We present a method to design driving protocols that achieve fast thermal equilibration of a system of interest using techniques inspired by machine learning training algorithms. For example, consider a Brownian particle manipulated by optical tweezers. The force on the particle can be controlled and adjusted over time, resulting in a driving protocol that transitions the particle from an initial state to a final state. Once the driving protocol has been completed, the system requires additional time to relax to thermal equilibrium. Designing driving protocols that bypass the relaxation period is of interest so that, at the end of the protocol, the system is either in thermal equilibrium or very close to it. Several studies have addressed this problem through reverse engineering methods, which involve prescribing a specific evolution for the probability density function of the system and then deducing the corresponding form of the driving protocol potential. Here we propose a new method that can be applied to more complex systems where reverse engineering is not feasible. We simulate the evolution of a large ensemble of trajectories while tracking the gradients with respect to a parametrization of the driving protocol. The final probability density function is compared to the target equilibrium one. Using machine learning libraries, the gradients are computed via backpropagation and the protocol is iteratively adjusted until the optimal protocol is achieved. We demonstrate the effectiveness of our approach with several examples.

Finite-width one hidden layer networks with multiple neurons in the readout layer display nontrivial output-output correlations that vanish in the lazy-training infinite-width limit. In this manuscript we leverage recent progress in the proportional limit of Bayesian deep learning (that is the limit where the size of the training set $P$, and the width of the hidden layers $N$ are taken to infinity keeping their ratio $\alpha =P/N$ finite) to rationalize this empirical evidence. In particular, we show that output-output correlations in finite fully connected networks are taken into account by a kernel shape renormalization of the infinite-width NNGP kernel, which naturally arises in the proportional limit. We perform accurate numerical experiments both to assess the predictive power of the Bayesian framework in terms of generalization, and to quantify output-output correlations in finite-width networks. By quantitatively matching our predictions with the observed correlations, we provide additional evidence that kernel shape renormalization is instrumental to explain the phenomenology observed in finite Bayesian one-hidden-layer networks.

The phase behavior of a quasi-one-dimensional fluid of hard spheres is studied in hard cylindrical and rectangular straight channels using the transfer operator method. The radius of the channel ($R$) is chosen such that only first-neighbor interactions are present, which allows only fluidlike and zigzag solidlike structures to form. It is managed to show that the fluid-solid structural change occurs approximately at $\rho =1/\langle {\sigma }_{\parallel }\rangle$, where $\rho$ is the one-dimensional density and $\langle {\sigma }_{\left|\right|}\rangle$ is the average of the longitudinal contact distance in the case of uniform particle distribution. At this density, the pressure of positionally disordered structure diverges, but the pressure ratio ($P/P_T$), where $P$ is the pressure of the system and $P_T$ is the pressure of an effective one-dimensional hard rod fluid, exhibits a peak. At the close packing, while the positional fluctuation $\left(\langle \mathrm{\Delta }{r}^2\rangle \right)$ vanishes with $P^{\tilde{\beta }}$, where $\tilde{\beta }=-2$ for any $R$ and cross section, $P/P_T$ (we call it $\tilde{\alpha }$) goes to 2 in the rectangular and 2.5 in the cylindrical channel. The most striking difference can be observed in the transversal position correlation length (ξ). It exhibits $exp\left(\mathrm{\Delta }P\right)$ dependence in the rectangular channel with $\mathrm{\Delta }$ being the extra length necessary to create a defect. However, ξ goes with $P^{\tilde{\gamma }}$ in the cylindrical channel, where $\tilde{\gamma }=1$. Interestingly, the cylindrically confined hard spheres obey the sum rules $\tilde{\alpha }+\tilde{\beta }=1/2$ and $\tilde{\beta }+\tilde{\gamma }=-1$ for any $R$.

