The phonon Boltzmann equation assigns each mode a frequency, group velocity, population, and lifetime. Heat current arises from the drift of those populations. This is the particle picture. Yet lattice vibrations are waves: distinct eigenmodes can retain phase relationships, hybridize, and exchange energy through off-diagonal coherences. When those coherences survive on the transport time scale, knowing only how many phonons occupy each mode is no longer sufficient.
The earlier articles treated the scale limits of Fourier’s law, Normal and Umklapp collisions, and the first-principles PBTE workflow. This article follows a different thread: when populations are insufficient, how does the Wigner equation place particle propagation and wave coherence in one kinetic framework?
For a periodic crystal, the harmonic dynamical matrix defines eigenmodes
\[\lambda=(\boldsymbol q,s),\]with wave vector $\boldsymbol q$, branch index $s$, frequency $\omega_{\boldsymbol q s}$, and group velocity
\[\boldsymbol v_{\boldsymbol q s} =\nabla_{\boldsymbol q}\omega_{\boldsymbol q s}.\]If the only dynamical variable is the population $n_{\boldsymbol q s}$, a phonon behaves as a quasiparticle with energy $\hbar\omega_{\boldsymbol q s}$ that propagates at its group velocity and is redistributed by collisions. The conventional PBTE solves for the departure of these populations from equilibrium.
The wave description retains more. With $\hat a_{\boldsymbol q s}$ the annihilation operator of a normal mode, define the single-particle density matrix
\[N_{ss'}(\boldsymbol q) =\langle \hat a_{\boldsymbol q s'}^{\dagger} \hat a_{\boldsymbol q s} \rangle.\]Its diagonal elements $N_{ss}$ are mode populations. Its off-diagonal elements $N_{ss’}$ encode phase correlations between vibrational branches. The particle picture is therefore not an alternative to wave mechanics; it is the reduced description obtained when the density matrix is effectively diagonal in the mode basis.
The useful question is not whether a phonon “really is” a particle or a wave. It is whether off-diagonal coherence can be eliminated before the heat current is formed at the material, temperature, and observation scale of interest.
Two modes with widely separated frequencies accumulate relative phase rapidly. Their off-diagonal contribution usually averages away over transport times, leaving distinguishable modes whose populations and lifetimes are effective kinetic variables.
The separation becomes ambiguous when the mode spacing is comparable to the linewidth. Define
\[\Delta\omega_{ss'} =|\omega_s-\omega_{s'}|, \qquad \Gamma_{ss'}=\Gamma_s+\Gamma_{s'}.\]A useful qualitative warning sign is
\[\Delta\omega_{ss'}\lesssim\Gamma_{ss'}.\]Within one scattering time, the system can no longer sharply resolve whether the excitation belongs to branch $s$ or $s’$. Large unit cells create dense phonon manifolds, local resonances produce flat or nearly degenerate bands, strong anharmonicity broadens the modes, and disorder weakens the identity of propagating quasiparticles. Each route can make coherence relevant to heat flow.
This does not mean that increasing linewidth always strengthens wave transport. Coherence reflects competition among mode mixing, frequency detuning, and dephasing. If linewidths are extremely narrow, separated modes average out through rapid relative phase rotation; if dephasing is too strong, coherence itself dies quickly. The crossover is most pronounced when spacing and broadening become comparable.
The phonon-gas model is thus natural for crystals with sparse, well-resolved bands and long-lived quasiparticles. Complex crystals and disordered solids may require populations and coherences together. This is a continuous spectral crossover, not a rigid division between classes of materials.
The Wigner description extends the density matrix into phase space through a matrix distribution
\[\boldsymbol N(\boldsymbol r,\boldsymbol q,t).\]For temperature fields that vary slowly on the scale of a unit cell, but without discarding interbranch coherence, the phonon Wigner transport equation can be written as
\[\frac{\partial\boldsymbol N}{\partial t} +i[\boldsymbol\Omega,\boldsymbol N] +\frac{1}{2}\sum_{\alpha} \left\{ \boldsymbol V^{\alpha}, \frac{\partial\boldsymbol N}{\partial r_{\alpha}} \right\} = \left.\frac{\partial\boldsymbol N}{\partial t}\right|_{\mathrm{coll}}.\]$\boldsymbol\Omega(\boldsymbol q)$ is the frequency matrix, $\boldsymbol V^{\alpha}(\boldsymbol q)$ is the velocity operator, and square and curly brackets denote a commutator and anticommutator.
The first term evolves the distribution in time. The commutator
\[i[\boldsymbol\Omega,\boldsymbol N]\]describes phase precession between modes of different frequency. The anticommutator transports energy in real space and contains both diagonal group velocities and off-diagonal velocity matrix elements. The collision term relaxes and repopulates the diagonal populations while also dephasing the coherences.
The heat current is likewise a matrix quantity. Schematically,
\[J^{\alpha} =\frac{\hbar}{2VN_q} \sum_{\boldsymbol q,ss'} (\omega_s+\omega_{s'}) V^{\alpha}_{ss'} N_{s's}.\]For $s=s’$, this is the familiar mode energy times group velocity times non-equilibrium population. For $s\ne s’$, heat current is carried through coherence between branches and off-diagonal velocity matrix elements. Both forms are present in the same equation.
In linear response, the Wigner solution can be organized as
\[\boldsymbol\kappa =\boldsymbol\kappa_{\mathrm P} +\boldsymbol\kappa_{\mathrm C},\]where $\boldsymbol\kappa_{\mathrm P}$ is the population contribution and $\boldsymbol\kappa_{\mathrm C}$ is the coherence contribution.
The population term reduces to the PBTE in its proper limit: heat capacities, group velocities, and collision-controlled mean free displacements determine the current. It represents energy carried by phonons propagating along bands between collisions.
The coherence term contains interbranch velocity matrix elements. Omitting convention-dependent prefactors, the contribution of a mode pair has the structure
\[\kappa_{\mathrm C}^{\alpha\beta} \sim \sum_{\boldsymbol q}\sum_{s\ne s'} A_{ss'}(T) V_{ss'}^{\alpha}V_{s's}^{\beta} \frac{\Gamma_s+\Gamma_{s'}} {(\omega_s-\omega_{s'})^2 +(\Gamma_s+\Gamma_{s'})^2/4},\]where $A_{ss’}(T)$ contains the thermal weights from mode energies and equilibrium populations. The Lorentzian-like factor makes the physical condition transparent: coherence needs nonzero off-diagonal velocity coupling and becomes important when frequency separation and linewidth are comparable.
In a simple crystal with few branches and well-separated frequencies, $\kappa_{\mathrm C}$ is often small and the Wigner equation naturally reduces to phonon Boltzmann transport. As unit cells become more complex, bands become denser, or disorder increases, $\kappa_{\mathrm P}$ can weaken while $\kappa_{\mathrm C}$ grows. In the corresponding limits, the framework connects propagating-phonon Boltzmann transport with Allen–Feldman-like intermode transfer in disordered solids.
Wigner coherence is off-diagonal in the branch indices near a given $\boldsymbol q$. Phonon hydrodynamics instead concerns collective drift of many populations under strong momentum-conserving Normal collisions. Hydrodynamic behavior can occur while the density matrix remains nearly diagonal in branch space. One problem concerns intermode phase; the other concerns collision invariants.
Ballistic transport is also not synonymous with wave transport. Ballistic means that carriers undergo few internal collisions across the device; a diagonal-population Boltzmann or Landauer model can describe it. Conversely, coherence can contribute in bulk materials with substantial scattering and finite linewidths.
Nanopillars, local resonators, and complex supercells create flat bands, avoided crossings, and near-degenerate modes, making Wigner analysis attractive. Yet observing a local resonance does not prove that $\kappa_{\mathrm C}$ dominates. One must examine frequency spacing, linewidths, off-diagonal velocity elements, and their thermal weights. Likewise, defining a “wave contribution” from the difference between molecular dynamics and particle Monte Carlo is an operational decomposition for a specific structure. It is related to, but not automatically identical with, the coherence conductivity $\kappa_{\mathrm C}$ of the Wigner framework.
A practical calculation should therefore follow a clear chain. The dynamical matrix provides frequencies, eigenvectors, and the full velocity matrix. Anharmonicity supplies linewidths or a more complete collision operator. The diagonal populations and off-diagonal coherences are then solved together, and $\kappa_{\mathrm P}$ and $\kappa_{\mathrm C}$ must each be converged. The scientifically useful result is not the statement that phonons have wave–particle duality, but which pairs of modes carry how much coherent heat at a given temperature and why that current cannot be absorbed into an independent-quasiparticle picture.
The value of the Wigner equation is precisely that it does not force a choice between “phonons are particles” and “phonons are waves.” Particle propagation is the diagonal dynamics of the density matrix; wave-like transport is its off-diagonal dynamics. They are two sectors of one quantum transport problem.
