Theory of Heat Transport of Normal Liquid 3He in Aerogel

The introduction of liquid 3He into silica aerogel provides us a with model system in which to study the effects of disorder on the properties of a strongly correlated Fermi liquid. The transport of heat, mass and spin exhibits cross-over behavior from a high temperature regime, where inelastic scattering dominates, to a low temperature regime dominated by elastic scattering off the aerogel. We report exact and approximate solutions to the Boltzmann-Landau transport equation for the thermal conductivity of liquid 3He, including elastic scattering of quasiparticles by the aerogel and inelastic quasiparticle collisions. These results provide quantitative predictions for the transport properties of liquid 3He in aerogel over a wide range of pressure, temperature and aerogel density. In particular, we obtain a scaling function, F(T/T*), for the normalized thermal conductivity, K/K_el, in terms of a reduced temperature, T/T*, where T* is a cross-over temperature defined by the elastic and inelastic collision rates. Theoretical results are compared with the available experimental data for the thermal conductivity.


I. INTRODUCTION
Aerogels are extremely low density solids formed as a rigid network of silica strands and clusters having a typical diameter of 30-50Å and porosities (̺) above 99%. 15 They turn out to be an ideal system for studying the effects of quenched random disorder on the otherwise pure, ordered phases of liquid 3 He. When impregnated with liquid 3 He, the aerogel is found to have dramatic effects on the transport properties and the phase diagram of liquid 3 He, although basic thermodynamic features characteristic of the Fermi liquid, such as the compressibility, magnetization and heat capacity, are essentially unchanged. 18 The low temperature transport of mass, heat and magnetization is substantially reduced. 28,30,31 In addition, the superfluid transition temperature as well as the superfluid order parameter are strongly suppressed relative to their bulk values. 28,36 In this paper we consider the effects of scattering of 3 He quasiparticles off a uniformly distributed random potential representing the aerogel structure, referred to as the "homogeneous scattering model" (HSM). 38 We obtain exact and approximate solutions to the Boltzmann-Landau transport equation for the thermal conductivity of liquid 3 He, including both inelastic collisions between quasiparticles and elastic scattering of quasiparticles by the random potential. The chief inadequacy of the HSM is its neglect of the inhomogeneous void-structure of the aerogel, or more generally mesoscopic correlations that are observed in static structure factor, and to which the superfluid transition temperature is sensitive. 27,34,38 However, the transport properties are limited by the mean free path for quasiparticles propagating ballistically within the aerogel, and hence are expected to be well accounted for in the framework of the HSM since the geometric mfp (ℓ) is typically much longer than the aerogel correlation length (ξ a ), e.g. ℓ/ξ a ≃ 3 for a 98% aerogel. Possible corrections to transport processes resulting from a small distribution of large voids or to fractal correlations on mesoscopic length scales ξ a within the aerogel are not included in the analysis presented in this work. However, the exact solution to the transport equation for the two-channel scattering model discussed in this paper should be of value in identifying observable corrections associated with correlated disorder or corrections to the two channel scattering theory. Liquid 3 He is a dense quantum liquid in which the interactions between Fermionic excitations (quasiparticles) are one to two orders of magnitude larger than the mean kinetic energy per particle. These interactions lead to strongly renormalized branches of Fermionic excitations, reflecting the correlated motion of many 3 He atoms, and the emergence of Bosonic excitations. The Fermionic excitations bear resemblance to 3 He atoms only in terms of their quantum numbers for spin (s = ±1/2) and fermion number (e = ±1). The Bosonic excitations come with and without spin and can be understood in terms of pairs of Fermionic excitations, e.g. the phonons of zero sound. Finally, the coupling between the Bosonic and Fermionic excitations leads to finite lifetimes for both types of excitations. 4,[21][22][23] Interactions between 3 He quasiparticles enhance the collision rate for quasiparticles near the Fermi surface, leading to a significant reduction in the lifetime of a quasiparticle at the Fermi surface. However, Fermi statistics rescues the lowenergy quasiparticles (as well as the Bosonic modes). 40 At low temperatures, T ≪ E f /k B ≈ 1 K, the number of exci-tations is low, n qp ≈ (k B T /E f )n. Similarly, binary collision processes are confined to a small region of phase space near the Fermi surface, ∆p ≈ (k B T /E f )p f . As a result the Pauli exclusion effect suppresses the quasiparticle collision rate, 4 1 where m ⋆ is the effective mass of a quasiparticle and W is the square of the transition matrix element for binary collisions averaged over the Fermi surface.
Furthermore, since the density of excitations is low, n qp ≪ n, transport coefficients are given by formulae familiar from gas kinetic theory. In particular, the transport of heat is dominated by thermally excited quasiparticles with the thermal conductivity given by 4 wherec v = 2π 2 3 N f k 2 B T is the low-temperature specific heat, N f is the quasiparticle density of states at the Fermi energy, v f is the Fermi velocity and v f τ κ is the transport meanfree-path for heat conduction, with τ κ ∼ τ in ∼ T −2 . Thus, heat transport becomes very efficient in pure 3 He with κ ∼ 1/T . 1,16 In aerogel, elastic collisions of 3 He quasiparticles with the silica strands lead to a temperature-independent contribution to the mean free path and hence the quasiparticle scattering rate. Thus, at sufficiently low temperatures, transport currents are limited by elastic scattering from the aerogel, whereas inelastic scattering of quasiparticles dominates at high temperatures. There is an intermediate regime where both mechanisms are important. The cross-over temperature separating these regimes is estimated from the inelastic collision rate in the pure 3 He in Eq. (1) and the mfp of the aerogel, ℓ, which provides an estimate for the elastic collision rate, Estimating τ κ from Eqs. (1) and (3) gives, The cross-over temperature, T ⋆ , defined by τ in (T ⋆ ) = τ el , is given by 29 where W is the dimensionless quasiparticle transition probability averaged over the Fermi surface (Eq. 171 of the appendix). For 98% porosity aerogel we estimate ℓ ≈ 1700Å, 38 , and thus T ⋆ ≈ 18 mK at p = 15 bar (see Fig. 1). The cross-over from the high-temperature regime dominated by inelastic scattering to the low-temperature regime dominated by elastic scattering off the aerogel is characteristic of all transport processes for 3 He in aerogel. 25,29,31,39 Below we report theoretical results for the heat transport coefficient of liquid 3 He-aerogel, in the normal state, over the broad temperature range, T c ≤ T ≪ E f /k B . Since the bulk properties of 3 He are well known, measurements of the transport coefficients can provide quantitative information on the effects of disorder on the transport of 3 He quasiparticles through aerogel.

