Analytic Mean Field Research Articles

Asynchronous and Coherent Dynamics in Balanced Excitatory-Inhibitory Spiking Networks.

Dynamic excitatory-inhibitory (E-I) balance is a paradigmatic mechanism invoked to explain the irregular low firing activity observed in the cortex. However, we will show that the E-I balance can be at the origin of other regimes observable in the brain. The analysis is performed by combining extensive simulations of sparse E-I networks composed of N spiking neurons with analytical investigations of low dimensional neural mass models. The bifurcation diagrams, derived for the neural mass model, allow us to classify the possible asynchronous and coherent behaviors emerging in balanced E-I networks with structural heterogeneity for any finite in-degree K. Analytic mean-field (MF) results show that both supra and sub-threshold balanced asynchronous regimes are observable in our system in the limit N >> K >> 1. Due to the heterogeneity, the asynchronous states are characterized at the microscopic level by the splitting of the neurons in to three groups: silent, fluctuation, and mean driven. These features are consistent with experimental observations reported for heterogeneous neural circuits. The coherent rhythms observed in our system can range from periodic and quasi-periodic collective oscillations (COs) to coherent chaos. These rhythms are characterized by regular or irregular temporal fluctuations joined to spatial coherence somehow similar to coherent fluctuations observed in the cortex over multiple spatial scales. The COs can emerge due to two different mechanisms. A first mechanism analogous to the pyramidal-interneuron gamma (PING), usually invoked for the emergence of γ-oscillations. The second mechanism is intimately related to the presence of current fluctuations, which sustain COs characterized by an essentially simultaneous bursting of the two populations. We observe period-doubling cascades involving the PING-like COs finally leading to the appearance of coherent chaos. Fluctuation driven COs are usually observable in our system as quasi-periodic collective motions characterized by two incommensurate frequencies. However, for sufficiently strong current fluctuations these collective rhythms can lock. This represents a novel mechanism of frequency locking in neural populations promoted by intrinsic fluctuations. COs are observable for any finite in-degree K, however, their existence in the limit N >> K >> 1 appears as uncertain.

Optimal collision avoidance in swarms of active Brownian particles

The effectiveness of collective navigation of biological or artificial agents requires to accommodate for contrasting requirements, such as staying in a group while avoiding close encounters and at the same time limiting the energy expenditure for maneuvering. Here, we address this problem by considering a system of active Brownian particles in a finite two-dimensional domain and ask what is the control that realizes the optimal tradeoff between collision avoidance and control expenditure. We couch this problem in the language of optimal stochastic control theory and by means of a mean-field game approach we derive an analytic mean-field solution, characterized by a second-order phase transition in the alignment order parameter. We find that a mean-field version of a classical model for collective motion based on alignment interactions (Vicsek model) performs remarkably close to the optimal control. Our results substantiate the view that observed group behaviors may be explained as the result of optimizing multiple objectives and offer a theoretical ground for biomimetic algorithms used for artificial agents.

Mean field homogenization methods for strand composites

Abstract A common method for dealing with in-process manufacturing waste is to cut the prepreg scrap into rectangular strands, and use these strands to produce composites. The properties of the strand composite depend on a number of factors, including the aspect ratio and orientation of the strand. Predictive simulations are required to determine the stiffness of such materials so that optimal composites can be fabricated without extensive experimental testing. In this work, we present a hybrid method for predicting the stiffness of strand composites by using a combination of finite element and analytic mean field homogenization techniques. The proposed model combines the versatility of finite element models and the ease of computation of analytic schemes. This method can be used to model strands of any shape and orientation distribution; it also presents possibilities for detailed parametric analysis. The proposed method yields predictions that are consistent with experimental measurements and provides insights about testing protocols and observed scatter in experimental results.

