Mean-field Approach Research Articles

Heteronuclear Magnetisms with Ultracold Spinor Bosonic Gases in Optical Lattices

Motivated by recent realizations of spin-1 NaRb mixtures in the experiments [Phys. Rev. Lett. 114, 255301 (2015); Phys. Rev. Lett. 128, 223201 (2022)], we investigate heteronuclear magnetism in the Mott-insulating regime. Different from the identical mixtures where the boson statistics only admits even parity states from angular momentum composition, for heteronuclear atoms in principle all angular momentum states are allowed, which can give rise to new magnetic phases. While various magnetic phases can be developed over these degenerate spaces, the concrete symmetry breaking phases depend on not only the degree of degeneracy but also the competitions from many-body interactions. We unveil these rich phases using the bosonic dynamical mean-field theory approach. These phases are characterized by various orders, including spontaneous magnetization order, spin magnitude order, singlet pairing order, and nematic order, which may coexist specially in the regime with odd parity. Finally we address the possible parameter regimes for observing these spin-ordered Mott phases.

Critical behavior of a mixed-spin Ising model on a square–octagon lattice

The magnetic behavior and phase diagrams of a mixed spin-5/2 and spin-3/2 Ising system on a two-dimensional square–octagon lattice were investigated using the mean-field approach based on the statistical Bogoliubov inequality. Two ground-state phase diagrams were plotted in two different planes. We also examined the effects of exchange interactions, crystal field, and external magnetic field on the magnetizations and phase diagrams of the system. First- and second-order phase transitions and isolated critical points were obtained for certain physical conditions. The influence of various physical system parameters on the blocking temperature was also clarified.

GAFF-Based Polarizable Force Field Development and Validation for Ionic Liquids.

Ionic liquids (ILs) have been used in many applications, including gas separations, electrochemistry, lubrication, and catalysis. Understanding how the different properties of ILs are related to their chemical structure and composition is crucial for these applications. Experimental investigations often provide limited insights and can be tedious in exploring a range of state points. Therefore, molecular simulations have emerged as a powerful tool that not only offers a microscopic perspective but also enables rapid screening and prediction of physical properties. The accuracy of these predictions, however, depends on the quality of the intermolecular potentials (force fields) used. The widely used classical fixed charge models, such as GAFF, OPLS, and CL&P, are popular due to their simplicity and computational efficiency. However, it has been shown that the use of integer charges with these classical models leads to sluggish dynamics. The use of scaled charge models can improve the dynamics, but these mean-field approaches are unable to account for polarization effects explicitly. Several different approaches have been proposed to include polarizability in IL force fields. In this work, we follow the protocol of the CL&Pol model to develop a Drude oscillator model based on the GAFF force field (Goloviznina, K., et al. J. Chem. Theory Comput. 2019, 15, 5858). We compare the performance of the model for eight imidazolium- and pyrrolidinium-based ILs against that of other models. We find that the new model provides reasonable estimations of density, self-diffusivity, and structural properties for these ILs and suggests a relatively simple way of extending the general GAFF model to more ILs.

Simulating selected magnetic properties of Tb x Pr1− x Al2, a magnetocaloric compound

Tb x Pr1−x Al2 are ferrimagnetic materials exhibiting magnetocaloric effect that have gained considerable attention due to their potential use as an alternative in refrigeration, magnetic sensors and in information storage technology. Here using the mean field approach numerical simulations were conducted for x = 0.1, 0.2, 0.3, 0.4, 0.5, and 0.75, to analyze selected physical properties, such as x-ray and neutron powder diffraction, magnetization and heat capacity. The simulations successfully reproduced the experimental data providing a comprehensive characterization and improved understanding of this family of compound.

Surface effect on the phase separation in thin liquid crystal film as seen from a mean-field approach

The effect of the interplay between attractive nonlocal surface interactions and attractive pair long-range intermolecular couplings on molecular structures of liquid crystals confined in thin cells with flat solid surfaces has been studied. It has been shown, in the framework of the extended McMillan′s mean field theory, that in the confined LC system a complex orientational and translational ordering of molecules can be induced. Several scenarios of the formation of local smectic A, nematic, and isotropic phases have been shown to coexist in a certain temperature range. Due to the nonlocality of the surface interaction, an unexpected complexity of the spatial arrangement of these local phases along the direction of the normal to the surface may exhibit itself.