在声子玻尔兹曼输运方程中，每个模式具有频率、群速度、布居和寿命，热流来自大量声子的漂移。这是粒子图像。可是晶格振动本质上是波，不同本征模式之间可以保持相位关系、发生杂化，并通过非对角相干传递能量。当这些相干不能在输运时间尺度内忽略时，只追踪每个模式“有多少声子”就不再充分。
前面的文章分别讨论了傅里叶定律的尺度极限、Normal 与 Umklapp 碰撞以及第一性原理 PBTE 工作流。本文只沿着另一条主线展开：当声子布居不足以描述热输运时，Wigner 方程怎样把粒子传播与波动相干放进同一个动力学框架？
对周期晶体，谐性动力学矩阵给出本征模式
\[\lambda=(\boldsymbol q,s),\]其中 $\boldsymbol q$ 是波矢，$s$ 是分支。模式具有频率 $\omega_{\boldsymbol q s}$ 和群速度
\[\boldsymbol v_{\boldsymbol q s} =\nabla_{\boldsymbol q}\omega_{\boldsymbol q s}.\]如果只关心模式布居 $n_{\boldsymbol q s}$，一个声子就表现为能量为 $\hbar\omega_{\boldsymbol q s}$、以群速度传播并被碰撞重新分配的准粒子。传统 PBTE 正是求解这些布居对平衡分布的偏离。
波动性包含更多信息。设 $\hat a_{\boldsymbol q s}$ 是模式的湮灭算符，则单粒子密度矩阵可以写为
\[N_{ss'}(\boldsymbol q) =\langle \hat a_{\boldsymbol q s'}^{\dagger} \hat a_{\boldsymbol q s} \rangle.\]对角元 $N_{ss}$ 就是模式布居；非对角元 $N_{ss’}$ 则记录两个振动分支之间的相位关联。粒子图像不是“错误的波动理论”，而是密度矩阵在模式表象中近似对角时得到的约化描述。
因此，真正的分界并不是问“声子究竟是粒子还是波”，而是问：在给定材料、温度和观测尺度下，非对角相干是否能在形成热流之前被安全消去？
若两个模式的频率相差很大，它们之间的相对相位会快速旋转，长时间平均后非对角贡献通常相互抵消。此时每个模式都可以被清楚区分，布居和寿命构成有效的动力学变量。
问题出现在频率间隔与线宽可比时。用
\[\Delta\omega_{ss'} =|\omega_s-\omega_{s'}|, \qquad \Gamma_{ss'}=\Gamma_s+\Gamma_{s'},\]分别表示模式间隔和总展宽，一个常用的定性判据是
\[\Delta\omega_{ss'}\lesssim\Gamma_{ss'}.\]此时“这是模式 $s$ 的声子还是模式 $s’$ 的声子”不再能在散射时间内被清楚分辨。复杂晶胞会产生密集声子支，局域共振会形成平坦或近简并模，强非谐性会增大线宽，无序则会进一步削弱传播模的独立性；这些因素都可能使模式相干进入热流。
这并不意味着线宽越大，波动贡献就一定越强。相干项来自模式混合、频率失谐与退相干之间的竞争：线宽太窄时，频率不同的模式会因快速相位旋转而平均掉；线宽过宽时，相干本身又会迅速衰减。最显著的交叉通常出现在间隔与展宽处于相近量级时。
从这个角度看，传统声子气体模型适合能带稀疏、模式可辨识且准粒子寿命足够长的晶体；复杂晶体和无序固体则可能需要同时追踪布居与相干。两者之间不是一道材料类别的硬边界，而是由声子谱和线宽共同控制的连续过渡。
Wigner 描述把密度矩阵从纯粹的模式空间扩展到相空间，定义随位置、波矢和时间变化的矩阵分布
\[\boldsymbol N(\boldsymbol r,\boldsymbol q,t).\]在温度变化尺度远大于晶胞、但又不预先丢弃分支间相干的条件下，声子 Wigner 输运方程可以写成
\[\frac{\partial\boldsymbol N}{\partial t} +i[\boldsymbol\Omega,\boldsymbol N] +\frac{1}{2}\sum_{\alpha} \left\{ \boldsymbol V^{\alpha}, \frac{\partial\boldsymbol N}{\partial r_{\alpha}} \right\} = \left.\frac{\partial\boldsymbol N}{\partial t}\right|_{\mathrm{coll}}.\]这里 $\boldsymbol\Omega(\boldsymbol q)$ 是频率矩阵，$\boldsymbol V^{\alpha}(\boldsymbol q)$ 是速度算符，方括号和花括号分别表示对易子与反对易子。
这个方程的逻辑非常紧凑。第一项描述分布随时间变化；对易子
\[i[\boldsymbol\Omega,\boldsymbol N]\]描述不同频率模式之间的相位进动；反对易子项描述能量在实空间中的传播，并同时包含对角群速度与非对角速度矩阵元；右端碰撞项则负责布居弛豫、模式再布居和相干退相干。
热流也不再只由对角布居给出。其矩阵形式可示意写为
\[J^{\alpha} =\frac{\hbar}{2VN_q} \sum_{\boldsymbol q,ss'} (\omega_s+\omega_{s'}) V^{\alpha}_{ss'} N_{s's}.\]当 $s=s’$ 时，它退化为熟悉的“模式能量乘群速度乘非平衡布居”；当 $s\neq s’$ 时，热流来自不同振动分支之间的相干及非对角速度矩阵元。这正是 Wigner 方程能够同时容纳粒子与波动贡献的原因。
在线性响应下，Wigner 方程给出的热导率可以组织为
\[\boldsymbol\kappa =\boldsymbol\kappa_{\mathrm P} +\boldsymbol\kappa_{\mathrm C},\]其中 $\boldsymbol\kappa_{\mathrm P}$ 来自密度矩阵对角元的布居传播，$\boldsymbol\kappa_{\mathrm C}$ 来自非对角元的相干传输。
布居项在适当极限下回到 PBTE：热容、群速度和由碰撞算符确定的平均自由位移共同产生热流。它强调声子沿能带传播，在碰撞之间携带能量。
相干项则包含不同分支之间的速度矩阵元。忽略具体归一化约定，它的模式对贡献具有如下结构：
\[\kappa_{\mathrm C}^{\alpha\beta} \sim \sum_{\boldsymbol q}\sum_{s\ne s'} A_{ss'}(T) V_{ss'}^{\alpha}V_{s's}^{\beta} \frac{\Gamma_s+\Gamma_{s'}} {(\omega_s-\omega_{s'})^2 +(\Gamma_s+\Gamma_{s'})^2/4},\]其中 $A_{ss’}(T)$ 汇集模式能量与平衡布居的热权重。这个类 Lorentz 结构清楚显示：相干贡献需要非零的非对角速度耦合，并在频率间隔与线宽可比时变得重要。
在简单晶体、支数较少且频率间隔远大于线宽时，$\kappa_{\mathrm C}$ 往往很小，Wigner 方程自然回到声子 Boltzmann 图像。随着晶胞变复杂、能带变密或无序增强，$\kappa_{\mathrm P}$ 可以减弱而 $\kappa_{\mathrm C}$ 增强；在相应极限下，这一框架能够连接传播声子的 Boltzmann 输运与无序固体中的 Allen–Feldman 型模间传输。
Wigner 相干描述的是同一 $\boldsymbol q$ 附近不同振动分支之间的非对角密度矩阵。声子流体动力学描述的则是大量声子布居在强 Normal 碰撞下形成的集体漂移；即使密度矩阵在分支表象中近似对角，这种集体效应仍然可以存在。前者关心模式相位，后者关心碰撞守恒量，它们是两个不同维度的问题。
弹道输运也不等于波动输运。弹道只表示载流子在器件尺度内很少发生内部散射，一个完全基于对角布居的 Boltzmann 或 Landauer 模型也可以描述弹道热流。反过来，相干贡献可以出现在具有明显散射和有限线宽的体材料中。
对带有纳米柱、局域共振单元或复杂超晶胞的结构，平坦带、避免交叉和近简并模使 Wigner 分析尤其有吸引力。但观察到局域共振，并不能自动推出 $\kappa_{\mathrm C}$ 占主导；还需要检查频率间隔、线宽、非对角速度矩阵元及其热权重。类似地，用分子动力学与粒子 Monte Carlo 的差异定义“波动贡献”是一种针对具体结构的操作性分解，它与 Wigner 热导率中的 $\kappa_{\mathrm C}$ 有联系，却不是无需验证便可直接等同的量。
实际计算因此应沿着一条明确逻辑展开：先由动力学矩阵得到频率、本征矢和完整速度矩阵，再由非谐性获得线宽或更完整的碰撞算符，最后同时求解对角布居与非对角相干，并分别检查 $\kappa_{\mathrm P}$ 和 $\kappa_{\mathrm C}$ 的收敛性。最值得报告的不是一句“声子具有波粒二象性”，而是哪些模式对在什么温度下产生了多大的相干热流，以及这一贡献为什么不能被独立准粒子图像吸收。
Wigner 输运方程的价值正在于此：它没有要求我们在“声子是粒子”与“声子是波”之间作选择，而是明确告诉我们，粒子传播是密度矩阵的对角动力学，波动传输是其非对角动力学；二者只是同一个量子输运问题的不同部分。
The central question is therefore not merely how frequently phonons collide, but which quantities a collision conserves and which slow collective motions it relaxes.
The previous article treated collective momentum only as an additional slow variable beyond Fourier’s law. Here the general non-Fourier criteria are not repeated; the focus is how that momentum is established, transported, and dissipated by collisions.
Anharmonicity allows phonons to be created and annihilated. For a three-phonon absorption process,
\[\lambda_1+\lambda_2\rightarrow\lambda_3,\]energy and crystal momentum satisfy
\[\omega_1+\omega_2=\omega_3,\] \[\boldsymbol q_1+\boldsymbol q_2 =\boldsymbol q_3+\boldsymbol G,\]where $\boldsymbol G$ is a reciprocal-lattice vector. A decay process obeys the corresponding relations with one incoming and two outgoing phonons.
The distinction between Normal and Umklapp processes is kinematic:
Because wave vectors that differ by $\boldsymbol G$ represent equivalent points in a periodic crystal, both processes conserve crystal momentum modulo a reciprocal-lattice vector. However, only the N process conserves the sum of phonon wave vectors inside the chosen first Brillouin zone.
The familiar picture of an Umklapp event “flipping” the momentum is a useful geometric mnemonic, not a literal mechanical collision between particles. A phonon wave vector is a label of a lattice eigenmode, and crystal momentum is defined only modulo $\boldsymbol G$.
Before asking whether momentum is conserved, crystal momentum must be distinguished from the mechanical momentum of the solid.
The total phonon crystal momentum may be written schematically as
\[\boldsymbol P_{\mathrm{cr}} =\frac{1}{V}\sum_\lambda \hbar\boldsymbol q_\lambda n_\lambda.\]It is a conserved quantity associated with discrete translational symmetry. It should not be confused with the mechanical momentum of the entire crystal, which also includes the motion of the lattice and its center of mass.
The heat current is a different moment of the distribution:
\[\boldsymbol J_Q =\frac{1}{V}\sum_\lambda \hbar\omega_\lambda \boldsymbol v_\lambda n_\lambda.\]Thus, conservation of $\sum_\lambda\hbar\boldsymbol q_\lambda n_\lambda$ does not generally imply conservation of $\boldsymbol J_Q$. The two become proportional only in special cases—for example, within a simplified isotropic model with a single linear acoustic branch. Real crystals contain nonlinear dispersions, several polarizations, optical branches, and anisotropic group velocities.
This is the first important qualification:
A Normal process conserves total crystal momentum, but it can still redistribute and partially relax heat current because heat current is not generally proportional to crystal momentum.
Normal collisions rapidly exchange energy and momentum among phonon modes while preserving the total energy and crystal momentum. Repeated N scattering does not drive the system directly toward the stationary Bose–Einstein distribution
\[n_\lambda^0 =\frac{1}{\exp(\hbar\omega_\lambda/k_BT)-1}.\]Instead, it drives phonons toward a displaced Bose–Einstein distribution,
\[n_\lambda^{d} =\frac{1}{ \exp\left[ (\hbar\omega_\lambda -\hbar\boldsymbol q_\lambda\cdot\boldsymbol u_d)/k_BT \right]-1},\]where $\boldsymbol u_d$ is a collective drift parameter. The displacement in reciprocal space represents a nonzero collective crystal momentum.
In this sense, N processes are analogous to momentum-conserving collisions among molecules in a fluid. Such collisions create local equilibrium and viscosity; they do not, by themselves, bring a moving fluid to rest.
This analogy also explains a subtle point: frequent N collisions can make individual phonon mean free paths short while allowing a collective heat-carrying drift to remain long-lived. A linewidth or single-mode lifetime is therefore not automatically a transport relaxation time.
Once Normal processes establish a drifting distribution, steady resistance is controlled by mechanisms that can destroy that drift.
A process is resistive if it relaxes the slow collective motion that overlaps with the heat current. Important mechanisms include
Umklapp processes are usually resistive because the phonon system transfers a reciprocal-lattice momentum to the lattice. Nevertheless, “U process” and “a fixed amount of thermal resistance” are not synonyms. The effect of an event depends on the participating modes, their heat-current contributions, and the full coupled collision operator.
Thermal resistance is most precisely connected to entropy production and decay of the driven non-equilibrium distribution—not merely to the number of scattering events.
After linearization, the steady phonon BTE can be written schematically as
\[\boldsymbol b=\boldsymbol\Omega\boldsymbol f,\]where $\boldsymbol b$ is the temperature-gradient driving term, $\boldsymbol f$ describes the non-equilibrium modal response, and $\boldsymbol\Omega$ is the linearized collision operator. Thermal conductivity has the structure
\[\kappa\sim \langle \boldsymbol b, \boldsymbol\Omega^{-1}\boldsymbol b\rangle.\]This expression contains the essential physics. The eigenvectors of $\boldsymbol\Omega$ are collective combinations of phonon populations. Conservation laws generate zero or slowly decaying eigenvectors. If the heat-current driving term overlaps strongly with one of these slow modes, thermal conductivity becomes large.
The single-mode relaxation-time approximation replaces $\boldsymbol\Omega$ by a diagonal matrix. It assigns each phonon an independent lifetime and therefore loses much of the repopulation and collective-drift physics created by off-diagonal couplings. This approximation can be adequate when resistive scattering dominates, but it may substantially underestimate conductivity when N scattering is strong.
The iterative BTE solution is not merely a numerical refinement. It restores the coupled response between modes.
The Callaway picture compresses this operator structure into two distinct relaxation tendencies.
The Callaway picture separates two tendencies:
\[\left.\frac{\partial n_\lambda}{\partial t}\right|_{N} \sim-\frac{n_\lambda-n_\lambda^d}{\tau_{N,\lambda}},\] \[\left.\frac{\partial n_\lambda}{\partial t}\right|_{R} \sim-\frac{n_\lambda-n_\lambda^0}{\tau_{R,\lambda}}.\]N processes relax the distribution toward the drifting state $n^d$, whereas resistive processes relax it toward the stationary state $n^0$. The competition between these tendencies determines whether phonons behave mainly as independent quasiparticles or as a collective fluid.
This model is physically illuminating, although a modern first-principles collision operator need not be exactly representable by two scalar relaxation times.
