Maxwellian Distribution Research Articles

CONVERGENCE RATE OF ALGORITHM FOR SOLVING LINEAR EQUATIONS BY QUANTUM ANNEALING

Various iterative algorithms for solving the linear equation ax = b using a quantum computer operating on the principle of quantum annealing are studied. Assuming that the result produced by the computer is described by the Boltzmann distribution, conditions under which these algorithms converge are obtained and an estimate of their convergence rate is provided. Application of this approach for algorithms that use an infinite number of qubits and a small number of qubits is considered.

Evaluating the Importance of Conformers for Understanding the Vacuum-Ultraviolet Spectra of Oxiranes: Experiment and Theory.

Vacuum-ultraviolet (VUV) absorption spectroscopy enables electronic transitions that offer the unambiguous identification of molecules. As target molecules become more complex, multifunctional species present a great challenge to both experimental and computational spectroscopy. This research reports both experimental and theoretical studies of oxiranes. Computationally, the nuclear ensemble approach has been used to accurately predict experimental spectra for a variety of molecules. However, this approach incurs great computational cost, as ensembles generally consist of thousands of geometries. The present study aims to drastically reduce the ensemble by evaluating the significance of the conformers to the predicted spectra. This approach was applied to 11 substituted oxiranes using the Conformer Rotamer Ensemble Sampling Tool (CREST) of Grimme to generate an ensemble of unique conformers determined by their Boltzmann populations. Five TD-DFT functionals (BMK, CAM-B3LYP, M06-2X, MN15, ωB97X-D) and EOM-CCSD were used to simulate the spectrum of each substituted oxirane ensemble. Computed spectra were then compared to the experiment using both qualitative and quantitative metrics. Based on these metrics, it was observed that certain conformers may not be necessary to characterize this set of oxiranes despite the temperature (323 K) of the experiment. A single conformer can then be used with TD-DFT and EOM-CCSD to replicate the experimental spectra of these medium-sized combustion species.

Cascades Towards Noise-Induced Transitions on Networks Revealed Using Information Flows

Complex networks, from neuronal assemblies to social systems, can exhibit abrupt, system-wide transitions without external forcing. These endogenously generated “noise-induced transitions” emerge from the intricate interplay between network structure and local dynamics, yet their underlying mechanisms remain elusive. Our study unveils two critical roles that nodes play in catalyzing these transitions within dynamical networks governed by the Boltzmann–Gibbs distribution. We introduce the concept of “initiator nodes”, which absorb and propagate short-lived fluctuations, temporarily destabilizing their neighbors. This process initiates a domino effect, where the stability of a node inversely correlates with the number of destabilized neighbors required to tip it. As the system approaches a tipping point, we identify “stabilizer nodes” that encode the system’s long-term memory, ultimately reversing the domino effect and settling the network into a new stable attractor. Through targeted interventions, we demonstrate how these roles can be manipulated to either promote or inhibit systemic transitions. Our findings provide a novel framework for understanding and potentially controlling endogenously generated metastable behavior in complex networks. This approach opens new avenues for predicting and managing critical transitions in diverse fields, from neuroscience to social dynamics and beyond.

Kinetic simulations of nuclear burning plasmas in inertial confinement fusions taking into account the large-angle collisions

Although ignition had been achieved at the National Ignition Facility (NIF), recent observations of the experiments indicate novel physics that beyond theoretical predictions emerge, e.g., the neutron analysis of experiments has revealed deviations from the Maxwellian distributions in ion relative kinetic energies of burning plasmas, with the surprising emergence of supra-thermal deuterium and tritium (DT) ions that fall outside the predictions of macroscopic statistical hydrodynamic models. Via our newly developed hybrid-particle-in-cell code, incorporating the newly-proposed model of large-angle collisions, we infer that this could be attributed to the increased significance of large-angle collisions among DT ions and α-particles in the burning plasma. Extensive and unprecedented kinetic investigations into the implications of large-angle collisions in the burning plasma have yield several key findings, including an ignition moment promotion by ∼10ps, the presence of supra-thermal ions below an energy threshold, and approximately twice the expected deposition of α-particles densities. The rationality of our findings is confirmed through the congruency between the neutron spectral moment analyses conducted by the NIF and our kinetic simulations, both highlighting progressively widening disparities between neutron spectral moment analyses and hydrodynamics predictions, which becomes more pronounced as the yield increases. Our kinetic simulations with large-angle collisions not only provide novel insights for experiment interpretation but also open new research opportunities for the largely unexplored regime of the nuclear burning plasmas, which are distinguished by their exceptionally high energy densities and hold immense potential for illuminating the intricate physics that underpins the evolution of the early universe.