Cubic elasticity of porous materials produced by additive manufacturing: experimental analyses, numerical and mean-field modelling

Although the elastic properties of porous materials depend mainly on the volume fraction of pores, the details of pore distribution within the material representative volume are also important and may be the subject of optimisation. To study their effect, experimental analyses were performed on samples made of a polymer material with a predefined distribution of spherical voids, but with various porosities due to different pore sizes. Three types of pore distribution with cubic symmetry were considered and the results of experimental analyses were confronted with mean-field estimates and numerical calculations. The mean-field ‘cluster’ model is used in which the mutual interactions between each of the two pores in the predefined volume are considered. As a result, the geometry of pore distribution is reflected in the anisotropic effective properties. The samples were produced using a 3D printing technique and tested in the regime of small strain to assess the elastic stiffness. The digital image correlation method was used to measure material response under compression. As a reference, the solid samples were also 3D printed and tested to evaluate the polymer matrix stiffness. The anisotropy of the elastic response of porous samples related to the arrangement of voids was assessed. Young’s moduli measured for the additively manufactured samples complied satisfactorily with modelling predictions for low and moderate pore sizes, while only qualitatively for larger porosities. Thus, the low-cost additive manufacturing techniques may be considered rather as preliminary tools to prototype porous materials and test mean-field approaches, while for the quantitative and detailed model validation, more accurate additive printing techniques should be considered. Research paves the way for using these computationally efficient models in optimising the microstructure of heterogeneous materials and composites.

Adsorption on nanoparticles with surface defects: mean field and energy level approaches.

In this work two theoretical approximations, the so-called theoretical approach of energy levels and an extension of the modified mean field approach (TAEL and MMFA, respectively) are applied to the study of surface decoration of modified nanostructures like crystalline nanoparticles. The surface of the nanoparticles is modified by the irreversible random deposition of defects consisting in isolated atoms. Such deposition is carried out until a certain surface density is reached, leaving the rest of the sites available for a second species to adsorb. Through the formulation of the integral equation, the theoretical approaches permit obtaining the adsorption isotherms and the compressibility of the adlayer. The main difference between the two approaches is the degree of details considered in their mathematical formulations: TAEL takes in account all the energy levels meanwhile MMFA only an average. The degree of precision and usefulness of both theories were evaluated in comparison with Monte Carlo simulations in the grand canonical assembly. Several cases were studied: attractive and repulsive lateral interactions and different fraction of defects. The effects of the nanoscale were considered for different types and sizes of nanoparticles. By calculating an integral error, we are able to affirm that TAEL reproduces all the properties of the analyzed quantities from the reference simulated curves. On the other hand, the MMFA performance is good only for a certain limited range of the parameters, however the strength is in the mathematical simplicity compared to TAEL.

Mixed quantum/classical theory (MQCT) approach to the dynamics of molecule-molecule collisions in complex systems.

We developed a general theoretical approach and a user-ready computer code that permit study of the dynamics of collisional energy transfer and ro-vibrational energy exchange in complex molecule-molecule collisions. The method is a mixture of classical and quantum mechanics. The internal ro-vibrational motion of collision partners is treated quantum mechanically using a time-dependent Schrödinger equation that captures many quantum phenomena including state quantization and zero-point energy, propensity and selection rules for state-to-state transitions, quantum symmetry and interference phenomena. A significant numerical speed up is obtained by describing the translational motion of collision partners classically, using the Ehrenfest mean-field trajectory approach. Within this framework a family of approximate methods for collision dynamics is developed. Several benchmark studies for diatomic and triatomic molecules, such as H2O and ND3 collided with He, H2 and D2, show that the results of MQCT are in good agreement with full-quantum calculations in a broad range of energies, especially at high collision energies where they become nearly identical to the full quantum results. Numerical efficiency of the method and massive parallelism of the MQCT code permit us to embrace some of the most complicated collisional systems ever studied, such as C6H6 + He, CH3COOH + He and H2O + H2O. Application of MQCT to the collisions of chiral molecules such as CH3CHCH2O + He, and to molecule-surface collisions is also possible and will be pursued in the future.

Multi-population dissolution in confined active fluids.

Autonomous out-of-equilibrium agents or cells in suspension are ubiquitous in biology and engineering. Turning chemical energy into mechanical stress, they generate activity in their environment, which may trigger spontaneous large-scale dynamics. Often, these systems are composed of multiple populations that may reflect the coexistence of multiple species, differing phenotypes, or chemically varying agents in engineered settings. Here, we present a new method for modeling such multi-population active fluids subject to confinement. We use a continuum multi-scale mean-field approach to represent each phase by its first three orientational moments and couple their evolution with those of the suspending fluid. The resulting coupled system is solved using a parallel adaptive level-set-based solver for high computational efficiency and maximal flexibility in the confinement geometry. Motivated by recent experimental work, we employ our method to study the spatiotemporal dynamics of confined bacterial suspensions and swarms dominated by fluid hydrodynamic effects. Our in silico explorations reproduce observed emergent collective patterns, including features of active dissolution in two-population active-passive swarms, with results clearly suggesting that hydrodynamic effects dominate dissolution dynamics. Our work lays the foundation for a systematic characterization and study of collective phenomena in natural or synthetic multi-population systems such as bacteria colonies, bird flocks, fish schools, colloid swimmers, or programmable active matter.

