We study the response of a two-dimensional classical $\mathit{XY}$ model to a finite (noninfinitesimal) twist of the boundary conditions. We use Monte Carlo simulations to evaluate the free energy difference between periodic and twisted-periodic boundary conditions and find deviations from the expected quadratic dependence on the twist angle. Consequently, the helicity modulus (spin stiffness) shows a nontrivial dependence on the twist angle. We show that the deviation from the expected behavior arises because of the mixing of states with opposite chirality which leads to an additional entropy contribution in the quasi-long-range ordered phase. We give an improved prescription for the numerical evaluation of the helicity modulus for a finite twist, and we discuss the spontaneous breaking of the chiral symmetry for the antiperiodic boundary conditions. We also discuss applications to discrete spin systems and some experimental scenarios where boundary conditions with finite twist are necessary.

The potential applications of boundary functionals of random processes, such as the extreme values of these processes, the moment of first reaching a fixed level, the value of the process at the moment of reaching the level, the moment of reaching extreme values, the time the process stays above a fixed level, and other functionals, are considered for the description of physical, chemical, and biological problems. Definitions of these functionals are provided, and characteristic functions are presented for the model with an exponential distribution of incoming demands. A generalization of these limitations is also considered. The potential uses of boundary functionals are demonstrated through examples such as a unicyclic network with affinity $A$, an asymmetric random walk, nonlinear diffusion, two-level model, Brownian motion, and multiple diffusing particles with reversible target-binding kinetics.

We elucidate the generic bifurcation behavior of local and global order in the nonreciprocal Ising model evolving under Glauber dynamics. We show that a critical magnitude of nearest-neighbor correlations within the respective lattices controls the emergence of coherent oscillations of global order as a result of frustration. Local order is maintained during these oscillations, implying nontrivial spatiotemporal correlations. Long-lived states emerge in the strong-interaction regime. The residence time in either of these states eventually diverges, giving rise to ordered nonequilibrium trapped states and a loss of ergodic behavior via a saddle-node-infinite-period bifurcation. Our work provides a comprehensive microscopic understanding of the nonreciprocal Ising model beyond the mean-field approximation.

The time-evolution operator corresponding to the fractional-time Schrödinger equation is nonunitary because it fails to preserve the norm of the vector state in the course of its evolution. However, in the context of the time-dependent non-Hermitian quantum formalism applied to the time-fractional dynamics, it has been demonstrated that a unitary evolution can be achieved for a traceless two-level Hamiltonian. This is accomplished by considering a dynamical Hilbert space embedding a time-dependent metric operator concerning which the system unitarily evolves in time. This allows for a suitable description of a quantum system consistent with the standard quantum mechanical principles. In this work, we investigate the Jaynes-Cummings model in the fractional-time scenario taking into account the fractional-order parameter $\alpha$ and its effect in unitary quantum dynamics. We analyze the well-known dynamical properties, such as the atomic population inversion and the atom-field entanglement, when the atom starts in its excited state and the field in a coherent state.

The three-body Lennard-Jones system on the plane has a transition state, which is the straight conformation located at a saddle point of the potential-energy landscape. We show that the transition state can be dynamically stabilized by excited vibration of particle distances. The stabilization mechanism is explained theoretically and is verified by performing molecular dynamics simulations. We also examine whether the dynamical stabilization gives an impact on the reaction rate between the two isomers of equilateral triangle conformations by comparing with the transition-state theory.

Stable vortex solitons (VSs) are objects of great interest for fundamental studies and various applications, including particle trapping, microscopy, data encoding, and matter-wave gyroscopes. However, three-dimensional (3D) VSs with high topological charges, supported by self-attractive nonlinearities, are unstable against fragmentation, which eventually leads to internal blowup (supercritical collapse) of the fragments. Here, we propose a scheme for realizing stable 3D VSs with topological charges up to 5 and 6 in the two components of a binary, Rydberg-dressed Bose-Einstein condensate with spin-orbit coupling (SOC). We show that, if the SOC strength exceeds a critical value, the rotational symmetry of the VSs in the transverse plane gets broken, resulting in a separation of the two components. Nevertheless, the VSs with the broken symmetry remain stable. The VS stability domains are identified in the system's parameter space. Moreover, application of torque to the stable VSs sets them in the state of robust gyroscopic precession.

