This work establishes links between the Ising model and elliptic curves via Mahler measures. First, we reformulate the partition function of the Ising model on the square, triangular, and honeycomb lattices in terms of the Mahler measure of a Laurent polynomial whose variety's projective closure defines an elliptic curve. Next, we obtain hypergeometric formulas for the partition functions on the triangular and honeycomb lattices and review the known series for the square lattice. Finally, at specific temperatures we express the free energy in terms of a Hasse-Weil $L$-function of an elliptic curve. At the critical point of the phase transition on all three lattices, we obtain the free energy more simply in terms of a Dirichlet $L$-function. These findings suggest that the connection between statistical mechanics and analytic number theory may run deeper than previously believed.

The problem of counting polymer coverings on rectangular lattices is investigated. In this model, a linear rigid polymer covers $k$ adjacent lattice sites such that no two polymers occupy a common site. Those unoccupied lattice sites are considered as monomers. We prove that for a given number of polymers ($k\text{-mers}$), the number of arrangements for the polymers on two-dimensional rectangular lattices satisfies simple recurrence relations. These recurrence relations are quite general and apply for arbitrary polymer length ($k$) and the width of the lattices ($n$). The well-studied monomer-dimer problem is a special case of the monomer-polymer model when $k=2$. It is known the enumeration of monomer-dimer configurations in planar lattices is #P complete. The recurrence relations shown here have the potential for hints for the solution of long-standing problems in this class of computational complexity.

The physical significance of the stochastic processes associated to the generalized Gibbs ensembles is scrutinized here with special attention to the thermodynamic fluctuations of small systems. Differently from the so-called stochastic thermodynamics, which starts from stochastic versions of the first and second law of thermodynamics and associates thermodynamic quantities to microscopic variables, here we consider stochastic variability directly in the macroscopic variables. By recognizing the potential structure of the Gibbs ensembles, when expressed as a function of the potential entropy generation, we obtain exact nonlinear thermodynamic Langevin equations (TLEs) for macroscopic variables, with drift expressed in terms of entropic forces. The analysis of the canonical ensemble for an ideal monoatomic gas and the related TLEs show that introducing currents leads to nonequilibrium heat transfer conditions with interesting bounds on entropy production but with no obvious thermodynamic limit. For a colloidal particle under constant force, the TLEs for macroscopic variables are different from those for the microscopic position, typically used in stochastic thermodynamics; while TLEs are consistent with the fundamental equation obtained from the Hamiltonian, stochastic thermodynamics requires isothermal conditions and entropy proportional to position.

The Langevin equation is ubiquitously employed to numerically simulate plasmas, colloids, and electrolytes. However, the usual assumption of white noise becomes untenable when the system is subject to an external ac electric field. This is because the charged particles in the system, which provide the thermal bath for the particle transport, become themselves responsive to the ac field and the thermal noise is field dependent and non-Markovian. We theoretically study the particle diffusivity in a Langevin transport model for a tagged charged particle immersed in a dense system of charged particles (plus also, possibly, other neutral particles) that act as the thermal bath, under an external ac electric field. This is done by properly accounting for the effects of the ac field on the thermal bath statistics. We analytically derive the time-dependent generalized diffusivity $D\left(t\right)$ for different initial conditions. The generalized diffusivity exhibits damped oscillatory-like behavior with initial very large peaks, where the generalized diffusion coefficient is enhanced by orders of magnitude with respect to the infinite-time steady-state value. The latter coincides with the Stokes-Einstein diffusivity in the absence of an external field. For initial conditions where the external field is already on at $t=0$ and the system is thermalized under dc conditions for $t\le 0$, the short-time behavior is hyperballistic, $\text{MSD}\sim t^4$ (where MSD is the mean-squared displacement), leading to giant enhancement of the particle transport. Finally, the theory elucidates the role of medium polarization on the local Lorentz field, and allows for estimates of the effective electric charge due to polarization by the surrounding charges.

We analytically study the dynamic behavior of a linear mechanical energy harvester nonlinearly coupled to a linear oscillating mode, driven by stochastic Gaussian forces. Using renormalization theory and Feynman diagrams, we determine the renormalization of three key system parameters: the natural frequencies of the oscillating components and the parameter associated with the driving force amplitude. Our results show that random forces can induce the well-known internal resonance state, where the renormalized quantities exhibit a nontrivial dependence on the working frequency. In this state, coherent synchronization between the two oscillatory systems leads to qualitative changes in their respective spectral densities, with direct implications for the harvested electrical energy. Finally, we numerically validate our analytical findings.

