An exact solution of the linearized boltzmann equation. We use this formulation for accelerating deviational monte carlo simulations of complex, multiscale problems. Simultaneous diagonalization of the collision kernel of that equation and a symmetry crystal class operator allow us to obtain a spectral. In appendix b we show how conservation laws result for the linearized functions in the new method. A phonon boltzmann transport equation approach is employed to describe anharmonic phonon. Novel feature in the formalism is that arbitrary finite concentration of electrons is admitted. The phonon boltzmann equation, properties and link to weakly anharmonic lattice dynamics. In the second order in the scattering strength and after a short period of forgetting details of initial conditions. This paper revisits the problem of the linearized boltzmann transport equation bte, or, equivalently, of the momentum relaxation time, momentum relaxation time mrt, for the calculation of low field mobility, which in previous works has been almost universally solved in approximated forms. Transport properties boltzmann equation current density.
Phonon dispersion relations and transition rates are obtained from density functional theory calculations. Pdf the linearized boltzmann equation for the pure phonon field may be solved formally in terms of the eigenvectors of the normalprocess. We present a deviational monte carlo method for solving the boltzmann equation for phonon transport subject to the linearized ab initio 3phonon scattering operator. The phonon boltzmann equation approach includes the exact ab initio solution of the linearized phonon boltzmann equation corresponding author. Phonon properties and thermal conductivity from first. A theory is presented for describing the lattice thermal conductivities of graphene and singlewalled carbon nanotubes. We propose an energy driven discretization method that allows an exact determination of the relaxation. An exact method is given for solving the linearized boltzmann equation for arbitrary magnetic field strengths in the presence of polar mode scattering and elastic scattering mechanisms.
Solution of boltzmanns equation for semiconductors using a spherical harmonic expansion. Graphene singlewalled carbon nanotubes and theory of. The phonon boltzmann equation, properties and link to weakly. Firstly the gray phonon boltzmanns transport equation is introduced. The boltzmann equation written in abstract form as df dt cf 2. Surfaces et spectroscopiesinternational audiencethe frequency dependent phonon boltzmann equation is transformed to an integral equation over the irreducible part of the brillouin zone. The variance reduction is achieved by simulating only the deviation from equilibrium. A predictive model for the size dependent thermal conductivity is. In this section some numerical results are presented, which might be of importance for mesoscopics. In chapter 2, we study the theoretical properties of the linearized and nonlinear pbe using standard. We have developed a formalism of the exact solution to the linearized phonon boltzmann transport equation bte for thermal conductivity calculations including three and fourphonon scattering. The former has no self interaction, just scattering with the medium, whereas the latter is the linearization of the fully nonlinear boltzmann equation.
Phonon transport conclusions e cient methods for solving the boltzmann equation for nanoscale transport applications nicolas g. A new form of projector is suggested leading to timeconvolutionless generalized master equations tclgme for the singleelectron density matrix or the wigner distribution function. An exact solution of the linearized boltzmann transport. The approach given may be used for any type of particle. The equivalence of the new linearized transport equations with the holstein equation is. In this paper we compare the nonlinear boltzmann equation appearing in the kinetic theory of gases, with its linearized version. Heat transport in twodimensional materials by directly. The phononcontributed part of the total thermal conductivity is the lattice thermal conductivity. Ballisticdiffusive phonon transport and size induced. The perturbed distribution function shows slope discontinuities at all multiples of the polar phonon energy.
The linearized boltzmann equation for the pure phonon field may be solved formally in terms of the eigenvectors of the normalprocess collision operator. Adjointbased deviational monte carlo methods for phonon. Boltzmann transport equation phonon boltzmann transport equation in the steady state. Efficient methods for solving the boltzmann equation for. Evaluating the thermoelectric properties of batis3 by. Boltzmann transport equations for nanoscience applications. Direct solution to the linearized phonon boltzmann equation. In this paper, we investigate the crossplane phonon transport from the viewpoint of the phonon boltzmann equation. Distribution function f t contains all the information about the transport in the system. Derivation of the linearized boltzmann equation from a. We do not assume from the reader any familiarity with boltzmanns equation. Get a printable copy pdf file of the complete article 253k. The phonon boltzmann equation, properties and link to. Linear alternative to the boltzmann equation without.