宽广温度压强下Rb<sub>x</sub>C<sub>60</sub> (x = 3,4,6)的热力学性质

本文基于自由体积理论(FVT)应用解析平均场近似的方法对RbxC60 (x = 3,4,6)的热力学性质进行研究。根据双指数势模型和文献中的压缩实验数据，拟合得到三套势参数，并对相应的势函数曲线进行了比较分析。在宽广温度和压强下对三种碱金属掺杂富勒烯RbxC60 (x = 3,4,6)的热力学量，包括热膨胀、体积模量、等容热容量、亥姆霍兹自由能进行了计算和分析。我们采用的方法不仅计算简单而且结果与文献中实验数据非常一致。 Based on the free volume theory (FVT), the thermodynamic properties of RbxC60 (x = 3,4,6) are studied by means of analytic mean field approximation. According to the double exponential po-tential model and the compression experimental data in the literature, three sets of potential pa-rameters are obtained, and the corresponding potential function curves are compared and ana-lyzed. The thermodynamic properties of three alkali metal doped fullerene RbxC60 (x = 3,4,6), in-cluding thermal expansion, bulk modulus, constant volume heat capacity and Helmholtz free en-ergy, were calculated and analyzed under wide temperature and pressure. Our method is not only simple in calculation, but also in good agreement with the experimental data in the literature.

Quantization of [formula omitted]-Fermi–Pasta–Ulam lattice with nearest and next-nearest neighbor interactions

We quantize the β-Fermi–Pasta–Ulam (FPU) model with nearest and next-nearest neighbor interactions using a number conserving approximation and a numerically exact diagonalization method. Our numerical mean field bi-phonon spectrum shows excellent agreement with the analytic mean field results of Ivić and Tsironis (2006), except for the wave vector at the midpoint of the Brillouin zone. We then relax the mean field approximation and calculate the eigenvalue spectrum of the full Hamiltonian. We show the existence of multi-phonon bound states and analyze the properties of these states by varying the system parameters. From the calculation of the spatial correlation function we then show that these multi-phonon bound states are particle like states with finite spatial correlation. Accordingly we identify these multi-phonon bound states as the quantum equivalent of the breather solutions of the corresponding classical FPU model. The four-phonon spectrum of the system is then obtained and its properties are studied. We then generalize the study to an extended range interaction and consider the quantization of the β-FPU model with next-nearest-neighbor interactions. We analyze the effect of the next-nearest-neighbor interactions on the eigenvalue spectrum and the correlation functions of the system.

Thermodynamic properties of solid face centered cubic Rb3C60 at high temperature and pressure

Analytic equation of state and thermodynamic quantities of solid fcc Rb3C60 are derived by using an analytic mean field potential method. For intermolecular forces, the double-exponential potential is utilized. Four potential parameters are determined by fitting experimental compression data of Rb3C60 up to 14 GPa at 296 K. Various physical quantities including isothermals, thermal expansion, isochoric heat capacity, Helmholtz free energy and internal energy are calculated and analyzed. Calculated results are consistent with available experimental data in literature. Furthermore, spinodal temperature for Rb3C60 is found to be 2,860 K. Results verify that analytic mean field potential method is a useful approach to consider the anharmonic effect at high temperatures. Numerous reasonable predictions and the change trend of the properties for Rb3C60 at high temperature and pressure have been given.

Analytic equation of state and thermodynamic properties for solid FCC C60 over a wide range of pressures and temperatures

As an analytic approximation to the free volume theory (FVT), the analytic mean field potential (AMFP) approach is applied to the multi-exponential model solid. The analytic expressions for the equation of state (EOS) and thermodynamic quantities are derived. The formalism for the case of the double-exponential (DE) model is applied to the solid fcc C60. The four potential parameters are determined by fitting the experimental data of cohesive energy, lattice constant and compression curve of solid C60. The numerical results calculated show that the Girifalco potential is too hard and gives compression curve prominently deviated from experiments at high pressure, whereas the DE potential can well describe the thermo-physical properties of C60 system within wide pressure range. The calculated variation of lattice constant versus temperature relationship and compression curve for solid C60 by using the DE potential agree well with experimental data available. Especially the AMFP with DE potential predicted that the fcc C60 solid is stable up to 2440 K, the spinodal temperature, in good agreement with the result from molecular dynamic (MD) simulation. The AMFP is a useful approach to consider the anharmonic effects at high temperature for solid C60.