We consider agents in a social network competing to be selected as partners in collaborative, mutually beneficial activities. We study this through a model in which an agent $i$ can initiate a limited number $k_i>0$ of games and selects partners from its one-hop neighborhood. Each agent can accept as many games offered by its neighbors. Each game signifies a productive joint activity, and the players attempt to maximize their individual utilities. Unsurprisingly, more trustworthy agents, as measured by the game-theoretic concept of limited-trust, are more desirable as partners. Agents learn about their neighbors' trustworthiness through interactions and their behaviors evolve in response. Empirical trials conducted on realistic social networks show that when given the option, many agents become highly trustworthy; most or all become highly trustworthy when knowledge of their neighbors' trustworthiness is based on past interactions rather than known a priori. This trustworthiness is not the result of altruism; instead, agents are intrinsically motivated to become trustworthy partners by competition. Two insights are presented: First, trustworthy behavior drives an increase in the utility of all agents, where maintaining a relatively minor level of trustworthiness may easily improve net utility by as much as 14.5%. If only one agent exhibits a small degree of trustworthiness among self-centered ones, then it can increase its personal utility by up to 25% in certain cases. Second, and counterintuitively, when partnership opportunities are abundant, agents become less trustworthy.

In this paper, using hydrodynamic entropy, we quantify multiscale disorder in Euler and hydrodynamic turbulence. These examples illustrate that the hydrodynamic entropy is not extensive because it is not proportional to the system size. Consequently, we cannot add hydrodynamic and thermodynamic entropies, which measure disorder at macroscopic and microscopic scales, respectively. In this paper, we also discuss the hydrodynamic entropy for the time-dependent Ginzburg-Landau equation and Ising spins.

We study one-point statistics of helical turbulent pulsations in the background of a three-dimensional large-scale vortex in a rotating fluid. Assuming that the helical flow is created by a statistically axially symmetric random force with broken mirror symmetry, we analytically calculate the velocity-vorticity mean including its magnitude and the anisotropy. For electrically conducting liquid, we examine the $\alpha$-effect in the system. The found elements of the $\alpha$-matrix strongly depend on the relation between Rossby $\mathrm{Ro}$ and magnetic Prandtl ${\mathrm{Pr}}_{\mathrm{m}}$ numbers in the considered region $\mathrm{Ro}\ll 1, {\mathrm{Pr}}_{\mathrm{m}}\ll 1$. We establish a criterion for the numbers when the $\alpha$-effect leads to instability of large-scale magnetic field.

Stochastic analytic continuation (SAC) of quantum Monte Carlo (QMC) imaginary-time correlation function data is a valuable tool in connecting many-body models to experimentally measurable dynamic response functions. Recent developments of the SAC method have allowed for spectral functions with sharp features, e.g., narrow peaks and divergent edges, to be resolved with unprecedented fidelity. Often it is not known what exact sharp features, if any, are present a priori, and, due to the ill-posed nature of the analytic continuation problem, multiple spectral representations may be acceptable. In this work we borrow from the machine learning and statistics literature and implement a cross validation technique to provide an unbiased method to identify the most likely spectrum among a set obtained with different spectral parametrizations and imposed constraints. We demonstrate the power of this method with examples using imaginary-time data generated by QMC simulations and synthetic data generated from artificial spectra. Our procedure, which can be considered a form of model selection, can be applied to a variety of numerical analytic continuation methods, beyond just SAC.

One of the key predictions of Parisi's broken replica symmetry theory of spin glasses is the existence of a phase transition in an applied field to a state with broken replica symmetry. This transition takes place at the de Almeida–Thouless (AT) line in the $h\text{−}T$ plane. We have studied this line in the power-law diluted Heisenberg spin glass in which the probability that two spins separated by a distance $r$ interact with each other falls as $1/r^{2\sigma }$. In the presence of a random vector field of variance $h_r^2$ the phase transition is in the universality class of the Ising spin glass in a field. Tuning $\sigma$ is equivalent to changing the dimension $d$ of the short-range system, with the relation being $d=2/(2\sigma -1)$ for $\sigma <2/3$. We have found by numerical simulations that $h_{\text{AT}}^2\sim (2/3-\sigma )$ implying that the AT line does not exist below six dimensions and that the Parisi scheme is not appropriate for spin glasses in three dimensions.

