Handbook of Materials Modeling pp 1-36 | Cite as

# Heat Transport in Insulators from Ab Initio Green-Kubo Theory

## Abstract

The Green-Kubo theory of thermal transport has long be considered incompatible with modern simulation methods based on electronic-structure theory, because it is based on such concepts as energy density and current, which are ill-defined at the quantum-mechanical level. Besides, experience with classical simulations indicates that the estimate of heat-transport coefficients requires analysing molecular trajectories that are more than one order of magnitude longer than deemed feasible using ab initio molecular dynamics. In this paper we report on recent theoretical advances that are allowing one to overcome these two obstacles. First, a general gauge invariance principle has been established, stating that thermal conductivity is insensitive to many details of the microscopic expression for the energy density and current from which it is derived, thus permitting to establish a rigorous expression for the energy flux from Density-Functional Theory, from which the conductivity can be computed in practice. Second, a novel data analysis method based on the statistical theory of time series has been proposed, which allows one to considerably reduce the simulation time required to achieve a target accuracy on the computed conductivity. These concepts are illustrated in detail, starting from a pedagogical introduction to the Green-Kubo theory of linear response and transport, and demonstrated with a few applications done with both classical and quantum-mechanical simulation methods.

Heat conduction in insulators is determined by the dynamics of the atomic nuclei, the electrons following adiabatically in their ground state: a regime that we will refer to as atomic or *adiabatic heat transport*. When a quasiparticle picture of the heat carriers (phonons) is adequate, a kinetic approach to heat conduction based on the Boltzmann transport equation (Peierls 1929) has been demonstrated (Broido et al. 2007) and successfully applied to crystalline insulators (Zhou et al. 2016), leveraging phonon frequencies and lifetimes from density-functional perturbation theory (Baroni et al. 1987, 2001; Giannozzi et al. 1991; Debernardi et al. 1995). As the temperature increases, anharmonic effects eventually make the quasiparticle picture break down well below melting (Turney et al. 2009), while the Boltzmann transport equation simply does not apply to glasses and liquids, where phonon dispersions are not even defined.

*κ*, can be expressed in terms of time correlations of the heat current,

*J*, as: where the brackets indicate ensemble averages over trajectories, which are accessible in principle to molecular dynamics (MD) simulations. In spite of the beauty, rigor, and broad scope of the GK theory, its adoption in the ab initio simulation of heat transport has stumbled at two seemingly insurmountable hurdles: first and foremost, the heat current entering Eq. (1) is ill-defined at the atomic scale, thus allegedly thwarting its implementation from first principles; second but not less important, experience in classical simulations, where a definition of the heat current is readily available in terms of suitably defined atomic energies (Irving and Kirkwood 1950), indicates that taming its statistical fluctuations requires so long MD simulations, as to be unaffordable using ab initio (AI) MD (Car and Parrinello 1985; Marx and Hutter 2009). As a matter of fact, no AIMD simulations of adiabatic heat transport based on GK theory have appeared until the spurious nature of these hurdles was recently revealed by Marcolongo et al. (2016) and Ercole et al. (2016, 2017).

The first difficulty was overcome by revealing a general *gauge invariance* principle for thermal transport, stemming from the hydrodynamic nature of energy fluctuations (see below), which makes transport coefficients independent of the microscopic expression of the energy density and current that are to a large extent ill-defined (Marcolongo et al. 2016; Ercole et al. 2016). Building on this principle, an expression for the energy current was derived from density-functional theory (Hohenberg and Kohn 1964; Kohn and Sham 1965), which allows simulating heat transport within the GK formalism, using equilibrium AIMD. The second difficulty was addressed using advanced statistical methods for the spectral analysis of stationary time series, which provide an asymptotically unbiased and consistent estimate of the power spectrum of the energy current, meaning that both the bias and the statistical error of the estimated conductivity can be made arbitrarily and controllably small in the limit of long simulation time (Ercole et al. 2017). The combination of these theoretical and methodological innovations makes the ab initio simulation of heat transport feasible in principle and affordable in practice. In this chapter, we review the efforts that have led to these achievements, starting with a brief review of the relevant theoretical concepts and illustrate them with relevant benchmarks and an application to a realistic model of liquid water.

## 1 Green-Kubo Theory of Heat Transport

### 1.1 Hydrodynamic Variables

*extensive variables*. By definition, the value that such a variable assumes for a system is the sum of the values it has for each of its subsystems. This property allows one to express an extensive variable,

*A*, as the integral of a suitably defined density,

*a*(

**r**), as:

**u**= {

*u*

_{α}} = {

*u*

_{1},

*u*

_{2},

*u*

_{3}}. When an extensive quantity is locally conserved, a current density,

*(*

**j****r**,

*t*), can be associated with its density in such a way that the two of them satisfy the continuity equation:

*indicates partial differentiation and the middle dot a scalar product (a divergence in this case). In the following, the densities and current densities of conserved quantities will be called*

**j***conserved densities*and

*conserved currents*for short. The space Fourier transform of Eq. (3) reads:

*adiabatic decoupling*in two distinct senses, depending on the context, to indicate the decoupling of electronic from nuclear degrees of freedom and that of hydrodynamic variables from fast atomic ones.

The long-wavelength Fourier components of conserved densities are called *hydrodynamic variables*. In macroscopically homogeneous systems, different wavelengths are decoupled from each other, while, as we have seen, the long wavelengths are adiabatically decoupled from all the other degrees of freedom. Let us suppose there are *Q* conserved extensive variables. In the case of a mono-atomic fluid, for instance, *Q* = 5, corresponding to mass (or particle number), energy, and the three components of the momentum. In order to simplify the notation, we set the value of the conserved quantities equal to zero, *A*^{i} = 0, so that their densities, *a*^{i}(**r**), directly refer to the departure from equilibrium, and we indicate by **j**^{i}(**r**, *t*) the corresponding currents. At equilibrium, all the conserved densities and currents vanish. Off equilibrium, it will be assumed that the wavelength and the time scale of the disturbances are so long that thermal equilibrium still holds *locally*. That is to say, a local temperature, pressure, and chemical potential can be defined, such that, when combined with the densities of extensive variable, they satisfy a local equation of state.

