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Journal articles on the topic 'Boltzmann-Fermi-Dirac equation'

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Published: 20 July 2024

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1

Mendl, Christian B. "Matrix-valued quantum lattice Boltzmann method." International Journal of Modern Physics C 26, no. 10 (June 24, 2015): 1550113. http://dx.doi.org/10.1142/s0129183115501132.

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We devise a lattice Boltzmann method (LBM) for a matrix-valued quantum Boltzmann equation, with the classical Maxwell distribution replaced by Fermi–Dirac functions. To accommodate the spin density matrix, the distribution functions become 2 × 2 matrix-valued. From an analytic perspective, the efficient, commonly used BGK approximation of the collision operator is valid in the present setting. The numerical scheme could leverage the principles of LBM for simulating complex spin systems, with applications to spintronics.

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2

Jiang, Ning, Linjie Xiong, and Kai Zhou. "The incompressible Navier-Stokes-Fourier limit from Boltzmann-Fermi-Dirac equation." Journal of Differential Equations 308 (January 2022): 77–129. http://dx.doi.org/10.1016/j.jde.2021.10.061.

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3

Jiang, Ning, and Kai Zhou. "The acoustic limit from the Boltzmann equation with Fermi-Dirac statistics." Journal of Differential Equations 398 (July 2024): 344–72. http://dx.doi.org/10.1016/j.jde.2024.04.014.

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4

Stańczy, R. "The existence of equilibria of many-particle systems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 139, no. 3 (May 26, 2009): 623–31. http://dx.doi.org/10.1017/s0308210508000413.

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In this paper we obtain the existence of a radial solution for some elliptic non-local problem with constraints. The problem arises from some mean field equation which models, among other things, a system of self-gravitating particles when one looks for its stationary solutions. We include the cases of Maxwell—Boltzmann, Fermi—Dirac and polytropic statistics.

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5

BENEDETTO, D., M. PULVIRENTI, F. CASTELLA, and R. ESPOSITO. "ON THE WEAK-COUPLING LIMIT FOR BOSONS AND FERMIONS." Mathematical Models and Methods in Applied Sciences 15, no. 12 (December 2005): 1811–43. http://dx.doi.org/10.1142/s0218202505000984.

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In this paper we consider a large system of bosons or fermions. We start with an initial datum which is compatible with the Bose–Einstein, respectively Fermi–Dirac, statistics. We let the system of interacting particles evolve in a weak-coupling regime. We show that, in the limit, and up to the second order in the potential, the perturbative expansion expressing the value of the one-particle Wigner function at time t, agrees with the analogous expansion for the solution to the Uehling–Uhlenbeck equation. This paper follows the same spirit as the companion work,2 where the authors investigated the weak-coupling limit for particles obeying the Maxwell–Boltzmann statistics: here, they proved a (much stronger) convergence result towards the solution of the Boltzmann equation.

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6

Dolbeault, J. "Kinetic models and quantum effects: A modified Boltzmann equation for Fermi-Dirac particles." Archive for Rational Mechanics and Analysis 127, no. 2 (1994): 101–31. http://dx.doi.org/10.1007/bf00377657.

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7

Allemand, Thibaut. "Existence and conservation laws for the Boltzmann–Fermi–Dirac equation in a general domain." Comptes Rendus Mathematique 348, no. 13-14 (July 2010): 763–67. http://dx.doi.org/10.1016/j.crma.2010.06.015.

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8

Lu, Xuguang, and Bernt Wennberg. "On Stability and Strong Convergence for the Spatially Homogeneous Boltzmann Equation for Fermi-Dirac Particles." Archive for Rational Mechanics and Analysis 168, no. 1 (June 1, 2003): 1–34. http://dx.doi.org/10.1007/s00205-003-0247-8.

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9

Figueiredo, José L., João P. S. Bizarro, and Hugo Terças. "Weyl–Wigner description of massless Dirac plasmas: ab initio quantum plasmonics for monolayer graphene." New Journal of Physics 24, no. 2 (February 1, 2022): 023026. http://dx.doi.org/10.1088/1367-2630/ac5132.

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Abstract We derive, from first principles and using the Weyl–Wigner formalism, a fully quantum kinetic model describing the dynamics in phase space of Dirac electrons in single-layer graphene. In the limit ℏ → 0, we recover the well-known semiclassical Boltzmann equation, widely used in graphene plasmonics. The polarizability function is calculated and, as a benchmark, we retrieve the result based on the random-phase approximation. By keeping all orders in ℏ, we use the newly derived kinetic equation to construct a fluid model for macroscopic variables written in the pseudospin space. As we show, the novel ℏ-dependent terms can be written as corrections to the average current and pressure tensor. Upon linearization of the fluid equations, we obtain a quantum correction to the plasmon dispersion relation, of order ℏ 2, akin to the Bohm term of quantum plasmas. In addition, the average variables provide a way to examine the value of the effective hydrodynamic mass of the carriers. For the latter, we find a relation in which Drude’s mass is multiplied by the square of a velocity-dependent, Lorentz-like factor, with the speed of light replaced by the Fermi velocity, a feature stemming from the quasi-relativistic nature of the Dirac fermions.