Define representative lengths
\[\ell_N=v\tau_N, \qquad \ell_R=v\tau_R,\]where $\ell_N$ is the distance needed to establish collective local equilibrium and $\ell_R$ is the distance over which collective momentum is destroyed. For a channel of width $W$, three regimes can be distinguished schematically:
For $W\ll\ell_N$, transport is close to ballistic.
Phonons reach the boundary before enough N collisions occur to establish a local drifting distribution. Boundary injection and transmission dominate.
For $\ell_N\ll W\ll\ell_R$, a hydrodynamic window opens.
Many N collisions establish local collective equilibrium, while resistive scattering remains weak across the device. Momentum is redistributed internally and is lost mainly at boundaries. A viscous, Poiseuille-like heat-flow profile can emerge.
For $W\gg\ell_R$, resistive processes restore diffusion.
Resistive events destroy the collective drift well inside the sample. Local Fourier behavior is recovered at sufficiently large scales.
These inequalities are a regime map rather than sharp universal boundaries. A real phonon spectrum contains many velocities and relaxation lengths, and boundary specularity can substantially shift the crossover.
The scale inequalities are not yet experimental conclusions; they must appear in observables.
Strong N scattering can produce phenomena that cannot be understood as a gas of independently relaxing phonons:
Poiseuille heat flow.
Momentum-conserving collisions redistribute momentum across a channel. With momentum loss at the walls, the heat-flux profile can become largest at the center and suppressed near the boundaries, analogous to viscous fluid flow.
Second sound.
If momentum relaxes slowly enough, temperature and heat flux can propagate as a damped wave rather than diffuse monotonically. N processes help establish the local drifting state that supports this collective mode; resistive processes damp it.
Non-monotonic size dependence.
Moving from ballistic to hydrodynamic transport can produce a Knudsen minimum or a faster-than-ballistic growth of effective conductivity with width over a limited range. The precise signature depends on geometry and scattering spectra.
None of these effects follows from the statement “N scattering is non-resistive” alone. They arise from the hierarchy of N, resistive, and boundary time scales.
“Every collision lowers thermal conductivity.”
False. Momentum-conserving collisions can establish collective local equilibrium and redistribute momentum without directly destroying the drift.
“Normal scattering never relaxes heat current.”
Also false in general. It conserves crystal momentum, not heat current. With nonlinear, multibranch dispersions, the two are not proportional.
“Umklapp rate is the transport resistance.”
Incomplete. U scattering is an important momentum-relaxing mechanism, but conductivity depends on how the complete collision operator acts on the driven distribution.
“The shortest phonon lifetime controls heat transport.”
Not necessarily. A short single-mode lifetime can result from rapid N redistribution while a collective drift remains long-lived.
After avoiding these misconceptions, the claim of collective transport still needs a testable chain of evidence.
A difference between RTA and the full PBTE is a clue, not proof of phonon hydrodynamics. A stronger case connects four types of evidence:
The purpose here is to identify collective flow, not to repeat the complete DFT-to-conductivity workflow. That workflow is developed in From First Principles to Lattice Thermal Conductivity: The PBTE Method.
The evidence returns us to the opening question: do phonon collisions always create thermal resistance? No.
Normal scattering conserves crystal momentum and can drive phonons toward a collectively drifting distribution. It often redistributes heat current rather than directly eliminating it. Resistive mechanisms are needed to destroy that drift and produce entropy associated with steady thermal resistance.
But the stronger statement “Normal scattering has no effect on thermal resistance” is also wrong. Heat current is not generally identical to crystal momentum, and N processes reshape the non-equilibrium distribution on which resistive processes act. They can therefore change thermal conductivity profoundly and even create an entirely different transport regime.
The correct hierarchy is
\[\text{collision kinematics} \rightarrow \text{conserved quantities} \rightarrow \text{slow collective modes} \rightarrow \text{thermal resistance}.\]Counting collisions is not enough. One must ask what the collisions conserve.
因此，真正关键的问题不只是声子碰撞得有多频繁，而是碰撞守恒什么物理量，又使哪些缓慢的集体运动发生弛豫。
上一篇文章只把集体动量视为傅里叶定律之外的一种额外慢变量；本文不再重复一般的非傅里叶判据，而是专门追问这种集体动量怎样在碰撞中建立、传递和耗散。
晶格非谐性允许声子的产生和湮灭。对于三声子吸收过程
\[\lambda_1+\lambda_2\rightarrow\lambda_3,\]能量和晶格动量满足
\[\omega_1+\omega_2=\omega_3,\] \[\boldsymbol q_1+\boldsymbol q_2 =\boldsymbol q_3+\boldsymbol G,\]其中 $\boldsymbol G$ 是倒格矢。一个声子衰变成两个声子的过程满足相应的能量和晶格动量关系。
Normal 与 Umklapp 过程的区别首先是运动学上的：
由于相差一个倒格矢的波矢在周期晶格中等价，两种过程都满足模倒格矢意义下的晶格动量守恒。但只有 N 过程守恒选定第一布里渊区内声子波矢之和。
常见的“Umklapp 使动量翻转”图像是一种有用的几何记忆方式，但不能把它理解成经典粒子之间的机械碰撞。声子波矢是晶格本征模式的标记，而晶格动量本来就只在模 $\boldsymbol G$ 的意义下定义。
但在讨论“动量是否守恒”之前，必须先分清晶格动量与机械动量。
声子总晶格动量可示意性地写成
\[\boldsymbol P_{\mathrm{cr}} =\frac{1}{V}\sum_\lambda \hbar\boldsymbol q_\lambda n_\lambda.\]它是与离散平移对称性相关的守恒量，不能与整个晶体的机械动量混淆。后者还包括晶格及其质心运动。
热流则是声子分布的另一个矩：
\[\boldsymbol J_Q =\frac{1}{V}\sum_\lambda \hbar\omega_\lambda \boldsymbol v_\lambda n_\lambda.\]因此，$\sum_\lambda\hbar\boldsymbol q_\lambda n_\lambda$ 守恒并不普遍意味着 $\boldsymbol J_Q$ 守恒。只有在一些特殊模型中，例如单一线性色散声学支的各向同性近似下，两者才近似成正比。真实晶体通常具有非线性色散、多种极化、光学支和各向异性群速度。
由此得到第一个重要限定：
Normal 过程守恒总晶格动量，但它仍然可能重新分配并部分弛豫热流，因为热流通常并不与晶格动量成正比。
Normal 碰撞在声子模式之间快速交换能量和动量，同时守恒总能量与晶格动量。反复发生的 N 散射不会直接把系统推向静止的玻色–爱因斯坦分布
\[n_\lambda^0 =\frac{1}{\exp(\hbar\omega_\lambda/k_BT)-1}.\]它趋向的是一个位移玻色–爱因斯坦分布：
\[n_\lambda^{d} =\frac{1}{ \exp\left[ (\hbar\omega_\lambda -\hbar\boldsymbol q_\lambda\cdot\boldsymbol u_d)/k_BT \right]-1},\]其中 $\boldsymbol u_d$ 是集体漂移参数。倒空间中的分布位移对应非零的集体晶格动量。
从这个意义上说，N 过程类似于流体分子之间守恒动量的碰撞。这类碰撞建立局域平衡和黏性，却不会单独使一个整体运动的流体停止。
这个类比也揭示了一个容易忽视的事实：频繁的 N 碰撞可以让单个声子模式的寿命很短，同时保持集体载热漂移具有很长的寿命。因此，谱学线宽或单模寿命不能自动等同于输运弛豫时间。
Normal 过程建立漂移分布之后，真正决定稳态热阻的是哪些机制能够破坏这一漂移。
如果一种过程能够弛豫与热流重叠的缓慢集体运动，它就是阻性过程。常见机制包括：
U 过程通常具有热阻性，因为声子系统会向晶格传递一个倒格矢对应的晶格动量。但“发生 U 过程”并不等于“产生一个固定大小的热阻”。一个散射事件的实际作用取决于参与模式、这些模式对热流的贡献，以及完整耦合碰撞算符的结构。
更精确地说，热阻与熵产生以及受驱动非平衡分布的衰减有关，而不只是与碰撞次数有关。
线性化后的稳态声子 BTE 可以示意性写成
\[\boldsymbol b=\boldsymbol\Omega\boldsymbol f,\]其中 $\boldsymbol b$ 是温度梯度产生的驱动项，$\boldsymbol f$ 表示各模式的非平衡响应，$\boldsymbol\Omega$ 是线性化碰撞算符。热导率具有如下结构：
\[\kappa\sim \langle\boldsymbol b, \boldsymbol\Omega^{-1}\boldsymbol b\rangle.\]这个表达式包含了问题的核心物理。$\boldsymbol\Omega$ 的本征矢是声子布居的集体组合；守恒定律会产生零衰减或缓慢衰减的本征模式。如果热流驱动与某个慢模高度重叠，热导率就会很大。
单模弛豫时间近似把 $\boldsymbol\Omega$ 替换成对角矩阵，给每个声子分配一个相互独立的寿命。这会丢失由非对角耦合产生的模式再布居和集体漂移。当阻性散射占主导时，这个近似可能足够；当 N 散射很强时，它可能明显低估热导率。
因此，迭代求解 BTE 不只是数值精度上的修正，它恢复了模式之间的耦合响应。
Callaway 图像可以把这个算符结构压缩成两种不同的弛豫趋势。
Callaway 图像把碰撞分成两种趋势：
\[\left.\frac{\partial n_\lambda}{\partial t}\right|_N \sim-\frac{n_\lambda-n_\lambda^d}{\tau_{N,\lambda}},\] \[\left.\frac{\partial n_\lambda}{\partial t}\right|_R \sim-\frac{n_\lambda-n_\lambda^0}{\tau_{R,\lambda}}.\]N 过程使分布趋向漂移态 $n^d$，阻性过程则使其趋向静止平衡态 $n^0$。两者之间的竞争决定声子主要表现为独立准粒子，还是形成集体流体。
这个模型非常适合解释物理，但现代第一性原理碰撞算符并不一定能够被两个标量弛豫时间完全表示。
定义代表性长度
\[\ell_N=v\tau_N, \qquad \ell_R=v\tau_R,\]其中 $\ell_N$ 是建立集体局域平衡所需的长度，$\ell_R$ 是集体动量被破坏的长度。对于宽度为 $W$ 的通道，可以示意性地区分三个区间。
当 $W\ll\ell_N$ 时，输运接近弹道。
在足够多的 N 碰撞建立漂移局域分布之前，声子已经到达边界，因此输运主要由边界注入和透射决定。
当 $\ell_N\ll W\ll\ell_R$ 时，流体动力学窗口打开。
大量 N 碰撞建立局域集体平衡，而阻性散射在器件尺度上仍然很弱。动量在声子内部重新分配，并主要在边界处损失，此时可能形成具有黏性的 Poiseuille 型热流。
当 $W\gg\ell_R$ 时，阻性过程恢复扩散行为。
阻性事件在样品内部就破坏了集体漂移，足够大的尺度上重新得到局域傅里叶输运。
这些不等式是输运区间图，而不是严格的普适分界。真实声子谱包含不同的速度和弛豫长度，边界镜面性也会显著改变过渡位置。
尺度不等式本身还不是实验结论，它必须落到可观测信号上。
强 N 散射可以产生独立声子气体图像无法解释的现象。
Poiseuille 热流。
守恒动量的碰撞会在通道横向重新分配动量。如果动量主要在边界处损失，热流在通道中心最大、靠近边界减小，类似黏性流体的速度分布。
第二声。
当动量弛豫足够慢时，温度和热流可以作为阻尼波传播，而不是单调扩散。N 过程帮助建立支持这种集体模式的漂移局域状态，阻性过程则使第二声衰减。
非单调尺寸效应。
从弹道区间进入流体动力学区间时，系统可能出现 Knudsen 极小值，或在有限尺寸范围内出现有效热导率随宽度超弹道增长的行为。具体特征取决于几何结构和散射谱。
这些现象并不能仅由“N 散射没有热阻”一句话推出，而来自 N、阻性散射与边界时间尺度之间的层级关系。
“每次碰撞都会降低热导率。”
错误。守恒动量的碰撞可以建立集体局域平衡并重新分配动量，而不直接破坏漂移。
“Normal 散射永远不会弛豫热流。”
一般情况下也不正确。N 过程守恒的是晶格动量，而不是热流。对于非线性、多支色散，两者并不成正比。
“Umklapp 散射率就是输运热阻。”
不完整。U 散射是重要的动量弛豫机制，但热导率取决于完整碰撞算符如何作用于受驱动分布。
“寿命最短的声子决定热输运。”
未必。短的单模寿命可能源于快速 N 再分配，而集体漂移仍然可以很长寿。
避免这些误区之后，还需要把“可能存在集体效应”提升为一条可以检验的证据链。
RTA 与完整 PBTE 的差异只能作为线索，不能单独证明声子流体动力学。更有说服力的判定需要把四类证据连起来：
这一节关心的是“如何识别集体流”，而不是重新介绍从 DFT、IFC 到热导率的完整计算流程；后者见《从第一性原理到晶格热导率：PBTE 方法》。
这些证据最终回到文章开头的问题：声子碰撞一定产生热阻吗？不一定。
Normal 散射守恒晶格动量，可以使声子趋向具有集体漂移的分布。它通常重新分配热流，而不是直接消除热流；必须存在阻性机制，才能破坏这种漂移并产生与稳态热阻相关的熵。
但“Normal 散射对热阻完全没有影响”同样错误。热流通常不等于晶格动量，而且 N 过程会重塑阻性过程所作用的非平衡分布。因此，它能够深刻改变热导率，甚至使系统进入完全不同的输运区间。
正确的物理层级是
\[\text{碰撞运动学} \rightarrow \text{守恒量} \rightarrow \text{缓慢集体模式} \rightarrow \text{热阻}.\]仅仅统计碰撞次数是不够的。真正需要问的是：这些碰撞守恒什么？
It says that the heat-flux density $\boldsymbol q$ is proportional and opposite to the local temperature gradient. The law works extraordinarily well at macroscopic scales, but it is not a fundamental conservation law. It is an emergent, near-equilibrium relation whose validity depends on length, time, scattering, and the possibility of defining a local temperature.