DEX Specs: A Mean-Field Approach to DeFi Currency Exchanges

DEX Specs: A Mean-Field Approach to DeFi Currency Exchanges

Mean-field theory approach to three-dimensional nematic phase transitions in microtubules.

Microtubules are dynamic intracellular fibers that have been observed experimentally to undergo spontaneous self-alignment. We formulate a three-dimensional (3D) mean-field theory model to analyze the nematic phase transition of microtubules growing and interacting within a 3D space, then make a comparison with computational simulations. We identify a control parameter G_{eff} and predict a unique critical value G_{eff}=1.56 for which a phase transition can occur. Furthermore, we show both analytically and using simulations that this predicted critical value does not depend on the presence of zippering. The mean-field theory developed here provides an analytical estimate of microtubule patterning characteristics without running time-consuming simulations and is a step towards bridging scales from microtubule behavior to multicellular simulations.

Effective electrical conductivity of random resistor networks generated using a Poisson–Voronoi tessellation

We studied the effective electrical conductivity of dense random resistor networks (RRNs) produced using a Voronoi tessellation when its seeds are generated by means of a homogeneous Poisson point process in the two-dimensional Euclidean space. Such RRNs are isotropic and in average homogeneous; however, local fluctuations of the number of edges per unit area are inevitable. These RRNs may mimic, e.g., crack-template-based transparent conductive films. The RRNs were treated within a mean-field approach. We found an analytical dependency of the effective electrical conductivity on the number of conductive edges (resistors) per unit area, nE. The effective electrical conductivity is proportional to nE when nE≫1.

Magnetic Anisotropy Tailoring by 5d-Doping in (Fe,Co)5SiB2 Alloys

Band-structure calculations were performed using the spin-polarized relativistic Korringa–Kohn–Rostoker (SPR-KKR) band-structure method, determining intrinsic magnetic properties, such as magnetic moments, magnetocrystalline anisotropy energy (MAE), and Curie temperatures, of Fe5−x−yCoxMySiB2 (M = Re, W) alloys. The general gradient approximation (GGA) for the exchange–correlation potential and the atomic sphere approximation (ASA) were used in the calculations. Previous studies have shown that for Fe5SiB2, the easy magnetization direction is in-plane, but it turns axial for Co-doping in the range 1 < x ≤ 2.5 (y = 0). Furthermore, studies have shown that 5d-doping enhances the MAE by enabling the strong spin–orbit coupling of Fe–3d and M–5d states. The aim of the present theoretical calculations was to find the dependence of the anisotropy constant K1 for combined Co- and M-doping, building a two-dimensional (2D) map of K1 for 0 ≤ x ≤ 2 and 0 ≤ y ≤ 1. Similar theoretical 2D maps for magnetization and Curie temperature vs. Co and M content (M = W and Re) were built, allowing for the selection of alloy compositions with enhanced values of uniaxial anisotropy, magnetization, and Curie temperature. The magnetic properties of the Fe4.1W0.9SiB2 alloy that meet the selection criteria for axial anisotropy K1 > 0.2 meV/f.u., Curie temperature Tc > 800 K determined by the mean-field approach, and magnetization µ0Ms > 1 T are discussed.

Synaptic Changes in Pallidostriatal Circuits Observed in the Parkinsonian Model Triggers Abnormal Beta Synchrony with Accurate Spatio-temporal Properties across the Basal Ganglia.

