Describing the collective dynamics of large neural populations using low-dimensional models for averaged variables has long been an attractive task in theoretical neuroscience. Recently developed reduction methods make it possible to derive such models directly from the microscopic dynamics of individual neurons. To simplify the reduction, the Cauchy distribution is usually assumed for heterogeneous network parameters. Here we extend the reduction method for a wider class of heterogeneities defined by the $q$-Gaussian distribution. The shape of this distribution depends on the Tsallis index $q$ and gradually changes from the Cauchy distribution to the normal Gaussian distribution as this index changes. We derive the mean-field equations for an inhibitory network of quadratic integrate-and-fire neurons with a $q$-Gaussian-distributed excitability parameter. It is shown that the dynamic modes of the network significantly depend on the form of the distribution determined by the Tsallis index. The results obtained from the mean-field equations are confirmed by numerical simulation of the microscopic model.

To develop the concept of evolutionary stability in a stochastic environment, we investigate the continuous-time dynamics of a two-phenotype linear evolutionary game with generally correlated random payoffs in pairwise interactions. By using the Gram-Schmidt orthogonalization procedure and Itô's formula, we deduce a stochastic differential equation for the phenotype frequencies that extends the replicator equation, called the stochastic replicator equation. We give conditions for stochastic stability of a fixation state or a constant interior equilibrium point with respect to the stochastic dynamics of the two phenotypes. We show that, if a fixation state is stochastically stable, then the pure strategy corresponding to this fixation state must be stochastically evolutionarily stable with respect to mixed strategies. However, this is not the case for a mixed strategy that corresponds to a stochastically stable constant interior equilibrium point with respect to the two phenotypes.

The internal dynamics of biomolecules, and hence their function, is governed by the structure of their free-energy landscape. Early flash-photolysis experiments on myoglobin suggested that the free-energy landscapes of proteins are hierarchically structured, with a characteristic distribution of free-energy barriers which gives rise to anomalous diffusion. Analytical results have been derived for one-dimensional or high-dimensional hierarchical free-energy landscapes. Recent improvements in methods and computer performance enable generating sufficiently long molecular dynamics (MD) trajectories to extract dynamics information covering many orders of magnitude, such that the broad distributions of energy barriers of proteins become accessible to quantitative studies of intermediate dimensions. In this work, we present a nonequilibrium method to estimate barrier height distributions from microsecond-long MD simulations. It infers barrier height distributions from anomalous diffusion exponents derived from principal component analysis and by comparison to simple hierarchical lattice models. These models are $d$-dimensional lattices of states separated by free-energy barriers, the heights of which are distributed as $p\left(\mathrm{\Delta }G\right)=1/\gamma exp(-\mathrm{\Delta }G/\gamma )$. The parameter $\gamma$ quantifies the “ruggedness” of the free-energy landscape in such models. We show that both parameters, i.e., ruggedness and effective dimensionality $d$, can be inferred from anomalous diffusion exponents. Assuming a similar dependency of anomalous diffusion exponents on $\gamma$ and $d$ for proteins, we estimate the ruggedness of the free-energy landscapes of 500 small, single-domain globular proteins between 15 and 20 kT per dimension with an estimated accuracy of 4.2 kT and dimensionality between 40 and 60 with an estimated accuracy of 10 dimensions. Remarkably, neither effective dimensionality nor the ruggedness correlates with protein size and both ruggedness and effective dimensionality are much smaller than the scatter of protein sizes. From this finding, we conclude that these two properties of the free-energy landscape of a protein are rather adapted to the particular function of each single protein and that, quite generally, are functionally relevant for globular proteins.

