We use renormalization-group and Monte Carlo calculations to investigate isotropically directed percolation on hierarchical lattices. We describe a general renormalization-group approach to exactly determine the critical exponents for these lattices. This approach is equivalent to, albeit simpler than, the method of cluster generating functions and can be applied to various hierarchical lattices. In the case of isotropically directed percolation, we need to determine two critical exponents, namely, ${\beta }_{\mathrm{scc}}$ and ${\beta }_{\mathrm{out}}$ associated, respectively, with the giant strongly connected component and the giant out component in the neighborhood of the percolation threshold. Our method is applied to different constructions of the hierarchical lattice, in particular, to a small-world hierarchical lattice which includes long-range bonds and exhibits a form of explosive percolation.

Game of Life is a two-state cellular automaton capable of displaying high complexity starting from simple rules. Taking inspiration from Game of Life, we introduce a new class of Cellular Automata with two types of alive states and one dead state. This broad class of cellular automata allows for more interesting behaviors, including replicators at relatively small scales. We develop an algorithm to search for rules that open for small-scale structures that exhibit self-replication. We also find replicators that emerge spontaneously from random initial conditions. We find several such rules, including some that even exhibit scale-free replication. This class of replicators, with repeated copying within growing structures, inspires origin-of-life discussions using diverse and cooperating building blocks.

We generalize the Krylov construction to periodically driven (Floquet) quantum systems using the theory of orthogonal polynomials on the unit circle. Compared to other approaches, our method works faster and maps any quantum dynamics to a one-dimensional tight-binding Krylov chain, which is efficiently simulated on both classical and quantum computers. We also suggest a classification of chaotic and integrable Floquet systems based on the asymptotic behavior of Krylov chain hopping parameters (Verblunsky coefficients). We illustrate this classification with random matrix ensembles, kicked top, and kicked Ising chain.

The physical mechanism of microbial motion near solid-liquid interfaces is crucial for understanding various biological phenomena and developing ecological applications. However, limited works have been conducted on the swimming behavior of $C.\mathit{reinhardtii}$, a typical “puller” type cell, near solid surfaces, particularly with varying and conflicting experimental observations. Here, we investigate the swimming behavior of $C.\mathit{reinhardtii}$ using a three-dimensional real-time tracking microscopy system both near a solid-liquid interface and in the fluid bulk region. We explore the relationships between the cell density, swimming speed and orientation with respect to the distance from the solid-liquid interface, confirming the phenomenon of $C.\mathit{reinhardtii}$ accumulation near the solid-liquid interface. Based on the traditional definitions of “pusher” and “puller” cells, we propose a simplified model consisting of a pair of mutually perpendicular force dipoles for $C.\mathit{reinhardtii}$. This model is employed to analyze the complex hydrodynamic interactions between $C.\mathit{reinhardtii}$ and the solid surface, providing a potential theoretical explanation for the observed accumulation phenomenon at the solid-liquid interface.

Liquid-liquid phase separation (LLPS) of biomacromolecules drives the formation of biomolecular condensates, which possess material properties crucial for various biological functions. While recent studies have primarily focused on LLPS driven by long-range, nonspecific interactions, the role of short-range, one-to-one specific interactions in sequence-dependent behavior still remains elusive. In this study, we combined theoretical analysis and coarse-grained molecular dynamics simulations to systematically investigate the sequence-dependent material properties of biomolecular condensates. By introducing the sequence descriptor ${\varphi }^{*}$, we identified strong correlations between ${\varphi }^{*}$ and key material or structural properties, such as the single-chain radius of gyration, critical temperature, density, surface tension, viscosity, and diffusion coefficient. Notably, near critical points, surface tension and viscosity exhibit distinct scaling relationships with temperature, with viscosity showing much greater sensitivity. Additionally, we found that sequences with high ${\varphi }^{*}$ may impede the efficient growth of droplets. Our findings provide a framework for understanding sequence-dependent material properties and offer valuable insights into designing biomolecular condensates with tailored stability and dynamic functionality.

We address the out-of-equilibrium critical dynamics of the three-dimensional lattice ${\mathbb{Z}}_2$ gauge model, and in particular the critical relaxational flows arising from instantaneous quenches to the critical point, driven by purely relaxational (single-spin-flip Metropolis) upgradings of the link ${\mathbb{Z}}_2$ gauge variables. We monitor the critical relaxational dynamics by computing the energy density, which is the simplest local gauge-invariant quantity that can be measured in a lattice gauge theory. The critical relaxational flow of the three-dimensional lattice ${\mathbb{Z}}_2$ gauge model is analyzed within an out-of-equilibrium finite-size scaling framework, which allows us to compute the dynamic critical exponent $z$ associated with the purely relaxational dynamics of the three-dimensional ${\mathbb{Z}}_2$ gauge universality class. We obtain $z=2.610\left(15\right)$, which significantly improves earlier results obtained by other methods, in particular those obtained by analyzing the equilibrium critical dynamics.

