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Journal articles on the topic 'Gaussian approximation potentials'

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Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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1

Bartók, Albert P., and Gábor Csányi. "Gaussian approximation potentials: A brief tutorial introduction." International Journal of Quantum Chemistry 115, no. 16 (April 27, 2015): 1051–57. http://dx.doi.org/10.1002/qua.24927.

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2

Klawohn, Sascha, James R. Kermode, and Albert P. Bartók. "Massively parallel fitting of Gaussian approximation potentials." Machine Learning: Science and Technology 4, no. 1 (February 16, 2023): 015020. http://dx.doi.org/10.1088/2632-2153/aca743.

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Abstract We present a data-parallel software package for fitting Gaussian approximation potentials (GAPs) on multiple nodes using the ScaLAPACK library with MPI and OpenMP. Until now the maximum training set size for GAP models has been limited by the available memory on a single compute node. In our new implementation, descriptor evaluation is carried out in parallel with no communication requirement. The subsequent linear solve required to determine the model coefficients is parallelised with ScaLAPACK. Our approach scales to thousands of cores, lifting the memory limitation and also delivering substantial speedups. This development expands the applicability of the GAP approach to more complex systems as well as opening up opportunities for efficiently embedding GAP model fitting within higher-level workflows such as committee models or hyperparameter optimisation.

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3

Bartók, Albert P., and Gábor Csányi. "Erratum: Gaussian approximation potentials: A brief tutorial introduction." International Journal of Quantum Chemistry 116, no. 13 (April 21, 2016): 1049. http://dx.doi.org/10.1002/qua.25140.

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4

Hiroshima, Fumio. "A Scaling Limit of a Hamiltonian of Many Nonrelativistic Particles Interacting with a Quantized Radiation Field." Reviews in Mathematical Physics 09, no. 02 (February 1997): 201–25. http://dx.doi.org/10.1142/s0129055x97000075.

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This paper presents a scaling limit of Hamiltonians which describe interactions of N-nonrelativistic charged particles in a scalar potential and a quantized radiation field in the Coulomb gauge with the dipole approximation. The scaling limit defines effective potentials. In one-nonrelativistic particle case, the effective potentials have been known to be Gaussian transformations of the scalar potential [J. Math. Phys.34 (1993) 4478–4518]. However it is shown that the effective potentials in the case of N-nonrelativistic particles are not necessary to be Gaussian transformations of the scalar potential.

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5

John, S. T., and Gábor Csányi. "Many-Body Coarse-Grained Interactions Using Gaussian Approximation Potentials." Journal of Physical Chemistry B 121, no. 48 (November 29, 2017): 10934–49. http://dx.doi.org/10.1021/acs.jpcb.7b09636.

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6

FUKUKAWA, K., Y. FUJIWARA, and Y. SUZUKI. "GAUSSIAN NONLOCAL POTENTIALS FOR THE QUARK-MODEL BARYON–BARYON INTERACTIONS." Modern Physics Letters A 24, no. 11n13 (April 30, 2009): 1035–38. http://dx.doi.org/10.1142/s021773230900053x.

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Gaussian nonlocal potentials for the quark-model baryon–baryon interactions are derived by using the Gauss-Legendre quadrature for the special functions. The reliability of the approximation is examined with respect to the phase shifts and the deuteron binding energy. The potential is accurate enough if one uses seven-point Gauss-Legendre quadrature.

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7

SÉNÉCHAL, DAVID. "CHAOS IN THE HERMITIAN ONE-MATRIX MODEL." International Journal of Modern Physics A 07, no. 07 (March 20, 1992): 1491–506. http://dx.doi.org/10.1142/s0217751x9200065x.

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The recursion coefficients Ri, which appear in the orthogonal polynomial method of solution for the Hermitian one-matrix model, are determined numerically for values of N up to a thousand. For some cases a chaotic behavior appears in some range, preventing a smooth flow from odd to even multicrititical models. This behavior is studied both for single-well and multiwell potentials. For multiwell potentials, the coefficients Ri generically tend toward more than one function in the N→∞ limit, and this structure is analyzed for small i using the Gaussian approximation.

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8

Demiroğlu, İlker, Yenal Karaaslan, Tuğbey Kocabaş, Murat Keçeli, Álvaro Vázquez-Mayagoitia, and Cem Sevik. "Computation of the Thermal Expansion Coefficient of Graphene with Gaussian Approximation Potentials." Journal of Physical Chemistry C 125, no. 26 (June 24, 2021): 14409–15. http://dx.doi.org/10.1021/acs.jpcc.1c01888.

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9

Exl, Lukas, Norbert J. Mauser, and Yong Zhang. "Accurate and efficient computation of nonlocal potentials based on Gaussian-sum approximation." Journal of Computational Physics 327 (December 2016): 629–42. http://dx.doi.org/10.1016/j.jcp.2016.09.045.

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10

OLSEN, R. A., and F. RAVNDAL. "EFFECTIVE POTENTIALS FOR ϕ4-THEORY IN 2+1 DIMENSIONS." Modern Physics Letters A 09, no. 28 (September 14, 1994): 2623–35. http://dx.doi.org/10.1142/s021773239400246x.

