The weight space of an artificial neural network can be systematically explored using tools from statistical mechanics. We employ a combination of a hybrid Monte Carlo algorithm which performs long exploration steps, a ratchet-based algorithm to investigate connectivity paths, and coupled replica models simulations to study subdominant flat regions. Our analysis focuses on one-hidden-layer networks and spans a range of energy levels and constrained density regimes. Near the interpolation threshold, the low-energy manifold shows a spiky topology. In the overparameterized regime, however, the low-energy manifold becomes entirely flat, forming an extended complex structure that is easy to sample. These numerical results are supported by an analytical study of the training error landscape, and we show numerically that the qualitative features of the loss landscape are robust across different data structures. Our study aims to provide new methodological insights for developing scalable methods for large networks.

In this work, we extensively explore the thermodynamic properties of Brownian particles moving in an underdamped medium. Extending our previous analysis [Phys. Rev. E 101, 012131 (2020)], we examine systems subjected to different thermal arrangements, such as quadratically and linearly decreasing temperature profiles, as well as piecewise constant-temperature distributions. By conducting rigorous derivations, we derived several thermodynamic relations that provide insights into nonequilibrium thermodynamics and the underlying mechanisms governing entropy production and extraction. Furthermore, we address a fundamental question that includes whether the vanishing entropy production or extraction rate necessarily implies a thermodynamic equilibrium. Our analytical findings reveal that for a Brownian particle in an underdamped medium (free from external forces or ratchet potentials), both the entropy production and extraction rates decrease to zero, even in the presence of a spatially varying temperature gradient. However, the total entropy production ($E_P>0$) and total entropy extraction ($H_d>0$) remain finite, indicating that the system retains intrinsic irreversibility driven by heat exchange via kinetic energy transfer. This also implies that the zero-entropy production rate does not signify an equilibrium. Our results further indicate that in the absence of external loads and potentials, most thermodynamic rates asymptotically decay to zero. However, the system's irreversibility persists because of the continuous heat flow from regions of higher to lower temperatures. We also consider the motion of a Brownian particle that moves in a ratchet potential coupled with a spatially varying temperature. Such particles move unidirectionally, even in the absence of force. By imposing asymmetry in the thermal arrangements, one can manipulate the direction of the particles, indicating that this finding has significant applications in microscale and nanoscale transport systems. We show that, because the velocity of a particle depends on its mass, barrier height, load, and noise intensity, the particles can be sorted along the reaction coordinate depending on their physical properties.

Despite the fact that a complete theoretical description of critical phenomena in connection with phase transitions has been well established through the renormalization-group theory, the microscopic nature of the phase transitions remains to be understood in a satisfactory way. For example, how does the interaction between particle/spins drive the system across distinct phases as parameters vary, and how do these particle/spins respond to parameter variations during this transition process? In this work, we investigate these questions through the paradigmatic example of quantum phase transitions (QPT) in the one-dimensional transverse Ising model (TIM). We first introduce $2L$ collective structures, referred to as patterns, for the TIM with $L$ ferromagnetically interacting spins, and then analyze the contributions of these patterns to the system's states, e.g., the ground state, the first excited state, and so on, from which the analog of the QPT process between the disordered phase in the weakly coupling regime and the ferromagnetic phase in the strongly coupling regime is clearly identified at the interaction strength $J_c=1$. We systematically explore this process for small lattice sizes of $L=6,8,10,12$, whose ground-state energies are identical to those obtained by directly numerical exact diagonalization. Increasing the system size up to $L=128$, the actual QPT point located at $J_c=1$ in the thermodynamical limit is gradually approached. Our results show that the pattern picture is not only able to provide a microscopic process of phase transitions, but also of practical interest in analyzing analogues of QPT in diverse quantum simulation platforms.

After the introduction of the sandpile model a number of different variants have been studied. In most of these models sand particles are indistinguishable. Here, we have painted the sand particles using a few distinct colors, and restrict them to move in linear trajectories only along their assigned lattice axes, one axis reserved for one color. Different colored particles interact among themselves through the toppling of unstable sand columns. Consequently, the avalanches or in general the self-organization processes in the sandpile have no overall preferred direction, though the individual particles execute directed motion. For such non-Abelian colored sandpiles the steady states are found to be different, as well as the avalanche size distributions. This sandpile so defined has a nontrivial spatial structure and belongs to a different universality class of sandpile models. The Dynamics of a granular heap with grains of different colors and properties may be described using this sandpile.

Slowly modulated nonlinear waves are accurately described in the framework of the conservative nonlinear Schrödinger equation (NLS). However, in many physical systems, the wave evolution is affected by energy gains and losses, therefore requiring introduction of forcing or damping in the NLS framework. Here, we analyze the idealized case in which the forcing term varies linearly with frequency such as the growth rate of positive components matches the decay rate of the negative ones. We reveal that the system is linearly unstable and prone to overall energy growth. The evolution of classical breather solutions is showcased in this framework. The most notable effects are the distortion of the breathers, which speed-up, and in the case of the Kuznetsov-Ma breather results in faster recurrence. Concurrently, spectral asymmetry between low and high frequency components develops.