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10

Muljadi, Bagus Putra, and Jaw-Yen Yang. "Simulation of shock wave diffraction by a square cylinder in gases of arbitrary statistics using a semiclassical Boltzmann–Bhatnagar–Gross–Krook equation solver." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2139 (November 2, 2011): 651–70. http://dx.doi.org/10.1098/rspa.2011.0275.

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The unsteady shock wave diffraction by a square cylinder in gases of arbitrary particle statistics is simulated using an accurate and direct algorithm for solving the semiclassical Boltzmann equation with relaxation time approximation in phase space. The numerical method is based on the discrete ordinate method for discretizing the velocity space of the distribution function and high-resolution method is used for evolving the solution in physical space and time. The specular reflection surface boundary condition is employed. The complete diffraction patterns including regular reflection, triple Mach reflection, slip lines, vortices and their complex nonlinear manifestations are recorded using various flow property contours. Different ranges of relaxation times corresponding to different flow regimes are considered, and the equilibrium Euler limit solution is also computed for comparison. The effects of gas particles that obey the Maxwell–Boltzmann, Bose–Einstein and Fermi—Dirac statistics are examined and depicted.

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11

Yang, Jaw-Yen, and Yu-Hsin Shi. "A kinetic beam scheme for ideal quantum gas dynamics." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 462, no. 2069 (February 14, 2006): 1553–72. http://dx.doi.org/10.1098/rspa.2005.1618.

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A novel kinetic beam scheme for the ideal quantum gas is presented for the computation of quantum gas dynamical flows. The quantum Boltzmann equation approach is adopted and the local thermodynamic equilibrium quantum distribution is assumed. Both Bose–Einstein and Fermi–Dirac gases are considered. Formulae for one spatial dimension is first derived and the resulting beam scheme is tested for shock tube flows. Implementation of high-order methods is also outlined. We only consider the system in the normal phase consisting of particles in excited states and both the classical limit and the nearly degenerate limit are computed. The flow structures can all be accurately captured by the present beam scheme. Formulations for multiple spatial dimensions are also included.

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12

Florkowski, Wojciech, and Ewa Maksymiuk. "Exact solution of the (0+1)-dimensional Boltzmann equation for massive Bose–Einstein and Fermi–Dirac gases." Journal of Physics G: Nuclear and Particle Physics 42, no. 4 (February 16, 2015): 045106. http://dx.doi.org/10.1088/0954-3899/42/4/045106.

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13

Lu, Xuguang. "On the Boltzmann equation for Fermi–Dirac particles with very soft potentials: Global existence of weak solutions." Journal of Differential Equations 245, no. 7 (October 2008): 1705–61. http://dx.doi.org/10.1016/j.jde.2008.06.028.

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14

Lu, Xuguang. "On the Boltzmann Equation for Fermi–Dirac Particles with Very Soft Potentials: Averaging Compactness of Weak Solutions." Journal of Statistical Physics 124, no. 2-4 (March 21, 2006): 517–47. http://dx.doi.org/10.1007/s10955-006-9039-5.

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15

Yang, Jaw-Yen, Bagus Putra Muljadi, Zhi-Hui Li, and Han-Xin Zhang. "A Direct Solver for Initial Value Problems of Rarefied Gas Flows of Arbitrary Statistics." Communications in Computational Physics 14, no. 1 (July 2013): 242–64. http://dx.doi.org/10.4208/cicp.290112.030812a.

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AbstractAn accurate and direct algorithm for solving the semiclassical Boltzmann equation with relaxation time approximation in phase space is presented for parallel treatment of rarefied gas flows of particles of three statistics. The discrete ordinate method is first applied to discretize the velocity space of the distribution function to render a set of scalar conservation laws with source term. The high order weighted essentially non-oscillatory scheme is then implemented to capture the time evolution of the discretized velocity distribution function in physical space and time. The method is developed for two space dimensions and implemented on gas particles that obey the Maxwell-Boltzmann, Bose-Einstein and Fermi-Dirac statistics. Computational examples in one- and two-dimensional initial value problems of rarefied gas flows are presented and the results indicating good resolution of the main flow features can be achieved. Flows of wide range of relaxation times and Knudsen numbers covering different flow regimes are computed to validate the robustness of the method. The recovery of quantum statistics to the classical limit is also tested for small fugacity values.