The out-of-equilibrium aggregation of dipolar particles, such as magnetized beads, leads to the formation of large structures composed of chains, loops, and eventually ribbons. In the present study, we focus on the evolution of these different substructures in a two-dimensional system confined within progressively shrinking environments. Using numerical simulations, we identify structural events as a function of the packing fraction. At low density, a percolation threshold ${\varphi }_p\approx 0.15$ is evidenced, where chainlike structures merge into a single large aggregate with significant voids. This gel-like structure then densifies as $\varphi$ increases. At large $\varphi$ values, crystallites of both square and hexagonal order phase appear, but they are far from extending over the whole system.

Granular fluids, as defined by a collection of moving solid particles, is a paradigm of a dissipative system out of equilibrium. Inelastic collisions between particles is the source of dissipation, and is the origin of a transition from a gas to a liquidlike state. This transition can be triggered by an increase of the solid fraction. Moreover, in compartmentalized systems, this condensation is driving the entire granular fluid into a Maxwell demon phenomenon, localizing most of the grains into a specific compartment. Classical approaches fail to capture these phenomena, thus motivating many experimental and numerical works. Herein, we demonstrate that the Onsager variational principle is able to predict accurately the coexistence of gas-liquid states in granular systems, opening ways to model other phenomena observed in such dissipative systems like segregation or the jamming transition.

Soot aerosols emitted during combustion can affect climate by scattering and absorbing the sunlight. Individual soot particles are fractal aggregates composed of elemental carbon. In the atmosphere, these aggregates acquire coatings by condensation and coagulation, resulting in significant compaction of the aggregates that changes the direct climate forcing of soot. Currently, no models exist to rigorously describe the process of soot restructuring, reducing prediction accuracy of atmospheric aerosol models. We develop a discrete element method contact model to simulate restructuring of fractal soot aggregates, represented as assemblies of spheres joined by cohesion and by sintered necks. The model is parametrized based on atomic force spectroscopy data and is used to simulate soot restructuring, showing that the fraction of necks in aggregates determines the restructuring pathway. Aggregates with fewer necks undergo local compaction, while aggregates with nearly full necking prefer global compaction. Additionally, full compaction occurs within tens of nanoseconds, orders of magnitude faster than the timescale of soot aging through condensation. An important implication is that the rate of condensation determines how many necks are fractured simultaneously, affecting the restructuring pathway, e.g., producing more compact soot at lower condensation rates than at faster condensation rates for comparable amounts of condensate formed on soot.

We examine the conditions for the emergence of self-organized criticality in the Olami-Feder-Christensen model by introducing a single defect under periodic boundary conditions. Our findings reveal that strong localized energy dissipation is crucial for self-organized criticality emergence, while weak localized or global energy dissipation leads to its disappearance in this model. Furthermore, slight dissipation perturbations to a system in a self-organized criticality reveal a novel state characterized by a limit cycle of distinct configurations. This newly discovered state offers significant insights into the fundamental mechanisms governing the emergence of self-organized criticality.

In 1998, O'Loan et al. introduced a simplified bus-route model to illustrate bus dynamics. However, fluctuations in passenger numbers make it challenging to achieve an exact solution for the model's stationary state, as these fluctuations can impact bus behavior. In this study, we present an exactly solvable model for a dual bus-route system that builds upon O'Loan et al.'s model. This dual model allows us to comprehensively analyze bus-route dynamics. Our model introduces additional parameters not previously considered by O'Loan et al. to account for neighboring effects and align with an observation in the original model, which can influence the average stationary current and velocity of buses. When the neighboring effect is weak, our model behaves similarly to O'Loan et al.'s bus-route model. However, with a strong neighboring effect, our model exhibits intriguing characteristics. Additionally, in some limiting cases, our model recovers well-known models such as the simple exclusion process, its generalization, the cooperative exclusion process, and the RNA polymerase model during the elongation stage.

The dynamics of initial long-wavelength excitations of the Fermi-Pasta-Ulam-Tsingou chain has been the subject of intense investigations since the pioneering work of Fermi and collaborators. We have recently found a regime where the spectrum of the Fourier modes decays with a power law and we have interpreted this regime as a transient turbulence associated with the Burgers equation. In this paper we present the full derivation of the latter equation from the lattice dynamics using an infinite-dimensional Hamiltonian perturbation theory. This theory allows us to relate the time evolution of the Fourier spectrum $E_k$ of the Burgers equation to that of the Fermi-Pasta-Ulam-Tsingou (FPUT) chain. As a consequence, we derive analytically both the shock time and the power law $-8/3$ of the spectrum at this time. Using the shock time as a unit, we follow numerically the time evolution of the spectrum and observe the persistence of the power $-2$ over an extensive time window. The exponent $-2$ has been widely discussed in the literature on the Burgers equation. The analysis of the Burgers equation in Fourier space also gives information on the time evolution of the energy of each single mode which, at short time, is also a power law depending on the $k\mathrm{th}$ wavenumber $E_k\sim t^{2k-2}$. This approach to the FPUT dynamics opens the way to a wider study of scaling regimes arising from more general initial conditions.