**q**, whereas their value at

**q**= 0 vanishes, because a nonvanishing value would imply a nonphysical long-range dependence of the currents on density fluctuations, in contrast with our assumption of local thermodynamic equilibrium. The long-wavelength low-frequency limit of the coupling constants can thus be assumed to be \( \tilde \Lambda ^{ij}(\mathbf {q},\omega) \sim q^2 \lambda ^{ij} \), so that the macroscopic (

**q**= 0) stationary (

*ω*= 0) components of the currents, \( {\mathbf {J}}^i = \frac {1}{\Omega} \int \boldsymbol j^i (\mathbf {r}) d\mathbf {r} \), are related to the corresponding components of the density gradients, \( {\mathbf {D}}^i=\frac {1}{\Omega}\int \nabla a^i(\mathbf {r})d\mathbf {r} \), through the equation:

*flux*.

*A*

^{i}, where

*S*is the system’s entropy and \( \chi ^{ij} = \frac {1}{\Omega} \frac {\partial A^i}{\partial x^j} \) the corresponding susceptibility. For instance, when

*A*

^{i}is the energy of the system, the corresponding conjugate variable is the inverse temperature,

*x*

^{i}= 1∕

*T*, while, when

*A*

^{i}represents the number of particles of a given species, one has

*x*

^{i}= −

*μ*

^{i}∕

*T*,

*μ*

^{i}being the corresponding chemical potential. The hypothesis of local thermodynamic equilibrium allows defining local values of the intensive variables, and we define

*thermodynamic forces*as their average gradients: \( {\mathbf {F}}^i= \frac {1}{\Omega} \int \nabla x^i(\mathbf {r})d\mathbf {r} \). The average density gradients are related to the thermodynamic forces through the susceptibility defined above, as:

*L*

^{ij}=∑

_{k}

*λ*

^{ik}

*χ*

^{kj}. Equation (9) expresses the linear relation between fluxes, the

**J**’s, and thermodynamic affinities, the

**F**s, for which Onsager derived his celebrated reciprocity relations (

*L*

^{ji}=

*L*

^{ij}) from microscopic reversibility (Onsager 1931a,b; Casimir 1945). Note that, according to our definition, both the

**J**s and the

**F**s in Eq. (9) do not depend on the size of the system.

### 1.2 Linear-Response Theory

*L*

^{ij}phenomenological coefficients appearing in Eq. (9), we consider a classical system of

*N*interacting atoms described by the Hamiltonian

*M*

_{n},

**R**

_{n}, and

**P**

_{n}are the masses, coordinates, and momenta of the

*n*-th particle, Γ = {

**R**

_{n},

**P**

_{n}} indicates the phase-space coordinates of the entire system, and

*V*is a generic many-body potential. Let us now suppose that the system is subject to an external perturbation that can be described as a linear combination of the conserved densities, {

*a*

^{i}(

**r**; Γ)}, as:

*a*(

**r**; Γ) is a phase-space function whose ensemble average is the conserved density:

*k*

_{B}the Boltzmann constant, and {

*v*

^{i}(

**r**,

*t*)} are time-dependent fields that couple to the conserved densities and vanish at

*t*= −

*∞*, when the system is assumed to be in thermal equilibrium at some temperature

*T*. Of course, conserved currents are also expected values of some phase-space functions,

*(*

**j****r**) = 〈

*(*

**j****r**;

*Γ*)〉. The phase-space functions whose expected values are conserved densities/currents will be referred to as

*phase-space samples*of the currents/densities. In the following, when the phase-space dependence of a conserved density/current is explicitly indicated, we will mean a phase-space sample; when it is not, a phase-space average will be implied. When a phase-space sample is evaluated along a dynamical trajectory,

*Γ*

_{t}, the sample function will depend on time and on the initial conditions of the trajectory. Averaging with respect to the initial conditions will result in a time-dependent expected value for the conserved densities (or currents):

*H*

^{∘}+

*V*

*′*. If it were driven by

*H*

^{∘}, evidently the value of

*a*would be time-independent. In the following, the notation Γ

_{t}will indicate an unperturbed time evolution. As an example, the phase-space sample of the particle density can be assumed to be

*n*(

**r**;

*Γ*) =∑

_{n}

*δ*(

**r**−

**R**

_{n}), the corresponding current is

*(*

**j****r**,

*Γ*) =∑

_{n}

*δ*(

**r**−

**R**

_{n})

**P**

_{n}∕

*M*

_{n}, and a local external potential is described by \( V^{\prime}(\varGamma ,t)=\sum _n v({\mathbf {R}}_n,t)=\int v^{\prime}(\mathbf {r},t) n(\mathbf {r};\varGamma) d \mathbf {r} \). Note that sample functions are not necessarily univocally defined. Different functions whose phase-space averages coincide in the long-wavelength limit sample the same hydrodynamical variable. More on this in Sect. 2.

*i*-th conserved current to the perturbation is:

**r**

*′*. The notation \( \partial ^{\prime}_\beta =\frac {\partial}{\partial r^{\prime}_\beta} \) has been used.

*L*

^{ij}=

*L*

^{ji}(Onsager 1931a,b), follow from Eq. (19) leveraging time-translational invariance, \( \langle J^i_\alpha (\Gamma _t) J^j_\beta (\Gamma _0) \rangle = \langle J^i_\alpha (\Gamma _0) J^j_\beta (\Gamma _{-t}) \rangle \), and micro-reversibility, \( \langle J^i_\alpha (\Gamma _t) J^j_\beta (\Gamma _0) \rangle = \langle J^i_\alpha (\Gamma _{-t}) J^j_\beta (\Gamma _0) \rangle \).

#### 1.2.1 Einstein-Helfand Expression for Transport Coefficients and the Wiener-Khintchine Theorem

The celebrated Einstein’s relation between the mean-square displacement of a diffusing particle and its velocity autocorrelation function is easily generalized to an arbitrary stochastic process and has in fact been utilized by Helfand (1960) to provide an “Einstein-like” expression for transport coefficients.