II. TRANSPORT THEORY
In normal liquid 3 He at low temperatures the transport of mass, energy and magnetization is carried predominantly by fermionic quasiparticles, whose distribution in phase space, n pσ , is governed by the Boltzmann-Landau transport equation, 5,9 ∂n p ∂t where p = (p, σ) denotes the momentum and spin of the quasiparticles. For spin-independent transport the quasiparticle energy is the sum of the equilibrium excitation energy, ǫ p , the coupling to an external scalar or vector potential, u ext (p, r, t), and the Landau molecular field energy. The latter arises from the interaction of a quasiparticle with the distribution of non-equilibrium quasiparticles, where δn p is the deviation of the distribution function from (global) equilibrium, For small disturbances from equilibrium the derivative of the equilibrium distribution function, confines the excitations to states that lie near the Fermi surface, ǫ p f = µ. The interaction energy between two quasiparticles is given by f p,p ′ , and in contrast to the distribution function, varies slowly with |p| in the vicinity of the Fermi surface. We can typically evaluate f p,p ′ , as well as the density of states, N (ǫ), on the Fermi surface, i.e.
. The latter equality defines the dimensionless Landau parameters.

A. Collision Integrals
The right side of the transport equation, I p , represents the change in the distribution function resulting from collision processes. We consider two scattering processes for 3 He in aerogel: (i) elastic collisions of quasiparticles with "impurities" representing the aerogel strands, and (ii) inelastic collisions between quasiparticles. The development of the transport theory for 3 He-aerogel presented below, particularly the reduction of the transport equation in the low temperature limit, parallels that development by Baym and Pethick in their review on transport in pure liquid 3 He, 8 and extends Brooker and Sykes' work on the transport coefficients of Fermi liquids. 37 In our case the effects of aerogel scattering enter through a contribution to the collision integral. For quasiparticle scattering by the aerogel strands where w(p 1 , p 2 ) is the transition rate for scattering of quasiparticles by the aerogel.
For inelastic quasiparticle-quasiparticle collisions at low temperatures, k B T ≪ E f , only binary collisions are important. We denote t(p 1 , p 2 ; p 3 , p 4 ) as the scattering amplitude for binary collisions between quasiparticles with momenta and spin p i = (p i , σ i ). The labels p 1 and p 2 refer to initial states while p 3 and p 4 refer to final states. Fermi's Golden Rule for the transition rate (p 1 , p 2 ) → (p 3 , p 4 ) is: For a translationally invariant system with spin-rotation invariant interactions between quasiparticles the transition rate includes momentum-and spin-conserving delta functions, where W is a smooth function of p i .
The collision integral for binary scattering includes the phase space factors for collisions that both increase and decrease the population of the state p 1 (scattering "in" and scattering "out"). In particular, The sum over final states (p 3 , p 4 ) is restricted to avoid double counting of equivalent states of identical particles related by exchange of p 3 ↔ p 4 .
The collision integral vanishes when evaluated with a local equilibrium distribution function, i.e. I[n l.e. p ] ≡ 0. For the elastic scattering contribution to the collision integral (Eq. 11) this identity is obvious. For the inelastic collision integral it is less so, but follows from the identity, where n 0 (ε) = 1/(e β(ε−µ) +1) is the Fermi distribution. This identity is a consequence of local equilibrium and the condition of detailed balance between the scattering "in" and scattering "out" contributions to the collision rate.
Although translational invariance is violated by the presence of the aerogel medium, the aerogel is sufficiently dilute that the scattering rate by the aerogel impurities is typically small compared to excitation energies in the normal state, i.e. /τ el ≪ k B T . 41 In this limit the effects of aerogel scattering on the intermediate states that enter the inelastic collision integral can be neglected. Thus, momentum conservation holds for the binary collision integral for normal 3 He in high porosity aerogels. At lower temperatures, e.g. in the superfluid phase,or for lower porosity aerogels, this approximation breaks down. This limit requires a microscopic treatment of the effects of aerogel scattering on inelastic collision processes which is outside the scope of the phenomenological Boltzmann-Landau transport theory.

B. Linearized Transport Equation
The transport coefficients of liquid 3 He in areogel are calculated from solutions of the linearized transport equation in steady-state. The particular solution depends on the nonequilibrium conditions that are established. For small deviations from equilibrium the nonequilibrium steady state is specified by a local equilibrium distribution function, parametrized by a local temperature, T (r), chemical potential, µ(r) and quasiparticle energy, ε p (r). The transport equation naturally separates by expanding about this local equilibrium distribution, The left side of Eq. (18) supplies the driving terms, e.g. ∇µ and ∇T , for the collision terms on the right side that act to restore equilibrium. Using Eq. (16) the linearized transport equation reduces to where δI p is the collision integral to linear order in δn p .

C. Quasiparticle Currents
The mass and heat currents are determined by the solution for δn p of Eq. (19). In particular, the mass current is given by where m * = p f /v f is the quasiparticle effective mass. This form for the mass current is applicable to interacting Fermi liquids which are Galilean invariant. 9 In pure liquid 3 He quasiparticle-quasiparticle interactions which give rise to the enhancement of the Fermionic mass are Galilean invariant. For liquid 3 He-aerogel Galilean invariance is violated by quasiparticle scattering off the aerogel. However, the non-Galilean contribution to the effective mass is of the order of concentration of scattering centers, n s /n ≪ 1, and thus negligible compared to the quasiparticle-quasiparticle effective mass enhancement.
Similarly, the quasiparticle heat current is given by the transport of excitations with energy, ξ p = ε p − µ,

D. Elastic Scattering Limit
Transport properties of normal 3 He in aerogel at sufficiently low temperatures, i.e. T ≪ T ⋆ , are limited by elastic scattering of quasiparticles by the aerogel structure. In this limit the transport equation is given by Eq. (19) with the collision term of Eq. (11). The integral equation for δn p is where The solutions to Eq. (22) are determined by the energy and momentum dependences of the driving term and are familiar from the theory of electron-impurity scattering in metals. 3 To proceed further we need the 3 He quasiparticleaerogel scattering probability, w(p, p ′ ).

