We study the entropy $S$ of longest increasing subsequences (LISs), i.e., the logarithm of the number of distinct LISs. We consider two ensembles of sequences, namely, random permutations of integers and sequences drawn independent and identically distributed (i.i.d.) from a limited number of distinct integers. Using sophisticated algorithms, we are able to exactly count the number of LISs for each given sequence. Furthermore, we are not only measuring averages and variances for the considered ensembles of sequences, but we sample very large parts of the probability distribution $p\left(S\right)$ with very high precision. Especially, we are able to observe the tails of extremely rare events which occur with probabilities smaller than ${10}^{-600}$. We show that the distribution of the entropy of the LISs is approximately Gaussian with deviations in the far tails, which might vanish in the limit of long sequences. Further, we propose a large-deviation rate function which fits best to our observed data.

We consider a simple model of a bistable system under the influence of multiplicative noise. We provide a path integral representation of the overdamped Langevin dynamics and compute conditional probabilities and escape rates in the weak noise approximation. The saddle-point solution of the functional integral is given by a diluted gas of instantons and anti-instantons, similar to the additive noise problem. However, in this case, the integration over fluctuations is more involved. We introduce a local time reparametrization that allows its computation in the form of usual Gaussian integrals. We found corrections to the Kramers escape rate produced by the diffusion function which governs the state-dependent diffusion for arbitrary values of the stochastic prescription parameter. Theoretical results are confirmed through numerical simulations.

Chimera states in one-dimensional nonlocally coupled phase oscillators are mostly assumed to be stationary, but breathing chimeras can occasionally appear, branching from the stationary chimeras via Hopf bifurcation. In this paper, we demonstrate two types of breathing chimeras: The type I breathing chimera looks the same as the stationary chimera at a glance, while the type II consists of multiple coherent regions with different average frequencies. Moreover, it is shown that the type I changes to the type II by increasing the breathing amplitude. Furthermore, we develop a self-consistent analysis of the local order parameter, which can be applied to breathing chimeras, and numerically demonstrate this analysis in the present system.

We investigate a nonlinear version of coevolving voter models, in which node states and network structure update as a coupled stochastic process. Most prior work on coevolving voter models has focused on linear update rules with fixed and homogeneous rewiring and adopting probabilities. By contrast, in our nonlinear version, the probability that a node rewires or adopts is a function of how well it “fits in” with the nodes in its neighborhood. To explore this idea, we incorporate a local-survey parameter ${\sigma }_i$ that encodes the fraction of neighbors of an updating node $i$ that share its opinion state. In an update, with probability ${\sigma }_i^q$ (for some nonlinearity parameter $q$), the updating node rewires; with complementary probability $1-{\sigma }_i^q$, the updating node adopts a new opinion state. We study this mechanism using three rewiring schemes: after an updating node deletes one of its discordant edges, it then either (1) “rewires-to-random” by choosing a new neighbor in a random process; (2) “rewires-to-same” by choosing a new neighbor in a random process from nodes that share its state; or (3) “rewires-to-none” by not rewiring at all (akin to “unfriending” on social media). We compare our nonlinear coevolving voter model to several existing linear coevolving voter models on various network architectures. Relative to those models, we find in our model that initial network topology plays a larger role in the dynamics and that the choice of rewiring mechanism plays a smaller role. A particularly interesting feature of our model is that, under certain conditions, the opinion state that is held initially by a minority of the nodes can effectively spread to almost every node in a network if the minority nodes view themselves as the majority. In light of this observation, we relate our results to recent work on the majority illusion in social networks.

The red blood cell membrane-cytoskeleton is a complex structure mainly responsible for giving the cell rigidity and shape. It also provides the erythrocyte with the ability to pass through narrow capillaries of the vertebrate blood circulatory system. Although the red blood cell viscoelastic properties have been extensively studied, reported experimental data differ by up to three orders of magnitude. This could be attributed to the natural cell variability, to the different techniques employed, and also to the models used for the cell response, which are highly dependent on cell geometry. Here, we use two methodologies based on optical tweezers to investigate the viscoelastic behavior of healthy human red blood cells, one applying small cell deformations (microrheology) and another imposing large deformations (tether extraction). We also establish a defocusing microscopy-based method to characterize the cell geometry and thus the erythrocyte form factor, an essential parameter that allows comparisons among the viscoelastic properties at different conditions. Moreover, for small deformations, a soft glassy rheology model is used to discuss the results, while for large deformations two surface shear moduli and one surface viscosity are determined, together with the surface tension and bending modulus of the erythrocyte membrane lipid component. We also show that F-actin is not detected in tethers, although the erythrocyte membrane has physical properties like those of other adherent cells, known to have tethers containing F-actin inside. Altogether, our results show good agreement with the reported literature and we argue that, to properly compare the viscoelastic properties of red blood cells in different situations, the task of cell geometry characterization must be accomplished. This may be especially important when the influence of agents, like the malaria parasite, induces changes in both the geometry and chemical constituents of the erythrocyte membrane. Together, the new methodologies and procedures used in this study would allow the erythrocyte community to better explore the mechanical behavior of red blood cells and may be useful to characterize erythrocyte viscoelasticity changes in several blood diseases.