Multi-lattice sampling of quantum field theories via neural operator-based flows

Abstract We consider the problem of sampling lattice field configurations on a lattice from the Boltzmann distribution corresponding to some action. Since such densities arise as approximationw of an underlying functional density, we frame the task as an instance of operator learning. We propose to approximate a time-dependent neural operator whose time integral provides a mapping between the functional distributions of the free and target theories.
 Once a particular lattice is chosen, the neural operator can be discretized to a finite-dimensional, time-dependent vector field which in turn induces a continuous normalizing flow between finite dimensional distributions over the chosen lattice. This flow can then be trained to be a diffeormorphism between the discretized free and target theories on the chosen lattice, and, by construction, can be evaluated on different discretizations of spacetime. We experimentally validate the proposal on the 2-dimensional $\phi^4$-theory to explore to what extent such operator-based flow architectures generalize to lattice sizes they were not trained on, and show that pretraining on smaller lattices can lead to a speedup over training directly on the target lattice size.

EWMA control chart framework for efficient Maxwell quality characteristic monitoring: An application to the aerospace industry

The Maxwell distribution is extensively employed in statistical modeling to analyze data, notably in the fields of chemistry, astrophysics, demography, actuarial sciences, economics, industry, and engineering. Recently, the application of the Maxwell data generating process (DGP) has focused extensively on the domain of statistical control charts. The V chart and VSQ chart are often used to identify unexpected changes in the distributional shift of the Maxwell process. The VSQ chart offers considerably better performance than the V chart for detecting moderate to large shifts in the scale parameter. However, the VSQ chart adopts the fundamental structure of the Shewhart monitoring scheme and is insensitive to small alterations in the target parameter. We propose a new control chart, namely, the Maxwell exponentially weighted moving average (MXEWMA) chart, for improved monitoring of quality attributes that are assumed to conform to Maxwell data generation. The factors used to design the parameters of the proposed chart are computed at different false alarm probabilities and across various sample sizes. The effectiveness of the suggested scheme is considered in terms of the different features of the run length (RL) distribution, including the average, median and standard deviation. A comparative study of the MXEWMA chart with the existing VSQ chart was performed across various sample sizes. The comparative analysis showed that the MXEWMA chart is an effective alternative and performs well in detecting reasonably small shifts in the parameter. Simulated data are employed to describe the computational procedure of the MXEWMA scheme. The simulation analysis demonstrated that the MXEWMA chart outperforms the existing method in identifying slight changes in the studied parameters. A real dataset is also considered to support the theoretical part of the work.

Implications of [formula omitted]-deformed statistics on stellar stability

Examination of Chandrasekhar’s equilibrium and stability condition for stars in the context of q-deformed Maxwell–Boltzmann statistics leads to the derivation of a new analytical formula that extends its classical case. This derivation assumes that the kinetic description of stellar matter conforms to a generalized Maxwell–Boltzmann distribution. It has been found that the maximum allowable radiation pressure at the center of a star for a given mass depends on the deformation parameter q. The classical Chandrasekhar condition is recovered in this framework when q equals 1.

Pulsational mode stability in complex EiBI-gravitating polarized astroclouds with (r,q)-distributed electrons

The pulsational mode of gravitational collapse (PMGC) originating from the combined gravito-electrostatic interaction in complex dust molecular clouds (DMCs) is a canonical mechanism leading to the onset of astronomical structure formation dynamics. A generalized semi-analytic model is formulated to explore the effects of the Eddington-inspired Born-Infeld (EiBI) gravity, non-thermal (r,q)-distributed electrons, and dust-polarization force on the PMGC stability concurrently. The thermal ions are treated thermo-statistically with the Maxwellian distribution law and the non-thermal electrons with the (r,q)-distribution law. The constitutive partially ionized dust grains are modeled in the fluid fabric. A spherical normal mode analysis yields a generalized linear PMGC dispersion relation. Its oscillatory and propagation characteristics are investigated in a judicious numerical platform. It is found that an increase in the polarization force and positive EiBI parameter significantly enhances the instability, causing the DMC collapse and vice versa. The electron non-thermality spectral parameters play as vital stabilizing factors, and so on. Its reliability and applicability are finally outlined in light of astronomical predictions previously reported in the literature.