Variancereduced particle simulation of the boltzmann. In this paper, we develop an adjoint formulation of the linearized boltzmann transport equation for phonon transport. Insert this expression into the linearized boltzmann equation, and remove the term with a. Theory of phonon thermal transport in singlewalled carbon nanotubes and graphene lucas lindsay advisor. This representation is particularly convenient as a basis for solutions, since in the isotropic dispersionless case the temperature deviation.
Derivation of the linearized boltzmann equation from a deterministic dynamics of hard spheres thierry bodineau joint works with isabelle gallagher, laure saintraymond. We note that the time coordinate, formally used for the purpose of the derivation, is not. C, s mechanical engineering, purdue university, west lafayette, in, u. In contrast, phonon boltzmann equation modeling is a mesoscopic approach avoiding or relieving the aforementioned three drawbacks 16,17. This can be easily verified by considering the ballistic and linearized. The collision terms in the boltzmann equation have several. Nonexistence of a gap in the eigenvalue spectrum of the. It is proved that the eigenvalue spectrum of the linearized collision operator of peierls phononboltzmann equation has no gap. Thermal networks form phonon boltzmanns transport equation. In the case of a gas of elastic sphere and in the absence of external forces, this equation takes the form. To motivate the work, we provide a thorough discussion of the poissonboltzmann equation, including derivation from a few basic assumptions, discussions of special case solutions, as well as common analytical approximation techniques.
We exhibit an intertwining operator for the two semigroups involved. Here, we evaluate the prospect of batis3 as a thermoelectric material by using the linearized electron and. The frequency dependent phonon boltzmann equation is transformed to an integral equation over the irreducible part of the brillouin zone. Phonon boltzmann equation abstract the effective thermal conductivity of nano. Other collisional invariants include energy when all sources are accounted for, spin total spin. Spectral gap and coercivity estimates for linearized. The boltzmann equation of phonon thermal transport solved in. A general theoretical treatment is given of the linearized boltzmann equation for flow in a bounded medium under conditions when the collision mean free path is of the order of the dimensions of the crosssection of the specimen. As a consequence the usual derivation of hydrodynamic equations from a boltzmann equation is not valid in the case of a phonon system. Iterative solution method for the linearized poissonboltzmann equation. An exact solution of the linearized boltzmann equation with applications to the hall mobility and hall factor of ngaas k fletcher and p n butcherl 1c. T and the heat current q are related to zeroeigenvalue eigenfunctions of this operator.
Solution of the linearized phonon boltzmann equation physical. To have the boltzmann linearized equation in the general case, let us. Later on we will discuss electronphonon scattering, which is inelastic. The boltzmann equation of phonon thermal transport solved in the relaxation time approximation ii data analysis amelia carolina sparavigna1 1 department of applied science and technology, politecnico di torino, torino, italy doi 10. Hadjiconstantinou, department of mechanical engineering, massachusetts institute of technology cambridge, ma 029, usa may 3, 2019 abstract we derive the continuum equations and. Introduction one must distinguish between the linear boltzmann equation and the linearized boltzmann equation. Lecture notes microscopic theory of transport nuclear. Solution of the linearized phonon boltzmann equation. Phonon properties including frequencies, velocities and scattering rates are largely determined by interatomic force constants ifcs. Solve linearized bte selfconsistently to determine occupations.
The boltzmann equation of phonon thermal transport solved in the relaxation time approximation i theory amelia carolina sparavigna1 1 department of applied science and technology, politecnico di torino, italy doi 10. Simultaneous diagonalization of the collision kernel of that equation and a symmetry crystal class operator allow us to obtain a spectral representation of the lattice thermal conductivity valid at finite frequency. Solution of boltzmanns equation for semiconductors using. A deviational monte carlo formulation of ab initio phonon. Hadjiconstantinou massachusetts institute of technology department of mechanical engineering 8 november 2011 acknowledgements. The boltzmann equation of phonon thermal transport solved. Boltzmanns equation describes the evolution of the oneparticle distribution function f fx, u, t, where the vector x, with components x 1, x 2, x 3, is the position vector, u, with components u 1, u 2, u 3, is the velocity vector, and t is the time. Pdf on the boltzmann equation of thermal transport for. Transport properties boltzmann equation relaxation time approximation equilibrium distribution relaxation time electrons in oscillating electric field small deviations linearized. The second approach, on the contrary, corrects the formula for thermal conductivity by solving the linearized boltzmann equation with given boundary conditions but uses bulk acoustic phonon dispersion and phonon mfp. Boltzmann equation an overview sciencedirect topics.
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