Advancements in materials design and manufacturing have allowed for the production of ordered and disordered metamaterials with diverse properties. Hyperuniform two-phase heterogeneous materials, which anomalously suppress density fluctuations on large length scales compared to typical disordered systems, and network materials are two classes of metamaterials that have desirable physical properties. Recent focus has been placed on the design of disordered hyperuniform network metamaterials that inherit the desirable properties of both of these metamaterial classes. In this work, we focus on determining the extent to which network structures derived from the spatial tessellations of hyperuniform point patterns inherit the hyperuniformity of the progenitor point patterns. In particular, we examine the Delaunay, Voronoi, Delaunay-centroidal, and Gabriel tessellations of nonhyperuniform and hyperuniform point patterns in two- and three-dimensional Euclidean space. We use the spectral density to characterize the density fluctuations of two-phase media created by thickening the edges of these tessellations in two dimensions and introduce a variance-based measurement to characterize the network structures directly in two and three dimensions. We find that, while none of the tessellations completely inherit the hyperuniformity of the progenitor point pattern, the degree to which the hyperuniformity is inherited is sensitive to the tessellation scheme and the short- and long-range translational disorder in the point pattern, but not to the choice of beam shape when mapping the networks into two-phase media.

We present a cooperative protocol to charge quantum spin networks up to the highest-energy configuration, in terms of the network's magnetization. The charging protocol leverages spin-spin interactions and the crossing of a phase transition's critical point to achieve superextensive charging precision. The cooperative protocol guarantees a precision advantage over any local charging protocol and leads to fluctuations (standard deviation) of the magnetization that scale as $1/N$, with $N$ the number of spins in the network, i.e., the size of the spin battery. We test our protocol on the D-Wave Advantage quantum processing unit by charging sublattices with sizes ranging from 40 to 5612 spins, achieving the highest-energy configuration with a sizable superextensive charging precision scaling and outperforming the local charging precision by four orders of magnitude.

We consider a free fermion chain with uniform nearest-neighbor hopping and let it evolve from an arbitrary initial state with a fixed macroscopic number of particles. We then prove that, at a sufficiently large and typical time, the measured coarse-grained density distribution is almost uniform with (quantum mechanical) probability extremely close to one. This establishes the emergence of irreversible behavior, i.e., a ballistic diffusion, in a system governed by quantum mechanical unitary time evolution. It is conceptually important that irreversibility from any initial state is proved here without introducing any randomness to the initial state or the Hamiltonian, while the known examples, both classical and quantum, rely on certain randomness or apply to limited classes of initial states. The essential new ingredient in the proof is the large deviation bound for every energy eigenstate, which is reminiscent of the strong ETH (energy eigenstate thermalization hypothesis).

We study the critical dynamics of the three-dimensional Heisenberg model with random cubic anisotropy in the out-of-equilibrium and equilibrium regimes. Analytical approaches based on field theory predict that the universality class of this model is that of the three-dimensional site-diluted Ising model. We have been able to estimate the dynamic critical exponent by working in the equilibrium regime and by computing the integrated autocorrelation times obtaining $z=2.50\left(5\right)$ (without taking into account scaling corrections) and $z=2.29\left(11\right)$ (by fixing the scaling corrections to that predicted by field theory). In the out-of-equilibrium regime we have focused on the study of the dynamic correlation length, allowing us to compute the dynamic critical exponent obtaining $z=2.38\left(2\right)$, which is compatible with the equilibrium ones. Finally, both estimates are also compatible with the most accurate prediction $z=2.35\left(2\right)$, from numerical simulations of the 3D site-diluted Ising model, in agreement with the predictions based on field theory.

We report on numerical experiments of thermalization between two reservoirs in which interspecies interactions are mediated via a Hamiltonian with a Gaussian dependence upon the distance between the particles. By using relatively high statistics, we are able to study deviations from the initially imprinted Maxwell-Boltzmann energy distributions as a function of the range and strength of the interaction, as well as the parameters of the two baths, such as masses and trapping frequencies, that strongly affect the spatial overlap of the two species. At large interaction strengths and for any range, deviations are attributable to the sudden conversion of latent interaction energy into kinetic energy of the particles. In addition, long-range interactions generate deviations even for interaction strengths in the weak regime, and nonuniform confinement introduces a further source of deviation from the Maxwell-Boltzmann energy distribution. This results in transient two-temperature Maxwell-Boltzmann distributions suggestive of an inverse cascading route from interparticle dynamics to thermalization. In the weakly interacting regime, a simple expression for the equilibrium temperature is also obtained and discussed. These findings may be of relevance in the dynamics of exothermic chemical reactions, in the physics of ultracold atomic mixtures, and in self-sustained burning plasmas.