This distinction matters. When Fourier’s law fails, energy is not violated. What fails is the assumption that the heat flux at one point and one instant is determined only by the temperature gradient at that same point and instant.
Local energy conservation is
\[\frac{\partial u}{\partial t}+\nabla\cdot\boldsymbol q=s,\]where $u$ is the energy density and $s$ is the volumetric heating rate. This equation does not specify how energy is transported. The missing physics is supplied by a constitutive relation for $\boldsymbol q$.
If the material is close to local equilibrium, $u$ can be written as a function of a local temperature. For a small temperature interval,
\[du=C_V\,dT,\]where $C_V$ is the volumetric heat capacity. Combining energy conservation with Fourier’s law gives the heat equation,
\[C_V\frac{\partial T}{\partial t} =\nabla\cdot(\boldsymbol\kappa\nabla T)+s.\]The logical chain is therefore
\[\text{energy conservation} +\text{local equilibrium} +\text{Fourier constitutive law} \Longrightarrow \text{heat equation}.\]Only the first term in this chain is fundamental. The other two require physical justification.
To see why that constitutive law has a limited domain, begin with the microscopic heat flux.
In a dielectric crystal, heat is carried mainly by collective lattice excitations. In the phonon quasiparticle picture, a mode $\lambda=(\boldsymbol q,j)$ has frequency $\omega_\lambda$, group velocity $\boldsymbol v_\lambda$, and occupation $n_\lambda$. The heat flux is
\[\boldsymbol q =\frac{1}{V}\sum_\lambda \hbar\omega_\lambda\boldsymbol v_\lambda\,\delta n_\lambda,\]where $\delta n_\lambda=n_\lambda-n_\lambda^0$ is the deviation from the equilibrium Bose–Einstein distribution.
The distribution is governed by the phonon Boltzmann transport equation (BTE),
\[\frac{\partial n_\lambda}{\partial t} +\boldsymbol v_\lambda\cdot\nabla n_\lambda =\left.\frac{\partial n_\lambda}{\partial t}\right|_{\mathrm{coll}}.\]The left-hand side describes drift through space; the collision operator describes scattering among phonons and with isotopes, defects, electrons, and boundaries. Fourier’s law appears only after the BTE is linearized near equilibrium and the microscopic non-equilibrium distribution is reduced to a response proportional to $\nabla T$.
In the relaxation-time approximation,
\[\boldsymbol\kappa =\frac{1}{V}\sum_\lambda C_\lambda\, \boldsymbol v_\lambda\otimes\boldsymbol v_\lambda\, \tau_\lambda,\]where $C_\lambda$ and $\tau_\lambda$ are the modal heat capacity and relaxation time. For an isotropic medium, the familiar kinetic estimate follows:
\[\kappa\simeq\frac{1}{3}C_V\bar v\Lambda,\]with a representative mean free path $\Lambda=\bar v\tau$. This equation is useful, but a real material has a broad spectrum of mean free paths rather than a single $\Lambda$.
Because carriers propagate over finite distances and retain memory, the more general response is nonlocal.
Fourier transport requires several forms of scale separation.
Local equilibrium.
A small volume must contain enough internal scattering to relax toward an equilibrium-like distribution. Only then does a single scalar temperature characterize its energy. A strongly non-equilibrium distribution may have a well-defined total energy while not having a unique thermodynamic temperature.
Separation of length scales.
Let $L$ be the characteristic scale over which temperature or geometry changes. The Knudsen number is
\[\mathrm{Kn}=\frac{\Lambda}{L}.\]When the heat-carrying mean free paths are much smaller than $L$, carriers lose memory of their previous locations before the macroscopic field changes appreciably. The flux is then approximately local.
Separation of time scales.
Let $t_{\mathrm{obs}}$ be the observation or heating time. If the relevant relaxation times satisfy
\[\tau_\lambda\ll t_{\mathrm{obs}},\]the carrier distribution can follow the changing temperature field quasi-statically. At ultrashort times, heat flux retains memory of earlier states.
Linear response.
Fourier’s law is the first-order response to a weak thermodynamic driving force. If the temperature changes substantially over one mean free path,
\[\frac{\Lambda|\nabla T|}{T}\not\ll 1,\]higher-order and nonlocal effects may become important.
These criteria are spectral: different modes have different $\Lambda_\lambda$ and $\tau_\lambda$. A sample can therefore be diffusive for some phonons and ballistic for others.
Diffusive regime.
For $\mathrm{Kn}\ll1$, many momentum-randomizing events occur across the characteristic length. The microscopic distribution is close to local equilibrium, thermal conductivity is approximately independent of sample size, and Fourier’s law is normally reliable.
Quasiballistic regime.
When $\mathrm{Kn}$ is no longer small for a significant part of the heat-carrying spectrum, boundaries and nonlocality suppress the contribution of long-mean-free-path modes. The measured effective conductivity becomes dependent on geometry and experimental length scale.
In this regime it is misleading to ask only, “What is the thermal conductivity of the material?” A better question is, “What conductivity does this material exhibit for this geometry, size, and heating profile?”
Ballistic regime.
For $\mathrm{Kn}\gg1$, carriers cross the system with few internal collisions. Transport is controlled by injection from contacts, transmission probabilities, and boundary scattering. A Landauer description is more natural than a local gradient law:
\[G=\frac{1}{2\pi}\sum_m\int \hbar\omega\, \mathcal T_m(\omega) \frac{\partial n^0}{\partial T},d\omega.\]Here $G$ is thermal conductance and $\mathcal T_m$ is the transmission of mode $m$. Conductance, rather than a size-independent conductivity, is the direct transport quantity.
Fourier’s law responds instantaneously to $\nabla T$. This mathematical idealization leads to a diffusion equation with nonzero response at arbitrarily large distance for any $t>0$. It does not imply physically infinite carrier speed; it indicates that microscopic relaxation time has been removed from the model.
A simple extension is the Cattaneo–Vernotte relation,
\[\tau_q\frac{\partial\boldsymbol q}{\partial t} +\boldsymbol q=-\boldsymbol\kappa\nabla T,\]which introduces heat-flux memory and produces a hyperbolic temperature equation. It can describe finite-time relaxation and wave-like temperature propagation phenomenologically. However, the fitted $\tau_q$ should not automatically be identified with one microscopic phonon lifetime; the microscopic BTE contains an entire spectrum of relaxation processes.
The criteria above assume that temperature is the only slow variable that must be retained. Two important classes of transport violate that assumption.
First, collective crystal momentum can relax much more slowly than local energy. Frequent collisions may then produce hydrodynamic rather than ordinary diffusive transport. The relevant question is what Normal and resistive processes conserve, not simply how often phonons collide. The collision-operator physics, displaced distributions, Poiseuille flow, and second sound are developed separately in Do Phonon Collisions Always Create Thermal Resistance?.
Second, intermode coherence may survive on the transport time scale. A population-only BTE may then need a density-matrix or Wigner extension. The point needed here is narrower: whether Fourier’s law emerges macroscopically and which microscopic variables are required to compute its conductivity are distinct questions.
Spatial nonlocality becomes especially visible near an interface, where incident carriers from the two sides generally have different non-equilibrium distributions.
At an interface, incident carriers from the two materials can have different non-equilibrium distributions. A temperature discontinuity may appear even in steady state:
\[\Delta T=R_K q_n,\]where $R_K$ is the thermal boundary resistance and $q_n$ is the normal heat flux. The jump does not violate energy conservation. It shows that a bulk local conductivity is insufficient to describe transmission across an atomically thin region.
Near a strongly scattering interface, the apparent temperature may also depend on how it is defined—from local energy, a probe response, or a fitted distribution. “Temperature at a point” is itself a coarse-grained concept outside local equilibrium.
Before applying Fourier’s law, ask four questions:
Can a local temperature be defined?
If internal equilibration is weak, solve for carrier distributions rather than assuming local equilibrium.
Are the important mean free paths much smaller than the device and heating length scales?
If not, include ballistic boundary transport or solve the BTE.
Are the relaxation times much shorter than the experimental time scale?
If not, include temporal memory or a frequency-dependent response.
Do momentum-conserving or coherent processes dominate?
If so, hydrodynamic or Wigner descriptions may be required.
As an illustration, suppose a group of heat-carrying modes has $\Lambda\sim100$ nm. In a $10\ \mu$m structure, its Knudsen number is about $10^{-2}$ and its contribution is likely diffusive. In a 100 nm structure, $\mathrm{Kn}\sim1$ and boundary-sensitive transport is expected. This estimate is diagnostic, not a universal cutoff, because the complete mean-free-path spectrum and boundary conditions matter.
The regime map leads to a final distinction: “failure of Fourier’s law” can mean several different things.
There are at least three distinct possibilities:
These should not be conflated. A material can have a meaningful local temperature but a nonlocal heat flux; it can also obey an effective Fourier law macroscopically even when its microscopic carriers are partly coherent.
The enduring value of Fourier’s law is therefore not that it is universally fundamental, but that it is the correct hydrodynamic limit of a wide class of microscopic transport processes. Understanding heat conduction means knowing both why that limit emerges and when the system has not yet reached it.