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16

SIGISMONDI, COSTANTINO, SIMONETTA FILIPPI, REMO RUFFINI, and LUIS ALBERTO SÁNCHEZ. "DAMPING TIME AND STABILITY OF DENSITY FERMION PERTURBATIONS IN THE EXPANDING UNIVERSE." International Journal of Modern Physics D 10, no. 05 (October 2001): 663–79. http://dx.doi.org/10.1142/s0218271801001190.

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The classic problem of the growth of density perturbations in an expanding Newtonian universe is revisited following the work of Bisnovatyi-Kogan and Zel'dovich. We propose a more general analytical approach: a system of free particles satisfying semidegenerate Fermi–Dirac statistics on the background of an exact expanding solution is examined in the linear approximation. This differs from the corresponding work of Bisnovatyi-Kogan and Zel'dovich where classical particles fulfilling Maxwell–Boltzmann statistics were considered. The solutions of the Boltzmann equation are obtained by the method of characteristics. An expression for the damping time of a decaying solution is discussed and a zone in which free streaming is hampered is found, corresponding to wavelengths less than the Jeans one. In the evolution of the system, due to the decrease of the Jeans length, those perturbations may lead to gravitational collapse. At variance with current opinions, we deduce that perturbations with λ≥λ J Max /1.48 are able to generate structures and the lower limit for substructures mass is M=M J max /(1.48)3≈M J max /3, where M J max is the maximum value of the Jeans mass.

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17

Biswas, Anirban, Dilip Kumar Ghosh, and Dibyendu Nanda. "Concealing Dirac neutrinos from cosmic microwave background." Journal of Cosmology and Astroparticle Physics 2022, no. 10 (October 1, 2022): 006. http://dx.doi.org/10.1088/1475-7516/2022/10/006.

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Abstract The existence of prolonged radiation domination prior to the Big Bang Nucleosynthesis (BBN), starting just after the inflationary epoch, is not yet established unanimously. If instead, the universe undergoes a non-standard cosmological phase, it will alter the Hubble expansion rate significantly and may also generate substantial entropy through non-adiabatic evolution. This leads to a thumping impact on the properties of relic species decoupled from the thermal bath before the revival of the standard radiation domination in the vicinity of the BBN. In this work, considering the Dirac nature of neutrinos, we have studied decoupling of ultra-relativistic right-handed neutrinos (νR s) in presence of two possible non-standard cosmological phases. While in both cases we have modified Hubble parameters causing faster expansions in the early universe, one of the situations predicts a non-adiabatic evolution and thereby a slower redshift of the photon temperature due to the expansion. Considering the most general form of the collision term with Fermi-Dirac distribution and Pauli blocking factors, we have solved the Boltzmann equation numerically to obtain ΔNeff for the three right-handed neutrinos. We have found that for a large portion of parameter space, the combined effect of early decoupling of νR as well as the slower redshift of photon bath can easily hide the signature of right-handed neutrinos, in spite of precise measurement of ΔNeff, at the next generation CMB experiments like CMB-S4, SPT-3G etc. This however will not be applicable for the scenarios with only fast expansion.

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18

CAVALLERI, GIANCARLO, ERNESTO TONNI, LEONARDO BOSI, and GIANFRANCO SPAVIERI. "VERY LONG DECAY TIME FOR ELECTRON VELOCITY DISTRIBUTION IN SEMICONDUCTORS, AND CONSEQUENT 1/f NOISE." Fluctuation and Noise Letters 07, no. 03 (September 2007): L193—L207. http://dx.doi.org/10.1142/s0219477507003842.

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The Boltzmann equation with electron-electron (e − e) interactions has been reduced to a Fokker-Planck equation (e − e FP ) in a previuos paper. In steady-state conditions, its solution q0(v) (where v is the electron speed) depends on the square of the acceleration a = eE/m. If we introduce the nonrenormalized zero-point field (ZPF) of QED, i.e., the one considered in stochastic electrodynamics, so that [Formula: see text], then q0(v) becomes similar to the Fermi-Dirac equation, and the two collision frequencies ν1(v) and ν2(v) appearing in the e − e FP become both proportional to 1/v in a small δv interval. The condition ν1(v) ∝ ν2(v) ∝ 1/v is at the threshold of the runaways. In the same δv range, the time-dependent solution q0(v,τ) of the e − e FP decays no longer exponentially but according to a power law ∝ τ− ɛ where 0.004 < ɛ < 0.006, until τ → ∞. That extremely long memory of a fluctuation implies the same dependence τ − ɛ for the conductance correlation function, hence a corresponding power-spectral noise S(f) ∝ fɛ−1 where f is the frequency. That behaviour is maintained even for a small sample because the back diffusion velocity of the electrons in the effective range δv, where they are in runaway conditions, is much larger than the drift velocity.