The standard formulation of thermostatistics, being based on the Boltzmann-Gibbs distribution and logarithmic Shannon entropy, describes idealized uncorrelated systems with extensive energies and short-range interactions. In this Rapid Communication, we use the fundamental principles of ergodicity (via Liouville's theorem), the self-similarity of correlations, and the existence of the thermodynamic limit to derive generalized forms of the equilibrium distribution for long-range-interacting systems. Significantly, our formalism provides a justification for the well-studied nonextensive thermostatistics characterized by the Tsallis distribution, which it includes as a special case. We also give the complementary maximum entropy derivation of the same distributions by constrained maximization of the Gibbs-Shannon entropy. The consistency between the ergodic and maximum entropy approaches clarifies the use of the latter in the study of correlations and nonextensive thermodynamics.

We study the motion of random walkers with residence time bias between first and subsequent visits to a site, as a model for synthetic molecular walkers composed of coupled DNAzyme legs known as molecular spiders. The mechanism of the transient superdiffusion has been explained via the emergence of a boundary between the new and the previously visited sites, and the tendency of the multilegged walker to cling to this boundary, provided residence time for a first visit to a site is longer than for subsequent visits. Using both kinetic Monte Carlo simulation and an analytical approach, we model a system that consists of unipedal walkers, each on its own one-dimensional track, connected by a tether, i.e., a kinematic constraint that no two walkers can be more than a certain distance apart. Even though a single unipedal walker does not at all exhibit directional, superdiffusive motion, we find that a team of unipedal walkers on parallel tracks, connected by a flexible tether, does enjoy a superdiffusive transient. Furthermore, unipedal walker teams exhibit a greater expected number of steps per boundary period and are able to diffuse more quickly than bipedal walker teams, which leads to longer periods of superdiffusion.

We report the grand-canonical solution of a ternary mixture of discrete hard spheres defined on a Husimi lattice built with cubes, which provides a mean-field approximation for this system on the cubic lattice. The mixture is composed of pointlike particles (0NN) and particles which exclude up to their first (1NN) and second neighbors (2NN), with activities $z_0, z_1$, and $z_2$, respectively. Our solution reveals a very rich thermodynamic behavior, with two solid phases associated with the ordering of 1NN ($S1$) or 2NN particles ($S2$), and two fluid phases, one being regular ($RF$) and the other characterized by a dominance of 0NN particles ($F0$ phase). However, in most part of the phase diagram these fluid ($F$) phases are indistinguishable. Discontinuous transitions are observed between all the four phases, yielding several coexistence surfaces in the system, among which a fluid-fluid and a solid-solid demixing surface. The former one is limited by a line of critical points and a line of triple points (where the phases $RF-F0-S2$ coexist), both meeting at a special point, after which the fluid-fluid coexistence becomes metastable. Another line of triple points is found, connecting the $F-S1, F-S2$, and $S1-S2$ coexistence surfaces. A critical $F-S1$ surface is also observed meeting the $F-S1$ coexistence one at a line of tricritical points. Furthermore, a thermodynamic anomaly characterized by minima in isobaric curves of the total density of particles is found, yielding three surfaces of minimal density in the activity space, depending on which activity is kept fixed during its calculation.

Single-file diffusion (SFD) in finite open nanopores is characterized by nonzero spatially varying tracer diffusion coefficients within a generalized hydrodynamic description. This contrasts with infinite SFD systems where tracer diffusivity vanishes. In standard tracer counterpermeation (TCP) analysis, two reservoirs, each containing a different species, are connected to opposite ends of a finite pore. We implement an extended TCP analysis to allow the two reservoirs to contain slightly different mixtures of the two species. Then, determination of diffusion fluxes through the pore allows extraction of diffusion coefficients for near-constant partial concentrations of the two species. This analysis is applied for a lattice-gas model describing two-component SFD through a finite linear pore represented by a one-dimensional array of cells. Two types of particles, A and B, can hop only to adjacent empty cells with generally different rates, $h_{\mathrm{A}}$ and $h_{\mathrm{B}}$. Particles are noninteracting other than exclusion of multiple cell occupancy. Results reveal generalized hydrodynamic tracer diffusion coefficients which adopt small values inversely proportional to pore length in the pore center, but which are strongly enhanced near pore openings.