Structure changes or transitions are common in growing networks (complex networks, graphs, etc.) and must be precisely determined. The introduced quantitative measure of the structural complexity of the network based on a procedure similar to the renormalization process allows one to reveal such changes. The proposed concept of the network structural complexity accounts for the difference between the actual and averaged network structures on different scales and corresponds to the qualitative comprehension of complexity. The structural complexity can be found for the weighted networks also. The structural complexities for various types of growing networks exhibiting transitions similar to phase transitions were found—the deterministic infinite and finite size artificial networks of different natures including percolation structures, and the time series of various types of cardiac rhythms mapped to complex networks using the parametric visibility graph algorithm. In all the cases the structural complexity of the growing network reaches a maximum near the transition point: the formation of a giant component in the graph or at the percolation threshold for two-dimensional and three-dimensional square lattices when a giant cluster having a fractal structure has emerged. Therefore, the structural complexity of the network allows us to detect and study processes similar to a second-order phase transition in complex networks. The structural complexity of a network node can serve as a kind of centrality index, auxiliary, or generalization to the local clustering coefficient. Such an index provides another new ranking manner for the network nodes. Being an easily computable measure, the network structural complexity might help to reveal different features of complex systems and processes of the real world.

Deep learning models have achieved high performance in a wide range of applications. Recently, however, there have been increasing concerns about the fragility of many of those models to adversarial approaches and out-of-distribution inputs. A way to investigate and potentially address model fragility is to develop the ability to provide interpretability to model predictions. To this end, input attribution approaches such as Grad-CAM and integrated gradients have been introduced to address model interpretability. Here, we combine adversarial and input attribution approaches in order to achieve two goals. The first is to investigate the impact of adversarial approaches on input attribution. The second is to benchmark competing input attribution approaches. In the context of the image classification task, we find that models trained with adversarial approaches yield dramatically different input attribution matrices from those obtained using standard techniques for all considered input attribution approaches. Additionally, by evaluating the signal-(typical input attribution of the foreground)-to-noise (typical input attribution of the background) ratio and correlating it to model confidence, we are able to identify the most reliable input attribution approaches and demonstrate that adversarial training does increase prediction robustness. Our approach can be easily extended to contexts other than the image classification task and enables users to increase their confidence in the reliability of deep learning models.

Taking inspiration from the crawling motion of biological cells on a substrate, we consider a physical model of self-propulsion where the spatiotemporal driving can involve both a mechanical actuation by active force couples and a chemical actuation through controlled mass turnover. When the material turnover is slow and the mechanical driving dominates, we find that the highest velocity at a given energetic cost is reached when the actuation takes the form of an active force configuration propagating as a traveling wave. As the rate of material turnover increases, and the chemical driving starts to dominate the mechanical one, such a peristalsis-type control progressively loses its efficacy, yielding to a standing-wave-type driving which involves an interplay between the mechanical and chemical actuation. Our analysis suggests a paradigm for the optimal design of crawling biomimetic robots where the conventional purely mechanical driving through distributed force actuators is complemented by a distributed chemical control of the material remodeling inside the force-transmitting machinery.

Platinum-coated Janus colloids exhibit self-propelled motion in aqueous solution via the catalytic decomposition of hydrogen peroxide. Here, we report their motion in a uniformly aligned nematic phase of lyotropic chromonic liquid crystal, disodium cromoglycate (DSCG). When active Janus colloids are placed in DSCG, we find that the anisotropy of the liquid crystal imposes a strong sense of direction to their motion; the Janus colloids tend to move parallel to the nematic director. Motion analysis over a range of timescales reveals a crossover from ballistic to anomalous diffusive behavior on timescales below the relaxation time for liquid crystal elastic distortions. Surprisingly we observe that smaller particles roll during ballistic motion, whereas larger particles do not. This result highlights the complexity of phoretically-driven particle motion, especially in an anisotropic fluid environment.