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Spontaneous symmetry breaking in ϕ4-theory in 2+1 dimensions is investigated using the Gaussian approximation. The theory stays in the symmetric phase at zero temperature as long as the bare coupling constant is below a critical value λc. When λ>λc the symmetric phase is again stable when the temperature is above a transition temperature T(λ). The obtained results are compared with the predictions of the standard one-loop effective potential.

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11

Delgadillo, Ricardo, Jianfeng Lu, and Xu Yang. "Gauge-Invariant Frozen Gaussian Approximation Method for the Schrödinger Equation with Periodic Potentials." SIAM Journal on Scientific Computing 38, no. 4 (January 2016): A2440—A2463. http://dx.doi.org/10.1137/15m1040384.

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12

Kim, M. H., and Dick K. P. Yue. "Sum-and Difference-Frequency Wave Loads on a Body in Unidirectional Gaussian Seas." Journal of Ship Research 35, no. 02 (June 1, 1991): 127–40. http://dx.doi.org/10.5957/jsr.1991.35.2.127.

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When nonlinear effects are included in the diffraction of waves by a large body, there are, at second order, interactions at the sums and differences of the component frequencies of the incident waves. In this paper, the complete deterministic and stochastic solutions of second-order (sum-and difference-frequency) wave loads in unidirectional Gaussian waves are considered. The deterministic result, namely the wave force quadratic transfer function (QTF) in bichromatic incident waves, is complete in the context of second-order diffraction theory in that all the relevant components including those due to the exact second-order potentials are obtained. Statistical properties of second-order wave excitations are then investigated using the QTF results and a two-term Volterra series model. For illustration, the exact second-order force spectra and probability distributions for the simple geometry of a truncated vertical cylinder are obtained and compared with those based on a number of existing approximation methods. It is found that second-order exciting force variances and probability of extreme values may be significantly underestimated by existing approximation methods.

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13

Koch, Wolfhard, Klaus Neymeyr, Markus Pernpointner, Barbara Schaper, and Klaus Strecker. "Simplified Non-Empirical Unrestricted Hartree-Fock Approximation (SUHF) for the Calculation of Electronic Ground State Properties of Molecules with Closed and Open Valence Shells. II. Diatomic Molecules." Zeitschrift für Naturforschung A 48, no. 7 (July 1, 1993): 834–40. http://dx.doi.org/10.1515/zna-1993-0715.

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Abstract A comparison of numerical Simplified Unrestricted Hartree-Fock (SUHF) results (electronic ground state configurations, total energies, first ionization potentials, atomic charges, atomic spin densities of diatomic molecules) with those of equivalent standard calculations suggests the applicability of the non-empirical but drastically simplified procedure. SUHF may even approach ab initio quality obtained with simple (STO-3G) contracted Gaussian basis sets.

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14

Fang, Di, and Lei Li. "Numerical approximation and fast evaluation of the overdamped generalized Langevin equation with fractional noise." ESAIM: Mathematical Modelling and Numerical Analysis 54, no. 2 (February 18, 2020): 431–63. http://dx.doi.org/10.1051/m2an/2019067.

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The generalized Langevin equation (GLE) is a stochastic integro-differential equation that has been used to describe the movement of microparticles with sub-diffusion phenomenon. It has been proved that with fractional Gaussian noise (fGn) mostly considered by biologists, the overdamped Generalized Langevin equation satisfying fluctuation dissipation theorem can be written as a fractional stochastic differential equation (FSDE). In this work, we present both a direct and a fast algorithm respectively for this FSDE model in order to numerically study ergodicity. The strong orders of convergence are proven for both schemes, where the role of the memory effects can be clearly observed. We verify the convergence theorems using linear forces, and then verify the convergence to Gibbs measure algebraically for the double well potentials in both 1D and 2D setups. Our work is new in numerical analysis of FSDEs and provides a useful tool for studying ergodicity. The idea can also be used for other stochastic models involving memory.

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15

Austen, Maggie A., Leif A. Eriksson, and Russell J. Boyd. "A density functional theory study of the free radicals NH2, NF2, NCl2, PH2, PF2, and PCl2." Canadian Journal of Chemistry 72, no. 3 (March 1, 1994): 695–704. http://dx.doi.org/10.1139/v94-094.

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The linear combination of Gaussian-type orbitals–density functional theory (LCGTO–DFT) approach is used to study geometries and hyperfine structures of a set of neutral radicals. Each of the title molecules is investigated by means of local density approximation calculations, and using the Becke–Perdew and Perdew–Wang–Perdew corrections to the exchange and correlation functionals. The effects of local vs. non-local potentials and of various basis sets are investigated. Total densities and unpaired spin densities are compared. The isotropic couplings are found to be very dependent on the type of exchange functional used, whereas the anisotropic couplings are relatively insensitive to the choice of basis set and functional. In most cases, the Perdew–Wang exchange corrections provide isotropic couplings in satisfactory agreement with experiment.

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16

James, Guillaume. "Travelling breathers and solitary waves in strongly nonlinear lattices." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2127 (July 23, 2018): 20170138. http://dx.doi.org/10.1098/rsta.2017.0138.

