Draft:Relaxon

A relaxon is an eigenmode of the symmetrized collision operator appearing in the phonon Boltzmann transport equation. Unlike a phonon, which is a normal mode of the harmonic lattice with a well-defined frequency and group velocity, a relaxon represents a collective deviation of many phonon populations from thermal equilibrium and has a well-defined relaxation time. The relaxon description of thermal transport was introduced by Andrea Cepellotti and Nicola Marzari in 2016.

Phonon Boltzmann equation

The thermal-conductivity tensor ${\boldsymbol {\kappa }}$ is defined by Fourier's law

$$\mathbf {J} =-{\boldsymbol {\kappa }}\nabla T,$$

where $\mathbf {J}$ is the heat-flux density. In a crystalline solid, a semiclassical microscopic description of heat transport is provided by the phonon Boltzmann equation, originally developed by Rudolf Peierls. It may be written as

$${\frac {\partial n_{\mu }}{\partial t}}+\mathbf {v} _{\mu }\cdot \nabla n_{\mu }=\left({\frac {\partial n_{\mu }}{\partial t}}\right)_{\mathrm {coll} },$$

where $\mu =(\mathbf {q} ,s)$ labels a phonon wave vector and branch, $n_{\mu }$ is its occupation number, and

$$\mathbf {v} _{\mu }=\nabla _{\mathbf {q} }\omega _{\mu }$$

is its group velocity. The term on the right describes changes in the phonon populations caused by scattering.

Close to local thermal equilibrium, the occupation is written as

$$n_{\mu }(\mathbf {r} ,t)={\bar {n}}_{\mu }{\bigl (}T(\mathbf {r} ,t){\bigr )}+\Delta n_{\mu }(\mathbf {r} ,t),$$

where

$${\bar {n}}_{\mu }(T)={\frac {1}{\exp \left(\hbar \omega _{\mu }/k_{\mathrm {B} }T\right)-1}}$$

is the equilibrium Bose–Einstein distribution and $\Delta n_{\mu }$ is a small nonequilibrium deviation.

For a stationary system subject to a small temperature gradient, linearization gives

$${\frac {\partial {\bar {n}}_{\mu }}{\partial T}}\,\mathbf {v} _{\mu }\cdot \nabla T=-\sum _{\mu '}\Omega _{\mu \mu '}\Delta n_{\mu '},$$

where $\Omega _{\mu \mu '}$ is the linearized phonon collision matrix. It may contain contributions from three-phonon and higher-order anharmonic interactions, isotope disorder, defects and other scattering mechanisms. In perturbative calculations, three-phonon transition probabilities are obtained from third-order interatomic force constants using Fermi's golden rule, subject to conservation of energy and crystal momentum.

Symmetrized collision operator

The collision matrix can be transformed into a real symmetric form by defining

$${\widetilde {\Omega }}_{\mu \mu '}=\Omega _{\mu \mu '}{\sqrt {\frac {{\bar {n}}_{\mu '}({\bar {n}}_{\mu '}+1)}{{\bar {n}}_{\mu }({\bar {n}}_{\mu }+1)}}},$$

together with the rescaled population deviation

$$\Delta {\widetilde {n}}_{\mu }={\frac {\Delta n_{\mu }}{\sqrt {{\bar {n}}_{\mu }({\bar {n}}_{\mu }+1)}}}.$$

The mode heat capacity is

$$C_{\mu }=\hbar \omega _{\mu }{\frac {\partial {\bar {n}}_{\mu }}{\partial T}}={\frac {(\hbar \omega _{\mu })^{2}}{k_{\mathrm {B} }T^{2}}}{\bar {n}}_{\mu }({\bar {n}}_{\mu }+1),$$

and the total heat capacity is

$$C=\sum _{\mu }C_{\mu }.$$

Introducing the normalized temperature vector

$$\theta _{\mu }^{0}={\sqrt {\frac {C_{\mu }}{C}}},$$

the steady-state linearized Boltzmann equation becomes

$${\sqrt {\frac {C}{k_{\mathrm {B} }T^{2}}}}\,\theta _{\mu }^{0}\,\mathbf {v} _{\mu }\cdot \nabla T=-\sum _{\mu '}{\widetilde {\Omega }}_{\mu \mu '}\Delta {\widetilde {n}}_{\mu '}.$$

The superscript $0$ in $\theta _{\mu }^{0}$ labels the temperature vector. This vector is not, in general, an eigenvector or a zero mode of the collision matrix.

For scattering processes satisfying the second law of thermodynamics locally, the symmetrized collision matrix is positive semidefinite,

$$\sum _{\mu \mu '}\Delta {\tilde {n}}_{\mu }{\widetilde {\Omega }}_{\mu \mu '}\Delta {\tilde {n}}_{\mu '}\geq 0,$$

so its eigenvalues are non-negative.

Relaxon basis

A relaxon $|\alpha \rangle$ is defined as an eigenvector of the symmetrized collision operator,

$${\hat {\widetilde {\Omega }}}|\alpha \rangle ={\frac {1}{\tau _{\alpha }}}|\alpha \rangle ,$$

where $\tau _{\alpha }$ is the relaxon relaxation time. In the phonon basis, this equation is

$$\sum _{\mu '}{\widetilde {\Omega }}_{\mu \mu '}\theta _{\mu '}^{\alpha }={\frac {1}{\tau _{\alpha }}}\theta _{\mu }^{\alpha },$$

where

$$\theta _{\mu }^{\alpha }=\langle \mu |\alpha \rangle .$$

Because the symmetrized collision matrix is real and symmetric, its eigenvectors can be chosen to form an orthonormal basis,

$$\sum _{\mu }\theta _{\mu }^{\alpha }\theta _{\mu }^{\beta }=\delta _{\alpha \beta },\qquad \sum _{\alpha }\theta _{\mu }^{\alpha }\theta _{\mu '}^{\alpha }=\delta _{\mu \mu '}.$$

The nonequilibrium phonon distribution can therefore be expanded as

$$\Delta {\widetilde {n}}_{\mu }=\sum _{\alpha }f_{\alpha }\theta _{\mu }^{\alpha },$$

where $f_{\alpha }$ gives the amplitude of relaxon $\alpha$.