它表示热流密度 $\boldsymbol q$ 与局域温度梯度成正比，方向与温度升高的方向相反。在宏观、缓慢变化并且接近平衡的系统中，这一关系极其有效，以至于我们很容易把它视为与能量守恒同等基本的定律。
但傅里叶定律并不是基本守恒定律。它实际上包含了一个很强的物理假设：
某一点、某一时刻的热流，可以只由同一点、同一时刻的温度梯度决定。
微观载流子却具有有限的传播速度、平均自由程和弛豫时间。它们可能来自其他位置，也可能保留此前状态的记忆。因此，傅里叶定律是否成立，取决于微观输运尺度与实验尺度之间是否存在充分的分离。
从这个角度看，所谓“傅里叶定律失效”并不意味着能量守恒失效，而是意味着局域温度、局域热导率或瞬时构成关系中的至少一个不再充分。
局域能量守恒可写为
\[\frac{\partial u}{\partial t} +\nabla\cdot\boldsymbol q =s,\]其中 $u$ 是能量密度，$\boldsymbol q$ 是热流密度，$s$ 是单位体积内的热源。
这个方程只说明：某一区域内能量的变化，必须来自流入、流出或内部热源。它并没有说明能量以什么方式传播，也没有给出热流与温度之间的关系。
要从能量守恒得到温度方程，还需要两个额外条件。
第一个条件是能量密度可以由局域温度描述，即
\[u=u(T).\]在温度变化不大时，
\[du=C_V\,dT,\]其中 $C_V$ 是体积热容。这个关系要求系统在局部接近平衡，使一个标量温度足以表征局域能量状态。
第二个条件是采用傅里叶构成关系：
\[\boldsymbol q=-\boldsymbol\kappa\nabla T.\]将二者代入能量守恒，才得到通常的热传导方程：
\[C_V\frac{\partial T}{\partial t} = \nabla\cdot\left(\boldsymbol\kappa\nabla T\right)+s.\]因此，逻辑顺序应当是
\[\text{能量守恒} +\text{局域平衡} +\text{局域热流构成关系} \Longrightarrow \text{傅里叶热传导方程}.\]其中只有能量守恒始终成立。局域温度是否存在、热流是否具有傅里叶形式，都必须根据具体输运条件判断。
要理解这个构成关系为什么有适用范围，需要先回到微观热流。
在绝缘晶体中，能量主要由晶格振动携带。在声子准粒子图像中，一个声子模式记为
\[\lambda=(\boldsymbol k,j),\]其中 $\boldsymbol k$ 是波矢，$j$ 是声子分支。每个模式具有频率 $\omega_\lambda$、群速度 $\boldsymbol v_\lambda$ 和占据数 $n_\lambda$。
相对于平衡分布的热流可写为
\[\boldsymbol q = \frac{1}{V} \sum_\lambda \hbar\omega_\lambda \boldsymbol v_\lambda \delta n_\lambda,\]其中
\[\delta n_\lambda=n_\lambda-n_\lambda^0\]是声子分布相对于平衡玻色–爱因斯坦分布 $n_\lambda^0$ 的偏离。
声子分布由玻尔兹曼输运方程控制：
\[\frac{\partial n_\lambda}{\partial t} + \boldsymbol v_\lambda\cdot\nabla n_\lambda = \left. \frac{\partial n_\lambda}{\partial t} \right|_{\mathrm{coll}}.\]左侧表示声子的时间演化与空间漂移，右侧碰撞项包含声子–声子、声子–同位素、声子–缺陷、声子–电子以及声子–边界散射。
傅里叶定律并不是直接写在 BTE 中的。它只会在以下步骤之后出现：
在弛豫时间近似下，可以写成
\[\delta n_\lambda \simeq -\tau_\lambda \boldsymbol v_\lambda\cdot\nabla T \frac{\partial n_\lambda^0}{\partial T}.\]代入热流表达式后得到
\[\boldsymbol\kappa = \frac{1}{V} \sum_\lambda C_\lambda \boldsymbol v_\lambda\otimes\boldsymbol v_\lambda \tau_\lambda,\]其中
\[C_\lambda = \hbar\omega_\lambda \frac{\partial n_\lambda^0}{\partial T}\]是模式热容。
对于各向同性介质，这一结果可近似写为熟悉的动理学形式：
\[\kappa \simeq \frac{1}{3}C_V\bar v\Lambda,\]其中 $\bar v$ 是代表性速度，$\Lambda=\bar v\tau$ 是代表性平均自由程。
这个公式非常有启发性，但也容易造成误解。真实晶体并不存在唯一的声子速度、寿命或平均自由程，而是具有宽广的模式分布。因此，一个系统是否处于傅里叶区间，通常不能只由单个平均自由程判断。
微观热流具有有限的传播距离和记忆，因此更一般的响应本来就是非局域的。
从更一般的角度看，热流可以写成对温度梯度的空间和时间非局域响应：
\[\boldsymbol q(\boldsymbol r,t) = - \int_{-\infty}^{t}dt' \int d\boldsymbol r'\, \boldsymbol{\mathcal K} \left( \boldsymbol r-\boldsymbol r', t-t' \right) \nabla T(\boldsymbol r',t').\]这里的响应核 $\boldsymbol{\mathcal K}$ 表示：当前位置的热流可能受到其他位置以及此前时刻温度梯度的影响。
傅里叶定律对应于这个响应核在空间和时间上都高度局域的极限：
\[\boldsymbol{\mathcal K} \left( \boldsymbol r-\boldsymbol r', t-t' \right) \rightarrow \boldsymbol\kappa\, \delta(\boldsymbol r-\boldsymbol r') \delta(t-t').\]于是得到
\[\boldsymbol q(\boldsymbol r,t) = -\boldsymbol\kappa \nabla T(\boldsymbol r,t).\]因此，傅里叶定律成立的本质不是载流子没有平均自由程或记忆，而是这些微观尺度相对于观测尺度足够小，以至于空间非局域性和时间记忆可以被忽略。
这一极限通常要求以下几种尺度分离。
局域平衡。
一个足够小但仍包含大量自由度的体积内，必须发生足够多的内部散射，使载流子分布能够由少数热力学变量描述。
只有在这种情况下，局域能量密度才能对应于唯一的温度。一个强烈非平衡的声子分布可以具有确定的总能量，却未必等价于任何玻色–爱因斯坦分布，因此也未必具有唯一的热力学温度。
空间尺度分离。
令 $L$ 表示温度场、加热区域或器件几何发生显著变化的特征长度。对于模式 $\lambda$，可定义模式克努森数
\[\mathrm{Kn}_\lambda = \frac{\Lambda_\lambda}{L}.\]当主要载热模式满足
\[\mathrm{Kn}_\lambda\ll1,\]声子在温度场显著变化之前已经经历多次散射，并逐渐丢失对先前位置和传播方向的记忆。热流因而可以近似看作局域响应。
时间尺度分离。
令 $t_{\mathrm{obs}}$ 表示加热、测量或温度变化的特征时间。如果相关声子模式满足
\[\tau_\lambda\ll t_{\mathrm{obs}},\]声子分布可以近似准静态地跟随温度场变化。
对于周期频率为 $\Omega$ 的热扰动，相应条件通常写为
\[\Omega\tau_\lambda\ll1.\]当 $\Omega\tau_\lambda$ 不再很小时，热流会表现出明显的相位滞后和时间记忆。
弱驱动与线性响应。
傅里叶定律是弱热力学驱动力下的一阶响应。一个简单的模式判据是
\[\frac{\Lambda_\lambda|\nabla T|}{T}\ll1.\]它表示声子在一个平均自由程内经历的相对温度变化很小。如果这一条件不满足，分布函数的高阶偏离以及温度依赖的非线性输运就可能变得重要。
这些条件都具有明显的模式依赖性。同一个样品可以对短平均自由程声子表现为扩散输运，同时对长平均自由程声子表现为准弹道输运。因此，傅里叶定律的适用性通常是逐渐丧失的，而不是在某个唯一尺度上突然失效。
讨论非傅里叶输运时，首先需要区分至少三类不同问题。
温度可能不再是充分的局域状态变量。
如果声子之间尚未建立局域平衡，那么相同的能量密度可以对应不同的模式分布。此时，用一个标量温度描述全部微观自由度可能是不充分的。
需要求解的对象不再只是 $T(\boldsymbol r,t)$，而是完整的载流子分布 $n_\lambda(\boldsymbol r,t)$，或者包含更多宏观变量的扩展模型。
热流可能具有空间或时间非局域性。
即使局域温度仍然可以定义，热流也未必只取决于当前位置、当前时刻的温度梯度。
长平均自由程产生空间非局域性，有限弛豫时间产生时间记忆。此时，傅里叶定律中的局域瞬时比例关系需要由积分核、广义热导率或更高阶输运方程替代。
热导率也可能不再是与实验条件无关的体材料常数。
当器件尺寸、加热长度或接触条件与载流子平均自由程相当时，测得的“有效热导率”会依赖几何结构、边界性质和测量方式。
这并不一定意味着温度不存在，而是意味着不能再把测量结果简单解释为无限体材料的内禀热导率。
这三类失效可以同时发生，也可以彼此独立。一个系统可能具有良好定义的局域温度，却表现出空间非局域热流；也可能仍然满足宏观傅里叶形式，但其热导率的微观来源已经不能由传统声子布居图像完整解释。
这些不同失效会随着平均自由程与器件尺度的相对大小，表现为扩散、准弹道与弹道输运之间的连续过渡。
扩散区间。
当主要载热模式满足
\[\mathrm{Kn}_\lambda\ll1,\]声子在穿越特征长度之前会经历多次使传播方向和动量随机化的散射。分布接近局域平衡，热流主要由局域温度梯度决定，热导率也基本不依赖样品尺寸。
这是傅里叶定律最典型的适用区间。
准弹道区间。
当一部分重要载热模式满足
\[\mathrm{Kn}_\lambda\sim1,\]这些模式在跨越加热区域或器件结构时只经历有限次数的散射。长平均自由程声子的贡献会受到边界、接触和加热尺度的抑制。
此时，热导率表现为一个依赖实验条件的有效量：
\[\kappa_{\mathrm{eff}} = \kappa_{\mathrm{eff}} (L,\text{geometry},\text{boundary},\text{heating}).\]因此，只问“这种材料的热导率是多少”已经不够。更完整的问题应当是：
在给定尺寸、边界条件和加热方式下，这个系统表现出怎样的有效热输运响应？
弹道区间。
当主要载热模式满足
\[\mathrm{Kn}_\lambda\gg1,\]载流子在穿过系统时很少经历内部散射。输运主要由两端接触的载流子注入、可用模式数、模式透射概率和边界散射决定。
在这种情况下，Landauer 描述通常比局域温度梯度定律更自然：
\[G = \frac{1}{2\pi} \sum_m \int \hbar\omega\, \mathcal T_m(\omega) \frac{\partial n^0}{\partial T} \,d\omega.\]其中 $G$ 是热导，$\mathcal T_m(\omega)$ 是模式 $m$ 的透射率。
在弹道极限中，直接而自然的输运量是热导 $G$，而不是假设与长度无关的体热导率 $\kappa$。如果仍然通过
\[\kappa_{\mathrm{eff}}=\frac{GL}{A}\]定义一个有效热导率，它通常会随长度 $L$ 改变。
傅里叶定律假设热流会立即响应温度梯度。将它代入能量守恒后得到扩散方程，而扩散方程在数学上会使任意局域扰动在所有空间位置立即产生非零响应。
这并不意味着真实声子具有无限传播速度，而是因为载流子的有限弛豫时间已经在傅里叶近似中被消去了。
一个常见的现象学扩展是 Cattaneo–Vernotte 关系：
\[\tau_q \frac{\partial\boldsymbol q}{\partial t} + \boldsymbol q = -\boldsymbol\kappa\nabla T.\]热流不再立即达到傅里叶值，而是在时间尺度 $\tau_q$ 上逐渐弛豫。将其与能量守恒结合，可以得到双曲型温度方程，并描述有限时间响应和波动式温度传播。
不过，拟合得到的 $\tau_q$ 通常是许多微观模式共同作用后的有效参数，不应简单等同于某一个声子的寿命。真实晶体具有完整的弛豫时间谱，因而其时间响应一般比单一指数弛豫更复杂。
从频域看，热导率也可以成为频率相关的复数量：
\[\boldsymbol q(\Omega) = -\boldsymbol\kappa(\Omega) \nabla T(\Omega).\]其中 $\boldsymbol\kappa(\Omega)$ 的实部描述耗散响应，虚部反映热流相对于温度梯度的相位滞后。
前面的尺度判据默认温度是唯一需要保留的慢变量。集体晶格动量或模式间相干如果弛豫得很慢，就需要扩展状态变量，但这不是本文要重复展开的主题。
当集体晶格动量比局域能量弛豫得慢时，频繁碰撞可能产生流体动力学行为。关键不再是碰撞次数，而是 Normal 与阻性过程分别守恒和弛豫什么。
本文只把这种情形标记为“需要额外慢变量”的分支。时间尺度、长度窗口、漂移分布、Poiseuille 热流和第二声将在《声子碰撞一定产生热阻吗？》中专门讨论。
如果近简并振动态之间的相干也不能快速消失，仅追踪独立声子布居同样可能不够，需要密度矩阵或 Wigner 输运。这里保留的核心结论是：宏观傅里叶定律能否成立，与计算热导率时需要保留哪些微观变量，是两个不同层次的问题。
空间非局域性在界面附近尤其直接，因为两侧入射声子通常具有不同的非平衡分布。
在两种材料的界面处，来自两侧的入射声子通常具有不同的模式分布。声子通过反射、透射和模式转换跨越界面，因此界面附近的分布可能显著偏离局域平衡。
即使系统处于稳态，界面两侧也可能出现温度跳跃：
\[\Delta T=R_Kq_n,\]其中 $R_K$ 是热边界电阻，$q_n$ 是法向热流。
这个温度跳跃并不违反能量守恒。稳态下，穿过界面的热流仍然连续；温度不连续反映的是界面对载流子透射的有限能力。
从微观角度看，原子尺度界面不能仅由两侧体材料的局域热导率描述。还需要知道：
在强非平衡界面附近，“温度”本身也可能依赖定义方式。由局域能量反演的温度、由某种实验探针测得的温度，以及通过拟合平衡分布得到的温度，未必完全相同。
因此，局域平衡之外的温度并不是一个无需说明的微观量，而是对复杂分布进行粗粒化后得到的有效描述。
把这些情形放在一起比较，可以看出它们并不是同一条轴上的简单排序。
扩散、弹道、流体动力学和相干输运有时被画成一条从“经典”到“非经典”的连续谱，但这种表示容易造成误解。
它们对应的并不是同一种物理条件：
| 输运区间 | 主要判据 | 需要保留的物理信息 | 典型描述 |
|---|---|---|---|
| 扩散 | $\mathrm{Kn}\lambda\ll1$，$\Omega\tau\lambda\ll1$ | 局域温度 | 傅里叶方程 |
| 准弹道 | 部分模式 $\mathrm{Kn}_\lambda\sim1$ | 模式分布和边界效应 | BTE、抑制函数 |
| 弹道 | 主要模式 $\mathrm{Kn}_\lambda\gg1$ | 注入与透射概率 | Landauer 方法 |
| 瞬态非傅里叶 | $\Omega\tau_\lambda\gtrsim1$ | 热流历史 | BTE、记忆核、Cattaneo 模型 |
| 流体动力学 | $\tau_N\ll\tau_R$，$\Lambda_N\ll L\ll\Lambda_R$ | 能量与集体动量 | Guyer–Krumhansl、流体方程 |
| 相干输运 | 模式间隔与线宽相当 | 密度矩阵非对角项 | Wigner 输运 |
例如，流体动力学输运并不是“比弹道输运更强的非傅里叶效应”。弹道输运意味着内部碰撞不足，而流体动力学恰恰需要频繁的动量守恒碰撞。
同样，相干输运描述的是模式之间的波动耦合，它与样品是否处于弹道区间也不是同一个问题。
在直接使用傅里叶定律之前，可以依次回答以下问题。
先问局域温度是否有意义。
局域载流子分布是否足够接近平衡？不同模式能否由同一个温度描述？
如果不能，就不应一开始便只求解温度场，而需要求解载流子分布或引入额外状态变量。
再比较主要载热模式的克努森数。
需要比较的不只是平均自由程与器件长度，还包括：
假设一组主要载热声子的平均自由程约为 $100$ nm。在 $10\ \mu\mathrm m$ 的缓慢变化结构中，
\[\mathrm{Kn}\sim10^{-2},\]它们通常接近扩散输运。
而在 $100$ nm 的器件或加热区域中，
\[\mathrm{Kn}\sim1,\]就应预期明显的准弹道效应和尺寸依赖。