Optimal mixing via tensorization for random independent sets on arbitrary trees

Abstract We study the mixing time of the single-site update Markov chain, known as the Glauber dynamics, for generating a random independent set of a tree. Our focus is obtaining optimal convergence results for arbitrary trees. We consider the more general problem of sampling from the Gibbs distribution in the hard-core model where independent sets are weighted by a parameter $\lambda \gt 0$ ; the special case $\lambda =1$ corresponds to the uniform distribution over all independent sets. Previous work of Martinelli, Sinclair and Weitz (2004) obtained optimal mixing time bounds for the complete $\Delta$ -regular tree for all $\lambda$ . However, Restrepo, Stefankovic, Vera, Vigoda, and Yang (2014) showed that for sufficiently large $\lambda$ there are bounded-degree trees where optimal mixing does not hold. Recent work of Eppstein and Frishberg (2022) proved a polynomial mixing time bound for the Glauber dynamics for arbitrary trees, and more generally for graphs of bounded tree-width. We establish an optimal bound on the relaxation time (i.e., inverse spectral gap) of $O(n)$ for the Glauber dynamics for unweighted independent sets on arbitrary trees. We stress that our results hold for arbitrary trees and there is no dependence on the maximum degree $\Delta$ . Interestingly, our results extend (far) beyond the uniqueness threshold which is on the order $\lambda =O(1/\Delta )$ . Our proof approach is inspired by recent work on spectral independence. In fact, we prove that spectral independence holds with a constant independent of the maximum degree for any tree, but this does not imply mixing for general trees as the optimal mixing results of Chen, Liu, and Vigoda (2021) only apply for bounded-degree graphs. We instead utilize the combinatorial nature of independent sets to directly prove approximate tensorization of variance via a non-trivial inductive proof.

A theoretical reaction rate model of a chemical exothermic decomposition surface from an external gas

In this study, kinetic theory was used to derive an equation of state for an exothermic decomposition surface caused by an external gas. The new model is physical, and its quantities have been physically interpreted. A new concept of interaction probability was used to describe the co-volume. This concept describes the interaction between the decomposition surface and the external gas to derive an expression for the linear reaction rate based on kinetic theory. The interaction probability is associated with the particle density of the gases provided by the Lennard-Jones potential and temperature. The Maxwell–Boltzmann distribution was used to establish the initial decomposition conditions based on the concepts of autoignition and activation energy.The aim of this study was to investigate when burning is a decomposition reaction in which the decomposing molecule contains oxygen and can be used as input to designing fuel cells, rocket motors, and propellants. Therefore, HMX and PETN were used as empirical data, and the new linear reaction rate model provided a good approximation and predicted the burn rate data. The model was compared with Vieille’s law vb=aPn for the normal pressure range. However, the model goes beyond the law and provides good predictions of burn rates with high pressures found in diamond anvil experiments.

Kr Adsorption in Porous Carbons: Temperature-Dependent Experimental and Computational Studies†.

The temperature dependence of the adsorption energy of krypton adsorption on activated carbon materials was studied by experiment and simulation. Adsorption isotherms were measured at temperatures from 250 to 330 K and analyzed with Henry's law. The adsorption energy determined from these measurements was found to weaken by more than 10% in this range. Slit pore widths for simulations in this work were modeled by the removal of integral numbers of planes in graphite. Vibrational dynamics of the krypton adsorbate and the carbon atom adsorbent were calculated with the stochastic temperature-dependent effective potential (sTDEP) method, using energetics from density functional theory (DFT) with the many-body dispersion energy method (MBD). Thermal displacements of carbon atoms had a negligible effect on the adsorption energy. The width of the slit pore had the greatest effect on the surface dynamics and the energies of the adsorbate atoms at different temperatures. Assuming a distribution of pore widths, the Boltzmann distribution of site occupancies causes a large weakening of the thermally averaged adsorption energy at higher temperatures.