A phonon is an eigenstate of the velocity operator,

$${\hat {\mathbf {v} }}|\mu \rangle =\mathbf {v} _{\mu }|\mu \rangle ,$$

but is not generally an eigenstate of the collision operator. Conversely, a relaxon has a well-defined relaxation time but is generally a superposition of many phonons.

In a spatially homogeneous steady-state problem, a transport velocity can be associated with a relaxon through

$$V_{\alpha }^{i}=\sum _{\mu }\theta _{\mu }^{0}v_{\mu }^{i}\theta _{\mu }^{\alpha }=\langle 0|{\hat {v}}_{i}|\alpha \rangle ,$$

where the vector $|0\rangle$ has components $\langle \mu |0\rangle =\theta _{\mu }^{0}$. Here $|0\rangle$ is the temperature vector and should not be confused with a zero-eigenvalue relaxon.

For spatially varying or time-dependent transport, the velocity operator is generally not diagonal in the relaxon basis. Its matrix elements are

$$V_{\alpha \beta }^{i}=\langle \alpha |{\hat {v}}_{i}|\beta \rangle =\sum _{\mu }\theta _{\mu }^{\alpha }v_{\mu }^{i}\theta _{\mu }^{\beta },$$

so spatial propagation can couple different relaxons.

Thermal conductivity

Within the harmonic expression for the lattice heat flux, the nonequilibrium heat-flux density is

$$J_{i}={\frac {1}{V}}\sum _{\mu }\hbar \omega _{\mu }v_{\mu }^{i}\Delta n_{\mu },$$

where $V$ is the crystal volume.

In operator notation, solving the steady-state linearized Boltzmann equation and comparing the resulting heat flux with Fourier's law gives

$$\kappa ^{ij}={\frac {C}{V}}\langle 0|{\hat {v}}_{i}{\hat {\widetilde {\Omega }}}^{+}{\hat {v}}_{j}|0\rangle ,$$

where ${\hat {\widetilde {\Omega }}}^{+}$ denotes the Moore–Penrose inverse of the symmetrized collision operator. If the collision operator has no null eigenvalues in the relevant transport subspace, its ordinary inverse may be used instead.

Inserting the relaxon resolution of the identity gives the spectral representation

$$\kappa ^{ij}={\frac {C}{V}}\sum _{\alpha }^{'}V_{\alpha }^{i}V_{\alpha }^{j}\tau _{\alpha }.$$

The prime indicates that only eigenmodes for which the inverse is defined are included. Exact zero modes require separate consideration: depending on their overlap with the heat-current operator and the available momentum-relaxing processes, they may correspond to a conserved quantity or to divergent ballistic transport.

This expression resembles the kinetic formula for independent carriers, but the carriers are collective relaxons rather than individual phonons. In this representation each relaxon is associated with the total heat capacity $C$, while its contribution is distinguished by its velocity and relaxation time.

Since ${\widetilde {\Omega }}$ is positive semidefinite, the finite conductivity obtained from its inverse or pseudoinverse is also positive semidefinite,

$$\sum _{ij}g_{i}\kappa ^{ij}g_{j}\geq 0$$

for any real vector $\mathbf {g}$.

Relaxation-time approximation

The relaxation-time approximation, also called the single-mode approximation in this context, neglects the off-diagonal elements of the collision matrix in the phonon basis,

$${\hat {\widetilde {\Omega }}}\simeq \sum _{\mu }|\mu \rangle {\frac {1}{\tau _{\mu }^{\mathrm {RTA} }}}\langle \mu |.$$

The resulting thermal conductivity is

$$\kappa _{\mathrm {RTA} }^{ij}={\frac {1}{V}}\sum _{\mu }C_{\mu }v_{\mu }^{i}v_{\mu }^{j}\tau _{\mu }^{\mathrm {RTA} }.$$

This assigns an independent lifetime to each phonon mode. In the full solution, however, an initial deviation of one phonon population is generally a superposition of several relaxons and therefore decays through several relaxation times. The relaxation-time approximation can consequently differ substantially from the full solution when off-diagonal scattering processes and collective phonon transport are important.

Direct numerical solution

Eigenvector methods for the linearized phonon Boltzmann equation predate the term relaxon. Guyer and Krumhansl developed an eigenfunction treatment of the collision operator in their study of phonon hydrodynamics.

Chaput later formulated a direct numerical solution by symmetrizing and diagonalizing the collision kernel. This produces a spectral representation of both static and frequency-dependent thermal conductivity.

The open-source package phono3py implements both relaxation-time and direct solutions of the phonon Boltzmann equation using harmonic and third-order interatomic force constants. In its direct solution, the symmetrized collision matrix is diagonalized and a numerical pseudoinverse is constructed from its eigenvalues. The eigenvectors obtained in this procedure are the same mathematical objects used as relaxons, subject to the normalization conventions used for the collision matrix.

See also

• Boltzmann transport equation
• Phonon
• Thermal conductivity and resistivity
• Thermal conduction
• Phonon scattering
• Second sound
• Hydrodynamic quantum analogs