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We study the existence of travelling breathers and solitary waves in the discrete p -Schrödinger (DpS) equation. This model consists of a one-dimensional discrete nonlinear Schrödinger (NLS) equation with strongly nonlinear inter-site coupling (a discrete p -Laplacian). The DpS equation describes the slow modulation in time of small amplitude oscillations in different types of nonlinear lattices, where linear oscillators are coupled to nearest-neighbours by strong nonlinearities. Such systems include granular chains made of discrete elements interacting through a Hertzian potential ( p = 5/2 for contacting spheres), with additional local potentials or resonators inducing local oscillations. We formally derive three amplitude PDEs from the DpS equation when the exponent of nonlinearity is close to (and above) unity, i.e. for p lying slightly above 2. Each model admits localized solutions approximating travelling breather solutions of the DpS equation. One model is the logarithmic NLS equation which admits Gaussian solutions, and the other is fully nonlinear degenerate NLS equations with compacton solutions. We compare these analytical approximations to travelling breather solutions computed numerically by an iterative method, and check the convergence of the approximations when . An extensive numerical exploration of travelling breather profiles for p = 5/2 suggests that these solutions are generally superposed on small amplitude non-vanishing oscillatory tails, except for particular parameter values where they become close to strictly localized solitary waves. In a vibro-impact limit where the parameter p becomes large, we compute an analytical approximation of solitary wave solutions of the DpS equation. This article is part of the theme issue ‘Nonlinear energy transfer in dynamical and acoustical systems’.

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17

Nguyen, Thuong T., Eszter Székely, Giulio Imbalzano, Jörg Behler, Gábor Csányi, Michele Ceriotti, Andreas W. Götz, and Francesco Paesani. "Comparison of permutationally invariant polynomials, neural networks, and Gaussian approximation potentials in representing water interactions through many-body expansions." Journal of Chemical Physics 148, no. 24 (June 28, 2018): 241725. http://dx.doi.org/10.1063/1.5024577.

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18

Staacke, Carsten G., Tabea Huss, Johannes T. Margraf, Karsten Reuter, and Christoph Scheurer. "Tackling Structural Complexity in Li2S-P2S5 Solid-State Electrolytes Using Machine Learning Potentials." Nanomaterials 12, no. 17 (August 26, 2022): 2950. http://dx.doi.org/10.3390/nano12172950.

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The lithium thiophosphate (LPS) material class provides promising candidates for solid-state electrolytes (SSEs) in lithium ion batteries due to high lithium ion conductivities, non-critical elements, and low material cost. LPS materials are characterized by complex thiophosphate microchemistry and structural disorder influencing the material performance. To overcome the length and time scale restrictions of ab initio calculations to industrially applicable LPS materials, we develop a near-universal machine-learning interatomic potential for the LPS material class. The trained Gaussian Approximation Potential (GAP) can likewise describe crystal and glassy materials and different P-S connectivities PmSn. We apply the GAP surrogate model to probe lithium ion conductivity and the influence of thiophosphate subunits on the latter. The materials studied are crystals (modifications of Li3PS4 and Li7P3S11), and glasses of the xLi2S–(100 – x)P2S5 type (x = 67, 70 and 75). The obtained material properties are well aligned with experimental findings and we underscore the role of anion dynamics on lithium ion conductivity in glassy LPS. The GAP surrogate approach allows for a variety of extensions and transferability to other SSEs.

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19

Merchant, A. C. "The generation of rotational bands by deep, diffuse potentials." Canadian Journal of Physics 66, no. 4 (April 1, 1988): 295–301. http://dx.doi.org/10.1139/p88-045.

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Numerical integration of the single-particle Schrödinger equation containing a deep, diffuse, spherically symmetric, attractive, local potential often gives rise to bound-state energy eigenvalues, which lie on almost perfect rotational bands. Each band is characterized by a constant value of (2n + l), where n is the number of interior nodes in the radial wave function and l is the orbital angular-momentum quantum number of a member state. The constants of near-proportionality between the state energies and l(l + 1) for the various bands are considered. An expression for them derived from the Bohr–Sommerfeld quantization rule is found to give numerical results in excellent agreement with those obtained by direct numerical integration of the Schrödinger equation. However, no analytic expressions could be obtained from it, except in the most trivial cases. Also, an approximate expression for these rotational parameters is derived using the Rayleigh–Schrödinger perturbation theory up to fourth order for the case of any potential that may be expanded in ascending even powers of the radial coordinate, r, for which the perturbation expansion converges. This approximation is excellent if (2n + l + 3/2)[Formula: see text], where m is the mass appearing in the Schrödinger equation, V0 is the depth of the attractive potential, and a is a length parameter characterizing its diffuseness. The general results are illustrated with applications to some deep attractive Gaussian potentials of various widths.

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20

GAO, XIANLONG, M. POLINI, M. P. TOSI, and B. TANATAR. "EFFECT OF DISORDER ON THE INTERACTING FERMI GASES IN A ONE-DIMENSIONAL OPTICAL LATTICE." International Journal of Modern Physics B 22, no. 25n26 (October 20, 2008): 4500–4510. http://dx.doi.org/10.1142/s0217979208050255.