Landau theory relates phase transitions to the minimization of the Landau functional (e.g., free energy functional), which is expressed as a power series of the order parameter. It has been shown that the critical behavior of certain physical systems can be described using Landau functionals that include nonanalytic terms, corresponding to odd or even noninteger powers of the absolute value of the order parameter. In particular, these nonanalytic terms can determine the order of the phase transition and the values of the critical exponents. Here, we show that such terms can also shape the finite-size scaling behavior of fluctuations of observables (e.g., of energy or magnetization) or the response functions (e.g., heat capacity or magnetic susceptibility) at the continuous phase transition point. We demonstrate this on two examples, the equilibrium molecular zipper and the nonequilibrium version of the Curie-Weiss model.

Artificial neural networks have revolutionized machine learning in recent years, but a complete theoretical framework for their learning process is still lacking. Substantial theoretical advances have been achieved for wide networks, within two disparate theoretical frameworks: the neural tangent kernel (NTK), which assumes linearized gradient descent dynamics, and the Bayesian neural network Gaussian process (NNGP) framework. Here we unify these two theories using gradient descent learning dynamics with an additional small noise in an ensemble of wide deep networks. We construct an exact analytical theory for the network input-output function and introduce a new time-dependent neural dynamical kernel (NDK) from which both NTK and NNGP kernels are derived. We identify two learning phases characterized by different time scales: an initial gradient-driven learning phase, dominated by deterministic minimization of the loss, in which the time scale is mainly governed by the variance of the weight initialization. It is followed by a slow diffusive learning stage, during which the network parameters sample the solution space, with a time constant that is determined by the noise level and the variance of the Bayesian prior. The two variance parameters can strongly affect the performance in the two regimes, particularly in sigmoidal neurons. In contrast to the exponential convergence of the mean predictor in the initial phase, the convergence to the final equilibrium is more complex and may exhibit nonmonotonic behavior. By characterizing the diffusive learning phase, our work sheds light on the phenomenon of representational drift in the brain, explaining how neural activity can exhibit continuous changes in internal representations without degrading performance, either by ongoing weak gradient signals that synchronize the drifts of different synapses or by architectural biases that generate invariant code, i.e., task-relevant information that is robust against the drift process. This work closes the gap between the NTK and NNGP theories, providing a comprehensive framework for understanding the learning process of deep wide neural networks and for analyzing learning dynamics in biological neural circuits.

The analysis of complex systems such as neural networks is made particularly difficult by the overwhelming number of their interacting components. In the absence of prior knowledge, identifying a small but informative subset of network nodes on which the analysis should focus is a rather challenging task. In this work, we address this problem in the context of a Hopfield model, which is observed through the lenses of low-resolution representations, or decimation mappings, consisting of subgroups of its neurons. The optimal, most informative mappings of the network are defined through a recently developed methodology, the mapping entropy optimization workflow (MEOW), which performs an unsupervised analysis of the states sampled by the network and identifies those subgroups of units whose configuration distribution is closest to that of the full, high-resolution model. Which neurons are retained in an optimal mapping is found to critically depend on the properties of the interaction matrix of the network and the level of detail employed to describe the system; by these means, it is thus possible to extract quantitative insight about the underlying properties of the high-resolution model through the analysis of its optimal low-resolution representations. These results show a tight and potentially fruitful relation between the level of detail at which the network is inspected and the type and amount of information that can be gathered from it, and showcase the MEOW approach as a practical, enabling tool for the study of complex systems.

A stretched DNA molecule that is also under- or overwound undergoes a buckling transition, forming intertwined looped domains called plectonemes. Here we develop a simple theory that extends the two-phase model of stretched supercoiled DNA, allowing for the coexistence of multiple plectonemic domains by including positional and length distribution entropies. Such a multiplectoneme phase is favored in long DNA molecules in which the gain of positional entropy compensates for the cost of nucleating a plectoneme along a stretched DNA segment. Despite its simplicity, the developed theory is shown to be in excellent agreement with Monte Carlo simulations of the twistable wormlike chain model. The theory predicts more plectonemes than experimentally observed, which we attribute to the limited resolution of experimental data. Since plectonemes are detected through fluorescence signals, those shorter than the observable threshold are likely missed.