随后比较实验时间与载流子弛豫时间。
对于稳定或缓慢变化的加热过程，声子分布通常能够跟随局域温度变化。
而在皮秒尺度激光加热、高频热调制或快速热脉冲中，部分模式可能满足
\[\Omega\tau_\lambda\gtrsim1,\]此时不能忽略热流的时间记忆。
最后检查是否存在额外的慢变量。
普通傅里叶理论默认能量或温度是唯一需要保留的慢变量。
如果声子总准动量也缓慢弛豫，就需要考虑流体动力学；如果模式相干在输运时间尺度内仍然存在，就需要保留密度矩阵的非对角部分。
因此，选择输运模型的关键并不是简单判断系统“多么非傅里叶”，而是识别：
在所研究的时间和长度尺度上，哪些微观信息尚未充分弛豫，因而不能被消去？
完成这些判断后，傅里叶定律的物理位置就变得清楚了。
傅里叶定律既不是对每一个声子的运动方程，也不是任意尺度下都成立的基本定律。它描述的是微观输运经过空间、时间和模式自由度粗粒化之后形成的宏观局域响应。
当以下条件同时近似满足时：
\[\Lambda_\lambda\ll L, \qquad \tau_\lambda\ll t_{\mathrm{obs}}, \qquad \frac{\Lambda_\lambda|\nabla T|}{T}\ll1,\]并且不存在需要额外保留的长寿命动量或相干变量，复杂的微观输运就会逐渐约化为
\[\boldsymbol q=-\boldsymbol\kappa\nabla T.\]因此，傅里叶定律的深刻之处不在于它在所有尺度上都正确，而在于大量具有不同频率、速度和寿命的微观载流子，在长时间和长长度尺度上能够共同形成如此简单的宏观规律。
当傅里叶定律失效时，真正需要追问的也不是“能量为什么不再守恒”，而是：
理解热传导，不仅要知道如何使用傅里叶定律，还要知道它是怎样从微观世界中出现的，以及一个具体系统为什么尚未到达这个长时间、长长度尺度下的扩散极限。
The first-principles phonon Boltzmann transport equation (PBTE) method connects three levels of description:
The method is powerful because it exposes microscopic transport mechanisms rather than returning only a fitted bulk number. It is not, however, exact or parameter-free in an absolute sense. Its predictions depend on the electronic-structure approximation, numerical convergence, the order at which anharmonicity is truncated, and whether the phonon quasiparticle picture is valid.
This article complements the earlier discussions of the emergence of Fourier’s law and Normal versus Umklapp scattering. Here the focus is methodological: how does an atomic structure become a prediction of $\boldsymbol\kappa$?
The conventional PBTE workflow is designed primarily for near-equilibrium lattice heat transport in periodic crystals whose vibrations can be described by reasonably well-defined phonon quasiparticles.
Its natural output is the intrinsic lattice thermal conductivity of a bulk crystal, possibly augmented by isotope, defect, electron–phonon, or boundary-scattering models. It is especially useful for answering questions such as:
The method does not automatically describe an interface, a finite device, or strongly nonlocal heat flow. Those problems require additional boundary conditions or transport formalisms even when the bulk phonon properties come from first principles.
Once the scope is clear, the next question is what “first principles” actually contributes to the transport calculation.
In this context, first-principles calculations do not simply “solve the Schrödinger equation.” In practice, density functional theory (DFT) provides an approximate electronic ground-state energy and atomic forces for a chosen exchange-correlation functional, pseudopotential or all-electron treatment, basis set, and Brillouin-zone sampling.
The Kohn–Sham equations are
\[\left[ -\frac{\hbar^2}{2m_e}\nabla^2 +V_{\mathrm{ext}}(\boldsymbol r) +V_{\mathrm H}[\rho](\boldsymbol r) +V_{\mathrm{xc}}[\rho](\boldsymbol r) \right] \phi_i(\boldsymbol r) =\varepsilon_i\phi_i(\boldsymbol r).\]For lattice dynamics, the most important result is not usually the electronic band structure itself, but an accurate local representation of the Born–Oppenheimer potential-energy surface near the relevant atomic configurations.
The phrase “without empirical fitting” should therefore be used carefully. Standard DFT-based PBTE calculations may not fit thermal conductivity to experiment, but they still contain modeling choices and approximations. The exchange-correlation functional, equilibrium volume, pseudopotential, treatment of long-range electrostatics, and magnetic or relativistic effects can all propagate into the phonon spectrum and scattering rates.
Let $u_{lb}^{\alpha}$ be the displacement of atom $b$ in unit cell $l$ along Cartesian direction $\alpha$. Expanding the potential energy around equilibrium gives
\[U=U_0 +\frac{1}{2!}\sum_{lb,l'b'}\sum_{\alpha\beta} \Phi_{lb,l'b'}^{\alpha\beta} u_{lb}^{\alpha}u_{l'b'}^{\beta} +\frac{1}{3!}\sum_{lb,l'b',l''b''}\sum_{\alpha\beta\gamma} \Psi_{lb,l'b',l''b''}^{\alpha\beta\gamma} u_{lb}^{\alpha}u_{l'b'}^{\beta}u_{l''b''}^{\gamma} +\mathcal O(u^4).\]The first-order term vanishes at a fully relaxed equilibrium structure. The coefficients are interatomic force constants (IFCs):
\[\Phi_{ij}^{\alpha\beta} =\frac{\partial^2 U} {\partial u_i^{\alpha}\partial u_j^{\beta}}, \qquad \Psi_{ijk}^{\alpha\beta\gamma} =\frac{\partial^3 U} {\partial u_i^{\alpha}\partial u_j^{\beta}\partial u_k^{\gamma}}.\]Second-order IFCs determine harmonic phonons. Third-order IFCs generate the leading three-phonon interaction. Fourth- and higher-order terms contribute to frequency renormalization and higher-order scattering, which can be important at elevated temperature or in strongly anharmonic materials.
Finite displacements.
In a supercell finite-displacement calculation, selected atoms are displaced and the resulting forces are evaluated. Symmetry reduces the number of required configurations, and the IFCs are obtained from finite differences or regression.
The displacement amplitude must be large enough to overcome numerical force noise but small enough to remain within the intended Taylor expansion. A value near $0.01$ Å is common for harmonic calculations, but it is not universal and should be tested.
Third- and higher-order IFCs require many more displacement patterns. Compressive sensing, systematic regression, and machine-learning potentials can reduce the cost, but they introduce their own training, regularization, and validation requirements.
Density functional perturbation theory.
Density functional perturbation theory (DFPT) computes the response to periodic perturbations by solving linearized self-consistent equations. It is particularly established for harmonic phonons, dielectric tensors, and Born effective charges in reciprocal space.
Higher-order response theory exists, but third-order IFC workflows are not equally available or equally convenient in every electronic-structure package. In practice, many PBTE calculations combine DFPT or finite displacements for harmonic IFCs with finite-displacement forces for third-order IFCs.
Once second-order IFCs are available, harmonic lattice dynamics converts real-space interactions into reciprocal-space phonon modes.
The harmonic equations of motion are
\[m_b\frac{d^2u_{lb}^{\alpha}}{dt^2} =-\sum_{l'b'\beta} \Phi_{lb,l'b'}^{\alpha\beta}u_{l'b'}^{\beta}.\]Using a plane-wave form leads to the eigenvalue problem
\[\sum_{b'\beta} D_{bb'}^{\alpha\beta}(\boldsymbol q) e_{b'\beta}^{\lambda} =\omega_{\lambda}^2 e_{b\alpha}^{\lambda},\]where $\lambda=(\boldsymbol q,j)$ labels a wave vector and phonon branch. The dynamical matrix is
\[D_{bb'}^{\alpha\beta}(\boldsymbol q) =\frac{1}{\sqrt{m_bm_{b'}}} \sum_{l'} \Phi_{0b,l'b'}^{\alpha\beta} \exp\!\left[i\boldsymbol q\cdot (\boldsymbol R_{l'}-\boldsymbol R_0)\right].\]Diagonalization gives the phonon frequencies $\omega_\lambda$ and eigenvectors $\boldsymbol e_\lambda$. The group velocity is
\[\boldsymbol v_\lambda=\nabla_{\boldsymbol q}\omega_\lambda.\]The modal heat capacity is
\[C_\lambda =\hbar\omega_\lambda \frac{\partial n_\lambda^0}{\partial T}, \qquad n_\lambda^0 =\frac{1}{\exp(\hbar\omega_\lambda/k_BT)-1}.\]Before any transport calculation, the harmonic model should reproduce basic physical constraints. Imaginary modes may indicate a genuinely unstable structure, an unconverged calculation, an inappropriate reference phase, or broken invariance in the fitted IFCs. They should not simply be removed without diagnosis.