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19

Zheng, Jin-Cheng. "Asymmetrical Transport Distribution Function: Skewness as a Key to Enhance Thermoelectric Performance." Research 2022 (July 15, 2022): 1–14. http://dx.doi.org/10.34133/2022/9867639.

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How to achieve high thermoelectric figure of merit is still a scientific challenge. By solving the Boltzmann transport equation, thermoelectric properties can be written as integrals of a single function, the transport distribution function (TDF). In this work, the shape effects of transport distribution function in various typical functional forms on thermoelectric properties of materials are systematically investigated. It is found that the asymmetry of TDF, characterized by skewness, can be used to describe universally the trend of thermoelectric properties. By defining symmetric and asymmetric TDF functions, a novel skewness is then constructed for thermoelectric applications. It is demonstrated, by comparison with ab initio calculations and experiments, that the proposed thermoelectric skewness not only perfectly captures the main feature of conventional skewness but also is able to predict the thermoelectric power accurately. This comparison confirms the unique feature of our proposed thermoelectric skewness, as well as its special role of connection between the statistics of TDF and thermoelectric properties of materials. It is also found that the thermoelectric performance can be enhanced by increasing the asymmetry of TDF. Finally, it is also interesting to find that the thermoelectric transport properties based on typical quantum statistics (Fermi-Dirac distributions) can be well described by typical shape parameter (skewness) for classical statistics.

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20

Qi, Yue. "(Invited) Modeling of the Electric Double Layer (EDL) at Li/SEI/Electrolyte Interfaces." ECS Meeting Abstracts MA2023-02, no. 5 (December 22, 2023): 881. http://dx.doi.org/10.1149/ma2023-025881mtgabs.

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Liquid electrolytes, consisting of salts, solvents, and additives, must form a stable solid electrolyte interphase (SEI) to ensure the performance and durability of lithium(Li)-ion batteries. However, the electric double layer (EDL) structure near charged surfaces is still unsolved, despite its importance in dictating the species being reduced for SEI formation near a negative electrode. Recently, we have developed an interactive Molecular Dynamics -Density Functional Theory -data statistics (MD-DFT-data) model to investigate the reduction reactions of multicomponent electrolytes within the EDL. We will illustrate the effect of EDL on SEI formation in two essential electrolytes, the carbonate-based electrolyte for Li-ion batteries and the ether-based electrolyte for batteries with Li-metal anodes. Our results reveal that the role of fluoroethylene carbonate (FEC) additive differs drastically in the two electrolytes as an SEI modifier to form the beneficial F-containing SEI component (e.g., LiF). The competition among the cations, anions, and various species in the solvents with a charged surface at different temperatures can all jointly determine the EDL structure and therefore the SEI compositions. [1] While the classical Poisson–Boltzmann EDL model developed for fully solvated ions face new challenges in high-concentration liquid electrolytes (HCE), localized high-concentration liquid electrolytes (LHCE) as well as solid electrolytes (SE), new theoretical developments are required. We will introduce a new DEL model in SE and SEI by solving the DFT-informed Poisson–Fermi–Dirac equation and demonstrate how it can be used for interlayer thickness design. [2] [1] Wu, Q.S, McDowell, M.T., & Qi, Y., Journal of the American Chemical Society 145 (4), 2473-2484 (2023) [2] Swift, M.W., Swift, J.W. & Qi, Y. Modeling the electrical double layer at solid-state electrochemical interfaces. Nat Comput Sci 1, 212–220 (2021)

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21

Barami, Soudeh, and Vahid Ghafarinia. "Calculation of the electric potential and surface oxygen ion density for planar and spherical metal oxide grains by numerical solution of the Poisson equation coupled with Boltzmann and Fermi-Dirac statistics." Sensors and Actuators B: Chemical 293 (August 2019): 31–40. http://dx.doi.org/10.1016/j.snb.2019.04.151.

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22

BROWN, S. R., and M. G. HAINES. "Transport in partially degenerate, magnetized plasmas. Part 1. Collision operators." Journal of Plasma Physics 58, no. 4 (December 1997): 577–600. http://dx.doi.org/10.1017/s0022377897006041.

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The quantum Boltzmann collision operator is expanded to yield a degenerate form of the Fokker–Planck collision operator. This is analysed using Rosenbluth potentials to give a degenerate analogue of the Shkarofsky operator. The distribution function is then expanded about an equilibrium Fermi–Dirac distribution function using a tensor perturbation formulation to give a zeroth-order and a first-order collision operator. These equations are shown to satisfy the relevant conservation equations. It is shown that the distribution function relaxes to a Fermi–Dirac form through electron–electron collisions.