We obtain the equations of fluctuating hydrodynamics for many-particle systems whose microscopic units have both translational and rotational motion. The orientational dynamics of each element are studied in terms of Langevin equations for the rotational motion of a corresponding fixed-length director $\mathbf{u}$. We consider the microscopic dynamics for two separate choices of basic variables: Brownian dynamics for position, Fokker-Planck dynamics for position, and momentum. In each case of the microscopic dynamics, the time evolution of a corresponding set of collective densities $\left\{\widehat{\psi }\right\}$ has been obtained as an exact representation. For the Brownian dynamics, noise in the Langevin equation for the director $\mathbf{u}$ is multiplicative. The corresponding equation of motion for the collective number-density $\widehat{\rho }$ has two different forms, respectively, for the $\mathrm{I}\widehat{\mathrm{t}}\mathrm{o}$ and Stratonvich interpretation of the multiplicative noise in the $\mathbf{u}$ equation. Without the $\mathbf{u}$ variable, both forms reduce to the standard Dean-Kawasaki form. We average the microscopic equations for the collective densities $\left\{\widehat{\psi }\right\}$ (which are, at this stage, a collection of Dirac δ functions) over the phase space variables and obtain a corresponding set of stochastic partial differential equations for the coarse-grained densities $\left\{\psi \right\}$ with smooth spatial and temporal dependence. For averaging, we use a general local-equilibrium distribution involving an extended set of dynamical variables for the rotational motion. The coarse-grained equations of motion for the collective densities $\left\{\psi \right\}$ constitute the fluctuating nonlinear hydrodynamics (FNH) for the fluid with both rotational and translational dynamics. From the stationary solution of the (deterministic) equation for the probability distribution $\mathcal{P}\left[\psi \right]$, we obtain a free-energy functional $\mathcal{F}\left[\psi \right]$. The $\mathcal{F}\left[\psi \right]\mathrm{s}$ for the different FNH descriptions with their corresponding set of $\left\{\psi \right\}$ are also worked out.

Self-propelled particles, like motile cells and artificial colloids, can spontaneously form macroscopic clusters. This phenomenon is called motility-induced phase separation (MIPS) and occurs even without attractive forces, provided that the self-propulsion direction fluctuates slowly. In addition to rotational noise, these particles may experience translational noise, not coupled to rotational noise, due to environmental fluctuations. We study the role of translational noise in the clustering of active Brownian hard disks. To tease apart the contribution of translational noise, we model excluded-volume interactions through a Monte-Carlo-like overlap rejection approach. We find that increasing translational diffusivity has a nonmonotonic effect on clustering. At low values, it makes clusters more compact and rounded (less filamentous), eventually promoting genuine MIPS. For sufficiently higher translational diffusivity, clusters evaporate. We develop a theory for the cluster mass distribution, and employ a hydrodynamic approach with parameters taken from the simulation, which explains the clustering phase diagram.

A parallel plate capacitor containing an electrolytic solution is the simplest model of a supercapacitor or electric double-layer capacitor. Using both analytical and numerical techniques, we solve the Poisson-Nernst-Planck equations for such a system, describing the mean-field charging dynamics of the capacitor, when a constant potential difference is abruptly applied to its plates. Working at constant total number of ions, we focus on the physical processes involved in the relaxation and, whenever possible, give its functional shape and exact time constants. We first review and study the case of a symmetric binary electrolyte, where we assume the two ionic species to have the same charges and diffusivities. We then relax these assumptions and present results for a generic strong (i.e., fully dissociated) binary electrolyte. At low electrolyte concentration, the relaxation is simple to understand, as the dynamics of positive and negative ions appear decoupled. At higher electrolyte concentration, we distinguish several regimes. In the linear regime (low voltages), relaxation is multiexponential, it starts by the buildup of the equilibrium charge profile and continues with neutral mass diffusion, and the relevant timescales feature both the average and the Nernst-Hartley diffusion coefficients. In the purely nonlinear regime (intermediate voltages), the initial relaxation is slowed down exponentially due to increased capacitance, while bulk effects become more and more evident. In the fully nonlinear regime (high voltages), the dynamics of charge and mass are completely entangled and, asymptotically, the relaxation is linear in time. We finally discuss nonideal behavior in real capacitors and provide conditions for which mean-field is expected to hold.

Some dynamical properties of nonlinear coupled systems can be described by the two-harmonic standard map, a two-dimensional area-preserving system with two parameters, where two distinct arbitrary resonant modes compete. Usually, the initial phase of the resonant modes is considered to be null. In this paper, we consider a non-null phase shift between the two competing isochronous modes that form the system. We observe that a nonzero phase shift alters the phase space, changing the stability and positions of the fixed points. Furthermore, the phase shift can change the dominant mode and create intermediate modes between the main ones. Lastly, we analyze the effect of the phase shift on the onset of secondary shearless curves in the phase space. Thus, different phase shifts result in various scenarios in which secondary shearless curves emerge in the phase space.