Radial Distribution of Electron Quasi-thermal Noise in the Inner Heliosphere

The electron population in the solar wind plasma can be described with three different components: a core, a halo, and a magnetic field aligned strahl. The electron quasi-thermal noise (QTN) is investigated by using an electron population model consisting of a core with a Maxwellian distribution and a halo with a kappa distribution, based on the empirical equations for electron density and temperature and the index for the kappa halo. The power spectra of the electron QTN are calculated at different heliocentric radial distances from 10 to 200 R s . The dependence of the QTN spectrum and effective Debye length on model parameters, including the ratio of the halo to the core for the density and temperature, the kappa index, and the antenna length, is further discussed. The results show that the electron QTN spectrum consists of a plateau in the low-frequency band f < f pt , a peak at the total plasma frequency f pt , and a rapidly decreasing part in the high-frequency band f > f pt . The QTN peak and plateau level continuously decrease as the radial distance increases, with the peak’s shape changing due to the variation of the kappa index. Although the model parameters are variable, the QTN plateau level presents less than an order of change with these parameters changing greatly, and only a monotonic change of the plateau is shown when the parameters are close to the practical situation. The results can provide a reference for future deep-space exploration in the inner heliosphere, and also for the design of detectors.

Accurate, Fast, and Non-Destructive Net Charge Measurement of Levitated Nanoresonators Based on Maxwell Speed Distribution Law

Nanoscale resonant devices based on optical tweezers are widely used in the field of precision sensing. In the process of driving the nanoresonator based on the Coulomb force, the real-time, precise regulation of the charge carried by the charged resonator is essential for continuous manipulation. However, the accuracy of the existing charge measurement methods for levitated particles is low, and these methods cannot meet the needs of precision sensing. In this study, a novel net charge measurement protocol for levitated particles based on spatial speed statistics is proposed. High-precision mass measurement based on Maxwell’s rate distribution law is the basis for improving the accuracy of charge measurement, and accurate measurement of net charge can be achieved by periodic electric field driving. The error of net charge measurement is less than 7.3% when the pressure is above 0.1 mbar, while it can be less than 0.76% at 10 mbar. This proposed method features real-time, high-precision, non-destructive, and in situ measurement of the net charge of particles in the medium vacuum, which provides new solutions for practical problems in the fields of high-precision sensing and nano-metrology based on levitated photodynamics.

BEGAN: Boltzmann-Reweighted Data Augmentation for Enhanced GAN-Based Molecule Design in Insect Pheromone Receptors.

Identifying small molecules that bind strongly to target proteins in rational molecular design is crucial. Machine learning techniques, such as generative adversarial networks (GAN), are now essential tools for generating such molecules. In this study, we present an enhanced method for molecule generation using objective-reinforced GANs. Specifically, we introduce BEGAN (Boltzmann-enhanced GAN), a novel approach that adjusts molecule occurrence frequencies during training based on the Boltzmann distribution exp(-ΔU/τ), where ΔU represents the estimated binding free energy derived from docking algorithms and τ is a temperature-related scaling hyperparameter. This Boltzmann reweighting process shifts the generation process toward molecules with higher binding affinities, allowing the GAN to explore molecular spaces with superior binding properties. The reweighting process can also be refined through multiple iterations without altering the overall distribution shape. To validate our approach, we apply it to the design of sex pheromone analogs targeting Spodoptera frugiperda pheromone receptor SfruOR16, illustrating that the Boltzmann reweighting significantly increases the likelihood of generating promising sex pheromone analogs with improved binding affinities to SfruOR16, further supported by atomistic molecular dynamics simulations. Furthermore, we conduct a comprehensive investigation into parameter dependencies and propose a reasonable range for the hyperparameter τ. Our method offers a promising approach for optimizing molecular generation for enhanced protein binding, potentially increasing the efficiency of molecular discovery pipelines.

1/f noise in semiconductors arising from the heterogeneous detrapping process of individual charge carriers

Abstract We propose a model of 1 / f noise in semiconductors based on the drift of individual charge carriers and their interaction with the trapping centers. We assume that the trapping centers are homogeneously distributed in the material. The trapping centers are assumed to be heterogeneous and have unique detrapping rates. We show that uniform detrapping rate distribution emerges as a natural consequence of the vacant trap depths following the Boltzmann distribution, and the detrapping process obeying Arrhenius law. When these laws apply, and if the trapping rate is low in comparison to the maximum detrapping rate, 1 / f noise in the form of Hooge’s relation is recovered. Hooge’s parameter, α H, is shown to be a ratio between the characteristic trapping rate and the maximum detrapping rate. The proposed model implies that 1 / f noise arises from the temporal charge carrier number fluctuations, not from the spatial mobility fluctuations.