Small-World Network Spectra in Mean-Field Theory

Collective dynamics on small-world networks emerge in a broad range of systems with their spectra characterizing fundamental asymptotic features. Here we derive analytic mean-field predictions for the spectra of small-world models that systematically interpolate between regular and random topologies by varying their randomness. These theoretical predictions agree well with the actual spectra (obtained by numerical diagonalization) for undirected and directed networks and from fully regular to strongly random topologies. These results may provide analytical insights to empirically found features of dynamics on small-world networks from various research fields, including biology, physics, engineering, and social science.

Pairing and coherence order parameters in a three-component ultracold Fermi gas

We investigate the mean-field ground state of a homogeneous three-component attractive Fermi gas at zero temperature. This is achieved by deriving a set of Bogoliubov--de Gennes (BdG) equations of the three-component system, including pairing both order parameters ${\ensuremath{\Delta}}_{\mathit{ij}}$ and coherence order parameters $\ensuremath{\langle}{\ensuremath{\psi}}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\psi}}_{j}\ensuremath{\rangle}$ ($i\ensuremath{\ne}j$), where ${\ensuremath{\psi}}_{j}$ is the field operator for spin level $j$. Ward-Takahashi identities are obtained to constrain these order parameters. In addition, we present an explicit analytic mean-field solution for symmetric systems and verify that the quasiparticle excitations consist of both gapped and gapless spectra, which correspond to the excitations of paired and unpaired atoms. We further point out that the omission of $\ensuremath{\langle}{\ensuremath{\psi}}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{\ensuremath{\psi}}_{j}\ensuremath{\rangle}$ in BdG equations could lead to an overestimation of ${\ensuremath{\Delta}}_{\mathit{ij}}$ in the strong coupling regime.

Kinetic rougheninglike transition with finite nucleation barrier

Recent observations of the growth of protein crystals have identified two different growth regimes. At low supersaturation, the surface of the crystal is smooth and increasing in size due to the nucleation of steps at defects and the subsequent growth of the steps. At high supersaturation, nucleation occurs at many places simultaneously, the crystal surface becomes rough, and the growth velocity increases more rapidly with increasing supersaturation than in the smooth regime. Kinetic roughening transitions are typically assumed to be due to the vanishing of the barrier for two-dimension nucleation on the surface of the crystal. We show here, by means of both analytic mean-field models and kinetic Monte Carlo simulations, that a transition between different growth modes reminiscent of kinetic roughening can also arise as a kinetic effect occurring at finite nucleation barriers.

Thermodynamic properties of cubic boron nitride based on an analytic mean field approach

We present a detailed derivation of the analytic expressions for the equation of state (EOS) and internal energy of Morse model solids based on an analytic mean field potential (AMFP) method. The formalism is applied to cubic boron nitride (c-BN). One set of potential parameters are determined by fitting the experimental P-V-T data of c-BN up to 160 GPa at 295 K and 80 GPa in the range 500–900 K. Various physical quantities including the isothermals, thermal expansion, isochoric heat capacity, Helmholtz free energy and internal energy are calculated and analyzed. The theoretical results are consistent with the available experimental data and those calculated by others. These results presented in this paper verify that the AMFP method is a useful approach to consider the anharmonic effect at high temperature. Numerous reasonable predictions and the change trend of the properties for c-BN at extreme conditions have been given.

Thermodynamic properties of solid FCC C84 based on an analytic mean field approach

The analytic mean field potential (AMFP) approach is applied to the poly-exponential model solid. The analytic expressions for the Helmholtz free energy, internal energy and equation of state (EOS) are derived. The formalism for the case of the double-exponential (DE) model is applied to fcc C84. One set of potential parameters are determined by fitting the experimental compression data of C84 up to 9.24 GPa at ambient temperature (297 K). The equilibrium distance and well depth for C60, C70 and C84 molecules are plotted. The thermophysical properties including the isothermals, thermal expansion, isochoric heat capacity, Helmholtz free energy and internal energy are calculated and analyzed. The theoretical results agree well with the experimental data available of C84. Basing the results of our calculations, we may also predict the behaviors of C84 at extreme conditions.