Various macroscopic oscillations, such as the heartbeat and the flashing of fireflies, are created by synchronizing oscillatory units (oscillators). To elucidate the mechanism of synchronization, several coupled oscillator models have been devised and extensively analyzed. Although parameter estimation of these models has also been actively investigated, most of the proposed methods are based on the data from individual oscillators, not from macroscopic quantities. In the present study, we propose a Bayesian framework to estimate the model parameters of coupled oscillator models, using the time series data of the Kuramoto order parameter as the only given data. We adopt the exchange Monte Carlo method for the efficient estimation of the posterior distribution and marginal likelihood. Numerical experiments are performed to confirm the validity of our method and examine the dependence of the estimation error on the observational noise and system size.

Naturally occurring materials are often disordered, with the dependence of their bulk properties on structure being challenging to predict due to the lack of underlying crystalline axes. In this paper, we develop a digital pipeline from algorithmically created configurations with tunable disorder to 3D printed materials, as a tool to aid in the study of such materials, using electrical resistance as a test case. The designed material begins with a random point cloud that is iteratively evolved using Lloyd's algorithm to approach uniformity, with the points being connected via a Delaunay triangulation to form a disordered network metamaterial. Utilizing laser powder bed fusion additive manufacturing with stainless steel 17-4 PH and titanium alloy Ti-6Al-4V, we are able to experimentally measure the bulk electrical resistivity as a function of the degree of disorder in the network. The effective resistance of the structure calculated from the combinatorial weighted graph Laplacian is in good agreement with experimental data. However, the effective resistance is sensitive to anisotropy and global network topology, preventing a single network statistic or disorder characterization from predicting global resistivity.

Locally resonant metamaterials are among the most studied types of elastic and acoustic metamaterials, with significant research focused on wave propagation in a periodic array of “meta-atoms.” Here we investigate the collision dynamics of two identical pendulum-suspended mass-in-mass resonators, essentially a two-sphere Newton's cradle, emphasizing the readily realizable scenario where the internal resonator frequency is much greater than the pendulum frequency. We first show that the dynamics of a collision can be described using effective parameters, similar to how previous metamaterials research has characterized wave propagation through effective material parameters. Nonconventional collision dynamics—observed in two colliding mass-in-mass systems where one is initially at rest—include behaviors such as the moving sphere rebounding as if from a fixed wall while the other remains essentially stationary, the spheres coupling and moving forward in near-unison, and the spheres recoiling in opposite directions. These responses can be achieved by tuning the effective parameters. We demonstrate that these parameters can take on values that differ significantly from those in a conventional Newton's cradle. Additionally, we investigate multiple collisions of the two spheres, revealing complex dynamics. This work paves the way for the development and study of new “collision-based metamaterial” structures.

Employing Monte Carlo sampling and a self-consistent multiscale numerical approach, we obtain a parametric simple formula for the effective electrical charge conductivity of composite systems containing randomly distributed intermetallic needle-shape precipitates. This formula, derived from extensive fits to computational results, accounts for various experimentally relevant parameters, including the externally applied voltage difference $V$ across the sample, the band gap $\mu$ and density $\varphi$ of the precipitates, the density of intermetallic materials, and the orientation of the needle-shape precipitates/order parameter $s$. Employing this formula, we determine the charge conductance threshold activation functionalities for each parameter versus the other parameters. This parametric formula can serve as a tool for analyzing and designing composite systems.

We investigate the dynamics of information scrambling in bosonic systems undergoing Gaussian unitary evolution associated with quadratic Hamiltonians. For initial Gaussian states, we observe the disappearance of the memory effect in the entanglement dynamics of disjoint blocks under Gaussian random local dynamics. In addition, we show that randomness in the Hamiltonian causes the tripartite mutual information to saturate at relatively large negative values. Therefore, despite being integrable, these systems exhibit information-scrambling diagnostics that mirror those observed in chaotic systems. We note, however, that random quadratic Hamiltonians can have a component exhibiting Wigner-Dyson energy-level statistics; for non-Gaussian states within the corresponding subspace, these systems can display chaotic behavior. Our results provide insight into the Gaussian dynamics of continuous-variable systems, which are useful and available resources for quantum information processing.

The coexistence of a multiplicity of vibrational states in a fiber laser is demonstrated for a specific set of laser parameters. These vibrational states consist of two dissipative solitons, whose distance and/or relative phase change periodically. More complex vibrations, involving either commensurate or incommensurate frequencies, are also reported. The multiplicity of available vibrations anticipates a much more intricate scenario when more than two solitons are involved. The case with three solitons is also examined.