A Unified Theoretical Framework for Phonon Heat Conduction (WTE)

In solids, the main carriers of heat are phonons. For decades, researchers have proposed two classic theories to explain phonon heat transport in different kinds of materials:

• In crystals, phonons behave like little particles that fly and scatter, and heat conduction is described by the Peierls–Boltzmann Equation (BTE).
• In glasses, phonons lose their “particle-like” character, vibrational modes become localized, and heat flow proceeds via coherent tunneling diffusion between different modes. This is the physical picture behind the Allen–Feldman (AF) equation.

However, in realistic complex materials, particle-like and wave-like mechanisms often coexist. Traditional BTE and AF models each apply only to one extreme limit and lack a unified description.
The Wigner Transport Equation (WTE) was introduced precisely to place both limits into a single quantum phase–space framework, thereby providing a unified description.

Peierls–Boltzmann Equation

BTE: Phonons propagate like little particles

Applicable systems: long-range ordered crystals with well-separated phonon bands and clearly defined modes.

Phonons can be viewed as localized wave packets, like little particles flying through space with a well-defined group velocity, undergoing scattering during their propagation.

Linearized BTE:

\[\partial_t f_{\mathbf q s} + \mathbf v_{\mathbf q s}\cdot\nabla_{\mathbf r} f_{\mathbf q s} = -\frac{f_{\mathbf q s} - f^{\text{eq}}_{\mathbf q s}}{\tau_{\mathbf q s}}\]

• $f_{\mathbf q s}$: phonon occupation function
• $\mathbf v_{\mathbf q s} = \nabla_{\mathbf q}\omega_{\mathbf q s}$: group velocity
• $\tau_{\mathbf q s}$: relaxation time

Thermal conductivity in RTA form:

\[\kappa_{\alpha\beta}^{\mathrm{BTE}} = \frac{1}{V}\sum_{\mathbf q s} C_{\mathbf q s}\, v_{\mathbf q s,\alpha}\, v_{\mathbf q s,\beta}\, \tau_{\mathbf q s},\]

Allen–Feldman Equation

AF: Energy is no longer carried by “flying particles”, but transferred by one “tuning fork” driving another into resonance.

Applicable systems: glasses and disordered solids.

In glasses, vibrational modes are highly localized and possess almost no effective group velocity. Energy is not transported via “phonon particles” but instead diffuses through tunneling/resonant coupling between different localized states:

• Energy spacing between vibrational states is extremely small (quasi-degenerate), allowing mutual coupling.
• This can be likened to “a row of tuning forks”: when frequencies are close, striking one tuning fork can drive the neighboring ones, and energy is passed along step by step.

Mathematical expression:

\[\kappa_{\text{AF}} = \frac{\pi}{V} \sum_{i\neq j} (n_i - n_j)\, |\langle i | \hat v | j \rangle|^2 \, \delta(\omega_i - \omega_j)\]

• $\hat v$: heat current (velocity) operator
• $\delta(\omega_i - \omega_j)$: energy conservation; only quasi-degenerate states can couple to each other

Wigner Transport Equation

WTE: The “director” that puts both “particles” and “tuning forks” on the same stage

Starting from the quantum mechanical one-body density matrix and performing a Wigner transformation, one obtains a distribution function in phase space (momentum $\mathbf q$ and real space $\mathbf R$):

\[n_{s,s'}(\mathbf q, \mathbf R, t)\]

Diagonal elements ($s=s’$): populations, corresponding to particle-like mechanisms, similar to BTE.
Off-diagonal elements ($s\neq s’$): coherences, corresponding to wave-like mechanisms, similar to AF.

The unique feature of WTE is that it tracks not only the number of phonons, but also the coherence between different phonon states.

The general form of WTE is:

\[\frac{\partial}{\partial t} n_{s,s'} + i(\omega_{\mathbf q s}-\omega_{\mathbf q s'})\,n_{s,s'} + \tfrac{1}{2}\{\mathbf v(\mathbf q), \nabla_{\mathbf R} n\}_{s,s'} = \left.\frac{\partial n}{\partial t}\right|_{\text{scatt}}\]

where

• First term: time evolution (dynamical change)
• Second term: coherent oscillation (wave-like mechanism)
• Third term: drift term (particle transport)
• Right-hand side: scattering term (anharmonic effects)

In the crystalline limit: band separations are large ($\Delta \omega \gg 1/\tau$), coherence terms oscillate rapidly and average out, and WTE → BTE.

In the glassy limit: modes are quasi-degenerate ($\Delta \omega \ll 1/\tau$), the particle term becomes ineffective, and WTE → AF.

In complex crystals / intermediate regimes: $\Delta \omega \sim 1/\tau$, particle and wave contributions are inseparable, and WTE yields

\[\kappa = \kappa_P + \kappa_C\]

• Particle term (populations): under weak interband-coupling approximation, it can be written in a form similar to BTE:

  \[\kappa_{\alpha\beta}^{P} \simeq \frac{1}{V}\sum_{\mathbf q s} C_{\mathbf q s}\, v_{\mathbf q s,\alpha}\, v_{\mathbf q s,\beta}\, \tilde{\tau}_{\mathbf q s},\]

  where $\tilde{\tau}{\mathbf q s}\approx \big(2\Gamma{\mathbf q s}\big)^{-1}$ is related to the spectral linewidth $\Gamma_{\mathbf q s}$ (arising from anharmonicity, isotope scattering, etc.), reflecting a spectroscopic correction to the “particle lifetime” in WTE.

• Coherent term (coherences): characterizes wave-like coupling/tunneling between quasi-degenerate modes. A commonly used Lorentzian-type expression in the frequency domain can be written as

  \[\kappa_{\alpha\beta}^{C} \simeq \frac{1}{V}\sum_{\mathbf q}\!\!\sum_{s\neq s'} \frac{\big[\hbar(\omega_{\mathbf q s}+\omega_{\mathbf q s'})/2\big]\, \partial_T n^{\mathrm{eq}}_{\bar{\omega}}} {(\omega_{\mathbf q s}-\omega_{\mathbf q s'})^{2} +(\Gamma_{\mathbf q s}+\Gamma_{\mathbf q s'})^{2}}\, (v_{\alpha})_{s s'}(v_{\beta})_{s' s}\, \big(\Gamma_{\mathbf q s}+\Gamma_{\mathbf q s'}\big),\]

  where $(v_{\alpha})_{ss’}$ denotes the off-diagonal elements of the velocity operator in the mode basis (determining the strength of coherent coupling), $\bar{\omega}$ is an appropriate average of the two mode frequencies, and $\Gamma$ controls the width associated with decoherence/dephasing.

This expression highlights that smaller band separations between modes, larger off-diagonal velocity matrix elements, and moderate linewidths all enhance coherent heat transport.

Relations and Unification of the Three Theories

• BTE limit: When band separations are much larger than linewidths, coherent effects can be neglected and WTE automatically reduces to BTE.
• AF limit: In the strongly disordered limit, WTE reduces to the AF equation.
• Complex crystals and intermediate regimes: When band spacings are comparable to phonon linewidths, neither BTE nor AF is sufficient. WTE simultaneously captures contributions from populations and coherent states and yields the full thermal conductivity.

Thus, WTE is not only a mathematical unification but also a physical bridge between these pictures.

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References:

[1] Nat. Phys. 15, 809–813 (2019)

[2] Phys. Rev. X 12, 041011 (2022)