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Interacting two-component Fermi gases loaded in a one-dimensional (1D) lattice and subjected to a harmonic trapping potential exhibit interesting compound phases in which fluid regions coexist with local Mott-insulator and/or band-insulator regions. Motivated by experiments on cold atoms inside disordered optical lattices, we present a theoretical study of the effects of a correlated random potential on these ground-state phases. We employ a lattice version of density-functional theory within the local-density approximation to determine the density distribution of fermions in these phases. The exchange-correlation potential is obtained from the Lieb-Wu exact solution of Fermi-Hubbard model. On-site disorder (with and without Gaussian correlations) and harmonic trap are treated as external potentials. We find that disorder has two main effects: (i) it destroys the local insulating regions if it is sufficiently strong compared with the on-site atom-atom repulsion, and (ii) it induces an anomaly in the inverse compressibility at low density from quenching of percolation. For sufficiently large disorder correlation length the enhancement in the inverse compressibility diminishes.

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21

Timmermann, Jakob, Yonghyuk Lee, Carsten G. Staacke, Johannes T. Margraf, Christoph Scheurer, and Karsten Reuter. "Data-efficient iterative training of Gaussian approximation potentials: Application to surface structure determination of rutile IrO₂ and RuO₂." Journal of Chemical Physics 155, no. 24 (December 28, 2021): 244107. http://dx.doi.org/10.1063/5.0071249.

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22

Alexandrov, Artem A., Alina U. Badamshina, and Stanislav L. Ogarkov. "Quantum Gases of Dipoles, Quadrupoles and Octupoles in Gross–Pitaevskii Formalism with Form Factor." Condensed Matter 5, no. 4 (October 15, 2020): 61. http://dx.doi.org/10.3390/condmat5040061.

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Here, classical and quantum field theory of dipolar, axisymmetric quadrupolar and octupolar Bose gases is considered within a general approach. Dipole, axisymmetric quadrupole and octupole interaction potentials in the momentum representation are calculated. These results clearly demonstrate attraction and repulsion areas in corresponding gases. Then the Gross–Pitaevskii (GP) equation, which plays a key role in the present paper, is derived from the corresponding functional. The zoology of the form factors appearing in the GP equation is studied in details. The proper classes for the description of spatially non-uniform condensates form factors are chosen. In the Thomas–Fermi approximation a general solution of the GP equation with a quasilocal form factor is obtained. This solution has an interesting form in terms of a double rapidly converging series that universally includes all the interactions considered. Plots of condensate density functions for the exponential-trigonometric form factor are given. For the sake of completeness, in this paper we consider the GP equation with an optical lattice potential in the limit of small condensate densities. This limit does not distinguish between dipolar, quadrupolar and octupolar gases. An important analysis of the condensate stability, in other words the study of condensate excitations, is also performed in this paper. In the Gaussian approximation (from the Gross–Pitaevskii functional), a functional describing the perturbations of the condensate is derived in detail. This problem is an analog of the Bogolubov transformation used in the study of quantum Bose gases in operator formalism. For a probe wave function in the form of a plane wave, a spectrum of (Bogoliubov) excitations was obtained, from which an equation describing the threshold momentum for the emergence of instability was derived. An important result of this paper is the dependence of the threshold on the momentum of a stationary condensate. For completeness of the presentation, the approximating expression in the form of a rapidly converging series is obtained for the corresponding dependence, and plots of the corresponding series for the exponential-trigonometric form factor are given. Finally, in the conclusion a quantum hydrodynamic theory for dipolar, axisymmetric quadrupolar and octupolar gases is briefly presented, giving a clue to the experimental determination of the form factors.

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23

Cea, Paolo. "Gaussian effective potential beyond the gaussian approximation." Physics Letters B 236, no. 2 (February 1990): 191–95. http://dx.doi.org/10.1016/0370-2693(90)90826-r.

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24

Bernardi, Davide, and Benjamin Lindner. "A frequency-resolved mutual information rate and its application to neural systems." Journal of Neurophysiology 113, no. 5 (March 1, 2015): 1342–57. http://dx.doi.org/10.1152/jn.00354.2014.

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The encoding and processing of time-dependent signals into sequences of action potentials of sensory neurons is still a challenging theoretical problem. Although, with some effort, it is possible to quantify the flow of information in the model-free framework of Shannon's information theory, this yields just a single number, the mutual information rate. This rate does not indicate which aspects of the stimulus are encoded. Several studies have identified mechanisms at the cellular and network level leading to low- or high-pass filtering of information, i.e., the selective coding of slow or fast stimulus components. However, these findings rely on an approximation, specifically, on the qualitative behavior of the coherence function, an approximate frequency-resolved measure of information flow, whose quality is generally unknown. Here, we develop an assumption-free method to measure a frequency-resolved information rate about a time-dependent Gaussian stimulus. We demonstrate its application for three paradigmatic descriptions of neural firing: an inhomogeneous Poisson process that carries a signal in its instantaneous firing rate; an integrator neuron (stochastic integrate-and-fire model) driven by a time-dependent stimulus; and the synchronous spikes fired by two commonly driven integrator neurons. In agreement with previous coherence-based estimates, we find that Poisson and integrate-and-fire neurons are broadband and low-pass filters of information, respectively. The band-pass information filtering observed in the coherence of synchronous spikes is confirmed by our frequency-resolved information measure in some but not all parameter configurations. Our results also explicitly show how the response-response coherence can fail as an upper bound on the information rate.

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25

Sokolovsky, A. I. "Realization of the Landau definitions of effective Hamiltonian and nonequilibrium free energy in microscopic theory." Journal of Physics and Electronics 28, no. 2 (December 9, 2020): 63–74. http://dx.doi.org/10.15421/332022.