Toggled field self-assembly is a useful way to circumvent kinetic arrest of the induced structures in field-directed self-assembly and, in particular, in magnetic suspensions such as magnetorheological (MR) fluids. During the field-off period, Brownian motion permits rearrangement of the particles and classical system-spanning chains (typically formed under uniaxial DC fields) collapse into dense ellipsoidal aggregates that are more energetically favorable. Here we show, through the use of experiments and particle-level simulations, that the process is very sensitive to the field configuration (e.g., frequency and strength), and, in particular, it can be dramatically accelerated when the magnetic field strength is increased. Moreover, when the field strength reaches sufficiently high values, the classical two-step aggregation process becomes a single one and morphological differences in the final structures emerge.

The self-organization of proteins into enriched compartments and the formation of complex patterns are crucial processes for life on the cellular level. Liquid-liquid phase separation is one mechanism for forming such enriched compartments. When phase-separating proteins are membrane-bound and locally disturb it, the mechanical response of the membrane mediates interactions between these proteins. How these membrane-mediated interactions influence the steady state of the protein density distribution is thus an important question to investigate in order to understand the rich diversity of protein and membrane-shape patterns present at the cellular level. This work starts with a widely used model for membrane-bound phase-separating proteins. We numerically solve our system to map out its phase space and perform a careful, systematic expansion of the model equations to characterize the phase transitions through linear stability analysis and free energy arguments. We observe that the membrane-mediated interactions, due to their long-range nature, are capable of qualitatively altering the equilibrium state of the proteins. This leads to arrested coarsening and length-scale selection instead of simple demixing and complete coarsening. In this study, we unambiguously show that long-range membrane-mediated interactions lead to pattern formation in a system that otherwise would not do so. This work provides a basis for further systematic study of membrane-bound pattern-forming systems.

Identifying model parameters from observed configurations poses a fundamental challenge in data science, especially with limited data. Recently, diffusion models have emerged as a novel paradigm in generative machine learning, capable of producing new samples that closely mimic observed data. These models learn the gradient of model probabilities, bypassing the need for cumbersome calculations of partition functions across all possible configurations. We explore whether diffusion models can enhance parameter inference by augmenting small datasets. Our findings demonstrate this potential through a synthetic task involving inverse Ising inference and a real-world application of reconstructing missing values in neural activity data. This study serves as a proof-of-concept for using diffusion models for data augmentation in physics-related problems, thereby opening new avenues in data science.

I construct a two dimensional monodisperse lattice gas model with selective geometric constraints featuring a fluid-solid continuous phase transition on the square lattice and a dimerized ground state with fourfold degeneracy. Finite-size scaling analysis confirms that the lattice gas belongs to the universality class of two-dimensional four-state Potts model, and since its thermodynamics is governed by entropy alone, this provides possibly the simplest statistical mechanical realization of an athermal tricritical point.

We investigate the growth dynamics of microbial populations, challenging the conventional logistic model. By analyzing empirical data from various biomes, we demonstrate that microbial growth is better described by a generalized logistic model, the $\theta$-logistic model. This accounts for different growth mechanisms and environmental fluctuations, leading to a generalized gamma distribution of abundance fluctuations. Our findings reveal that microbial growth is never sublinear, so they cannot endorse—at least in the microbial world—the recent proposal of this mechanism as a stability enhancer of highly diverse communities. These results have significant implications for understanding macroecological patterns and the stability of microbial ecosystems.

Graph-theoretical methods have proven valuable for investigating alterations in both anatomical and functional brain connectivity networks during Alzheimer's disease (AD). Recent studies suggest that representing brain networks in a suitable geometric space can better capture their connectivity structure. This study introduces a novel approach to characterize brain connectivity changes using low-dimensional, informative representations of networks in a latent geometric space. Specifically, the networks are embedded in a polar representation of the hyperbolic plane, the hyperbolic disk. Here, we used a geometric score, entirely based on the computation of distances between nodes in the latent space, to measure the effect of a perturbation on the nodes. Precisely, the score is a local measure of distortion in the geometric neighborhood of a node following a perturbation. The method is applied to a brain network dataset of patients with AD and healthy participants, derived from diffusion-weighted (DWI) and functional magnetic resonance (fMRI) imaging scans. We show that, compared with standard graph measures, our method more accurately identifies the brain regions most affected by neurodegeneration. Notably, the abnormalities detected in memory-related and frontal areas are robust across multiple brain parcellation scales. Finally, our findings suggest that the geometric perturbation score could serve as a potential biomarker for characterizing the progression of the disease.

What differences in behavior can one draw between the measurable quantities in fixed-force and fixed-extension experiments on semiflexible polymers (such as DNA), stretched by applying a force? The marked difference between the two types of experiment, represented by two different scaling curves as a function of a reduced stretching force and reduced extension, is revealed here. Potential regions of the parameter space that could benefit from further experimental investigation are also suggested.