We study a variant of the one-dimensional swarmalator model where the units' interactions have a controllable length scale or range. We tune the model from the long-range regime, which is well studied, into the short-range regime, which is insufficiently studied, and find diverse collective states: sync dots, where the swarmalators arrange themselves into $k>1$ delta points of perfect synchrony; $q$-waves, where the swarmalators form spatiotemporal waves with winding number $q>1$; and an active state where unsteady oscillations are found. We present the phase diagram and derive most of the threshold boundaries analytically. These states may be observable in real-world swarmalator systems with low-range coupling, such as biological microswimmers or active colloids.

We introduce a language competition model that is based on the Abrams-Strogatz model and incorporates the effects of memory and learning in the dynamics of language shift. On a coarse-grained timescale, the effects of memory and learning can be expressed as thresholds on the speakers' fractions of the competing languages. In its simplest form, the resulting model is exactly solvable. Besides the consensus on one of the two languages, the model describes additional equilibrium states that are not present in the Abrams-Strogatz model: a stable dynamical coexistence of the two languages and a frozen state coinciding with the initial state. We show numerically that these results are preserved for threshold functions of a more general shape. The comparison of the model predictions with historical datasets demonstrates that while the Abrams-Strogatz model fails to describe some relevant language competition situations, the proposed model provides a good fitting.

Dynamic environments shape diverse dynamics in evolutionary game systems. We introduce spatial heterogeneity of resources into the prisoner's dilemma game model to explore coevolutionary game dynamics with environmental feedback. The availability of resources significantly affects the survival competitiveness of surrounding individuals. Feedback between individuals' strategies and the resources they can use leads to the oscillating dynamic known as the “oscillatory tragedy of the commons.” Our findings indicate that when the influence of individuals' strategies on the update rate of resources is significantly high in systems characterized by environmental heterogeneity, they can attain an equilibrium state that avoids the oscillatory tragedy. In contrast to the numerical results obtained in well-mixed structures, self-organized clustered patterns emerge in simulations utilizing square lattices, further enhancing the stability of the system. We discuss critical phenomena in detail, demonstrating that the aforementioned transition is robust across various system parameters, including the strength of cooperators in restoring the environment, initial distributions of cooperators, system size and structures, and noise.

Understanding how cooperation emerges in public goods games is crucial for addressing societal challenges. While optional participation can establish cooperation without identifying cooperators, it relies on specific assumptions—that individuals abstain and receive a non-negative payoff, or that nonparticipants cause damage to public goods—which limits our understanding of its broader role. We generalize this mechanism by considering nonparticipants' payoffs and their potential direct influence on public goods, allowing us to examine how various strategic motives for nonparticipation affect cooperation. Using replicator dynamics, we find that cooperation thrives only when nonparticipants are motivated by individualistic or prosocial values, with individualistic motivations yielding optimal cooperation. These findings are robust to mutation, which slightly enlarges the region where cooperation can be maintained through cyclic dominance among strategies. Our results suggest that while optional participation can benefit cooperation, its effectiveness is limited and highlights the limitations of bottom-up schemes in supporting public goods.

Chemical gradients provide the primordial energy for biological functions by driving the mechanical movement of microscopic engines. Their thermodynamic properties remain elusive, especially concerning the dynamic change in energy demand in biological systems. In this article, we derive a constraint relation between the output power and the conversion efficiency for a chemically fueled steady-state rotary motor analogous to the ${\mathrm{F}}_0$ motor of ATPase. We find that the efficiency at maximum power is half of the maximum quasistatic efficiency. These findings shall aid in the understanding of natural chemical engines and inspire the manual design and control of chemically fueled microscale engines.

The study of interactions between biomimetic membranes and micron-sized particles is crucial for understanding various biological processes. Here, we control microparticle spontaneous engulfment by giant lipid vesicles by tuning particle surface charge, exploring regimes from negligible to strong adhesion. We focus our attention on dissipative phenomena at the micron- and nanoscales, occurring when a particle is wrapped by a lipid vesicle bilayer or when the particle diffuses at the lipid-monolayer interface of a droplet. For particles wrapped by membrane bilayers, we highlight the influence of the particle penetration depth and the impact of substructures on particle friction. Our work is complemented by hydrodynamic simulations that take into account the effects of the shape of the membrane wrapping the particle and the water gap separating the lipid bilayer membrane from the particle on translational particle drag. We show, however, that a purely hydrodynamic model is not suitable to describe the friction of a particle diffusing at the interface of an aqueous microdroplet in oil, stabilized by a single lipid layer. In hydrodynamic models, dissipation is solely described by the surface shear viscosity of the interface and the bulk fluid viscosity, but in this partial wetting configuration, an additional source of dissipation is required to account for fluctuations at the contact line. Hence, through experimental and numerical studies, we demonstrate that the dissipation contributions for the two geometries are fundamentally different.