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23

Troy, William C. "Low temperature properties of the Fermi–Dirac, Boltzmann and Bose–Einstein equations." Physics Letters A 376, no. 45 (October 2012): 2887–93. http://dx.doi.org/10.1016/j.physleta.2012.10.003.

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24

Suárez, Alberto, and Jean Pierre Boon. "Nonlinear Hydrodynamics of Lattice-Gas Automata with Semi-Detailed Balance." International Journal of Modern Physics C 08, no. 04 (August 1997): 653–74. http://dx.doi.org/10.1142/s0129183197000564.

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Equations governing the evolution of the hydrodynamic variables in a lattice-gas automaton, arbitrarily far from equilibrium, are derived from the micro-dynamical description of the automaton, under the condition that the local collision rules satisfy semi-detailed balance. This condition guarantees that a factorized local equilibrium distribution (for each node) of the Fermi–Dirac form is invariant under the collision step but not under propagation. The main result is the set of fully nonlinear hydrodynamic equations for the automaton in the lattice-Boltzmann approximation; these equations have a validity domain extending beyond the region close to equilibrium. Linearization of the hydrodynamic equations derived here leads to Green–Kubo formulae for the transport coefficients.

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25

Trakhtenberg, L. I., O. J. Ilegbusi, and M. A. Kozhushner. "Comments on the article “Calculation of the electric potential and surface oxygen ion density for planar and spherical metal oxide grains by numerical solution of the Poisson equation coupled with Boltzmann and Fermi-Dirac statistics” (Sensors and Actuators B: Chemical, 293 (2019) 31–40)." Sensors and Actuators B: Chemical 302 (January 2020): 126986. http://dx.doi.org/10.1016/j.snb.2019.126986.

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26

Ghafarinia, Vahid, and Soudeh Barami. "Reply to comments on the article “Calculation of the electric potential and surface oxygen ion density for planar and spherical metal oxide grains by numerical solution of the Poisson equation coupled with Boltzmann and Fermi-Dirac statistics” (Sensors and Actuators B: Chemical, 293 (2019))." Sensors and Actuators B: Chemical 321 (October 2020): 128545. http://dx.doi.org/10.1016/j.snb.2020.128545.

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27

Gajewski, Herbert, and Konarad Gröger. "Semiconductor Equations for variable Mobilities Based on Boltzmann Statistics or Fermi-Dirac Statistics." Mathematische Nachrichten 140, no. 1 (1989): 7–36. http://dx.doi.org/10.1002/mana.19891400102.

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28

SYROS, C. "PRINCIPLES OF A NEW QUANTUM THEORY." Modern Physics Letters A 13, no. 21 (July 10, 1998): 1675–88. http://dx.doi.org/10.1142/s0217732398001753.

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The principles of a random quantum theory (R-QT) which is alternatively time-asymmetric or time-symmetric if the quantization is Fermi–Dirac or Bose–Enstein, respectively are presented. Bohr's quantization rule is applied on the field-action integral. A time topological space, [Formula: see text], is mathematically defined in which the paradoxes in standard quantum theory are solved. The time "quantum", is created as a regular, positive into-map of an observed observable's change resulting from a fundamental interaction process. [Formula: see text] is constructed as the union of time elements and can be embedded disconnectedly in the continuous Newtonian universal time, [Formula: see text]. Six axioms are formulated characterizing the space–time and R-QT. The disconnectedness of the (κ×λκ)-fold time-space, [Formula: see text], imparts a kind of disconnectedness to the κ×λκ-fold space–times, [Formula: see text], and induces the chrono-topology. In chrono-topology the unitary, U, or non-measure preserving, R, dynamics, is implemented by means of a time evolution, "complex" operator, [Formula: see text]. It breaks down by means of Bohr quantization into: [Formula: see text][Formula: see text] coincides formally — apart from the spontaneous renormalization — with the time evolution operator in the standard QFT. [Formula: see text] is a novum and produces the Maxwell–Boltzmann energy level distribution in a non-Euclidean QFT. Compatibility between time-reversal invariance of the standard QT equations and irreversibility of some phenomena both in microcosmos and macrocosmos is obtained. The [Formula: see text]-evolution leads to a time's arrow on quantum-scale systems.

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29

Borsoni, Thomas. "Extending Cercignani’s Conjecture Results from Boltzmann to Boltzmann–Fermi–Dirac Equation." Journal of Statistical Physics 191, no. 5 (April 27, 2024). http://dx.doi.org/10.1007/s10955-024-03262-3.

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30

Jiang, Ning, and Kai Zhou. "Global well-posedness of Boltzmann-Fermi-Dirac equation for hard potential." Kinetic and Related Models, 2024, 0. http://dx.doi.org/10.3934/krm.2024014.