Excessive oscillatory activity across basal ganglia (BG) nuclei in the β frequencies (12-30 Hz) is a hallmark of Parkinson's disease (PD). While the link between oscillations and symptoms remains debated, exaggerated β oscillations constitute an important biomarker for therapeutic effectiveness in PD. The neuronal mechanisms of β-oscillation generation however remain unknown. Many existing models rely on a central role of the subthalamic nucleus (STN) or cortical inputs to BG. Contrarily, neural recordings and optogenetic manipulations in normal and parkinsonian rats recently highlighted the central role of the external pallidum (GPe) in abnormal β oscillations, while showing that the integrity of STN or motor cortex is not required. Here, we evaluate the mechanisms for the generation of abnormal β oscillations in a BG network model where neuronal and synaptic time constants, connectivity, and firing rate distributions are strongly constrained by experimental data. Guided by a mean-field approach, we show in a spiking neural network that several BG sub-circuits can drive oscillations. Strong recurrent STN-GPe connections or collateral intra-GPe connections drive γ oscillations (>40 Hz), whereas strong pallidostriatal loops drive low-β (10-15 Hz) oscillations. We show that pathophysiological strengthening of striatal and pallidal synapses following dopamine depletion leads to the emergence of synchronized oscillatory activity in the mid-β range with spike-phase relationships between BG neuronal populations in-line with experiments. Furthermore, inhibition of GPe, contrary to STN, abolishes oscillations. Our modeling study uncovers the neural mechanisms underlying PD β oscillations and may thereby guide the future development of therapeutic strategies.

Message-passing approach to higher-order percolation

Hypergraph describes real-world networks widely because it captures pairwise and multiple nodes’ interactions. Various kinds of damages, such as network attacks, hardware malfunctions, and communication disruptions, may impair the function of those real-world systems. We propose a generalized higher-order percolation model to investigate the robustness of the hypergraph, in which the nodes and hyperedges were randomly removed with probabilities. An accurate approach to studying the higher-order percolation model should overcome non-local tree-like structures and higher-order interactions, which makes the classical mean-field approach invalid. To this end, we develop a message-passing approach in which we first transform the hypergraph into a factor graph then develop a message-passing approach on the factor graph. Through extensive experimental studies on both artificial and real-world hypergraphs, our theory can accurately predict numerical results. The experimental data demonstrate that our theory achieves average accuracy rates in calculating giant connected component (GCC) size of 99.87% for artificial loopless hypergraphs, 99.24% for artificial hypergraphs with loops, and 99.65% for real-world hypergraphs. Our findings provide another way to understand the robustness of hypergraphs, and also provide certain ideas for studying complex systems in various fields.

Effects of optogenetic astroglia activation in modulation of synaptic transmission and rhythmogenesis of the hippocampus

In the field of neuronal signal transduction research, astroglia have become increasingly important. Astrocytes, or astroglial cells, are recognized as the third element in synaptic transmission regulation. This complex is conventionally known as a tripartite synapse. A multitude of mathematical models are being developed to investigate the function of astroglia in the tripartite synapse. Neurobiological experiments were carried out to confirm the dynamics observed in the previously proposed mean field activity mathematical model [1]. The experiments focused on evaluating the effect of optogenetic activation of astrocytes on synaptic transmission in C57BL/6 inbred mouse hippocampal slices. One month prior to the experiments, the AAV GFAP ChR2 eYFP virus was injected into the lateral ventricles of the brains of the experimental mice. This procedure was crucial for the expression of astrocyte-specific photosensitive channels, also known as channel rhodopsin. The patch-clamp method was used to conduct experiments on both the experimental mice and a control group of mice that did not receive virus injections. Spontaneous neuronal activity (local field potentials) and synaptic currents (GABA currents) were simultaneously recorded in the experiments. After activating astrocytes that expressed the photosensitive channel, an increase in GABAergic currents became evident during the transmission of synaptic signals between neurons. This suggests that astrocytes likely modulate synaptic transmission by releasing a gliotransmitter into the synaptic cleft. Thus, the study confirmed the presence of mean field activity patterns in a phenomenological model that describes the dynamics of a population of neurons. This model was developed based on the Tsodyks–Markram model and considers the primary attributes of neuron-glial interaction through a tripartite synapse. The model includes the short-term synaptic plasticity of the Tsodyks–Markram model and the astrocyte potential of synaptic transmission. The activation of astrocytes results in a diverse range of dynamic modes that describe various patterns of network activity under the mean field approach framework. Currently, multiple experimental hypotheses exist regarding astrocytes’ release of a gliotransfer into the synaptic cleft and its specific type. Alternatively, a complex cascade of sequential activation of glutamateergic and GABAergic receptors may occur. Additional experimental work is necessary to assess the pharmacological contribution of channels and transporters involved in modulating synaptic transmission during astrocyte optogenetic activation. The obtained results will refine the mathematical model of mean field neuronal activity to increase its biological plausibility. The methodology used involves identifying sparse nonlinear dynamical systems from data by solving for the system’s equations of motion [2]. These equations are reconstructed from noisy measurement data, allowing for a more accurate representation of the dynamic system. The only assumption regarding the arrangement of a dynamical system is that there exist only a handful of significant factors regulating the dynamics, leading to equations that are sparse within the realm of potential functions. Sparse regression is used to ascertain the minimum number of terms required in dynamic equations for precise data representation. This strategy enables the construction of mathematical models that are both as accurate as feasible, and maximally uncomplicated, eliminating the need for retraining. Notably, this method is suitable for parameterized systems and systems subjected to external influences or changes over time. The two approaches were used to forecast the dynamics of the mean field of a neural population. The accuracy of the forecast was subsequently evaluated.