Rigorous theories connecting physical properties of a heterogeneous material to its microstructure offer a promising avenue to guide the computational material design and optimization. The spectral density function ${\tilde{\chi }}_{{}_V}\left(k\right)$, which can be obtained experimentally from scattering data, enables accurate determination of various transport and wave propagation characteristics, including the time-dependent diffusion spreadability $\mathcal{S}\left(t\right)$ and effective dynamic dielectric constant ${\epsilon }_e$ for electromagnetic wave propagation. Moreover, ${\tilde{\chi }}_{{}_V}\left(k\right)$ determines rigorous upper bounds on the fluid permeability $\mathcal{K}$. Given the importance of ${\tilde{\chi }}_{{}_V}\left(k\right)$, we present here an efficient Fourier-space based computational framework to construct three-dimensional (3D) statistically isotropic two-phase heterogeneous materials corresponding to targeted spectral density functions. In particular, we employ a variety of analytical functional forms for ${\tilde{\chi }}_{{}_V}\left(k\right)$ that satisfy all known necessary conditions to construct disordered stealthy hyperuniform, standard hyperuniform, nonhyperuniform, and antihyperuniform two-phase heterogeneous material systems at varying phase volume fractions. We show that by tuning the correlations in the system across length scales via the targeted functions, one can generate a rich spectrum of distinct structures within each of the above classes of materials. Importantly, we present the first realization of antihyperuniform two-phase heterogeneous materials in 3D, which are characterized by autocovariance function ${\chi }_{{}_V}\left(r\right)$ with a power-law tail, resulting in microstructures that contain clusters of dramatically different sizes and morphologies. We also determine the diffusion spreadability $\mathcal{S}\left(t\right)$ and estimate the fluid permeability $\mathcal{K}$ associated with all of the constructed materials directly from the corresponding spectral densities. Although it is well established that the long-time asymptotic scaling behavior of $\mathcal{S}\left(t\right)$ only depends on the functional form of ${\tilde{\chi }}_{{}_V}\left(k\right)$, with the stealthy hyperuniform and antihyperuniform media, respectively, achieving the most and least efficient transport, we find that varying the length-scale parameter within each class of ${\tilde{\chi }}_{{}_V}\left(k\right)$ functions can also lead to orders of magnitude variation of $\mathcal{S}\left(t\right)$ at intermediate and long time scales. Moreover, we find that increasing the solid volume fraction ${\varphi }_1$ and correlation length $a$ in the constructed media generally leads to a decrease in the dimensionless fluid permeability $\mathcal{K}/a^2$, while the antihyperuniform media possess the largest $\mathcal{K}/a^2$ among the four classes of materials with the same ${\varphi }_1$ and $a$. These results indicate the feasibility of employing parameterized spectral densities for designing composites with targeted transport properties.

Single-cell experiments revealed substantial variability in generation times, growth rates, but also in birth and division sizes between genetically identical cells. Understanding how these fluctuations determine the fitness of the population, i.e., its growth rate, is necessary in any quantitative theory of evolution. Here, we develop a biologically relevant model which accounts for the stochasticity in single-cell growth rates, birth sizes, and division sizes. We derive expressions for the population growth rate and mean birth size in the population in terms of single-cell fluctuations. Allowing division sizes to fluctuate reveals how the mechanism of cell size control (timer, sizer, and adder) influences population growth. Surprisingly, we find that fluctuations in single-cell growth rates can be beneficial for population growth when slow-growing cells tend to divide at smaller sizes than fast-growing cells. Our framework is not limited to exponentially growing cells like Escherichia coli, and we derive similar expressions for cells with linear and bilinear growth laws, such as Mycobacterium tuberculosis and fission yeast Schizosaccharomyces pombe, respectively.

The quasilinear diffusion theory is generalized to include the resonance broadening effect of weak turbulence on relativistic electrons in Earth's radiation belts. In contrast with quasilinear resonant diffusion, in which cases the resonant condition should be rigorously satisfied, the second-order theory demonstrates that the wave-driven scattering of relativistic electrons could be broadened into an area of velocity space adjacent to the resonant region. It is shown that a finite departure time, resulted from the second-order wave-particle interaction, is the key factor controlling the occurrence and effectiveness of such a resonance broadening. The formula to solve the departure time has been proposed. Adopting the solved departure time, the resultant model can correctly reproduce results of test particle simulations. Our result indicates the importance of resonance broadening effect in the radiation belt modeling, and in understanding the weak turbulence dynamics in a magnetized plasma system.