*X*

_{t}be a stationary stochastic process. One has:

When the stochastic process is the velocity of a Brownian particle, Eq. (20) allows one to establish a relation between the diffusion constant of the particle, temperature, and the autocorrelation time of the velocity. When *X*_{t} is the heat flux of a macroscopic body, Eq. (20) allows one to estimate the thermal conductivity, as given by Eq. (1), from the asymptotic behavior of the “energy displacement” \( \mathcal {D}(\tau) = \int _0^\tau \mathbf {J}(\Gamma _t) dt \).

*power spectral density*:

*power spectrum*. In the following, the suffix \( \mathcal {T} \) will be neglected for simplicity and its value assumed to be sufficiently large as to be considered infinite. More generally, when several conserved currents interact with each other, one can define the

*cross-spectrum*of the conserved fluxes as the Fourier transform of the cross-time correlation functions:

### 1.3 Heat Transport

*T*(

**r**) =

*T*+ Δ

*T*(

**r**) (| Δ

*T*|≪

*T*), the effects of this inhomogeneity can be formally described by the distribution function:

*e*(

**r**; Γ) is an energy (Hamiltonian) density, such that \( \int e(\mathbf {r};\varGamma) d\mathbf {r}= H^\circ (\varGamma) \). Equation (11) becomes:

**r**and integrate by parts to obtain:

**F**

_{n}is the force acting on the

*n*-th atom and \( {\mathbf {V}}_n=\frac {{\mathbf {P}}_n}{M_n} \) its velocity.

The manipulations leading from the continuity equation, Eq. (3), to Eq. (32) deserve some further comments, as they imply neglecting a boundary term, \( {\mathbf {J}}_{\partial \Omega} = \frac {1}{\Omega} \int _{\partial \Omega} \left (\boldsymbol {j}(\mathbf {r}) \cdot \hat {\mathbf {n}} \right) \mathbf {r} \,d\mathbf {r} \) (where *∂* Ω is the boundary of the integration volume and \( \hat {\mathbf {n}} \) the normal to it), which in general does not vanish in the thermodynamic limit and is ill-defined in periodic boundary conditions (PBC). The correct way of addressing this problem is to work with the Taylor expansion of the space Fourier transform of the continuity equation, Eq. (4), and to perform the thermodynamic limit at finite wavelength. The leading nonvanishing term in the Taylor expansion yields Eq. (31) without any boundary term in the way.

#### 1.3.1 Energy Flux from Classical Force Fields

*V*(

**R**

_{1},

**R**

_{2}, ⋯

**R**

_{N}), an energy density can be defined in terms of local atomic energies as:

*v*

_{n}s are a set of atomic potential energies whose sum is the total potential energy of the system, ∑

_{n}

*v*

_{n}=

*V*, with a short-range dependence on the coordinates of the other atoms. In the presence of long-range forces, this condition is effectively guaranteed by local charge neutrality, which we will assume throughout. By inserting Eq. (33) into Eq. (32), the energy flux can be cast into the form:

*m*-th atom to the force acting on the

*n*-th atom, ∑

_{m}

**F**

_{nm}=

**F**

_{n}, and

**F**

_{nm}= −

**F**

_{mn}. When the interaction among atoms can be expressed in terms of two-body potentials, one has \( v_m=\frac {1}{2}\sum _n v({\mathbf {R}}_n- {\mathbf {R}}_m) \) and \( {\mathbf {F}}_{n m} = - \frac {1}{2} \nabla _{{\mathbf {R}}_n} v({\mathbf {R}}_n- {\mathbf {R}}_m) \). Here we implicitly assumed that the interaction energy is equally partitioned between atoms

*m*and

*n*. In Sect. 2, we shall see this is not the only possible choice, with far-reaching consequences on the theory of heat transport.

The first term on the right-hand side of Eq. (35) is often called *convective* and the second *virial*. We feel that the wording “convective” is somewhat misleading in this context, as the convective current, as well as its contribution to heat conductivity, may not vanish even in the absence of convection.

#### 1.3.2 Multicomponent Fluids

*Q*− 1,

*Q*being the number of atomic species, so that the resulting dimension of the matrix of Onsager coefficients,

*L*, is

*Q*×

*Q*. The heat flux is defined as the non-convective component of the energy flux, i.e., the value of the latter in the absence of mass transport, that is to say when all the particle fluxes vanish. It is unfortunate, but inevitable due to common usage, that this definition of non-convective flux clashes with a different definition given above while commenting Eq. (35). By imposing this condition in Eq. (9), with

**J**

^{1}≡

**J**

^{E}, and

**J**

^{q}(

*q*= 2, …

*Q*) being independent particle fluxes, the thermal conductivity, defined as the ratio of the heat flux over the temperature gradient, is given by:

*any*non-singular linear transformation of the independent particle fluxes. For instance, in the case of a two-component liquid, energy and particle currents are coupled as in:

**J**

^{Q}is the particle current of one of the two species (say, the second) and

*μ*the corresponding chemical potential (Sindzingre and Gillan 1990). By imposing that the particle current vanishes, the resulting thermal conductivity is:

## 2 Gauge Invariance of Heat Transport Coefficients

*E*(Ω

_{i}) are the energies of the two isolated subsystems,

*W*

_{12}their interaction energy, and

*λ*an arbitrary constant. In the thermodynamic limit, when all the subsystems’ energies are much larger than the interaction between any pairs of them, the value of the

*λ*constant is irrelevant. When it comes to defining energy densities (i.e., energies of infinitesimal portions of a system) or atomic energies, instead, the magnitude of the interaction between different subsystems is comparable to their energies, which become therefore intrinsically ill-defined.