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31

Jiang, Ning, and Kai Zhou. "The Compressible Euler and Acoustic Limits from Quantum Boltzmann Equation with Fermi–Dirac Statistics." Communications in Mathematical Physics 405, no. 2 (January 30, 2024). http://dx.doi.org/10.1007/s00220-023-04883-7.

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AbstractThis paper justifies the compressible Euler and acoustic limits from quantum Boltzmann equation with Fermi–Dirac statistics rigorously. By employing Hilbert expansion, in particular analyzing the nonlinear implicit transformation between the classical form of compressible Euler equations and the one obtained directly from BFD, and some new type of Grad–Caflisch type decay estimate of the linearized collision operator, we establish the compressible Euler limit from scaled BFD equation, which was formally derived by Zakrevskiy in (Kinetic models in the near-equilibrium regime. Thesis at Polytechnique, 2015) by moment method. Consequently, the acoustic limit is obtained in optimal scaling with respect to Knudsen number.

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32

Potting, Robertus. "The Boltzmann equation and equilibrium thermodynamics in Lorentz-violating theories." European Physical Journal Plus 138, no. 4 (April 18, 2023). http://dx.doi.org/10.1140/epjp/s13360-023-03889-3.

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AbstractIn this work, we adapt the foundations of relativistic kinetic theory and the Boltzmann equation to particles with Lorentz-violating dispersion relations. The latter are taken to be those associated to two commonly considered sets of coefficients in the minimal Standard-Model Extension. We treat both the cases of classical (Maxwell–Boltzmann) and quantum (Fermi–Dirac and Bose–Einstein) statistics. It is shown that with the appropriate definition of the entropy current, Boltzmann’s H-theorem continues to hold. We derive the equilibrium solutions and then identify the Lorentz-violating effects for various thermodynamic variables, as well as for Bose–Einstein condensation. Finally, a scenario with nonelastic collisions between multiple species of particles corresponding to chemical or nuclear reactions is considered.

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33

Raynaud, C., J. L. Autran, P. Masson, M. Bidaud, and A. Poncet. "Analysis of MOS Device Capacitance-Voltage Characteristics Based on the Self-Consistent Solution of the Schrödinger and Poisson Equations." MRS Proceedings 592 (1999). http://dx.doi.org/10.1557/proc-592-159.

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ABSTRACTThe one-dimensional Schridinger and Poisson equations have been numerically solved in metal-oxide-semiconductor devices using a three-point finite difference scheme with a non-uniform mesh size. The capacitance-voltage characteristic of the structure has been calculated via this self-consistent approach and results have been compared with data obtained from the resolution of Poisson equation using different approximated methods based on the Boltzmann statistic with and without a first order quantum effect correction or the exact Fermi-Dirac statistic. The present work permits to evaluate and quantify the errors made by these approximations in determining the thickness of ultra-thin oxides.

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34

Li, Zongguang. "Existence and uniqueness of solutions to the Fermi-Dirac Boltzmann equation for soft potentials." Quarterly of Applied Mathematics, October 27, 2023. http://dx.doi.org/10.1090/qam/1681.

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In this paper we consider a modified quantum Boltzmann equation with the quantum effect measured by a continuous parameter δ \delta that can decrease from δ = 1 \delta =1 for the Fermi-Dirac particles to δ = 0 \delta =0 for the classical particles. In case of soft potentials, for the corresponding Cauchy problem in the whole space or in the torus, we establish the global existence and uniqueness of non-negative mild solutions in the function space L T ∞ L v , x ∞ ∩ L T ∞ L x ∞ L v 1 L^{\infty }_{T}L^{\infty }_{v,x}\cap L^{\infty }_{T}L^{\infty }_{x}L^1_v with small defect mass, energy and entropy but allowed to have large amplitude up to the possibly maximum upper bound F ( t , x , v ) ≤ 1 δ F(t,x,v)\leq \frac {1}{\delta } . The key point is that the obtained estimates are uniform in the quantum parameter 0 > δ ≤ 1 0> \delta \leq 1 .

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35

Anwasia, Benjamin, and Diogo Arsénio. "Quantized collision invariants on the sphere." Communications in Mathematics Volume 32 (2024), Issue 3... (April 25, 2024). http://dx.doi.org/10.46298/cm.12766.