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Equilibrium fluctuations of some set of parameters in the states described by the canonical Gibbs distribution are investigated. In the theory of phase transitions of the second kind, these parameters are components of the order parameter. The microscopic realization of the Landau definition of the effective Hamiltonian of the system for studying the equilibrium fluctuations of the specified system of parameters is discussed in the terms of the probability density of their values. A general formula for this function is obtained and it is expressed through the equilibrium correlation functions of these parameters. An expression for the effective Hamiltonian in terms of deviations of the parameters from their equilibrium values is obtained. The deviations are considered small for conducting the calculations. The possibility of calculating the exact free energy of the system using the found effective Hamiltonian is discussed. In the microscopic theory, the implementation of the Landau definition of nonequilibrium thermodynamic potentials introduced in his phenomenological theory of phase transitions of the second kind is investigated. Nonequilibrium states of a fluctuating system described with some sets of parameters are considered. A general formula for nonequilibrium free energy expressed through the correlation functions of these parameters is obtained as for the effective Hamiltonian above. Like the previous case, the free energy expression via parameter deviations from the equilibrium values is obtained and small deviations are considered for calculations. The idea of the identity of the effective Hamiltonian of the system and its nonequilibrium free energy is discussed in connection with the Boltzmann distribution. The Gaussian approximation of both developed formalisms is considered. A generalization of the constructed theory for the case of spatially inhomogeneous states and the study of long-wave fluctuations are developed.

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26

Burkitt, A. N., and G. M. Clark. "Calculation of Interspike Intervals for Integrate-and-Fire Neurons with Poisson Distribution of Synaptic Inputs." Neural Computation 12, no. 8 (August 1, 2000): 1789–820. http://dx.doi.org/10.1162/089976600300015141.

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We present a new technique for calculating the interspike intervals of integrate-and-fire neurons. There are two new components to this technique. First, the probability density of the summed potential is calculated by integrating over the distribution of arrival times of the afferent post-synaptic potentials (PSPs), rather than using conventional stochastic differential equation techniques. A general formulation of this technique is given in terms of the probability distribution of the inputs and the time course of the postsynaptic response. The expressions are evaluated in the gaussian approximation, which gives results that become more accurate for large numbers of small-amplitude PSPs. Second, the probability density of output spikes, which are generated when the potential reaches threshold, is given in terms of an integral involving a conditional probability density. This expression is a generalization of the renewal equation, but it holds for both leaky neurons and situations in which there is no time-translational invariance. The conditional probability density of the potential is calculated using the same technique of integrating over the distribution of arrival times of the afferent PSPs. For inputs with a Poisson distribution, the known analytic solutions for both the perfect integrator model and the Stein model (which incorporates membrane potential leakage) in the diffusion limit are obtained. The interspike interval distribution may also be calculated numerically for models that incorporate both membrane potential leakage and a finite rise time of the postsynaptic response. Plots of the relationship between input and output firing rates, as well as the coefficient of variation, are given, and inputs with varying rates and amplitudes, including inhibitory inputs, are analyzed. The results indicate that neurons functioning near their critical threshold, where the inputs are just sufficient to cause firing, display a large variability in their spike timings.

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27

COTĂESCU, ION I., PAUL GRĂVILĂ, and MARIUS PAULESCU. "PSEUDO-GAUSSIAN OSCILLATORS." International Journal of Modern Physics C 19, no. 10 (October 2008): 1607–15. http://dx.doi.org/10.1142/s012918310801314x.

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A new family of one-dimensional quantum models is proposed in terms of new potentials with a Gaussian asymptotic behavior but approaching to the potential of the harmonic oscillator when x → 0. These models give rise to approximative equidistant energy levels of bound states and resonances as it is shown using numerical calculations based on an efficient method of generating functionals.

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28

Lu, Wen-Fa, Chul Koo Kim, and Kyun Nahm. "Sine-Gordon effective potential beyond Gaussian approximation." Physics Letters B 546, no. 1-2 (October 2002): 177–88. http://dx.doi.org/10.1016/s0370-2693(02)02659-x.

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29

YAHYA, G. A. "DEFORMED OPTICAL POTENTIAL ANALYSES FOR P+12C." International Journal of Modern Physics E 13, no. 04 (August 2004): 851–66. http://dx.doi.org/10.1142/s0218301304002508.

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The high energy single folding optical potential approximation is studied to calculate the differential cross-section for proton elastic scattering of 12 C at 156 MeV and 1440 MeV and 12 C for state 2+ (4.44 MeV) at 1440 MeV. A Gaussian nuclear density distribution was used for the proton and Gaussian and Brink nuclear density distributions for the 12 C target. We used the following three effects to derive twelve different methods for the central optical potential: (i) Love and Franey and the Gaussian amplitudes, with the Brink and one-term Gaussian nuclear density distributions, (ii) Pauli correlation in the Gaussian amplitude with these densities, (iii) coupling channels on the differential cross-sections in proton elastic scattering of 12 C at 1440 MeV with single channel calculations using these amplitudes, nuclear density distributions and Pauli correlation in the Gaussian amplitude. A new numerical technique was performed to solve the deformed optical potential equations using computational programs.

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30

Bingham, N., and Tasmin Symons. "Gaussian random fields: with and without covariances." Theory of Probability and Mathematical Statistics 106 (May 16, 2022): 27–40. http://dx.doi.org/10.1090/tpms/1163.