This section keeps only the ingredients needed to construct the numerical collision matrix. The distinct resistive roles of Normal and Umklapp processes are developed in the companion article on phonon collisions and collective heat flow and are not repeated here.
Third-order IFCs determine matrix elements $V_{\lambda\lambda’\lambda’’}$ for three-phonon absorption and decay. Allowed processes satisfy energy conservation and crystal-momentum selection rules:
\[\omega_\lambda+\omega_{\lambda'}=\omega_{\lambda''}, \qquad \boldsymbol q+\boldsymbol q' =\boldsymbol q''+\boldsymbol G,\]or
\[\omega_\lambda=\omega_{\lambda'}+\omega_{\lambda''}, \qquad \boldsymbol q =\boldsymbol q'+\boldsymbol q''+\boldsymbol G.\]Fermi’s golden rule gives rates with the schematic structure
\[\Gamma^{(3)} \propto |V_{\lambda\lambda'\lambda''}|^2 \times\text{occupation factors} \times\delta(\Delta\omega) \times\Delta_{\boldsymbol q,\boldsymbol G}.\]The energy delta function must be integrated numerically on a discrete $\boldsymbol q$ mesh, using a broadening scheme or a tetrahedron-like treatment. The result can be sensitive to mesh density and integration parameters.
Isotope scattering is commonly treated as elastic mass-disorder scattering. Defects, boundaries, electrons, and four-phonon interactions require additional models or matrix elements. Adding rates with Matthiessen’s rule can be useful, but it can obscure mode coupling and correlations when the underlying collision mechanisms are not independent.
With scattering matrix elements in hand, the problem moves from an equilibrium spectrum to the non-equilibrium response driven by a temperature gradient.
Under a small temperature gradient, write the phonon distribution as
\[n_\lambda =n_\lambda^0 -\frac{\partial n_\lambda^0}{\partial T} \boldsymbol F_\lambda\cdot\nabla T,\]where $\boldsymbol F_\lambda$ is the unknown vector mean-free-displacement response. The heat flux is determined by the non-equilibrium part:
\[\boldsymbol J_Q =-\frac{1}{V}\sum_\lambda C_\lambda\boldsymbol v_\lambda (\boldsymbol F_\lambda\cdot\nabla T).\]Comparison with $J_Q^{\alpha}=-\kappa^{\alpha\beta}\nabla_\beta T$ gives
\[\kappa^{\alpha\beta} =\frac{1}{V}\sum_\lambda C_\lambda v_\lambda^{\alpha}F_\lambda^{\beta}.\]The response $\boldsymbol F$ is found from a linear system whose schematic form is
\[\sum_{\lambda'} \Omega_{\lambda\lambda'} \boldsymbol F_{\lambda'} =\boldsymbol v_\lambda.\]$\boldsymbol\Omega$ is the linearized collision operator. Its diagonal terms describe loss from a mode; its off-diagonal terms describe repopulation of other modes. The precise normalization of $\boldsymbol\Omega$, the driving term, and $\boldsymbol F$ varies among derivations and software packages, but the physical content is the same: thermal conductivity is controlled by the inverse collision operator projected onto the heat-current-carrying response.
Solving that linear system requires a direct modeling choice: whether to retain or discard repopulation between modes.
In the single-mode relaxation-time approximation (RTA), the off-diagonal mode coupling is neglected:
\[\boldsymbol F_\lambda^{\mathrm{RTA}} =\tau_\lambda\boldsymbol v_\lambda.\]The conductivity becomes
\[\kappa_{\mathrm{RTA}}^{\alpha\beta} =\frac{1}{V}\sum_\lambda C_\lambda v_\lambda^{\alpha} v_\lambda^{\beta}\tau_\lambda.\]For an isotropic material this reduces to the familiar kinetic form
\[\kappa_{\mathrm{RTA}} =\frac{1}{3V}\sum_\lambda C_\lambda |\boldsymbol v_\lambda|^2\tau_\lambda.\]RTA is inexpensive and useful for decomposing modal contributions, but it discards off-diagonal repopulation in the collision operator. Iterative, variational, or direct solvers retain that coupling, so the difference between the two solutions is a numerical and modeling choice that should be reported. Deciding whether the difference signals collective transport requires separate evidence from slow modes, scale windows, and geometry rather than another discussion inside this workflow article.
A reliable PBTE calculation is a sequence of convergence and validation problems, not a single command.
First relax the reference structure.
Converge the equilibrium volume, cell shape, atomic positions, and—where relevant—magnetic state. Residual stress or forces alter the harmonic spectrum and can strongly affect low-frequency modes.
Then converge the electronic calculation.
Test the basis cutoff, electronic $\boldsymbol k$ mesh, smearing scheme, self-consistency threshold, pseudopotential or PAW dataset, and exchange-correlation functional. Convergence should be judged using forces, stress, and phonon-sensitive observables, not total energy alone.
Next determine the harmonic IFCs.
Choose a supercell or reciprocal-space grid that captures the interaction range. Check translational and rotational invariance, acoustic modes near $\Gamma$, non-analytic corrections in polar materials, and agreement with measured or independently calculated dispersions when available.
Then determine the anharmonic IFCs.
Converge supercell size, displacement magnitude, interaction cutoff, and regression settings. Enforce symmetry and sum rules carefully: corrections should reduce numerical noise without concealing an inadequate supercell or incomplete model.
Now construct the collision operator.
Include the scattering mechanisms required by the material and temperature range. Three-phonon plus isotope scattering is a common baseline, not a universal endpoint. Four-phonon scattering, temperature-renormalized phonons, electron–phonon scattering, defects, or boundaries may be necessary.
Finally solve and converge the PBTE.
Converge the phonon $\boldsymbol q$ mesh, energy-conservation integration, and iterative-solver tolerance. Compare RTA and full solutions, and inspect tensor symmetry as well as scalar averages.
Validation must go beyond total conductivity.
Agreement in total $\kappa$ can result from compensating errors. Whenever possible, compare several quantities:
PBTE results should be reported with a convergence narrative. The dominant uncertainty can arise at several levels.
| Level | Typical source of uncertainty | Useful diagnostic |
|---|---|---|
| Electronic structure | Functional, volume, pseudopotential, $\boldsymbol k$ mesh | Forces, stress, elastic constants, phonon frequencies |
| Harmonic IFCs | Supercell, long-range electrostatics, sum rules | Acoustic modes, dispersion, group velocities |
| Anharmonic IFCs | Displacement size, cutoff, regression, supercell | Grüneisen parameters, linewidth stability |
| BZ integration | $\boldsymbol q$ mesh, broadening or tetrahedra | $\kappa$ versus mesh and integration parameter |
| Collision physics | Missing four-phonon, electron, defect, or boundary terms | Temperature trend and mode-resolved rates |
| Transport model | RTA, iterative PBTE, coherence neglected | RTA/full comparison and mode-overlap analysis |
Reporting only the final value—for example, “$\kappa=150$ W m$^{-1}$ K$^{-1}$”—does not establish predictive accuracy. A convincing result shows that the important intermediate physics is stable and interpretable.
Numerical convergence does not guarantee a complete physical model, so the limits of the conventional PBTE must also be tested.
The standard phonon-gas PBTE relies on separable quasiparticle populations. It may need extension when:
Finite devices may require spatially resolved BTE, Monte Carlo, deterministic transport, or Landauer approaches. Strongly anharmonic crystals may require self-consistent phonons or temperature-dependent effective potentials. Complex crystals and disordered solids may require a Wigner or density-matrix treatment that includes off-diagonal coherences.
The appropriate question is therefore not “Is PBTE accurate?” in isolation, but “Does the chosen PBTE contain the slow variables and scattering mechanisms relevant to this material, temperature, and geometry?”
Widely used tools include phonopy/phono3py, ShengBTE, and related interfaces to electronic-structure codes such as VASP, Quantum ESPRESSO, ABINIT, and others. They automate important algebra and data handling, but they cannot decide whether an IFC cutoff is adequate, an imaginary mode is physical, or an omitted scattering mechanism matters.
A reproducible calculation should record at least:
The goal is not merely to rerun the workflow, but to make the physical approximations auditable.
The reason to record the full workflow is not to accumulate input parameters, but to make the final physical inference traceable. The first-principles PBTE is best understood as a chain of controlled reductions:
\[\text{electronic ground state} \longrightarrow \text{energy derivatives and IFCs} \longrightarrow \text{phonon modes and interactions} \longrightarrow \text{collision operator} \longrightarrow \text{non-equilibrium distribution} \longrightarrow \boldsymbol\kappa.\]Each arrow introduces numerical choices and physical assumptions. The strength of the method is that those choices can be tested systematically and that the final conductivity can be decomposed into interpretable microscopic contributions.
The deepest result is not a single value of $\kappa$. It is an explanation of which excitations carry heat, which collisions destroy or redistribute their collective motion, and why a macroscopic transport coefficient emerges from the atomic-scale dynamics.