The self-adaptive mechanisms inherently in numerous real-world processes are fundamental for the emergence of collective dynamics. In this study, we establish a general framework elucidating universal phase transitions in adaptive networks of coupled phase oscillators, in which the interactions within a single layer include an adaptive strategy characterized by the feedback between the coupling strength and the magnitude of the order parameter, mediated through a power-law function with different correlation exponents. We analytically demonstrate that there exists a critical correlation exponent demarcating the first-order and second-order phase transitions of the system, with synchronization transition attributes exhibiting no dependence of the adaptive fractions on either side of this critical threshold. Nevertheless, we mathematically substantiate that, within the critical scenario characterized by the correlation exponent, the manifestation of either abrupt or continuous phase transition is contingent upon the product of the adaptive weights. Crucially, we reveal that the bifurcations governing various phase transitions—manifested by a spectrum of interrelated scaling laws of the order parameters—are comprehensively appreciated in terms of the perturbation theory. Our findings contribute profound insights into the dynamical nature of phase transitions and offer enhanced methodologies for the controlling and manipulating of synchronization transitions in adaptive networks.

We present analytical results for the effect of preferential node deletion on the structure of networks that evolve via node addition and preferential attachment. To this end, we consider a preferential-attachment-preferential-deletion model, in which at each time step, with probability $P_{\mathrm{add}}$ there is a growth step where an isolated node is added to the network, followed by the addition of $m$ edges, where each edge connects a node selected uniformly at random to a node selected preferentially in proportion to its degree. Alternatively, with probability $P_{\mathrm{del}}=1-P_{\mathrm{add}}$ there is a contraction step, in which a preferentially selected node is deleted and its links are erased. The balance between the growth and contraction processes is captured by the growth/contraction rate $\eta =P_{\mathrm{add}}-P_{\mathrm{del}}$. For $0<\eta \le 1$ the overall process is of network growth, while for $-1\le \eta <0$ the overall process is of network contraction. Using the master equation and the generating function formalism, we study the time-dependent degree distribution $P_t\left(k\right)$. It is found that for each value of $m>0$ there is a critical value ${\eta }_c\left(m\right)=-(m-2)/(m+2)$ such that for ${\eta }_c\left(m\right)<\eta \le 1$ the degree distribution $P_t\left(k\right)$ converges toward a stationary distribution $P_{\mathrm{st}}\left(k\right)$. In the special case of pure growth, where $\eta =1$, the model is reduced to a preferential attachment growth model and $P_{\mathrm{st}}\left(k\right)$ exhibits a power-law tail, which is a characteristic of scale-free networks. In contrast, for ${\eta }_c\left(m\right)<\eta <1$ the distribution $P_{\mathrm{st}}\left(k\right)$ exhibits an exponential tail, which has a well-defined scale. This implies a phase transition at $\eta =1$, in contrast with the preferential-attachment-random-deletion model [Budnick et al., J. Stat. Mech. (2025) 013401], in which the power-law tail remains intact as long as $\eta >0$. These results illustrate the sensitivity of evolving networks to preferential node deletion, in contrast with their robustness to random node deletion. While for $\eta \ge max\{{\eta }_c\left(m\right),0\}$ the stationary degree distribution $P_{\mathrm{st}}\left(k\right)$ lasts indefinitely, for ${\eta }_c\left(m\right)<\eta <0$ (and $m>2$) it persists for a finite lifetime, until the network vanishes. It is also found that in the regime of $-1\le \eta \le {\eta }_c\left(m\right)$ the time-dependent degree distribution $P_t\left(k\right)$ does not converge toward a stationary form, but continues to evolve until the network is reduced to a set of isolated nodes. These results provide insight on the structure of transient social networks, such as dating networks and job-seeking platforms, in which user turnover is intrinsically high.

Spatial metapopulation models are fundamental to theoretical ecology, enabling us to study how landscape structure influences global species dynamics. Traditional models, including recent generalizations, often rely on the deterministic limit of stochastic processes, assuming large population sizes. However, stochasticity, arising from dispersal events and population fluctuations, profoundly shapes ecological dynamics. In this work, we extend the classical metapopulation framework to account for finite populations, examining the impact of stochasticity on species persistence and dynamics. Specifically, we analyze how the limited capacity of local habitats influences survival, deriving analytical expressions for the finite-size scaling of the survival probability near the critical transition between survival and extinction. Crucially, we demonstrate that the deterministic metapopulation capacity plays a fundamental role in the statistics of survival probability and extinction time moments. These results provide a robust foundation for integrating demographic stochasticity into classical metapopulation models and their extensions.