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We begin with isotropic Gaussian random fields, and show how the Bochner–Godement theorem gives a natural way to describe their covariance structure. We continue with a study of Matérn processes on Euclidean space, spheres, manifolds and graphs, using Bessel potentials and stochastic partial differential equations (SPDEs). We then turn from this continuous setting to approximating discrete settings, Gaussian Markov random fields (GMRFs), and the computational advantages they bring in handling large data sets, by exploiting the sparseness properties of the relevant precision (concentration) matrices.

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31

Zhang, Cunzhi, and Qiang Sun. "Gaussian approximation potential for studying the thermal conductivity of silicene." Journal of Applied Physics 126, no. 10 (September 9, 2019): 105103. http://dx.doi.org/10.1063/1.5119281.

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32

Fujikake, So, Volker L. Deringer, Tae Hoon Lee, Marcin Krynski, Stephen R. Elliott, and Gábor Csányi. "Gaussian approximation potential modeling of lithium intercalation in carbon nanostructures." Journal of Chemical Physics 148, no. 24 (June 28, 2018): 241714. http://dx.doi.org/10.1063/1.5016317.

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33

George, Janine, Geoffroy Hautier, Albert P. Bartók, Gábor Csányi, and Volker L. Deringer. "Combining phonon accuracy with high transferability in Gaussian approximation potential models." Journal of Chemical Physics 153, no. 4 (July 28, 2020): 044104. http://dx.doi.org/10.1063/5.0013826.

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34

Leboeuf, M., A. M. Köster, and D. R. Salahub. "Approximation of the molecular electrostatic potential in a gaussian density functional method." Theoretical Chemistry Accounts: Theory, Computation, and Modeling (Theoretica Chimica Acta) 96, no. 1 (April 28, 1997): 23–30. http://dx.doi.org/10.1007/s002140050199.

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35

Harris, John G. "The Wavefield Radiated Into an Elastic Half-Space by a Transducer of Large Aperture." Journal of Applied Mechanics 55, no. 2 (June 1, 1988): 398–404. http://dx.doi.org/10.1115/1.3173689.

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The wavefield radiated into an elastic half-space by an ultrasonic transducer, as well as the radiation admittance of the transducer coupled to the half-space, are studied. Two models for the transducer are used. In one an axisymmetric, Gaussian distribution of normal traction is imposed upon the surface, while in the other a uniform distribution of normal traction is imposed upon a circular region of the surface, leaving the remainder free of traction. To calculate the wavefield, each wave emitted by the transducer is expressed as a plane wave multiplied by an asymptotic power series in inverse powers of the aperture’s (scaled) radius. This reduces the wave equations satisfied by the compressional and shear potentials to their parabolic approximations. The approximations to the radiated waves are accurate at a depth where the wavefield remains well collimated.

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36

Lahkar, Jugal, Rashidul Hoque, and D. K. Choudhury. "Masses of heavy–light mesons with two-loop static potential in a variational approach." Modern Physics Letters A 34, no. 14 (May 10, 2019): 1950106. http://dx.doi.org/10.1142/s0217732319501062.

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In this work, we investigate the masses of a few heavy–light mesons with variational method, taking into account the two-loop effects in the static Cornell potential. Specifically, we consider two wave functions, Gaussian and Coulomb. Our analysis suggests that phenomenologically such approach is more successful in the heavy flavored meson sector than the stationary state perturbation theory and other approximation methods.

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37

Boukahil, A., and D. L. Huber. "Coherent potential approximation for the absorption spectra of a two-dimensional Frenkel exciton system with Gaussian diagonal disorder." Modern Physics Letters B 28, no. 32 (December 30, 2014): 1450251. http://dx.doi.org/10.1142/s0217984914502510.

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We investigate the optical absorption and the density of states of a Frenkel exciton system on a square lattice with nearest-neighbor interactions and a Gaussian distribution of transition frequencies (i.e. Gaussian diagonal disorder). Results are presented for the absorption and the density of states of direct and indirect edge systems for a range of variances. There is reasonable agreement with the corresponding finite array calculations of Schreiber and Toyozawa. The existence of an Urbach tail is also investigated.

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38

Sharma, A., and M. Sheinman. "Analytical approximation for chemical potential in organic materials with Gaussian density of states." Journal of Physics D: Applied Physics 46, no. 12 (February 21, 2013): 125106. http://dx.doi.org/10.1088/0022-3727/46/12/125106.

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39

Fabčič, Tomaž, Jörg Main, and Günter Wunner. "Fictitious-Time Wave-Packet Dynamics in Atomic Systems." Zeitschrift für Naturforschung A 65, no. 10 (October 1, 2010): 871–76. http://dx.doi.org/10.1515/zna-2010-1015.

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Gaussian wave packets (GWPs) are well suited as basis functions to describe the time evolution of arbitrary wave functions in systems with non-singular smooth potentials. They are rare in atomic systems on account of the singular behaviour of the Coulomb potential.We present a time-dependent variational method that makes the use of GWPs possible in the description of propagation of quantum states also in these systems. We use a regularization of the Coulomb potential and introduce a fictitious-time coordinate in which the evolution of an initial state can be calculated exactly and analytically for a pure Coulomb potential. Therefore, in perturbed atomic systems variational approximations only arise from those parts of the potentials which deviate from the Coulomb potential. The method is applied to the hydrogen atom in external magnetic and electric fields. It can be adapted to systems with definite symmetries and thus allows for a wide range of applications.