In this work we study the relationship between the softness of a mean-field caging potential and dynamics at the local level. We first describe the local softness, which shows a distribution, thus identifying structural heterogeneity. We show that the lifetime of the softness parameter is connected to the lifetime of the well-known cage structure in supercooled liquids. Finally, our theory predicts that the local softness and the local dynamics is causal below the onset temperature where there is a decoupling between the short and long time dynamics, thus allowing a static description of the cage. With the decrease in temperature, the correlation between structure and dynamics increases. The study shows that at lower temperatures, the structural heterogeneity increases, and since the structure becomes a better predictor of the dynamics, it leads to an increase in the dynamical heterogeneity. We also find that the softness of a hard, immobile region evolves with time and becomes soft and eventually mobile due to the rearrangements in the neighborhood, confirming the well-known facilitation effect.

A two-dimensional or quasi-two-dimensional nematic liquid crystal refers to a surface-confined system. When such a system is further confined by external line boundaries or excluded from internal line boundaries, the nematic directors form a deformed texture that may display defect points or defect lines, for which winding numbers can be clearly defined. Here, a particular attention is paid to the case when the liquid crystal molecules prefer to form a boundary nematic texture in parallel to the wall surface (i.e., following the homogeneous boundary condition). A general theory, based on geometric argument, is presented for the relationship between the sum of all winding numbers in the system (the total winding number) and the type of confinement angles and curved segments. The conclusion is validated by comparing the theoretical defect rule with existing nematic textures observed experimentally and theoretically in recent years.

The exceptional properties of the two-dimensional material graphene make it attractive for multiple functional applications, whose large-area samples are typically polycrystalline. Here, we study the mechanical properties of graphene in computer simulations and connect these to the experimentally relevant mechanical properties. In particular, we study the fluctuations in the lateral dimensions of the periodic simulation cell. We show that over short timescales, both the area $A$ and the aspect ratio $B$ of the rectangular periodic box show diffusive behavior under zero external field during dynamical evolution, with diffusion coefficients $D_A$ and $D_B$ that are related to each other. At longer times, fluctuations in $A$ are bounded, while those in $B$ are not. This makes the direct determination of $D_B$ much more accurate, from which $D_A$ can then be derived indirectly. We then show that the dynamic behavior of polycrystalline graphene under external forces can also be derived from $D_A$ and $D_B$ via the Nernst-Einstein relation. Additionally, we study how the diffusion coefficients depend on structural properties of the polycrystalline graphene, in particular, the density of defects.

We show that the coarse-grained dynamics model with the time-dependent and fluctuating potential (transient potential) can be derived from the microscopic Hamiltonian dynamics. The concept of the transient potential was first introduced rather phenomenologically, and its relation to the underlying microscopic dynamics has not been clarified yet. This is in contrast to the generalized Langevin equation, the relation of which to the microscopic dynamics is well-established. In this work, we show that the dynamic equations with the transient potential can be derived for the coupled oscillator model, without any approximations. It is known that the dynamics of the coupled oscillator model can be exactly described by the generalized Langevin-type equations. This fact implies that the dynamic equations with the transient potential can be utilized as a coarse-grained dynamics model in a similar way to the generalized Langevin equation. Then we show that the dynamic equations for the transient potential can also be formally derived for the microscopic Hamiltonian dynamics, without any approximations. We use the projection operator method for the coarse-grained variables and transient potential. The dynamic equations for the coarse-grained positions and momenta are similar to those in the Hamiltonian dynamics, but the interaction potential is replaced by the transient potential. The dynamic equation for the transient potential is the generalized Langevin equation with the memory effect. Our result justifies the use of the transient potential to describe the coarse-grained dynamics. We propose several approximations to obtain the simplified dynamics model. We show that, under several approximations, the dynamic equation for the transient potential reduces to the relatively simple Markovian dynamic equation for the potential parameters. We also show that with several additional approximations, the approximate dynamics model further reduces to the Markovian Langevin-type equations with the transient potential.