*v*(|

**R**

_{n}−

**R**

_{m}|), and define the atomic energies as (Marcolongo 2014; Ercole et al. 2016):

*γ*

_{nm}= −

*γ*

_{mn}is

*any*antisymmetric matrix. As the interatomic potential appearing in Eq. (39) is symmetric with respect to the atomic indices, it is clear that the sum of all the atomic energies does not depend on

*γ*, thus making any choice of

*γ*equally permissible. This trivial observation has deep consequences on the theory of thermal fluctuations and transport, because the value of the macroscopic energy flux, instead, depends explicitly on

*γ*, thus making one fear that the resulting transport coefficients would depend on

*γ*as well. Using the same manipulations that lead from Eqs. (33) and (34) to Eq. (35), for any choice of the

*γ*matrix in Eq. (39), a corresponding expression for the macroscopic energy flux can be found, reading (Marcolongo 2014; Ercole et al. 2016):

*v*

_{nm}=

*v*(|

**R**

_{n}−

**R**

_{m}|).

*ρ*= 0.925

*σ*

^{−3}. In Fig. 2a, we display the resulting macroscopic energy-flux autocorrelation function corresponding to different choices of the

*γ*matrix in Eqs. (39) and (40). Figure 2a clearly shows that the \( \langle {\mathbf {J}}^{\scriptscriptstyle E}_{\gamma}(t) \cdot {\mathbf {J}}^{\scriptscriptstyle E}_{\gamma}(0) \rangle \) correlation functions dramatically depend on the

*γ*matrices in Eqs. (39) and (40). Notwithstanding, the integrals of all these time correlation functions tend to the same limit at large integration times, as shown in Fig. 2b.

*e′*(

**r**,

*t*) and

*e*(

**r**,

*t*), whose integrals over a macroscopic volume differ by a quantity that scales as the volume boundary, should be considered as equivalent. This equivalence can be expressed by the condition that two equivalent densities differ by the divergence of a (bounded) vector field:

*gauges*of the same scalar field. Energy is also conserved: because of this, for any given gauge of the energy density,

*e*(

**r**,

*t*), an energy current density can be defined,

*(*

**j****r**,

*t*), so as to satisfy the continuity equation, Eq. (3). By combining Eqs. (42) and (3), we see that energy current densities and macroscopic fluxes transform under a gauge transformation as:

*e*and

*e′*, differing by a bounded total time derivative, as in Eq. (44), result in the same heat conductivity, as given by the Green-Kubo formula, Eq. (29). More generally, the Onsager coefficients coupling two fluxes,

**J**

^{1}and

**J**

^{2}, do not depend on the gauge of either one of them. In fact, let \( \left ({\mathbf {J}}^1\right)^{\prime} = {\mathbf {J}}^1 + \dot {\mathbf {P}} \); one has:

*κ′*=

*κ*.

### 2.1 Molecular Fluids

In a one-component molecular fluid such as liquid water or, say, ethanol, there are in general *Q* fluxes interacting with each other through Onsagers’ Eq. (9), where *Q* is the number of atomic species in a molecule. The requirement that atoms are bound in molecules of fixed composition, however, sets a number of constraints that substantially simplify the treatment of heat transport, making the molecular case similar to the one-component one.

*A*,

*B*, ⋯ indicate atomic species and

*N*

_{A},

*N*

_{B}, ⋯ the corresponding atomic stoichiometric indices. For each atomic species, we define the normalized number flux as:

*M*

_{X}the atomic mass of species

*X*, momentum conservation requires that ∑

_{X}

*M*

_{X}

*N*

_{X}

**J**

^{X}= 0 in the center-of-mass reference frame. The flux

**J**

^{XY}=

**J**

^{X}−

**J**

^{Y}is the total time derivative of a bounded vector, because its integral is the sum over all the molecules of the difference between the average atomic positions of either species within a same molecule, which is obviously bounded if molecules do not dissociate. As any number flux

**J**

^{X}can be expressed as a linear combination of the total momentum and of several

**J**

^{XY}fluxes, each of them is the total time derivative of a bounded vector. Therefore, the Onsager coefficient coupling any of these atomic fluxes with any other, or with the energy flux, vanishes. We conclude that energy is the only conserved quantity relevant for heat transport in a molecular fluid and that the energy-flux autocorrelation function directly yields the thermal conductivity, as in Eq. (1).

## 3 Density-Functional Theory of Adiabatic Heat Transport

Quantum simulation methods based on density-functional theory (DFT) have long been thought to be incompatible with the GK theory of thermal transport *because in first-principles calculations, it is impossible to uniquely decompose the total energy into individual contributions from each atom* (Stackhouse et al. 2010). For this reason, ab initio simulations of heat transport have often been performed using non-equilibrium approaches.

Stackhouse et al. (2010), for instance, computed the thermal conductivity of periclase MgO using a method devised by Müller-Plathe (1997). In this approach, a net heat flux, rather than a temperature gradient, is imposed to the simulated system, and the thermal conductivity is evaluated as the ratio between the heat flux and the resulting temperature gradient.

In the so-called *approach to equilibrium* methodology of Lampin et al. (2013), the system is first prepared in an out-of-equilibrium state characterized by an inhomogeneous temperature distribution, and the thermal conductivity is evaluated from the time it takes for the system to relax to equilibrium. This technique has been combined with AIMD to simulate thermal transport in a GeTe_{4} glass by Bouzid et al. (2017) and further generalized and applied to crystalline and nanostructured MgO by Puligheddu et al. (2017).

Recently, there have been several attempts to combine the GK approach to heat transport with ab initio techniques based on electronic-structure theory, by adopting some ad hoc definitions for the energy flux. Kang and Wang (2017), for instance, derived an expression for the energy flux from a (rather arbitrary) quantum-mechanical definition of the atomic energies and used a modified MD integration algorithm to cope with the difficulties ensuing from the implementation of their expression in PBC. Carbogno et al. (2017) gave a different expression for the energy flux, based on a normal-mode decomposition of the atomic coordinates and forces, which, while allowing to reduce the effects of thermal fluctuations, can only be applied to crystalline solids.

In spite of the undoubted ingenuity of these proposals, the problem still remains as of how it is possible that a rather arbitrary definition of the heat flux results in an allegedly well-defined value for the thermal conductivity. The gauge-invariance principle introduced in Sect. 2 not only provides a solution to this conundrum, but it also gives a rigorous way of deriving an expression for the energy flux directly from DFT, without introducing any ad hoc ingredients.