Laguerre-Gaussian (LG) laser beams have many interesting applications due to their unique field structure. However, the generation of intense LG beams remains challenging because of the limited damage threshold of intensity for optical elements used for the conversion from normal Gaussian beams to LG beams. Here, we propose a plasma-based scheme to amplify a LG pulse through energy transfer from copropagating multiple Gaussian pulses as pumps. It is found that efficient energy transfer can be realized by properly choosing the relative phase difference between the LG pulse and the Gaussian pulses, where the intensity of the LG pulse can be enhanced by nearly two orders of magnitude. This scheme works for either weakly or highly relativistic intense lasers, as well as for LG pulses with different topological charges. Furthermore, the properties of the amplified LG pulse can be manipulated by controlling the Gaussian pulses, such as the pump laser beam number, laser intensity, and propagation direction. Our studies provide a way of using plasma optics to amplify LG lasers to relativistic high intensity and manipulate their spatial structure for applications.

We employ well-known concepts from statistical physics, quantum field theories, and general topology to study magnetic reconnection and topology change and their connection in incompressible flows in the context of an effective field theory without appealing to magnetic field lines. We consider the dynamical system corresponding to wave packets moving with Alfvén velocity $\dot{\mathbf{x}}\left(t\right):={\mathbf{V}}_A(\mathbf{x},t)$ whose trajectories $\mathbf{x}\left(t\right)$ define pathlines, which naturally provides a mathematical way to estimate the rate of magnetic topology change. A considerable simplification is attained, in fact, by directly employing well-known concepts from hydrodynamic turbulence without appealing to the complicated notion of magnetic field lines moving through plasma, which may prove even more useful in the relativistic regime. Continuity conditions for magnetic field allow rapid but continuous divergence of pathlines, shown to imply reconnection, but not discontinuous divergence, which would change topology. Thus, topology can change only due to time-reversal symmetry breaking, e.g., by dissipative effects. In laminar and even chaotic flows, the separation of pathlines at all times remains proportional to their initial separation, argued to correspond to slow reconnection, and topology changes by dissipation with a rate proportional to resistivity. In turbulence, pathlines diverge superlinearly with time independent of their initial separation, i.e., fast reconnection, and magnetic topology changes by turbulent dissipation with a rate independent of small-scale plasma effects. The crucial role of turbulence in enhancing topology change and reconnection rates originates from its ability to break time-reversal invariance and make the flow superchaotic. In fact, due to the loss of Lipschitz continuity of the magnetic field in turbulence, pathlines separate superlinearly even if their initial separation tends to vanish, unlike deterministic chaos. This superchaotic behavior is an example of spontaneous stochasticity in statistical physics, sometimes called the real butterfly effect in chaos theory to distinguish it from the butterfly effect, in which trajectories can diverge exponentially only if initial separation remains finite. If 3D reconnection is defined as magnetic topology change, it can be fast only in turbulence where both reconnection and topology change are driven by spontaneous stochasticity, independent of any plasma effects. Our results strongly support the Lazarian-Vishniac theory of turbulent reconnection.

Landau-damped oscillations in collisionless plasmas, described by van Kampen and Case, are quasimodes, representing a continuous superposition of singular eigenfunctions, not true eigenmodes. Recent work by Ng et al. shows that even rare collisions replace these singular modes with discrete regular modes having complex eigenvalues for the phase velocity (or frequency), approaching Landau eigenvalues in the collisionless limit. We analytically derive approximate expressions for the eigenvalue correction due to rare collisions and for the shape of the eigenfunction describing distribution function perturbations in velocity space, demonstrating its increasing oscillations in the resonance region as the collision frequency tends to zero. We also obtain approximate expressions for the resonance region's width and peak value, and the oscillation period within it. We validate these analytical results with high-precision numerical calculations using a standard linear matrix eigenvalue problem approach.

