition, the inertial ratchets are synchronously modulated by the feedback control protocol in time. Here, we analyze the transport performance by studying the average velocity and energy conversion efficiency of the dimer induced by friction symmetry breaking and external forces. Furthermore, we can also identify the enhancement of the centre-of-mass mean velocity and energy conversion efficiency of inertial frictional ratchets for intermediate values of the driving amplitude, coupling strength and damping force. Remarkably, in the weak bias case, the directed transport of inertial Brownian particles can be reversed twice by modulating the suitable friction of the dimer. In particular, the frictional ratchets can acquire a series of resonant steps under the influence of harmonic force. These conclusions of reliable transport in noisy environments are expected to provide insights into the performance of natural molecular motors.

Thermalization in many-body systems, especially with strong interactions, is a central question in physics. In this work, we present a novel framework for the thermalization of interacting wave systems, distinguishing between trivial (no momentum exchange) and nontrivial interactions (significant energy redistribution). This distinction leads to a statistically equivalent model with weakened interactions. By applying this to FPUT-like models, we identify a unique double scaling of thermalization times. Crucially, our findings suggest the persistence of prethermalization in strong interactions.

A multi-relaxation-time discrete Boltzmann model (DBM) with split collision is proposed for both subsonic and supersonic compressible reacting flows, where chemical reactions take place among various components. The physical model is based on a unified set of discrete Boltzmann equations that describes the evolution of each chemical species with adjustable acceleration, specific heat ratio, and Prandtl number. On the right-hand side of discrete Boltzmann equations, the collision, force, and reaction terms denote the change rates of distribution functions due to self- and cross-collisions, external forces, and chemical reactions, respectively. The source terms can be calculated in three ways, among which the matrix inversion method possesses the highest physical accuracy and computational efficiency. Through Chapman–Enskog analysis, it is proved that the DBM is consistent with the reactive Navier–Stokes equations, Fick's law and the Stefan–Maxwell diffusion equation in the hydrodynamic limit. Compared with the one-step-relaxation model, the split collision model offers a detailed and precise description of hydrodynamic, thermodynamic, and chemical nonequilibrium effects. Finally, the model is validated by six benchmarks, including multicomponent diffusion, mixture in the force field, Kelvin–Helmholtz instability, flame at constant pressure, opposing chemical reaction, and steady detonation.

The emerging concept of multi-caloric effects, introduced in 2010, entails the application of multiple interplay fields to a thermodynamic system. While multi-caloric effects are the main focus of experimental endeavors, theoretical considerations fall short of providing a thorough understanding. This paper introduces a comprehensive presentation on multi-caloric effects, employing the method and theory of exterior derivative formations. It addresses every aspect of thermodynamic systems, showcasing its applicability to multi-caloric materials (both single-phase and multi-phase materials), and its adaptability to different scenarios (either in single or multiple force fields). The formulation of Maxwell relationships, characterized by their generality and universality, enables a clear prediction in entropy and temperature, facilitating a distinct identification between independent and interdependent contributions from multi-caloric effects. These insights hold significant importance in designing and developing specialized thermodynamic materials, optimizing functional performances and exploring innovative mechanisms.

In this work, we study the interaction between two colloidal particles in a liquid crystal that is in the isotropic phase. The interaction is caused by surface-induced polarization of the liquid crystal molecules in the vicinity of the particles. We find that the interaction is short-ranged in both 2D and 3D geometry. Under symmetric homeotropic anchoring conditions, the interaction is repulsive. While under anti-symmetric homeotropic anchoring conditions, the interaction is repulsive at short distances but attractive at intermediate distances. The particle size has a strong impact on the effective interaction force.