*ε*

_{v}, eigenfunctions

*ϕ*

_{v}(

**r**), and density

*n*(

**r**) =∑

_{v}|

*ϕ*

_{v}(

**r**)|

^{2}(Martin 2008):

*𝜖*

_{XC}[

*n*](

**r**) is a local exchange-correlation (XC) energy per particle defined by the relation \( \int \epsilon _{\scriptscriptstyle XC}[n](\mathbf {r})n(\mathbf {r}) d\mathbf {r}=E_{\scriptscriptstyle XC} [n] \), the latter being the total XC energy of the system; and \( \mu _{\scriptscriptstyle XC} (\mathbf {r}) = \frac {\delta E_{\scriptscriptstyle XC}}{\delta n(\mathbf {r})} \) is the XC potential. The DFT total energy can be readily written as the integral of a DFT energy density (Chetty and Martin 1992):

*H*

_{KS}is the instantaneous self-consistent Kohn-Sham Hamiltonian and \( v_{\scriptscriptstyle H} = {\mathtt {e}^2} \int d\mathbf {r}^{\prime} \frac {n(\mathbf {r}^{\prime})}{|\mathbf {r}-\mathbf {r}^{\prime}|} \) is the Hartree potential. An explicit expression for the DFT energy flux is obtained by computing the first moment of the time derivative of the energy density, Eqs. (48), (49), (50), and (51), as indicated in Eq. (31), resulting in a number of terms, some of which are either infinite or ill-defined in PBC. Casting the result in a regular, boundary-insensitive, expression requires a careful breakup and refactoring of the various harmful terms, as explained by Marcolongo (2014) and in the online version of Marcolongo et al. (2016). The final result reads:

*v*-th occupied orbital, Eq. (58), and of its adiabatic time derivative (Giannozzi et al. 2017), Eq. (59), \( \hat P_v \) and \( \hat P_c = 1 - \hat P_v \) being the projector operators over the occupied- and empty-states manifolds, respectively. Both these functions are well defined in PBC and can be computed, explicitly or implicitly, using standard density-functional perturbation theory (Baroni et al. 2001).

## 4 Data Analysis

The MD evaluation of the GK integral, Eq. (1), usually proceeds in two steps. One first evaluates the integrand as a running average of the time-lagged current products, \( \langle J^i(\tau)J^j(0)\rangle \sim \frac {1}{\mathcal {T}-\tau} \int _0^{\mathcal {T}-\tau} J^i(t+\tau)J^j(t)dt \), where \( \mathcal {T} \) is the length of the MD trajectory. The matrix defined in Eq. (19) is then estimated as a function of the upper limit of integration: \( L^{ij}(\mathcal {T})\propto \frac {\Omega}{k_B} \int _0^{\mathcal {T}} \langle J^i(\tau)J^j(0)\rangle \,d\tau \). One then recovers, via Eq. (36), an estimate for the thermal conductivity depending on \( \mathcal {T} \): \( \kappa (\mathcal {T})\propto \frac {1}{(L^{-1}(\mathcal {T}))^{{\scriptscriptstyle 1}{\scriptscriptstyle 1}}} \). This function is usually very noisy: in fact, at times greater than the correlation time between *J*^{i} and *J*^{j}, the correlation function 〈*J*^{i}(*τ*)*J*^{j}(0)〉 approaches zero; hence \( L^{ij}(\mathcal {T}) \) starts integrating noise and behaves like the distance traveled by a random walk, whose variance grows linearly with the upper integration limit. The evaluation of transport coefficients thus requires averaging over multiple trajectories (possibly multiple segments of a same long trajectory) and estimating the resulting uncertainty as a function of both the length of each trajectory and the upper limit of integration. This is a cumbersome task that often leads to a poor estimate of the statistical and systematic errors on the computed conductivity, all the more so when the signal is inherently oscillatory, due to the existence of high-frequency features in the power spectrum of the energy flux, possibly due to intramolecular oscillations that meddle with the noise. Some authors try to overcome these problems by either fitting the autocorrelation function or the GK integral with a multi-exponential function (Schelling et al. 2002; Zhang et al. 2015) or by extrapolating the power spectrum of the energy flux to the zero-frequency limit (Volz and Chen 2000). Others have attempted an error analysis of the MD estimate of the GK integral, based on either heuristic or rigorous arguments (Jones and Mandadapu 2012; Wang et al. 2017; Oliveira and Greaney 2017), but they all require an estimate of an optimal value for the upper limit of integration, which determines a bias in the estimate and which is in general difficult to obtain. Different classes of systems require different approaches to error analysis, but it is widely believed that all of them always require so long simulation times as to be unaffordable with accurate but expensive AIMD techniques (Carbogno et al. 2017). In order to solve this problem, Ercole et al. (2017) considered it in the light of the statistical theory of stationary time series.

### 4.1 Solids and One-Component Fluids

*time series*),

*J*

_{n}=

*J*(

*n𝜖*), 0 ≤

*n*≤

*N*− 1, where

*𝜖*is the sampling period of the flux and

*N*the length of the time series, that we assume to be even. As was shown in Sect. 1.2.1, the Wiener-Khintchine theorem allows one to express the heat conductivity in terms of the zero-frequency value of the power spectrum of the energy flux (see Eqs. (21), (22), (23), and (24)):

*k*≤

*N*− 1. Here, the convention for the sign in the exponential of the time-to-frequency Fourier transform is opposite to what is adopted in Ercole et al. (2017) and in most of the signal analysis literature, in order to comply with the convention for the space-time Fourier transforms usually adopted in the Physics literature and in Eqs. (4) and (5). The

*sample spectrum*\( \hat S_k \), aka

*periodogram*, is defined as:

*N*, it is an unbiased estimator of the power spectrum of the process, as defined in Eq. (21), evaluated at \( \omega _k=2\pi \frac {k}{N\epsilon} \), namely: \( \langle \hat S_k \rangle = S(\omega _k) \). The reality of the \( \hat J \)s implies that \( \tilde J_k=\tilde J^*_{N-k} \) and \( \hat S_k=\hat S_{N-k} \), so that periodograms are usually reported for \( 0\leq k\leq \frac {N}{2} \) and their Fourier transforms evaluated as discrete cosine transforms.