Ehrenfest modeling of cavity vacuum fluctuations and how to achieve emission from a three-level atom

A much-needed solution for the efficient modeling of strong coupling between matter and optical cavity modes is offered by mean-field mixed quantum–classical dynamics, where a classical cavity field interacts self-consistently with quantum states of matter through Ehrenfest’s theorem. We previously introduced a modified mean-field approach, referred to as decoupled mean-field (DC-MF) dynamics, wherein vacuum fluctuations of the cavity field are decoupled from the quantum-mechanical ground state as a means to resolve an unphysical drawing of energy from the vacuum fluctuations by a two-level atom. Here, we generalize DC-MF dynamics for an arbitrary number of (nondegenerate) atomic levels and show that it resolves an unphysical lack of emission from a three-level atom predicted by conventional mean-field dynamics. We furthermore show DC-MF to provide an improved description of reabsorption and (resonant) two-photon emission processes.

A thermodynamical model of non-deterministic computation in cortical neural networks

Neuronal populations in the cerebral cortex engage in probabilistic coding, effectively encoding the state of the surrounding environment with high accuracy and extraordinary energy efficiency. A new approach models the inherently probabilistic nature of cortical neuron signaling outcomes as a thermodynamic process of non-deterministic computation. A mean field approach is used, with the trial Hamiltonian maximizing available free energy and minimizing the net quantity of entropy, compared with a reference Hamiltonian. Thermodynamic quantities are always conserved during the computation; free energy must be expended to produce information, and free energy is released during information compression, as correlations are identified between the encoding system and its surrounding environment. Due to the relationship between the Gibbs free energy equation and the Nernst equation, any increase in free energy is paired with a local decrease in membrane potential. As a result, this process of thermodynamic computation adjusts the likelihood of each neuron firing an action potential. This model shows that non-deterministic signaling outcomes can be achieved by noisy cortical neurons, through an energy-efficient computational process that involves optimally redistributing a Hamiltonian over some time evolution. Calculations demonstrate that the energy efficiency of the human brain is consistent with this model of non-deterministic computation, with net entropy production far too low to retain the assumptions of a classical system.

Correlation functions for confined wormlike chains

Polymer models describing the statistics of biomolecules under confinement have applications to a wide range of single-molecule experimental techniques and give insight into biologically relevant processes in vivo. In this paper, we determine the transverse position and bending correlation functions for a wormlike chain confined within slits and cylinders (with one and two confined dimensions, respectively) using a mean-field approach that enforces rigid constraints on average. We show the theoretical predictions accurately capture the statistics of a wormlike chain from Monte Carlo simulations in both confining geometries for both weak and strong confinement. We also show that the longitudinal correlation function is accurately computed for a chain confined to a slit and leverages the accuracy of the model to suggest an experimental technique to infer the (often unobservable) transverse statistics from the (directly observable) longitudinal end-to-end distance.

Quantum reaction-limited reaction-diffusion dynamics of annihilation processes.

We investigate the quantum reaction-diffusion dynamics of fermionic particles which coherently hop in a one-dimensional lattice and undergo annihilation reactions. The latter are modelled as dissipative processes which involve losses of pairs 2A→∅, triplets 3A→∅, and quadruplets 4A→∅ of neighboring particles. When considering classical particles, the corresponding decay of their density in time follows an asymptotic power-law behavior. The associated exponent in one dimension is different from the mean-field prediction whenever diffusive mixing is not too strong and spatial correlations are relevant. This specifically applies to 2A→∅, while the mean-field power-law prediction just acquires a logarithmic correction for 3A→∅ and is exact for 4A→∅. A mean-field approach is also valid, for all the three processes, when the diffusive mixing is strong, i.e., in the so-called reaction-limited regime. Here we show that the picture is different for quantum systems. We consider the quantum reaction-limited regime and we show that for all the three processes power-law behavior beyond mean field is present as a consequence of quantum coherences, which are not related to space dimensionality. The decay in 3A→∅ is further, highly intricate, since the power-law behavior therein only appears within an intermediate time window, while at long times the density decay is not power law. Our results show that emergent critical behavior in quantum dynamics has a markedly different origin, based on quantum coherences, to that applying to classical critical phenomena, which is, instead, solely determined by the relevance of spatial correlations.