Lévy walks are continuous-time random-walk processes with a spatiotemporal coupling of jump lengths and waiting times. We here apply the Hermite polynomial method to study the behavior of LWs with power-law walking time density for four different cases. First we show that the known result for the infinite density of an unconfined, unbiased LW is consistently recovered. We then derive the asymptotic behavior of the probability density function (PDF) for LWs in a constant force field, and we obtain the corresponding $q\mathrm{th}$-order moments. In a harmonic external potential we derive the relaxation dynamic of the LW. For the case of a Poissonian walking time an exponential relaxation behavior is shown to emerge. Conversely, a power-law decay is obtained when the mean walking time diverges. Finally, we consider the case of an unconfined, unbiased LW with decaying speed $v\left(\tau \right)=v_0/\sqrt{\tau }$. When the mean walking time is finite, a universal Gaussian law for the position-PDF of the walker is obtained explicitly.

The statistics of the first-encounter time of diffusing particles changes drastically when they are placed under confinement. In the present work, we make use of Monte Carlo simulations to study the behavior of a two-particle system in two- and three-dimensional domains with reflecting boundaries. Based on the outcome of the simulations, we give a comprehensive overview of the behavior of the survival probability $S\left(t\right)$ and the associated first-encounter time probability density $H\left(t\right)$ over a broad time range spanning several decades. In addition, we provide numerical estimates and empirical formulas for the mean first-encounter time $\langle \mathcal{T}\rangle$, as well as for the decay time $T$ characterizing the monoexponential long-time decay of the survival probability. Based on the distance between the boundary and the center of mass of two particles, we obtain an empirical lower bound $t_B$ for the time at which $S\left(t\right)$ starts to significantly deviate from its counterpart for the no boundary case. Surprisingly, for small-sized particles, the dominant contribution to $T$ depends only on the total diffusivity $D=D_1+D_2$, in sharp contrast to the one-dimensional case. This contribution can be related to the Wiener sausage generated by a fictitious Brownian particle with diffusivity $D$. In two dimensions, the first subleading contribution to $T$ is found to depend weakly on the ratio $D_1/D_2$. We also investigate the slow-diffusion limit when $D_2\ll D_1$, and we discuss the transition to the limit when one particle is a fixed target. Finally, we give some indications to anticipate when $T$ can be expected to be a good approximation for $\langle \mathcal{T}\rangle$.

We study a quantum Otto cycle that uses a 2-qubit working substance whose Hamiltonian does not commute with itself at different times during unitary strokes. We investigate how the cycle responds to the loss of quantum adiabaticity when these strokes are operated with a finite duration. We find that qualitative features such as the possibility of counter-rotating cycles operating as heat engines, or a cycle efficiency that can increase with a decrease in the temperature difference between the baths, are resilient even to highly nonadiabatic strokes. However, cycle efficiency rapidly decreases, although it can still remain above the standard Otto value for small degrees of quantum nonadiabaticity.

Symmetry breaking ubiquitously occurs across complex systems, from phase transition in statistical physics to self-organized lane formation in pedestrian dynamics. Here, we uncover spontaneous symmetry breaking in a simple model of ride-sharing adoption. We analyze how collective interactions among ride-sharing users to avoid detours in shared rides give rise to spontaneous symmetry breaking and pattern formation in the adoption dynamics. These dynamics result in bistability of high homogeneous and partial heterogeneous adoption states, potentially limiting the population-wide adoption of ride sharing. Our results provide a framework to understand real-world adoption patterns of ride sharing in complex urban settings and support the (re)design of ride-sharing services and incentives for sustainable shared mobility.