第一性原理声子玻尔兹曼输运方程（PBTE）连接了三个描述层次：
这一方法的优势在于，它不仅给出一个拟合的体材料数值，还能揭示微观输运机制。但它并非绝对意义上的精确或“无参数”方法。预测结果依赖电子结构近似、数值收敛、非谐展开的截断阶数，以及声子准粒子图像是否成立。
本文承接前面关于傅里叶定律如何从微观输运中出现以及Normal 与 Umklapp 散射的讨论。这里关注的是方法本身：一个原子结构究竟怎样变成对 $\boldsymbol\kappa$ 的预测？
传统 PBTE 工作流主要用于周期晶体中的近平衡晶格热输运，并假设晶格振动能够由相对明确的声子准粒子描述。
它最自然的输出是体晶体的内禀晶格热导率，也可以进一步加入同位素、缺陷、电子–声子或边界散射模型。PBTE 特别适合回答以下问题：
这一方法不会自动描述界面、有限尺寸器件或强非局域热输运。即使体材料声子性质来自第一性原理，这些问题仍然需要额外的边界条件或其他输运理论。
明确适用范围后，下一步才是说明“第一性原理”究竟向输运计算提供什么。
在这里，第一性原理计算不能被简单描述为“求解薛定谔方程”。实际计算通常使用密度泛函理论（DFT），在给定交换关联泛函、赝势或全电子方法、基组以及布里渊区采样的条件下，近似得到电子基态能量和原子受力。
Kohn–Sham 方程为
\[\left[ -\frac{\hbar^2}{2m_e}\nabla^2 +V_{\mathrm{ext}}(\boldsymbol r) +V_{\mathrm H}[\rho](\boldsymbol r) +V_{\mathrm{xc}}[\rho](\boldsymbol r) \right] \phi_i(\boldsymbol r) =\varepsilon_i\phi_i(\boldsymbol r).\]对于晶格动力学，最重要的结果通常不是电子能带本身，而是平衡构型附近 Born–Oppenheimer 势能面的准确局域表示。
因此，“不需要经验拟合”这一说法也应谨慎使用。标准 DFT-PBTE 计算通常不会用实验热导率拟合模型，但仍然包含许多建模选择与近似。交换关联泛函、平衡体积、赝势、长程静电相互作用以及磁性或相对论效应，都可能继续影响声子谱和散射率。
令 $u_{lb}^{\alpha}$ 表示第 $l$ 个晶胞中原子 $b$ 沿笛卡尔方向 $\alpha$ 的位移。在平衡位置附近展开势能：
\[U=U_0 +\frac{1}{2!}\sum_{lb,l'b'}\sum_{\alpha\beta} \Phi_{lb,l'b'}^{\alpha\beta} u_{lb}^{\alpha}u_{l'b'}^{\beta} +\frac{1}{3!}\sum_{lb,l'b',l''b''}\sum_{\alpha\beta\gamma} \Psi_{lb,l'b',l''b''}^{\alpha\beta\gamma} u_{lb}^{\alpha}u_{l'b'}^{\beta}u_{l''b''}^{\gamma} +\mathcal O(u^4).\]对于充分弛豫的平衡结构，一阶项为零。展开系数就是原子间力常数（IFC）：
\[\Phi_{ij}^{\alpha\beta} =\frac{\partial^2U} {\partial u_i^{\alpha}\partial u_j^{\beta}}, \qquad \Psi_{ijk}^{\alpha\beta\gamma} =\frac{\partial^3U} {\partial u_i^{\alpha}\partial u_j^{\beta}\partial u_k^{\gamma}}.\]二阶 IFC 决定谐性声子，三阶 IFC 产生最低阶三声子相互作用。四阶及更高阶项会影响频率重整化和高阶散射，在高温或强非谐材料中可能非常重要。
有限位移法。
在超胞有限位移计算中，对选定原子施加位移并计算对应受力。晶体对称性可以减少所需构型数量，随后通过有限差分或回归得到 IFC。
位移幅度必须足够大，使受力信号超过数值噪声，同时又必须足够小，以保证处于目标 Taylor 展开的适用区间。谐性计算经常使用接近 $0.01$ Å 的位移，但这不是普适数值，仍应进行测试。
三阶及更高阶 IFC 需要更多位移构型。压缩感知、系统回归和机器学习势可以降低计算成本，但也会引入训练集、正则化和验证等新的要求。
密度泛函微扰理论。
密度泛函微扰理论（DFPT）通过求解线性化自洽方程，计算系统对周期微扰的响应。它在谐性声子、介电张量和 Born 有效电荷等倒空间计算中尤其成熟。
虽然也存在高阶响应理论，但不同电子结构软件对三阶 IFC 的支持程度和使用便利性并不相同。实际 PBTE 工作流通常采用 DFPT 或有限位移获得二阶 IFC，再通过有限位移受力提取三阶 IFC。
得到二阶 IFC 后，谐性晶格动力学把实空间相互作用转换为倒空间声子模式。
谐性运动方程为
\[m_b\frac{d^2u_{lb}^{\alpha}}{dt^2} =-\sum_{l'b'\beta} \Phi_{lb,l'b'}^{\alpha\beta}u_{l'b'}^{\beta}.\]代入平面波形式后得到本征值方程：
\[\sum_{b'\beta} D_{bb'}^{\alpha\beta}(\boldsymbol q) e_{b'\beta}^{\lambda} =\omega_\lambda^2e_{b\alpha}^{\lambda},\]其中 $\lambda=(\boldsymbol q,j)$ 表示波矢和声子分支。动力学矩阵为
\[D_{bb'}^{\alpha\beta}(\boldsymbol q) =\frac{1}{\sqrt{m_bm_{b'}}} \sum_{l'} \Phi_{0b,l'b'}^{\alpha\beta} \exp\!\left[i\boldsymbol q\cdot (\boldsymbol R_{l'}-\boldsymbol R_0)\right].\]对动力学矩阵进行对角化，可以得到声子频率 $\omega_\lambda$ 和本征矢 $\boldsymbol e_\lambda$。群速度定义为
\[\boldsymbol v_\lambda=\nabla_{\boldsymbol q}\omega_\lambda.\]模式热容为
\[C_\lambda =\hbar\omega_\lambda \frac{\partial n_\lambda^0}{\partial T}, \qquad n_\lambda^0 =\frac{1}{\exp(\hbar\omega_\lambda/k_BT)-1}.\]在进行输运计算之前，谐性模型应首先满足基本物理约束。虚频可能意味着结构确实不稳定，也可能来自计算尚未收敛、参考相选择不当或拟合 IFC 破坏了不变性。不能在没有诊断原因的情况下简单删除虚频。
本节只讨论构造数值碰撞矩阵所需的输入与选择定则；N/U 过程为何具有不同热阻作用，已在碰撞物理专题中展开，这里不再重复。
三阶 IFC 决定三声子吸收和衰变过程的矩阵元 $V_{\lambda\lambda’\lambda’’}$。允许过程必须满足能量守恒和晶格动量选择定则：
\[\omega_\lambda+\omega_{\lambda'}=\omega_{\lambda''}, \qquad \boldsymbol q+\boldsymbol q' =\boldsymbol q''+\boldsymbol G,\]或者
\[\omega_\lambda=\omega_{\lambda'}+\omega_{\lambda''}, \qquad \boldsymbol q =\boldsymbol q'+\boldsymbol q''+\boldsymbol G.\]由费米黄金规则得到的散射率具有如下示意结构：
\[\Gamma^{(3)} \propto |V_{\lambda\lambda'\lambda''}|^2 \times\text{布居因子} \times\delta(\Delta\omega) \times\Delta_{\boldsymbol q,\boldsymbol G}.\]在离散 $\boldsymbol q$ 网格上，能量 delta 函数必须通过展宽方法或类似四面体积分的方法进行数值处理。计算结果可能对网格密度和积分参数十分敏感。
同位素散射通常作为弹性质量无序散射处理。缺陷、边界、电子以及四声子相互作用则需要额外模型或矩阵元。利用 Matthiessen 规则直接相加散射率有时很实用，但当不同碰撞机制并非独立时，它也可能掩盖模式耦合和相关性。
有了散射矩阵元之后，问题才从平衡声子谱进入受温度梯度驱动的非平衡响应。
在较小温度梯度下，将声子分布写成
\[n_\lambda =n_\lambda^0 -\frac{\partial n_\lambda^0}{\partial T} \boldsymbol F_\lambda\cdot\nabla T,\]其中 $\boldsymbol F_\lambda$ 是待求的矢量平均自由位移响应。热流由非平衡部分决定：
\[\boldsymbol J_Q =-\frac{1}{V}\sum_\lambda C_\lambda\boldsymbol v_\lambda (\boldsymbol F_\lambda\cdot\nabla T).\]与 $J_Q^{\alpha}=-\kappa^{\alpha\beta}\nabla_\beta T$ 比较，得到
\[\kappa^{\alpha\beta} =\frac{1}{V}\sum_\lambda C_\lambda v_\lambda^{\alpha}F_\lambda^{\beta}.\]$\boldsymbol F$ 由一个线性方程组确定，其示意形式为
\[\sum_{\lambda'} \Omega_{\lambda\lambda'} \boldsymbol F_{\lambda'} =\boldsymbol v_\lambda.\]$\boldsymbol\Omega$ 是线性化碰撞算符。其对角项描述声子离开某一模式，非对角项描述其他模式的再布居。不同推导和软件对 $\boldsymbol\Omega$、驱动项以及 $\boldsymbol F$ 的归一化定义可能不同，但物理含义一致：热导率由碰撞算符的逆在载热响应方向上的投影决定。
求解这个线性系统时，最直接的模型选择是保留还是忽略模式之间的再布居。
在单模弛豫时间近似（RTA）中，模式之间的非对角耦合被忽略：
\[\boldsymbol F_\lambda^{\mathrm{RTA}} =\tau_\lambda\boldsymbol v_\lambda.\]于是热导率为
\[\kappa_{\mathrm{RTA}}^{\alpha\beta} =\frac{1}{V}\sum_\lambda C_\lambda v_\lambda^{\alpha} v_\lambda^{\beta}\tau_\lambda.\]对于各向同性材料，可约化为熟悉的动理学形式：
\[\kappa_{\mathrm{RTA}} =\frac{1}{3V}\sum_\lambda C_\lambda|\boldsymbol v_\lambda|^2\tau_\lambda.\]RTA 计算成本低，也便于分解模式贡献，但会丢弃碰撞算符的非对角再布居。迭代、变分或直接求解方法则保留这些耦合，因此两种解的差异是一个必须报告的数值与模型选择。至于这种差异何时意味着集体输运，需要结合慢模、尺度窗口和几何证据判断，而不是在本文的工作流部分重复讨论。
可信的 PBTE 计算是一系列收敛与验证问题，而不是一条命令。
首先弛豫参考结构。
需要收敛平衡体积、晶胞形状、原子位置以及必要时的磁性状态。残余应力和受力会改变谐性声子谱，并可能显著影响低频模式。
随后收敛电子结构计算。
测试基组截断能、电子 $\boldsymbol k$ 网格、展宽方案、自洽阈值、赝势或 PAW 数据集以及交换关联泛函。收敛性应根据受力、应力和声子敏感量判断，而不应只检查总能量。
接着确定谐性 IFC。
选择能够覆盖相互作用范围的超胞或倒空间网格。检查平移与转动不变性、$\Gamma$ 点附近的声学模式、极性材料中的非解析修正，并在可能时与实验或独立计算的声子色散进行比较。
再确定非谐 IFC。
收敛超胞尺寸、位移幅度、相互作用截断和回归设置。应谨慎施加对称性和求和规则：修正可以降低数值噪声，但不能用来掩盖过小超胞或不完整模型。
然后构造碰撞算符。
根据材料和温度范围加入必要的散射机制。三声子加同位素散射是常见起点，而不是普适终点。四声子散射、温度重整化声子、电子–声子散射、缺陷或边界效应都可能需要考虑。
最后求解并收敛 PBTE。
收敛声子 $\boldsymbol q$ 网格、能量守恒积分以及迭代求解器阈值。比较 RTA 和完整解，并检查热导率张量对称性，而不只是标量平均值。
验证不能停在总热导率。
总热导率与实验吻合也可能来自相互抵消的误差。条件允许时，应同时比较：
PBTE 结果应当附带完整的收敛说明。主要不确定性可能来自多个层次。
| 层次 | 典型不确定性来源 | 有效诊断量 |
|---|---|---|
| 电子结构 | 泛函、体积、赝势、$\boldsymbol k$ 网格 | 受力、应力、弹性常数、声子频率 |
| 谐性 IFC | 超胞、长程静电作用、求和规则 | 声学模式、色散、群速度 |
| 非谐 IFC | 位移幅度、截断、回归、超胞 | Grüneisen 参数、线宽稳定性 |
| 布里渊区积分 | $\boldsymbol q$ 网格、展宽或四面体 | $\kappa$ 随网格及积分参数的变化 |
| 碰撞物理 | 遗漏四声子、电子、缺陷或边界项 | 温度趋势和模式分辨散射率 |
| 输运模型 | RTA、迭代 PBTE、忽略相干 | RTA/完整解比较及模式重叠分析 |
仅报告最终结果，例如“$\kappa=150$ W m$^{-1}$ K$^{-1}$”，并不能证明预测可靠。可信的研究应说明关键中间物理量已经稳定，并具有可解释性。
数值收敛并不等于物理模型完整，因此还必须判断传统 PBTE 的边界。
标准声子气体 PBTE 依赖可分离的准粒子布居。当出现以下情况时，通常需要扩展：
有限尺寸器件可能需要空间分辨 BTE、Monte Carlo、确定性输运或 Landauer 方法。强非谐晶体可能需要自洽声子或温度相关有效势。复杂晶体和无序固体则可能需要包含非对角相干项的 Wigner 或密度矩阵方法。
因此，更合适的问题不是孤立地问“PBTE 是否准确”，而是问：“当前 PBTE 是否包含了这个材料、温度和几何条件下真正重要的慢变量与散射机制？”
常用工具包括 phonopy/phono3py、ShengBTE，以及它们与 VASP、Quantum ESPRESSO、ABINIT 等电子结构软件的接口。这些软件可以自动完成大量代数运算和数据处理，但不能替研究者判断 IFC 截断是否充分、虚频是否真实，或者遗漏的散射机制是否重要。
可复现计算至少应记录：
可复现性的目标不仅是重新运行同一工作流，还要使其中的物理近似能够被检查。
记录完整工作流的目的不是堆积输入参数，而是让最终物理推断可以被追溯。第一性原理 PBTE 最适合被理解为一系列受控约化：
\[\text{电子基态} \longrightarrow \text{能量导数与 IFC} \longrightarrow \text{声子模式及相互作用} \longrightarrow \text{碰撞算符} \longrightarrow \text{非平衡分布} \longrightarrow \boldsymbol\kappa.\]每一个箭头都会引入数值选择和物理假设。该方法的优势在于，这些选择可以被系统检验，并且最终热导率可以被分解成具有明确意义的微观贡献。
最重要的结果并不是某一个 $\kappa$ 数值，而是解释哪些激发携带热量、哪些碰撞破坏或重新分配它们的集体运动，以及宏观输运系数为什么会从原子尺度动力学中出现。