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40

Fouokeng, Georges Collince, Martin Tchoffo, Lukong Cornelius Fai, Ngwa Engelbert Afuoti, J. C. Ngana Kuetche, and A. M. Temgoua Nouaze. "The quenching field effect on the motion of an electron in an electromagnetic field potential." Modern Physics Letters B 28, no. 08 (March 26, 2014): 1450058. http://dx.doi.org/10.1142/s0217984914500584.

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In this paper, the Feynman path integral approach based on the Gaussian wave packet approximation is employed to evaluate the coherence factor and the probability distribution for a single electron subjected to a magnetic field tailored by a variable on/off force. The quenching field effect is observed to create decoherence in the motion of the electron. The coherence factor is observed to decrease while the distribution probability function shows dephasing characteristics for weak and strong field strengths.

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41

Senff, UE, and PC Burton. "A CEPA2 Study of the H2?H2 Isotropic Potential Function." Australian Journal of Physics 42, no. 1 (1989): 47. http://dx.doi.org/10.1071/ph890047.

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A CEPA2 study of the DDl (Koide and Kihara 1974) conformation of two hydrogen molecules has been carried out using a basis set consisting of 102 independent gaussian functions to describe the interacting system. Calculations using a smal\er basis set indicate that this conformation provides a close approximation to the isotropic Vooo potential function. The study was carried out for intermolecular separations ranging from R = 2�0 to 15�0 a.u. keeping the H2 bond lengths fixed at r = 1 �449 a.u. The wel\ depth was determined to be 104� 73/ih at Rm = 6�597 a.u. and the zero point crossing at Ro = 5�825 a.u.

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42

BHAGWAT, K. V., S. V. LAWANDE, and D. C. KHANDEKAR. "ON THE DENSITY OF ELECTRONIC STATES OF TWO-DIMENSIONAL POSITIONALLY DISORDERED SYSTEMS WITH WEAK ELECTRON-ION POTENTIAL." International Journal of Modern Physics B 01, no. 05n06 (October 1987): 1321–28. http://dx.doi.org/10.1142/s0217979287001742.

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The path-integral formulation of Edwards and Gulyaev has been used to investigate the behaviour of electronic density of states in the two-dimensional positionally disordered systems. The average electron propagator is evaluated within the first cumulant approximation. It is shown that in general the two-dimensional system possesses a non-zero density of states in the band-tail region. In particular an expression for the density of states for a Gaussian model of a two-dimensional disordered system is derived. Numerical values of density of states for this model for some typical values of parameters are also given.

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43

Maignan, Aude, and Tony Scott. "A Comprehensive Analysis of Quantum Clustering : Finding All the Potential Minima." International Journal of Data Mining & Knowledge Management Process 11, no. 1 (January 31, 2021): 33–54. http://dx.doi.org/10.5121/ijdkp.2021.11103.

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Quantum clustering (QC), is a data clustering algorithm based on quantum mechanics which is accomplished by substituting each point in a given dataset with a Gaussian. The width of the Gaussian is a σ value, a hyper-parameter which can be manually defined and manipulated to suit the application. Numerical methods are used to find all the minima of the quantum potential as they correspond to cluster centers. Herein, we investigate the mathematical task of expressing and finding all the roots of the exponential polynomial corresponding to the minima of a two-dimensional quantum potential. This is an outstanding task because normally such expressions are impossible to solve analytically. However, we prove that if the points are all included in a square region of size σ, there is only one minimum. This bound is not only useful in the number of solutions to look for, by numerical means, it allows to to propose a new numerical approach “per block”. This technique decreases the number of particles by approximating some groups of particles to weighted particles. These findings are not only useful to the quantum clustering problem but also for the exponential polynomials encountered in quantum chemistry, Solid-state Physics and other applications.

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44

Karki, Bhishma, Saddam Husain Dhobi, Jeevan Jyoti Nakarmi, and Kishori Yadav. "Energy Eigenvalue and Thermodynamic Properties of q-deformed Hulthen Potential." BIBECHANA 19, no. 1-2 (July 28, 2022): 165–69. http://dx.doi.org/10.3126/bibechana.v19i1-2.46416.

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The objective of this work is to calculate the energy eigenvalue for q-deformed Hulthen potential using a Gaussian hypergeometric function with centrifugal approximation factor and related thermodynamical properties. For this, we develop a mathematical model using the Schrodinger wave equation to find the energy eigenvalue. In addition, the thermodynamic parameters were also calculated for q-deformed Hulthen potential using the partition function. The energy eigenvalue for quantum numbers n=1 to n=5 was observed for screening parameters 0.1, 0.5, and 1. In between, 0.1 to 1 and n=1 to n=2 the energy eigenvalue ranges from -1.048 to -208.572. The energy eigenvalue for considering potential shows that decrease with increasing the quantum number which means electron are loosely bounded nucleus in an atom. Also, the vibrational mean energy, vibrational mean free energy, vibrational specific heat capacity, and vibrational entropy are obtained but due to complexity, the further development of the equation is the limitation of this work. BIBECHANA 19(2022)165-169

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45

Wang, Yaqin, Yi Luo, and Muhammad Shafeeque. "Using a Gaussian Function to Describe the Seasonal Courses of Monthly Precipitation and Potential Evapotranspiration across the Yellow River Basin, China." Journal of Hydrometeorology 20, no. 11 (November 1, 2019): 2185–201. http://dx.doi.org/10.1175/jhm-d-19-0019.1.