E. Elastic Scattering Model
We model the aerogel as a distribution of local scattering centers represented by the potential, U (r) = i u(r − R i ). The terms u(r − R i ) represent the potential provided by the aerogel scattering centers at the fixed positions, {R i |i = 1...N s }. For a random distribution of uncorrelated scattering centers the rate is proportional to the mean number density of scattering centers, n s = N s /V . In the Born approximation the transition rate is related to the matrix elements of u, For stronger scattering the potential u is replaced by the t-matrix for quasiparticle scattering by aerogel strands. 34 We shall assume that the scattering by the aerogel is nonmagnetic. This should be sufficient for describing transport processes in zero field, particularly if the aerogel strands are "coated" with a layer of solid 4 He. However, it is known that 3 He atoms form a highly polarizable solid layer on the surface of the aerogel strands and that these nuclear spins exhibit a Curie-like spin susceptibility. 36 Thus, spin-exchange scattering of 3 He quasiparticles by localized and polarizable 3 He spins may be relevant to magnetic transport processes and transport in relatively low magnetic fields. The simplest scattering model for 3 He-aerogel assumes the 3 He quasiparticles interact with the aerogel via an isotropic scattering potential. 38 There is no preferred direction within the aerogel and the scattering probability depends on the relative orientation of the initial and final quasiparticle momenta. In this case, where w l (ξ p ) is the scattering probability for quasiparticles with relative orbital angular momentum l, and P l (x) is the corresponding Legendre polynomial. Note that |p| = |p ′ | for elastic scattering, and we have parametrized the functional dependence of w(p, p ′ ) on |p| by the energy ξ p = v f (|p|−p f ) measured relative to the Fermi surface. The probabilities for scattering in the orbital channels are proportional to, which vary smoothly with ξ p on the scale of E f .
For an isotropic scattering medium we make the ansatz, The momentum sum is represented as The terms (∂n 0 /∂ξ p ) and δ(ξ where . .
. We introduce the scattering rate for orbital channel l, and express the transport scattering rate in terms of the l = 0 and l = 1 (s-and p-wave) scattering rates, Note that the density of states, N (ξ p ), and the scattering probabilities, w l (ξ p ), vary slowly with excitation energy on the scale of the Fermi energy. Thus, for many cases of interest we can safely neglect the energy dependence of the scattering rate and evaluate τ el (ξ p ) ≃ τ el (0) ≡ τ el .
In particular, the mass current induced by a pressure gradient at constant temperature is determined by the quasiparticle mobility defined by j m = −ν∇µ. For k B T ≪ E f Eq. (10) is sharply peaked at the Fermi energy, and the quasiparticle mobility calculated from Eqs. (20) and (27) is to leading order in T /E f , which has the intuitive interpretation of transport of momentum p f over a distance of order the mfp, ℓ = v f τ el , within the aerogel. Similarly, the heat current, Eq. (21), induced by an temperature gradient, j q = −κ∇T , defines the thermal conductivity, which to leading order in T /E f is given by, and also has the simple interpretation as the flux of thermal energy v f k B T transported over a distance of order ℓ.