A major challenge within material science is the proper modeling of force transmission through fragmenting materials under compression. A particularly demanding material is sea ice, which on small scales is an anisotropic material with quasibrittle characteristics under failure. Here we use the particle-based model HiDEM and laboratory-scale experiments on saline ice to develop a material model for fragmenting ice. The material behavior of the HiDEM model-ice, and the experiments are compatible on force transmission and fragmentation if: (i) the typical HiDEM glacier-scale particle size of meters is brought down to millimeters corresponding to the grain size of the laboratory ice, (ii) the often used HiDEM lattice structure is replaced by a planar random structure with an anisotropy in the direction normal to the randomized plane, and (iii) the instant tensile and bending failure criterion, used in HiDEM on glacier scale, is replaced by a cohesive softening failure potential for energy dissipation. The main outcomes of this exercise is that many of the, more or less, traditional ice modeling schemes are proven to be incomplete. In particular, local crushing of ice is not valid as a generic failure mode for fragmented ice under compression. Rather, shear failure, as described by Mohr-Coulomb theory is demonstrated to be the dominant failure mode.

Rayleigh-Taylor mixing in the presence of a third component with intermediate density is investigated through three-dimensional large-eddy simulation (LES) with a high-order compact finite-difference code. Two configurations are considered: (1) a symmetric configuration in which the Atwood number between the heavy and intermediate components matches the Atwood number between the intermediate and light components and (2) an asymmetric configuration in which the Atwood number between the heavy and intermediate components is an order of magnitude greater than the Atwood number between the intermediate and light components. Mass fraction covariances are extracted, and proposed Reynolds-averaged Navier-Stokes (RANS) closures for density-specific-volume and density-mass-fraction covariances are evaluated in an a priori fashion. In addition, a multicomponent extension of the $k-\varphi -L-a-V$ RANS model [Morgan, Phys. Rev. E 104, 015107 (2021)] is presented which includes model equations for the upper-triangular elements of the mass fraction covariance matrix. This model, referred to as the $k-\varphi -L-a-\mathcal{C}$ model, is compared against results from LES and against other RANS models. Profiles of average mass fraction, mass-fraction covariance, and density-specific-volume covariance obtained with the $k-\varphi -L-a-\mathcal{C}$ model are found to agree well with LES data. Finally, the impact of three-component turbulent mixing on average reaction rate is investigated in both premixed and nonpremixed cases by heating the mixing layer and allowing it to undergo thermonuclear (TN) burn. A closure model for average reaction rate is proposed for use with the $k-\varphi -L-a-\mathcal{C}$ model, and when this model is applied, improved agreement is obtained between LES and RANS in total TN neutron production.

The exchange-correlation of electrons, as a fundamental effect in quantum mechanics, plays an important role in the collective motions of electrons in warm dense matter. We derive the quantum kinetic equations based on the time-dependent Kohn-Sham equation. By using a temperature-dependent functional for the exchange correlation, the excitations of electrostatic waves are analyzed under the adiabatic local density approximation (ALDA). We find that the influences of the exchange-correlation effect on the group velocity of electrostatic waves can be as high as $10\%$ when both the density and temperature are low. Moreover, we also compare the results obtained by using ALDA-based kinetic theory, exchange kinetic theory, and quantum hydrodynamics, and discuss the differences among them.

Semi-Markov processes generalize Markov processes by adding temporal memory effects as expressed by a semi-Markov kernel. We recall the path weight for a semi-Markov trajectory and the fact that thermodynamic consistency in equilibrium imposes a crucial condition called direction-time independence for which we present an alternative derivation. We prove a thermodynamic uncertainty relation that formally resembles the one for a discrete-time Markov process. The result relates the entropy production of the semi-Markov process to mean and variance of steady-state currents. We prove a further thermodynamic uncertainty relation valid for semi-Markov descriptions of coarse-grained Markov processes that emerge by grouping states together. A violation of this inequality can be used as an inference tool to conclude that a given semi-Markov process cannot result from coarse graining an underlying Markov one. We illustrate these results with representative examples.