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Abstract Seasonal variations in precipitation (P) and potential evapotranspiration (ET0) are critical for regional hydrometeorological studies and water resource management. The sinusoidal function is widely used to describe the seasonal pattern of P and ET0. However, high errors occur either in the arid places or in places with hyperseasonal precipitation. These limitations are intrinsic properties of the sinusoidal climate descriptor and remain a barrier to provide insight into regional water–energy partitions and hydrologic similarity and predictability. In this study, we used a Gaussian framework as an alternative to describe seasonal variations in P and ET0 regimes in the Yellow River basin (YRB). The results show that the Gaussian framework provides a good approximation to the seasonal pattern of P and has a strong regional applicability for reproducing the monthly P and ET0. This allows us to assess the climate seasonality characterizing the regional balance between water supply and energy availability using δP, δET0, and aridity index. The climate seasonality indicates that the balance between water supply and energy availability has a switch in about 32% of the grid cells during the seasonal cycle from 1982 to 2015. These grid cells are mostly located in regions with average annual precipitation above 550 mm. In the northwest region of the YRB, which has a dry climate, the amount of potential evapotranspiration always exceeds the precipitation. We argue that the Gaussian function provides a quantitative conceptual framework for accurate assessment of regional water supply and energy availability and offers a penetrating insight into hydrometeorology.

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46

Landscheid, Ulf, and Werner Kutzelnigg. "On the accuracy of conventional calculations of potential energy surfaces." Collection of Czechoslovak Chemical Communications 53, no. 9 (1988): 1953–60. http://dx.doi.org/10.1135/cccc19881953.

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It is demonstrated for the example of H2+ that with the conventional basis expansion method (in terms of Gaussians) with fixed non-linear parameters the error of a computed potential curve varies strongly with the geometry. If the non-linear parameters are optimized for the equilibrium geometry, much larger errors arise for other geometries. This is very pronounced if only one polarization function of a given l is added to the basis, and qualitatively the same problem arises for any new l. The dependence of the error on the distance may also be oscillatory. For the example of Li2 in the SCF or an MC-SCF approximation essentially the same effects are observed. Here in addition substantial errors for large R may arise in the SCF curve if the basis is not able to describe the spurious ionic contributions.

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47

Mocanu, F. C., K. Konstantinou, and S. R. Elliott. "Quench-rate and size-dependent behaviour in glassy Ge2Sb2Te5 models simulated with a machine-learned Gaussian approximation potential." Journal of Physics D: Applied Physics 53, no. 24 (April 16, 2020): 244002. http://dx.doi.org/10.1088/1361-6463/ab77de.

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48

Zhang, Linfeng, Han Wang, Maria Carolina Muniz, Athanassios Z. Panagiotopoulos, Roberto Car, and Weinan E. "A deep potential model with long-range electrostatic interactions." Journal of Chemical Physics 156, no. 12 (March 28, 2022): 124107. http://dx.doi.org/10.1063/5.0083669.

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Machine learning models for the potential energy of multi-atomic systems, such as the deep potential (DP) model, make molecular simulations with the accuracy of quantum mechanical density functional theory possible at a cost only moderately higher than that of empirical force fields. However, the majority of these models lack explicit long-range interactions and fail to describe properties that derive from the Coulombic tail of the forces. To overcome this limitation, we extend the DP model by approximating the long-range electrostatic interaction between ions (nuclei + core electrons) and valence electrons with that of distributions of spherical Gaussian charges located at ionic and electronic sites. The latter are rigorously defined in terms of the centers of the maximally localized Wannier distributions, whose dependence on the local atomic environment is modeled accurately by a deep neural network. In the DP long-range (DPLR) model, the electrostatic energy of the Gaussian charge system is added to short-range interactions that are represented as in the standard DP model. The resulting potential energy surface is smooth and possesses analytical forces and virial. Missing effects in the standard DP scheme are recovered, improving on accuracy and predictive power. By including long-range electrostatics, DPLR correctly extrapolates to large systems the potential energy surface learned from quantum mechanical calculations on smaller systems. We illustrate the approach with three examples: the potential energy profile of the water dimer, the free energy of interaction of a water molecule with a liquid water slab, and the phonon dispersion curves of the NaCl crystal.

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49

Delgadillo, Ricardo, and Di Liu. "A Spectral Algorithm for the Time-Dependent Kohn--Sham Equations: Accurately Treating External Potentials Based on Frozen Gaussian Approximations." SIAM Journal on Scientific Computing 42, no. 3 (January 2020): B656—B674. http://dx.doi.org/10.1137/19m1245104.

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50

Nakamura, Issei, and V. Dmitrašinović. "Linear model in the Gaussian wave functional approximation II: analyticity of the S-matrix and the effective potential/action." Nuclear Physics A 713, no. 1-2 (January 2003): 133–47. http://dx.doi.org/10.1016/s0375-9474(02)01293-9.

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