Active matter, exemplified by adaptive living materials such as the actomyosin cytoskeleton, can navigate material parameter space, leading to unconventional mechanical responses. In particular, it can self-drive toward elastic spinodal regimes, where inhomogeneous floppy modes induce elastic degeneracy and enable a controlled interplay between rigidity loss and recovery. Proximity to such marginal states leads to stress localization and the formation of force chains that can be actively assembled and disassembled. Here we extend the classical notion of spinodal states to active solids and demonstrate how these extreme mechanical regimes can be actively accessed. Moreover, we show that in a nonlinear setting, crossing elastic spinodals generates new energy wells and makes force channeling an intrinsic feature of the emerging microstructure.

Making sense of complex inhomogeneous systems composed of many interacting species is a grand challenge that pervades basically all natural sciences. Phase separation and pattern formation in reaction-diffusion systems have been largely studied as two separate paradigms. Here we show that in reaction-diffusion systems composed of many species, the presence of a conservation law constrains the evolution of the conserved quantity to be governed by a Cahn-Hilliard-like equation. This establishes a direct link with the paradigm of coexistence and recent “active” field theories. Hence, even for complex many-species systems a dramatically simplified but accurate description emerges over coarse spatiotemporal scales. Using the nullcline (the line of homogeneous steady states) as the central motif, we develop a geometrical framework which endows chemical space with a basis and suitable coordinates. This framework allows us to capture and understand the effect of eliminating fast nonconserved degrees of freedom and to explicitly construct coefficients of the coarse field theory. We expect that the theory we develop here will be particularly relevant to advance our understanding of biomolecular condensates.

Results are given for the exact energy and momentum of a family of oscillatory pulses. The pulses are characterized by a length $a$ and a dominant wave number $K$. Pulses with angular momentum about the direction of propagation are also discussed. The total energy, momentum, and angular momentum of all pulses are simply expressed as integrals over a wave-number weight function, and evaluated explicitly for the $Ka$ family. For large $Ka$ the speed of sound times the net momentum tends to the total energy from below.

A system of interacting walkers is considered in a two-dimensional hypothetical space, where the dynamics of each walker are governed by the opinion states of the agents of a fully connected three-state opinion dynamics model. Such walks, studied in different models of statistical physics, are usually considered in one-dimensional virtual spaces. Here, the mapping is done in such a way that the walk is directed along the $Y$ axis while it can move either way along the $X$ axis. The walk shows that there are three distinct regions as the noise parameter, responsible for driving a continuous phase transition in the model, is varied. In absence of any noise, the scaling properties and the form of the distribution along either axis do not follow any conventional form. For any finite noise below the critical point the bivariate distribution of the displacements is found to be a modified biased Gaussian function while above it, only the marginal distribution along one direction is Gaussian. The marginal probability distributions can be extracted and the scaling forms of different quantities, showing power-law behavior, are obtained. The directed nature of the walk is reflected in the marginal distributions as well as in the exponents.

We consider Taylor dispersion in periodic but highly corrugated channels. Exact analytical expressions for the long-time diffusion constant and drift along the channel are derived to next-to-leading order in the limit of a small channel period. Using these results, we show how an effective model for Taylor dispersion in porous media with tortuous pores can be framed in terms of dispersion in a uniform channel with absorption/desorption at its surface, an effective slip length for the flow at the surface, and an effective, universal, diffusion constant on the surface. This work thus extends the concept of an effective slip length for hydrodynamic flows to Taylor dispersion by those flows. The analytical results are confirmed by numerical calculations, and they present a robust method to understand and upscale the transport properties of flows in porelike geometries.

We combine machine-learning techniques with Monte Carlo (MC) simulations and finite-size scaling (FSS) to study continuous and first-order phase transitions in Ising, Blume-Capel, and Ising-metamagnet spin models. We go beyond earlier studies that had concentrated on obtaining the correlation-length exponent $\nu$. In particular, we show (a) how to combine neural networks (NNs), trained with data from MC simulations of Ising-type spin models on finite lattices, with FSS to obtain both thermal magnetic exponents $y_t=1/\nu$ and $y_h$, respectively, at both critical and tricritical points, (b) how to obtain the NN counterpart of two-scale-factor universality at an Ising-type critical point, and (c) how to get FSS at a first-order transition. We also obtain the FSS forms for the output of our trained NNs as functions of both the temperature and the magnetic field.