Analytic equation of state and thermodynamic properties of diamond based on an analytic mean field approach

The analytic mean field potential (AMFP) method is applied to diamond. The analytic expressions for the Helmholtz free energy, internal energy and equation of state (EOS) are derived. The formalism for the case of the Morse potential is used in this work. The six potential parameters are determined by fitting the compression experimental data of diamond at ambient temperature. The thermophysical properties including the isothermals, thermal expansion, isochoric heat capacity, Helmholtz free energy and internal energy are calculated and analyzed. The theoretical results agree well with the experimental data available for diamond. These results presented in this paper verify that the AMFP method is a useful approach to consider the anharmonic effects at high temperature. Numerous reasonable predictions and the change trend of the properties for diamond at extreme conditions have been given.

Analytic Equation of State and Thermodynamic Properties for α-, β-, and γ-Si3N4Based on Analytic Mean Field Approach

Analytic Equation of State and Thermodynamic Properties for α-, β-, and γ-Si₃N₄Based on Analytic Mean Field Approach

Analytic equation of state for solids with multi-exponential potential based on analytic mean field potential approach and applied to the epsilon phase of solid oxygen

The thermodynamic properties of the ε phase of solid oxygen are studied by using the analytic mean field approach (AMFP). Analytic expressions for the Helmholtz free energy, internal energy and equation of state of solid oxygen have been derived based on the multi-exponential potential. The formulism for the case of double-exponential (DE) model is applied to the ε phase of solid oxygen. Its four potential parameters are determined through fitting the experimental compression data of the ε phase of solid oxygen. Numerical results of the pressure dependence of the volume calculated by using the AMFP are in good agreement with the original experimental data. This suggests that the AMFP is a useful approach to study the thermodynamic properties of the ε phase of solid oxygen. Furthermore, we predict the variation of the volume, lattice parameters and intermolecular distances with pressure, and some thermodynamic quantities versus volume, at several higher temperatures.

Analytic Equation of State and Thermodynamic Properties of Solid FCC C61D2 Based on an Analytic Mean Field Approach

Analytic expressions for the equation of state and internal energy of poly-exponential solids are derived based on the analytic mean field potential (AMFP) method. The formalism is applied to fcc C61D2. Two sets of potential parameters are determined by fitting the experimental compression data of C61D2 up to 1 GPa at 343 and 307 K, respectively. The difference between the two sets of parameters is small. Whereas the difference between the potential of the C61D2 molecules and that of the C60 molecules is fairly prominent, the conclusion is different from that in the literature, and the reason is unclear at present. The thermo-physical properties including the isothermals, thermal expansion, isochoric heat capacity, Helmholtz free energy and internal energy are calculated and analyzed. The theoretical results agree well with the experimental data available for solid C61D2.

Fiber Depolymerization

Depolymerization is, by definition, a crucial process in the reversible assembly of various biopolymers. It may also be an important factor in the pathology of sickle cell disease. If sickle hemoglobin fibers fail to depolymerize fully during passage through the lungs then they will reintroduce aggregates into the systemic circulation and eliminate or shorten the protective delay (nucleation) time for the subsequent growth of fibers. We study how depolymerization depends on the rates of end- and side-depolymerization, k end and k side, which are, respectively, the rates at which fiber length is lost at each end and the rate at which new breaks appear per unit fiber length. We present both an analytic mean field theory and supporting simulations showing that the characteristic fiber depolymerization time τ = 1 / k end k side depends on both rates, but not on the fiber length L, in a large intermediate regime 1 ≪ k side L 2/ k end ≪ ( L/ d) 2, with d the fiber diameter. We present new experimental data which confirms that both mechanisms are important and shows how the rate of side depolymerization depends strongly on the concentration of CO, acting as a proxy for oxygen. Our theory remains rather general and could be applied to the depolymerization of an entire class of linear aggregates, not just sickle hemoglobin fibers.