*N*limit, are uncorrelated among themselves and have variances proportional to the power spectrum evaluated at

*ω*

_{k}. For

*k*= 0 or \( k=\frac {N}{2} \), \( \tilde J_k \) is real and \( {\sim}\mathcal {N}\left (0, \frac {N}{\epsilon}S({\omega _k}) \right) \); for \( k\notin \left \{0,\frac {N}{2}\right \} \), \( \mathfrak {Re}\tilde {J}_k \) and \( \mathfrak {Im}\tilde {J}_k \) are independent and both \( {\sim}\mathcal {N}\left (0, \frac {N}{2 \epsilon}S({\omega}_k) \right) \), where \( \mathcal {N} (\mu ,\sigma ^2) \) indicates a normal deviate with mean

*μ*and variance

*σ*

^{2}. We conclude that in the large-

*N*limit the sample spectrum of the heat flux time series reads:

*ξ*s are independent random variables distributed as a \( \chi _1^2 \) variate for

*k*= 0 or \( k=\frac {N}{2} \) and as one half a \( \chi _2^2 \) variate, otherwise. Here and in the following \( \chi ^2_\nu \) indicates the chi-square distribution with

*ν*degrees of freedom. For the sake of simplicity, we make as though all the

*ξ*s were identically distributed, \( \xi _k \sim \frac {1}{2} \chi _2^2 \) for all values of

*k*, thus making an error of order \( \mathcal {O}(1/N) \), which vanishes in the longtime limit that is being assumed throughout this section.

*p*= 1, ⋯

*ℓ*. For instance, in equilibrium MD, a same trajectory delivers one independent time series per Cartesian component of the heat flux, all of which are obviously equivalent in isotropic systems. In these cases, it is expedient to define a mean sample spectrum by averaging over the

*ℓ*different realizations: where the Open image in new windows are \( \chi _{2\ell}^2 \) variates, divided by the number of degrees of freedom: Open image in new window \( \bigl (\text{for}\, k \notin \{0,\frac {N}{2} \}\bigr) \).

Equations (63) and (64) show that Open image in new window is an unbiased estimator of the zero-frequency value of the power spectrum, Open image in new window, and through Eq. (60), of the transport coefficients we are after. However, this estimator is not consistent, i.e., its variance does not vanish in the large-*N* limit. This is so because a longer time series increases the number of discrete frequencies at which the power spectrum is sampled, rather than its accuracy at any one of them.

*ψ*(

*z*) is the digamma function (Weisstein PolyGamma). The variance of the Open image in new window variables is \( \sigma _{\ell}^{2} =\psi ^{\prime}(\ell) \), where

*ψ′*(

*z*) is the tri-gamma function (Weisstein PolyGamma).

*cepstrum*” of the time series as the inverse Fourier transform of its sample log-spectrum (Childers et al. 1977): A generalized central-limit theorem for Fourier transforms of stationary time series ensures that, in the large-

*N*limit, these coefficients are a set of independent (almost) identically distributed zero-mean normal deviates (Anderson 1994; Peligrad and Wu 2010). It follows that: where Open image in new window are independent zero-mean

*normal*deviates with variances Open image in new window \( =\frac {1}{N}\sigma _\ell \) for \( n\notin \left \{0,\frac {N}{2}\right \} \) and Open image in new window otherwise. Figure 4 displays the cepstral coefficients of the low-frequency region of the spectrum of water (marked in Fig. 3), showing that only the first few coefficients are substantially different from zero.

*P*

^{∗}the smallest integer such that

*C*

_{n}≈ 0 for

*P*

^{∗}≤

*n*≤

*N*−

*P*

^{∗}. By limiting the Fourier transform of the sample cepstrum, Eq. (66), to

*P*

^{∗}coefficients, we obtain an efficient estimator of the zero-frequency component of the log-spectrum as: Inspection of Eq. (68) shows that Open image in new window is a normal estimator whose expectation and variance are:

*only*through the number of these coefficients,

*P*

^{∗}. Notice that the absolute error on the logarithm of the conductivity directly and nicely yields the relative error on the conductivity itself.

*P*

^{∗}using the Akaike’s information criterion (Akaike 1974), but other more advanced

*model selection*approaches (Claeskens and Hjort 2008) may be more effective. This method consists in choosing

*P*

^{∗}as the one that minimizes the function:

*P*

^{∗}:

### 4.2 Multicomponent Fluids

In Sect. 1.3.2, we have seen that in a fluid made of *Q* atomic species, there are in general *Q* macroscopic fluxes interacting with each other through Onsager’s phenomenological equations, Eq. (9), not counting the different Cartesian components that do not interact among themselves because of space isotropy. A MD simulation thus samples *Q* stochastic processes, one for each interacting flux, that we suppose to be stationary. These processes can be thought of as different components of a same multivariate process (Bertossa et al. 2018). As in Sect. 4.1, for the sake of generality we suppose to have *ℓ* independent samples of such a process, described by a multivariate time series of length *N*: Open image in new window; *p* = 1, …*ℓ*; *i* = 1, …*Q*; *n* = 0, …*N* − 1. Stationarity implies that \( \langle {J}^i_n\rangle \) does not depend on *n* and that \( \langle {J}^i_n {J}^j_m \rangle \) only depends on *n* − *m*. We will further assume that \( \langle {J}^i_n\rangle =0 \) and that \( \langle {J}^i_n {J}^j_0 \rangle \) is an even function of *n*, which is the case when *J*^{i} and *J*^{j} have the same signature under time-reversal. By combining Eq. (36) with Eq. (24), we see that in order to evaluate the thermal conductivity in the multicomponent case, we need an efficient estimator for \( \left (S^{-1}_0\right)^{11} \), where \( S^{kl}_0=S^{kl}(\omega =0) \) is the zero-frequency cross-spectrum of the relevant fluxes, ordered in such a way that the energy one is the first.