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We show that a measurable function $g:\mathbb{S}^{d-1}\to\mathbb{R}$, with $d\geq 3$, satisfies the functional relation \begin{equation*} g(\omega)+g(\omega_*)=g(\omega')+g(\omega_*'), \end{equation*} for all admissible $\omega,\omega_*,\omega',\omega_*'\in\mathbb{S}^{d-1}$ in the sense that \begin{equation*} \omega+\omega_*=\omega'+\omega_*', \end{equation*} if and only if it can be written as \begin{equation*} g(\omega)=A+B\cdot\omega, \end{equation*} for some constants $A\in \mathbb{R}$ and $B\in\mathbb{R}^d$. Such functions form a family of quantized collision invariants which play a fundamental role in the study of hydrodynamic regimes of the Boltzmann--Fermi--Dirac equation near Fermionic condensates, i.e., at low temperatures. In particular, they characterize the elastic collisional dynamics of Fermions near a statistical equilibrium where quantum effects are predominant.

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36

Wang, Jinrong, and Lulu Ren. "Global existence and stability of solutions of spatially homogeneous Boltzmann equation for Fermi-Dirac particles." Journal of Functional Analysis, October 2022, 109737. http://dx.doi.org/10.1016/j.jfa.2022.109737.

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37

Liu, Bocheng, and Xuguang Lu. "On the Convergence to Equilibrium for the Spatially Homogeneous Boltzmann Equation for Fermi–Dirac Particles." Journal of Statistical Physics 190, no. 8 (August 8, 2023). http://dx.doi.org/10.1007/s10955-023-03152-0.

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38

Kapusta, Joseph I. "Perspective on Tsallis statistics for nuclear and particle physics." International Journal of Modern Physics E, August 16, 2021, 2130006. http://dx.doi.org/10.1142/s021830132130006x.

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This is a concise introduction to the topic of nonextensive Tsallis statistics meant especially for those interested in its relation to high-energy proton–proton, proton–nucleus and nucleus–nucleus collisions. The three types of Tsallis statistics are reviewed. Only one of them is consistent with the fundamental hypothesis of equilibrium statistical mechanics. The single-particle distributions associated with it, namely Boltzmann, Fermi–Dirac and Bose–Einstein, are derived. These are not equilibrium solutions to the conventional Boltzmann transport equation which must be modified in a rather nonintuitive manner for them to be so. Nevertheless, the Boltzmann limit of the Tsallis distribution is extremely efficient in representing a wide variety of single-particle distributions in high-energy proton–proton, proton–nucleus and nucleus–nucleus collisions with only three parameters, one of them being the so-called nonextensitivity parameter [Formula: see text]. This distribution interpolates between an exponential at low transverse energy, reflecting thermal equilibrium, to a power law at high transverse energy, reflecting the asymptotic freedom of Quantum Chromodynamics (QCD). It should not be viewed as a fundamental new parameter representing nonextensive behavior in these collisions.

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Ludwick, Kevin J., and Holston Sebaugh. "Deriving the dark matter-dark energy interaction term in the continuity equation from the Boltzmann equation." Modern Physics Letters A, May 25, 2021, 2150122. http://dx.doi.org/10.1142/s0217732321501224.

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Dark energy and dark matter are two of the biggest mysteries of modern cosmology, and our understanding of their fundamental nature is incomplete. Many parametrizations of couplings between the two in the continuity equation have been studied in the literature, and observational data from the growth of perturbations can constrain these parametrizations. Assuming standard general relativity with a simple Yukawa-type coupling between dark energy and dark matter fields in the Lagrangian, we use the Boltzmann equation to analytically express and calculate the interaction kernel Q in the continuity equation and compare it to that of a typical parametrization. We arrive at a comparably very small result, as expected. Since the interaction is a function of the dark matter mass, other observational data sets can be used to constrain the mass. This calculation can be modified to account for other couplings of the dark energy and dark matter fields. This calculation required obtaining a distribution function for dark energy that leads to an equation of state parameter that is negative, which neither Bose–Einstein nor Fermi–Dirac statistics can supply, and this is the main result of this paper. Treating dark energy as a quantum scalar field, we use adiabatic subtraction to obtain a finite analytic approximation for its distribution function that assumes the FLRW metric and nothing more.

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Suwa, Yudai, Hiroaki W. H. Tahara, and Eiichiro Komatsu. "Kompaneets equation for neutrinos: Application to neutrino heating in supernova explosions." Progress of Theoretical and Experimental Physics 2019, no. 8 (August 1, 2019). http://dx.doi.org/10.1093/ptep/ptz087.

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Abstract We derive a “Kompaneets equation” for neutrinos, which describes how the distribution function of neutrinos interacting with matter deviates from a Fermi–Dirac distribution with zero chemical potential. To this end, we expand the collision integral in the Boltzmann equation of neutrinos up to the second order in energy transfer between matter and neutrinos. The distortion of the neutrino distribution function changes the rate at which neutrinos heat matter, as the rate is proportional to the mean square energy of neutrinos, $E_\nu^2$. For electron-type neutrinos the enhancement in $E_\nu^2$ over its thermal value is given approximately by $E_\nu^2/E_{\nu,\rm thermal}^2=1+0.086(V/0.1)^2$, where $V$ is the bulk velocity of nucleons, while for the other neutrino species the enhancement is $(1+\delta_v)^3$, where $\delta_v=mV^2/3k_{\rm B}T$ is the kinetic energy of nucleons divided by the thermal energy. This enhancement has a significant implication for supernova explosions, as it would aid neutrino-driven explosions.