Spatio-temporal community dynamics induced by frequency dependent interactions

A mathematical model incorporating the effects of possibly asymmetric frequency dependent interactions is proposed. Model predictions for an idealized two-species annual plant community with asymmetric linear frequency dependence are explored using (i) analytic mean field equilibrium predictions, (ii) deterministic, discrete-time, finite-population, mean field predictions, and (iii) stochastic, discrete-time, cellular automata predictions for a variety of sizes of the spatial interaction and dispersal neighborhoods. We define species interaction factors, ranging from 0 to 1, which incorporate both frequency independent and frequency dependent terms. The maximum competitive ability of a species is reduced unless species frequency is optimal based on species-specific frequency dependence coefficients, ranging from −1 to +1. Assuming that maximum competitive ability is identical for two species, they can coexist indefinitely when they have equal absolute magnitude or both have sufficiently negative frequency dependence. Although smaller scales of spatial interactions reduce the region of the parameter space in which stable coexistence is predicted, the time to extinction of one species can be significantly increased or decreased by the locality of interactions, depending on whether the losing species has positive or negative frequency dependence, respectively. The sensitivity to initial conditions in the community at large is dramatically reduced as the spatial scale of interactions is decreased. As a consequence, smaller spatial interaction neighborhoods increase the ability of introduced species to invade established communities in regions of the parameter space not predicted by mean field approximations. In the “loser positive, winner positive” regions, smaller scales of interaction dramatically increased invasiveness. In the “loser positive, winner negative” regions of the parameter space, invasion success decreases, but time to extinction of the resident species during successful invasions increases, with an increase in the spatial scale of interactions. The “loser negative, winner positive” regions were relatively insensitive to initial conditions, so invasion success was relatively high at a variety of spatial scales. Surprisingly, invasions in parts of this region are most often successful with intermediate neighborhood sizes, although the maximum time that the losing species could persist before being driven to extinction increases with an increase in the spatial scale of interactions. These results are explained by understanding cluster formation and density and the relative local inter-specific dynamics in cluster interiors, exteriors, and boundaries. In summary, frequency dependent interactions, and the spatial scale on which these interactions occur, can have a big impact on spatio-temporal community dynamics, with implications regarding species coexistence and invasiveness. The model proposed herein provides a theoretical framework for studying frequency dependent interactions that may shed light on spatio-temporal dynamics in real ecological communities.

Statistical mechanics of the self-gravitating gas: Thermodynamic limit, instabilities and phase diagrams

We show that the self-gravitating gas at thermal equilibrium has an infinite volume limit in the three ensembles (GCE, CE, MCE) when ( N , V ) → ∞ , keeping N / V 1 / 3 fixed, that is, with η ≡ G m 2 N V 1 / 3 T fixed. We develop Monte Carlo simulations, analytic mean field methods (MF) and low density expansions. We compute the equation of state and find it to be locally p ( r → ) = T ρ V ( r → ) , that is a local ideal gas equation of state. The system is in a gaseous phase for η < η T = 1.51024 … and collapses into a very dense object for η > η T in the CE with the pressure becoming large and negative. The isothermal compressibility diverges at η = η T . We compute the fluctuations around mean field for the three ensembles. We show that the particle distribution can be described by a Haussdorf dimension 1 < D < 3 . To cite this article: H.J. de Vega, N.G. Sanchez, C. R. Physique 7 (2006).

Three-dimensional normal grain growth: Monte Carlo Potts model simulation and analytical mean field theory

Abstract By means of extensive Monte Carlo Potts model simulations it is shown that three-dimensional normal grain growth can adequately be described by an average scaled growth law, which is a quadratic function of the relative grain size. A generalized analytic mean-field theory based on this growth law yields a scaled grain size distribution function that is in excellent agreement with the simulation results.