*cross-periodogram*) as: By discretizing Eq. (23), we see that Open image in new window is an unbiased estimator of the cross-spectrum, Open image in new window. As it was the case for univariate processes, in the large

*N*limit the real and imaginary parts of \( \tilde J^i_k \) are normal deviates that are uncorrelated for

*k*≠

*k′*. We conclude that the cross-periodogram is a random matrix distributed as a complex Wishart deviate (Goodman 1963a,b): The notation \( {\mathcal {CW}_Q} \left (S, \ell \right) \) in Eq. (74) indicates the distribution of the

*Q*×

*Q*Hermitian matrix Open image in new window, where Open image in new window (

*p*= 1, ⋯

*ℓ*,

*i*= 1, ⋯

*Q*) are

*ℓ*samples of an

*Q*-dimensional zero-mean normal variate whose covariance is

*S*

^{ij}= 〈

*X*

^{i}

*X*

^{j∗}〉.

*R*is a real lower triangular random matrix of the form:

*R*is independent of the specific covariance matrix and only depends upon

*ℓ*and

*Q*. In particular, it is independent of the ordering of the fluxes

*J*

^{i}. By expressing the

*ξ*

_{k}are independent random (with respect to

*k*) random variables, distributed as:

*χ*

^{2}distribution, that becomes 2(

*ℓ*−

*Q*+ 1), and a different factor in front of the result. Figure 6 shows an example of multi-component power spectrum for a solution of water and ethanol.

*κ*, obtained via Eq. (38), in the case of a water-ethanol solution, as a function of the upper time-integration limit \( \mathcal {T} \) (Bertossa et al. 2018). Both the Green-Kubo and the Einstein-Helfand definitions of the finite-time expression of Onsager’s coefficients (see Eq. (20)) are displayed. Due to thermal fluctuations, the integral of the correlation function becomes a random walk as soon as the latter vanishes, eventually assuming any value. Therefore, there will be a set of times (see Fig. 7) where the term

*L*

^{QQ}at the denominator in Eq. (38) vanishes, leading to divergences in the evaluation of

*κ*; an issue not affecting the one-component case. Hence, in such a formulation of the multicomponent case, the mean value of the thermal conductivity estimator

*in the time domain*does not exist. On the contrary, the multicomponent frequency-domain approach presented in this section, and built on sound statistical basis, provides a well-defined expression for the estimator of

*κ*and its statistical error.

### 4.3 Data Analysis Workflow

*cepstral analysis*method, in order to highlight the simplicity of its practical implementation.

- 1.
From a MD simulation compute the heat flux time series \( J_n^1 \) and the independent particle fluxes \( J_n^q \),

*q*= 2, …,*Q*. - 2.
Compute the discrete Fourier transform of the fluxes, \( \tilde {J}^{i}_k \), and the element \( 1/(\hat {S}^{-1})^{{\scriptscriptstyle 1}{\scriptscriptstyle 1}} \). In practice, only a selected low-frequency region shall be used (see Ercole et al. (2017) for a a detailed discussion). To lighten the notation, we drop the left superscripts of the variables in this subsection.

- 3.
Calculate \( \log {}\left [1/(\hat {S}^{-1})^{{\scriptscriptstyle 1}{\scriptscriptstyle 1}}\right ] \).

- 4.
Compute the inverse discrete Fourier transform of the result to obtain the cepstral coefficients \( \hat {C}_n \).

- 5.
Apply the Akaike Information Criterion, Eq. (71), to estimate the number of cepstral coefficients to retain,

*P*^{∗}. - 6.Finally apply Eq. (68) to obtain \( \hat {L}_0^* \), and evaluate the thermal conductivity asand its statistical error as:$$ \displaystyle \begin{aligned} \kappa = \frac{\Omega}{2k_B T^2} \exp\left[\hat{L}_0^* - {\psi(\ell - Q+1) + \log{}(\ell -Q+1)} \right], \end{aligned} $$(80)$$ \displaystyle \begin{aligned} \frac{\Delta\kappa}{\kappa} = \sqrt{{\psi^{\prime}(\ell -Q+1)} \frac{4P^{*}-2}{N}}. \end{aligned} $$(81)

## 5 A Few Representative Results

Calculations of the thermal conductivity based on the Green-Kubo formalism combined with first-principles molecular dynamics are quite recent. The first benchmarks from Marcolongo et al. (2016) have been performed on liquid Argon and heavy water at ambient conditions, as reported below.

### 5.1 A Benchmark on a Model Mono-atomic Fluid

As a first test, liquid Argon was simulated by Marcolongo et al. (2016) using a local LDA functional neglecting dispersion forces. The resulting fictitious system, dubbed LDA-Argon, is a hard-core weakly interacting fluid whose dynamics is expected to be easily mimicked by a simple two-body potential, which can be engineered by standard force-matching techniques. This observation allows one to effectively test the ideas developed in Sect. 3: even if the energy density in LDA-Argon and in its fitted classical counterpart will likely be different, the resulting thermal conductivity is expected to coincide within the quality of the classical fit.

*Å*, corresponding to a density of 1.34 g cm

^{−3}. Trajectories were sampled in the NVE ensemble for 100 ps, and the classical model was fitted with a pair potential of the form

*V*(

*r*) =

*P*

_{2}(

*r*)e

^{−αr},

*P*

_{2}(

*r*) being a second-order polynomial. In Fig. 8, we show the resulting autocorrelation functions at a representative temperature of 400 K. The DFT autocorrelation function shows a more structured behavior than that of the classical potential. Nevertheless, when considering the long time limit of the Green-Kubo integral, the thermal conductivities indeed coincide within statistical uncertainty, as predicted by theory.

### 5.2 Heavy Water at Ambient Conditions

Marcolongo et al. (2016) also computed the thermal conductivity of heavy water at ambient conditions. Simulations were performed using the PBE exchange correlation energy functional (Perdew et al. 1996), which is known to predict qualitatively the self-diffusion coefficient of water at ambient conditions only for a simulation temperature of about 400 K (Sit and Marzari 2005): the latter condition was imposed accordingly. A system of 64 heavy water molecules was sampled for 90 ps in the NVE ensemble using a cubic cell corresponding to the experimental density of 1.11 g cm^{−3}.