We investigate two complementary field-theoretical models describing the flat phase of polymerized—phantom—membranes by means of a two-loop, weak-coupling, perturbative approach performed near the upper critical dimension $D_{\mathrm{uc}}=4$, extending the one-loop computation of Aronovitz and Lubensky [Phys. Rev. Lett. 60, 2634 (1988)]. We derive the renormalization group equations within the modified minimal substraction scheme, then analyze the corrections coming from two-loop with a particular attention paid to the anomalous dimension and the asymptotic infrared properties of the renormalization group flow. We finally compare our results to those provided by nonperturbative techniques used to investigate these two models.

We investigate the phase diagram and finite-temperature transitions of three-dimensional scalar $\mathrm{SO}\left(N_c\right)$ gauge theories with $N_f\ge 2$ scalar flavors. These models are constructed starting from a maximally O($N$)-symmetric multicomponent scalar model ($N=N_c{N}_f$), whose symmetry is partially gauged to obtain an $\mathrm{SO}\left(N_c\right)$ gauge theory, with O($N_f$) or $\mathrm{U}\left(N_f\right)$ global symmetry for $N_c\ge 3$ or $N_c=2$, respectively. These systems undergo finite-temperature transitions, where the global symmetry is broken. Their nature is discussed using the Landau-Ginzburg-Wilson (LGW) approach, based on a gauge-invariant order parameter, and the continuum scalar $\mathrm{SO}\left(N_c\right)$ gauge theory. The LGW approach predicts that the transition is of first order for $N_f\ge 3$. For $N_f=2$ the transition is predicted to be continuous: It belongs to the O(3) vector universality class for $N_c=2$ and to the $XY$ universality class for any $N_c\ge 3$. We perform numerical simulations for $N_c=3$ and $N_f=2,3$. The numerical results are in agreement with the LGW predictions.

Thermodynamic uncertainty relations (TURs) are the inequalities which give lower bounds on the entropy production rate using only the mean and the variance of fluctuating currents. Since the TURs do not refer to the full details of the stochastic dynamics, it would be promising to apply the TURs for estimating the entropy production rate from a limited set of trajectory data corresponding to the dynamics. Here we investigate a theoretical framework for estimation of the entropy production rate using the TURs along with machine learning techniques without prior knowledge of the parameters of the stochastic dynamics. Specifically, we derive a TUR for the short-time region and prove that it can provide the exact value, not only a lower bound, of the entropy production rate for Langevin dynamics, if the observed current is optimally chosen. This formulation naturally includes a generalization of the TURs with the partial entropy production of subsystems under autonomous interaction, which reveals the hierarchical structure of the estimation. We then construct estimators on the basis of the short-time TUR and machine learning techniques such as the gradient ascent. By performing numerical experiments, we demonstrate that our learning protocol performs well even in nonlinear Langevin dynamics. We also discuss the case of Markov jump processes, where the exact estimation is shown to be impossible in general. Our result provides a platform that can be applied to a broad class of stochastic dynamics out of equilibrium, including biological systems.

We consider accurate investigation of the energy current and its components, heat and work, in some boundary-driven quantum spin systems. The expressions for the currents, as well as the associated Lindblad master equation, are obtained via a repeated interaction scheme. We consider small systems in order to analytically compute the steady distribution to study the current in the steady state. Asymmetrical $XXZ$ and quantum Ising models are detailed analyzed. For the $XXZ$ chain we present cases in which different compositions of heat and work currents, obtained via the repeated interaction protocol, lead to the same energy current, which may be obtained via the Lindblad master equation. For the quantum Ising chain, we describe a case of zero energy current and novanishing heat and work currents. Our findings make clear that to talk about heat in these boundary-driven spin quantum systems we must go beyond an investigation involving only the Lindblad master equation.

We study the kinetic roughening of the single-step (SS) growth model with a tunable parameter $p$ in $1+1$ and $2+1$ dimensions by performing extensive numerical simulations. We show that there exists a very slow crossover from an intermediate regime dominated by the Edwards-Wilkinson class to an asymptotic regime dominated by the Kardar-Parisi-Zhang (KPZ) class for any $p<\frac{1}{2}$. We also identify the crossover time, the nonlinear coupling constant, and some nonuniversal parameters in the KPZ equation as a function $p$. The effective nonuniversal parameters are continuously decreasing with $p$ but not in a linear fashion. Our results provide complete and conclusive evidence that the SS model for $p\ne \frac{1}{2}$ belongs to the KPZ universality class in $2+1$ dimensions.