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41

"A theoretical justification for the application of the Arrhenius equation to kinetics of solid state reactions (mainly ionic crystals)." Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 450, no. 1940 (September 8, 1995): 501–12. http://dx.doi.org/10.1098/rspa.1995.0097.

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Although the Arrhenius equation has been widely and successfully applied to innumerable solid state reactions, this use lacks a theoretical justification because the energy distribution amongst the immobilized constituents of a crystalline reactant is not represented by the Maxwell-Boltzmann equation. The present analysis focuses attention on the role of the reactant-product interface, the active zone within which chemical changes preferentially proceed in many solid state rate processes. We identify interface energy levels, that are the precursors to the bond redistribution step, as extensions to the band structure of the solid into the structurally less-regular reaction zone. These interface energy levels are analogous to impurity levels. Electron reorganization requires a locally high energy so that interface levels are appreciably above the Fermi level of the crystalline reactant (and product). Occupancy is determined by energy distribution functions based on Fermi-Dirac statistics for electrons and Bose-Einstein statistics for phonons. For the highest energies, necessary for reaction, both distributions approximate to the exponential energy term, thereby providing a theoretical justification for the application of the Arrhenius equation to reactions of solids. The treatment given here has been largely developed from the theory applicable to ionic solids and the conclusions are most directly relevant to reactions of this class of substance. It is intended, however, that the approach should be of value in extending theoretical understanding of all rate processes involving solids which require the preinvestment of energy in an electron reorganization step.

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42

Mouton, Alexandre, and Thomas Rey. "On Deterministic Numerical Methods for the Quantum Boltzmann-Nordheim Equation. I. Spectrally Accurate Approximations, Bose-Einstein Condensation, Fermi-Dirac Saturation." SSRN Electronic Journal, 2021. http://dx.doi.org/10.2139/ssrn.3954908.

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43

Mouton, Alexandre, and Thomas Rey. "On Deterministic Numerical Methods for the quantum Boltzmann-Nordheim Equation. I. Spectrally accurate approximations, Bose-Einstein condensation, Fermi-Dirac saturation." Journal of Computational Physics, May 2023, 112197. http://dx.doi.org/10.1016/j.jcp.2023.112197.

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44

Muscato, Orazio, Giovanni Nastasi, Vittorio Romano, and Giorgia Vitanza. "Optimized quantum drift diffusion model for a resonant tunneling diode." Journal of Non-Equilibrium Thermodynamics, January 23, 2024. http://dx.doi.org/10.1515/jnet-2023-0059.

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Abstract The main aim of this work is to optimize a Quantum Drift Diffusion model (QDD) (V. Romano, M. Torrisi, and R. Tracinà, “Approximate solutions to the quantum drift-diffusion model of semiconductors,” J. Math. Phys., vol. 48, p. 023501, 2007; A. El Ayyadi and A. Jüngel, “Semiconductor simulations using a coupled quantum drift-diffusion schrödinger-Poisson model,” SIAM J. Appl. Math., vol. 66, no. 2, pp. 554–572, 2005; L. Barletti and C. Cintolesi, “Derivation of isothermal quantum fluid equations with Fermi-Dirac and bose-einstein statistics,” J. Stat. Phys., vol. 148, pp. 353–386, 2012) by comparing it with the Boltzmann-Wigner Transport Equation (BWTE) (O. Muscato, “Wigner ensemble Monte Carlo simulation without splitting error of a GaAs resonant tunneling diode,” J. Comput. Electron., vol. 20, pp. 2062–2069, 2021) solved using a signed Monte Carlo method (M. Nedjalkov, H. Kosina, S. Selberherr, C. Ringhofer, and D. K. Ferry, “Unified particle approach to Wigner-Boltzmann transport in small semiconductor devices,” Phys. Rev. B, vol. 70, pp. 115–319, 2004). A situation of high non equilibrium regime is investigated: electron transport in a Resonant Tunneling Diode (RTD) made of GaAs with two potential barriers in GaAlAs. The range of the suitable voltage bias applied to the RTD is analyzed. We find an acceptable agreement between QDD model and BWTE when the applied bias is low or moderate with a threshold of about 0.225 V over a length of 150 nm; it is found out that the use of a field dependent mobility is crucial for getting a good description of the negative differential conductivity in such a range. At higher bias voltages, we expect that QDD model loses accuracy.

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