The ab initio treatment of heat transport in molecular fluids requires some care in order to eliminate non-diffusive components of the energy flux that, while not contributing to the conductivity, do increase the noise of the flux time series to a level that may compromise its analysis. To see where the problem comes from, let us split the potential energy of the system into the sum of non-interacting atomic energies plus an interaction energy as: \( V(\{{\mathbf {R}}_n\}) = \sum _n\epsilon ^\circ _n + V_{int}(\{{\mathbf {R}}_n\}) \), where \( \epsilon _n^\circ \) is the energy of the *n*-th atom when it is isolated from the rest. In classical simulations, the energy of isolated atoms never enters the description of the system, and the \( \epsilon ^\circ _n \)’s can be simply set to zero. In quantum simulations, instead, atomic and interaction energies enter on a same footing, and the former give a large and fluctuating contribution to the total energy flux, \( {\mathbf {J}}^\circ =\sum _X\epsilon ^\circ _X {\mathbf {J}}^X \), where **J**^{X} is the flux defined in Eq. (46). In a mono-atomic fluid, **J**^{X} is constant because of momentum conservation, and it is actually equal to zero in the center-of-mass reference frame. In molecular fluids, the **J**^{X} do not vanish, but, as we have seen in Sect. 2.1, they are non-diffusive and hence do not contribute to the heat conductivity while adding considerable noise to the energy-flux time series. In order to remove them, instead of estimating **J**^{∘} from the noninteracting atomic energies, we prefer to implement a *decorrelation technique*, as described below.

**J**

^{1}and

**J**

^{2}be two macroscopic fluxes and

**J**

^{12}=

**J**

^{1}+

**J**

^{2}their sum. The corresponding conductivities

*κ*

^{1},

*κ*

^{2}, and

*κ*

^{12}then satisfy \( |{\kappa ^{12}}-\kappa ^1-\kappa ^2| \le 2 \sqrt {\kappa ^1 \kappa ^2} \). As a consequence, when

*κ*

^{2}vanishes,

*κ*

^{12}coincides with

*κ*

^{1}. Let us now suppose that a set of fluxes {

**Y**

^{u}},

*u*= 1, …

*U*is known to exhibit a non-diffusive behavior. The above argument shows that the auxiliary flux defined as:

**J**. Optimal values of the {

*λ*

^{u}} coefficients can then be determined by imposing that the new time series

**J**

*′*is uncorrelated with respect to the non-diffusive ones, namely:

**Y**

^{u}fluxes give a slowly converging contribution to the Green-Kubo integral, which is thus difficult to evaluate numerically.

**Y**

^{1}=

**J**

^{H}+

**J**

^{O}, i.e., the sum of hydrogen and oxygen average velocities and

**Y**

^{2}=

**J**

^{el}, the adiabatic electronic current. Note that the two time series

**J**

^{H}and

**J**

^{O}are trivially related, because of momentum conservation. Therefore

**J**

^{H},

**J**

^{O}, or

**J**

^{H}+

**J**

^{O}would all be equivalent choices. The latter is defined, following the same notation of Sect. 3, as:

**J**

^{el}is non-diffusive and can thus be used to decorrelate the heat current. In the original paper the thermal conductivity was evaluated from the slope of the energy displacement \( \mathcal {D}(\tau)=\int _0^\tau \mathbf {J}(t)dt \) (see Eq. (20)) and the corresponding error obtained from a standard block analysis, resulting in a nonoptimal estimate of both (see Fig. 9a). The same data have been reanalyzed with the novel cepstral technique presented in Sect. 4. We denote by \( \left ({\mathbf {J}}^1\right)^{\prime} \) and \( \left ({\mathbf {J}}^{12} \right)^{\prime} \) the currents decorrelated with respect to

**Y**

^{1}alone and with respect to both

**Y**

^{1}and

**Y**

^{2}, respectively. The power spectra of the two currents are plotted in Fig. 9b. Only the low-frequency region of the spectra (up to ∼9.0 THz) was used for the cepstral analysis (see Ercole et al. (2017) for the technical details). The minimization of Eq. (71) suggests a cutoff \( P^*_1=17 \) and \( P^*_{12}=15 \), but a value 1.5 times larger was actually been used to reduce the bias possibly due to the fast variation of the spectrum at frequency close to zero. The resulting thermal conductivities are

*κ*

^{1}= 0.80 ± 0.12 W∕mK for the \( \left ({\mathbf {J}}^1 \right)^{\prime} \) flux and

*κ*

^{12}= 0.93 ± 0.14 W∕mK for \( \left ({\mathbf {J}}^{12} \right)^{\prime} \), compatible with each other. By comparison, experiments give a value

*κ*≈ 0.6 W∕mK (Matsunaga and Nagashima 1983; Ramires et al. 1995). In this case, data analysis would not have yielded any meaningful results failing a proper decorrelation of the heat flux time series.

## 6 Outlook

We believe that the ideas presented in this chapter will pave the way to new developments and applications in the field of heat transport, particularly for strongly anharmonic and/or disordered systems, where approaches based on the Boltzmann transport equation do not apply or are bound to fail. The general concept of gauge invariance of heat conductivity will likely apply to other transport properties as well, such as ionic conduction, viscosity, and many others, and/or simulation methodologies, such as those based on a neural-network representation of interatomic potentials, which hold the promise of a strong and long-lasting impact on molecular simulations. The applicability of this concept would not be as broad if not assisted by the powerful data-analysis methods which have also been described in this chapter. Here again, we believe that there is ample room for improvement, leveraging more general (possibly non-Fourier) representations of the log-spectrum of the currents to be analyzed and more advanced statistical-inference techniques to estimate the parameters of the spectral models resulting from these representations. From the applicative point of view, we expect that these methodological advances will have a strong impact in all those cases where heat, mass, and charge transport occurs in conditions that cannot be adequately described at the atomistic level by simple force fields, e.g., in complex materials; systems at extreme external conditions, such as those occurring in the planetary interiors; complex or reactive fluids; and many others.

## 7 Cross-References

## Notes

### Acknowledgements

This work was supported in part by the MaX EU Centre of Excellence, grant no 676598. SB, LE, and FG are grateful to Davide Donadio for insightful discussions all over the Summer of 2017 and beyond.

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