Systems with discrete symmetries are highly important in quantum mechanics. We consider a two-dimensional quantum billiard with fourfold rotational symmetry, where the eigenenergies and eigenstates can be grouped into four symmetry subspaces. Unlike the threefold rotational symmetry case, here the interference of the scarring states on the fundamental domain orbits (FDO) is clean, that they either interfere constructively or annihilate completely. We shall show the complex behavior of the interference revealed in the length spectra for eigenenergies that belong to a particular symmetry subspace and combinations of different symmetry subspaces. We then provide detailed analysis of phase accumulation along the FDOs, which are the keys to determine the interference and could explain the enhancement or annihilation of the peaks well. The quantization condition for the scarring states belonging to different symmetry subspaces is discussed and used to reveal the time-reversal symmetry broken for a particular subspace. An experimental scheme to observe such complex behaviors is also proposed.

One of the computational models proposed to study emerging phenomena in decentralized systems is the swarmalator model [Nat. Commun. 8, 1504 (2017)], where only long-range interactions are considered. But in living systems many of the collective behaviors observed arise from the combined effect of long- and short-range interactions. In this work we present an extension to the swarmalator model which includes a Gaussian short-range repulsive term of the type $\frac{1}{\sigma }{e}^{-{\left|x\right|}^2/\sigma }$ along with the results of numerical simulations. Using several order parameters we can distinguish between static and dynamic aggregations and between synchronous and asynchronous states. We have found six long-term collective states, some of them not previously reported and the most remarkable one showing oscillatory collective behavior. The results obtained show a multiplicity of complex behaviors that extend the applicability of the swarmalator model.

The human brain is composed of distinct regions that are each associated with particular functions and distinct propensities for the control of neural dynamics. However, the relation between these functions and control profiles is poorly understood, as is the variation in this relation across diverse scales of space and time. Here we probe the relation between control and dynamics in brain networks constructed from diffusion tensor imaging data in a large community sample of young adults. Specifically, we probe the control properties of each brain region and investigate their relationship with dynamics across various spatial scales using the Laplacian eigenspectrum. In addition, through analysis of regional modal controllability and partitioning of modes, we determine whether the associated dynamics are fast or slow, as well as whether they are alternating or monotone. We find that brain regions that facilitate the control of energetically easy transitions are associated with activity on short length scales and slow timescales. Conversely, brain regions that facilitate control of difficult transitions are associated with activity on long length scales and fast timescales. Built on linear dynamical models, our results offer parsimonious explanations for the activity propagation and network control profiles supported by regions of differing neuroanatomical structure.

Network data sets are often constructed by some kind of thresholding procedure. The resulting networks frequently possess properties such as heavy-tailed degree distributions, clustering, large connected components, and short average shortest path lengths. These properties are considered typical of complex networks and appear in many contexts, prompting consideration of their universality. Here we introduce a simple model for correlated relational data and study the network ensemble obtained by thresholding it. We find that some, but not all, of the properties associated with complex networks can be seen after thresholding the correlated data, even though the underlying data are not “complex.” In particular, we observe heavy-tailed degree distributions, a large numbers of triangles, and short path lengths, while we do not observe nonvanishing clustering or community structure.

Previous research in biology and physics speculates that high-frequency electromagnetic fields may be an unexplored method of cellular and subcellular communication. The predominant theory for generating electric fields in the cell is mechanical vibration of charged or polar biomolecules such as cell membranes or microtubules. The challenge to this theory is explaining how high-frequency vibrations would not be overdamped by surrounding biological media. As many of these suspected resonators are too large for atomistic molecular dynamics simulations, accurately modeling biological resonators remains an ongoing challenge. While many resonators have been studied and simulated, the general limitations on communication imposed by energy transfer arguments have not been considered. Starting with energy transfer expressions from coupled-mode theory, we derive expressions for the minimum quality factor ($Q$ factor) required to sustain communication for both near- and far-field interactions. We compare previous simulation studies and our theory. We determine the flexing mode of microtubules as an identified resonance in the literature which meets our criteria. Our results suggest the major obstacle to meeting our criteria for effective electromagnetic communication is the trade-off between the $Q$ factor and the plasma frequency: Resonators must be large enough to have a large $Q$ factor, but small enough to resonate at frequencies greater than the plasma frequency.