We derive a version of the so-called “best Fokker-Planck approximation” (BFPA) to describe the spatial properties of interacting active Ornstein-Uhlenbeck particles in arbitrary spatial dimensions. In doing so, we also take into account the odd-diffusive contribution of the Lorentz force acting on a charged particle in a spatially dependent magnetic field, sticking to the overdamped limit. While the BFPA itself does not turn out to be widely useful, our general approach allows us to deduce an appropriate generalization of the Fox approximation, which we use to characterize the stationary behavior of a single active particle in an external potential by deriving analytic expressions for configurational probability distributions (or effective potentials). In agreement with computer simulations, our theory predicts that the Lorentz force reduces the effective attraction and thus the probability to find an active particle in the vicinity of a repulsive wall. Even for an inhomogeneous magnetic field, our theoretical findings provide useful qualitative insights, specifically regarding the location of accumulation regions.

The pressure prediction of nanoconfined gases is crucial for various industrial applications, yet existing equations of state (EOS) often compromise between accuracy and usability. This study derives a theoretical EOS in analytical form, combining precision and practicality through statistical physics. It begins by examining the pressure drop mechanism due to adsorbed molecules and refines molecular pairing and potential energy interactions. The resulting EOS accurately predicts the pressure of gases in nanoconfined spaces, especially with an average prediction error of about $5\%$ for strongly adsorbing gases like nitrogen, oxygen, and argon. Additionally, it aligns with the Van der Waals EOS under macroscopic conditions, offering a unified framework across scales. This model enhances both engineering applications and the understanding of nanoconfined EOS.

Polymer vesicles consist of a thin bilayer membrane, the shape of which can be precisely controlled by adjusting the composition of block copolymers. These self-assembled structures provide a powerful method for mimicking biological cell membranes. A key characteristic of these membranes is their elastic moduli, which play a crucial role in determining their mechanical properties. Traditionally, these moduli are derived by fitting the free energy of different membrane shapes to Helfrich's model. In this paper, we present a curvature expansion method that yields an analytical expression for the moduli within the framework of self-consistent field theory. A significant advantage of our approach is its ability to easily extract higher-order moduli, a task that presents substantial challenges for existing methods.

We numerically investigate the stress distribution in pastes after horizontal rotation by using an elastoplastic model. Residual stress remains as a memory of rotation. The stress in the circumferential direction increases after the rotation, whereas that in the radial direction decreases. The residual stress is analytically related to the plastic deformation induced by the rotation. Based on the time evolution of plastic deformation, we theoretically describe the mechanism of the changes in the stress distribution.

Integrable partial differential equation (PDE) systems are of great interest in natural science, but are exceedingly rare and difficult to discover. To this end, we introduce OptPDE, a numerical approach that Optimizes PDEs' coefficients via automatic differentiation to maximize their number of conserved quantities, $n_{\mathrm{CQ}}$, and thus discover candidates for integrable systems. The algorithm discovers four families of PDEs with at least one nontrivial conserved quantity. We analytically study one new candidate $u_t=u_x^3$, revealing its interesting properties. It is worth mentioning that our approach aims to discover systems with as many conserved quantities as possible, which does not guarantee integrability. Despite this limitation, our humble effort suggests that numerical algorithms that leverage automatic differentiation are potentially useful for discovering candidates for integrable systems.

Estimating free energy is a fundamental problem in statistical mechanics. Recently, machine-learning-based methods, particularly the variational autoregressive networks (VANs) have been proposed to minimize variational free energy and to approximate the Boltzmann distribution. VAN enjoys notable advantages, including the exact computation of the normalized joint distribution and fast sampling, which are critical features often missing in Markov chain Monte Carlo algorithms. However, VAN also faces significant computational challenges. These include difficulties in the optimization of variational free energy in a complicated parameter space and slow convergence of learning. In this work, we introduce an optimization technique based on natural gradients to the VAN framework, namely ng-VAN, to enhance the learning efficiency and accuracy of the conventional VAN. The method has computational complexity cubic in the batch size rather than in the number of model parameters, hence it can be efficiently implemented for a large VAN model. We carried out extensive numerical experiments on the Sherrington-Kirkpatrick model, spin glasses on random graphs, and the two-dimensional Ising model. Our results indicate that compared with the conventional VAN, ng-VAN significantly improves the accuracy in estimating free energy and converges much faster with shorter learning time. This allows extending the VAN framework's applicability to challenging statistical mechanics problems that were previously not accessible.