A Consistent Kinetic Fokker–Planck Model for Gas Mixtures

We propose a general multi-species Fokker–Planck model. We prove consistency of our model: conservation properties, positivity of all temperatures, H-Theorem and the shape of equilibrium as Maxwell distributions with the same mean velocity and temperature. Moreover, we derive the usual macroscopic equations from the kinetic two-species BGK model and compute explicitly the exchange terms of momentum and energy.

Fragmented perspective of self-organized criticality and disorder in log gravity

We use a statistical model to discuss nonequilibrium fragmentation phenomena taking place in the stochastic dynamics of the log sector in log gravity. From the canonical Gibbs model, a combinatorial analysis reveals an important aspect of the n-particle evolution previously shown to generate a collection of random partitions according to the Ewens distribution realized in a disconnected double Hurwitz number in genus zero. By treating each possible partition as a member of an ensemble of fragmentations, and ensemble averaging over all partitions with the Hurwitz number as a special case of the Gibbs distribution, a resulting distribution of cluster sizes appears to fall as a power of the size of the cluster. Dynamical systems that exhibit a distribution of sizes giving rise to a scale-invariant power-law behavior at a critical point possess an important property called self-organized criticality. As a corollary, the log sector of log gravity is a self-organized critical system at the critical point μl = 1. A similarity between self-organized critical systems, spin glass models and the dynamics of the log sector which exhibits aging behavior reminiscent of glassy systems is pointed out by means of the Pòlya distribution, also known to classify various models of (randomly fragmented) disordered systems, and by presenting the cluster distribution in the log sector of log gravity as a distinguished member of this probability distribution. We bring arguments from a probabilistic perspective to discuss the disorder in log gravity, largely anticipated through the conjectured AdS3/LCFT2 correspondence.

On the Definition of Velocity in Discrete-Time, Stochastic Langevin Simulations

We systematically develop beneficial and practical velocity measures for accurate and efficient statistical simulations of the Langevin equation with direct applications to computational statistical mechanics and molecular dynamics sampling. Recognizing that the existing velocity measures for the most statistically accurate discrete-time Verlet-type algorithms are inconsistent with the simulated configurational coordinate, we seek to create and analyze new velocity companions that both improve existing methods as well as offer practical options for implementation in existing computer codes. The work is based on the set of GJ methods that, of all methods, for any time step within the stability criteria correctly reproduces the most basic statistical features of a Langevin system; namely correct Boltzmann distribution for harmonic potentials and correct transport in the form of drift and diffusion for linear potentials. Several new and improved velocities exhibiting correct drift are identified, and we expand on an earlier conclusion that, generally, only half-step velocities can exhibit correct, time-step independent Maxwell–Boltzmann distributions. Specific practical and efficient algorithms are given in familiar forms, and these are used to numerically validate the analytically derived expectations. One especially simple algorithm is highlighted, and the ability of one of the new on-site velocities to produce statistically correct averages for a particular damping value is specified.

A Particle Method for the Multispecies Landau Equation.

The multispecies Landau collision operator describes the two-particle, small scattering angle or grazing collisions in a plasma made up of different species of particles such as electrons and ions. Recently, a structure preserving deterministic particle method (Carrillo et al. in J. Comput. Phys. 7:100066, 2020) has been developed for the single species spatially homogeneous Landau equation. This method relies on a regularization of the Landau collision operator so that an approximate solution, which is a linear combination of Dirac delta distributions, is well-defined. Based on a weak form of the regularized Landau equation, the time dependent locations of the Dirac delta functions satisfy a system of ordinary differential equations. In this work, we extend this particle method to the multispecies case, and examine its conservation of mass, momentum, and energy, and decay of entropy properties. We show that the equilibrium distribution of the regularized multispecies Landau equation is a Maxwellian distribution, and state a critical condition on the regularization parameters that guarantees a species independent equilibrium temperature. A convergence study comparing an exact multispecies Bobylev-Krook-Wu (BKW) solution to the particle solution shows approximately 2nd order accuracy. Important physical properties such as conservation, decay of entropy, and equilibrium distribution of the particle method are demonstrated with several numerical examples.

Contour deformations for nonholomorphic actions

We show how contour deformations may be used to control the sign problem of lattice Monte Carlo calculations with nonholomorphic Boltzmann factors. Such actions arise naturally in quantum mechanical scattering problems. The approach is demonstrated in conjunction with the holomorphic gradient flow. As our central example we compute the real-time evolution of a particle in a one-dimensional analog of the Yukawa potential. Published by the American Physical Society 2024