The spin-1/2 Ising-Heisenberg model on martini and martini-diced lattices is exactly solved using a star-triangle transformation, which affords an exact mapping correspondence to an effective spin-1/2 Ising model on a triangular lattice. The ground-state phase diagram of both investigated quantum spin models display two spontaneously ordered ferromagnetic phases and one macroscopically degenerate disordered phase. In contrast to a classical ferromagnetic phase where the spontaneous magnetization of the Ising as well as Heisenberg spins acquire fully saturated values, the spontaneous magnetization of the Heisenberg spins is subject to a quantum reduction to one-third of its saturated value within a quantum ferromagnetic phase. The spontaneous magnetization and logarithmic divergence of the specific heat as the most essential features of both ferromagnetic phases disappear whenever the investigated quantum spin model is driven to the highly degenerate disordered phase. The disordered phase with nonzero residual entropy originates either from a geometric spin frustration caused by antiferromagnetic interactions or more strikingly it may also alternatively arise from a competition of the ferromagnetic Ising and Heisenberg interactions of easy-axis and easy-plane type, respectively. All three available ground states coexist together at a single triple point, around which anomalous magnetic and thermodynamic behavior can be detected.

Complex nonlinear dynamics are ubiquitous in many fields. Moreover, we rarely have access to all of the relevant state variables governing the dynamics. Delay embedding allows us, in principle, to account for unobserved state variables. Here we provide an algebraic approach to delay embedding that permits explicit approximation of error. We also provide the asymptotic dependence of the first-order approximation error on the system size. More importantly, this formulation of delay embedding can be directly implemented using a recurrent neural network (RNN). This observation expands the interpretability of both delay embedding and the RNN and facilitates principled incorporation of structure and other constraints into these approaches.

In the framework of physical reservoir computing (RC), machine learning algorithms designed for digital computers are executed using analog computerlike nonlinear physical systems that can provide energy-efficient computational power for predicting time-dependent quantities that can be found using nonlinear differential equations. We suggest a bubble-based RC (BRC) system that combines the nonlinearity of an acoustic response of a cluster of oscillating gas bubbles in water with a standard echo state network (ESN) algorithm that is well suited to forecast chaotic time series. We confirm the plausibility of the BRC system by numerically demonstrating its ability to forecast certain chaotic time series similarly to or even more accurately than ESN.

The focus of structural balance theory is dedicated to social benefits, while in a real network individual benefits sometimes are the focus as well. The Strauss's model addresses individual benefits besides the social one with a simple assumption that all individual benefits are equivalent. Therefore, the results show that the competition between two terms leads to a phase transition between individual and social benefits and there is a critical point $\mathrm{CP}$ that represents a first-order phase transition in the network. Concerning a real network of relations, individuals adjust the strength of their relationships based on the benefits they acquire from them. Hence, by addressing heterogeneity in the individual interactions, we study a modified version of Strauss's model in which the first term represents the heterogeneous individual benefits by ${\theta }_{ij}$, and the coefficient of the second term, $\alpha$, measures the strength of social benefits. Our studies show that there is a region where the triangles are in a crumpled state rather than being dispersed in the network; furthermore, increasing the heterogeneity of individual benefits results in the narrower region of the crumpled state. Outside of the mentioned region, the network is a mixture of links and triangles and the value of $\alpha$ determines whether the individual benefit or social benefit dominates. For the small value of $\alpha$, the individual benefit dominates, whereas in the large value of $\alpha$, the social benefit dominates.

Identifying influential subpopulations in metapopulation epidemic models has far-reaching potential implications for surveillance and intervention policies of a global pandemic. However, there is a lack of methods to determine influential nodes in metapopulation models based on a rigorous mathematical background. In this study, we derive the message-passing theory for metapopulation modeling and propose a method to determine influential spreaders. Based on our analysis, we identify the most dangerous city as a potential seed of a pandemic when applied to real-world data. Moreover, we particularly assess the relative importance of various sources of heterogeneity at the subpopulation level, e.g., the number of connections and mobility patterns, to determine properties of spreading processes. We validate our theory with extensive numerical simulations on empirical and synthetic networks considering various mobility and transmission probabilities. We confirm that our theory can accurately predict influential subpopulations in metapopulation models.