F. Two Channel Collision Integral
For higher temperatures, i.e. T T ⋆ , both elastic and inelastic collision processes limit transport currents. The transport coefficients are then calculated from where the right side of Eq. (34) contains the linearized collision integrals for both inelastic and elastic scattering. The elastic collision integral follows immediately from Eq. (11), Since the driving terms in Eq. (34) are confined to excitation energies within k B T of the Fermi energy we express where ψ pi measures the deviation of the equlibrium distribution for quasiparticles with excitation energy ξ pi at the point p i on the Fermi surface.
For |ξ p1 | ≪ E f energy conservation, combined with the phase-space restriction required by the Fermi distribution factors in Eq. (35), forces the scattered excitation energy to be confined to the low-energy shell, i.e. |ξ p2 | k B T . Thus, slowly varying functions of p i can be be evaluated with momenta, in close vicinity of the Fermi momentum. The scattering rate w reduces to a function of the directions of the momenta for incident and scattered excitations with excitation energies near the Fermi energy, w(p 1 , p 2 ) w(p 1 ,p 2 ; ξ p1 ). For the isotropic scattering model and a driving term proportional to v pi · ∇T we set in which case the elastic collision integral becomes, Linearizing the inelastic collision rate in Eq. (14), and making use of Eq. (15) yields, Here we consider an un-polarized Fermi liquid in which the only spin-dependent interactions are those that arise from exchange symmetry. In this case, ξ pi = ξ pi and the distribution functions are independent of σ i . Thus, we can carry out the sum over the spin states σ 2,3,4 . We can also eliminate one of the momentum sums, resulting in, where is the spin-averaged scattering rate; W ↑↑ is the scattering rate for σ 1 = σ 2 =↑ and W ↑↓ is the rate for σ 1 = −σ 2 =↑. The weight factors take into account the restriction to avoid double counting of equivalent states, so the remaining momentum sums over p 3 and p 4 are unrestricted. 9 For |ξ p1 | ≪ E f the energy and momentum conservation laws, combined with the phase-space restrictions required by the Fermi distribution factors in Eq. (40), force all excitation energies to be confined to the low-energy shell, i.e. |ξ pi | k B T . In this limit slowly varying functions of p i can be evaluated on the Fermi surface. Thus, the scattering rate W becomes a function of the directions of the momenta for quasiparticles on the Fermi surface, and the momentum sums can be approximated by To carry out the angular integrations we adopt Abrikosov and Khalatnikov's parametrization 5 of W in terms of the angle θ between the two incoming momenta, and φ, the scattering angle between the planes defined by n =p 1 ×p 2 and n ′ =p 3 ×p 4 . The integration over the directionp 3 is expressed in terms of angles relative to the conserved direction of the total momentum, P = p 1 + p 2 , Since p 4 = P 2 + p 2 3 − 2P · p 3 we have dp 4 = (p 3 /p 4 )P d(cos α), and for momenta near the Fermi surface, P ≃ 2p f cos(θ/2). Thus, Also, the azimuthal angle φ 3 is the scattering angle up to a fixed but arbitrary constant, thus, dφ 3 = dφ. In the case of the integration over the incoming momentum directionp 2 we choose the remaining momentum direction p 1 as the polar axis, The binary collision integral then reduces to The inhomogeneous terms of the linearized transport equation dictate the symmetry of the solution with respect to excitation energy, ξ p1 → −ξ p1 , and momentum direction, p 1 . We separate the angular and energy dependences of the non-equilibrium distribution function with the ansatz, where ϕ (±) (ξ pi ) = ±ϕ (±) (−ξ pi ). We can now carry out the integration over φ 2 for each term in Eq. (47), where the direction cosines, x i =p i · p 1 for i = 2, 3, 4, are simply related to (θ, φ). The angular integrations decouple from the energy integrations which are confined to the lowenergy shell near the Fermi surface. We define the average scattering rate as well as the weighted averages, r i = x i W / W . Changing ξ p3,p4 → −ξ p3,p4 the resulting collision integral reduces to where and λ (±) are given by, It is then convenient to measure the excitation energy in units of T , i.e. set t i ≡ ξ pi /T , for i = 1, 2, 3, 4. The resulting inelastic collision integrals reduce to where 1/τ in is the quasiparticle-quasiparticle collision rate given in Eq. (1). These same transformations applied to the elastic collision integral in Eq. (38) yield, where 1/τ el is the rate for quasiparticles on the Fermi surface scattering elastically off the aerogel. Finally, the left-hand side of the linearized transport equation provides the driving terms that determine the particular solution for the nonequilbrium distribution functions. The driving terms which are even and odd under t 1 → −t 1 are We simplify the even and odd components of the transport equation by an additional transformation of the distribution function, The linearized transport equation with both inelastic and elastic channels included in the collision integral then reduces to the linear integral equations, 35 Physical solutions for ζ (±) are non-vanishing only in the lowenergy region near the Fermi level, i.e. |t| ≪ 1. Thus, we can Fourier transform, and convert the integral equation for ζ (±) (t) into a linear differential equation forζ (±) (z), where We cast this differential equation into standard form defined on the domain x ∈ [−1, 1] with the transformation, and the differential operator, Thus, the nonequilibrium distribution function is obtained as the solution of an inhomogeneous linear differential equation, where G. Thermal Conductivity The heat current for example can be expressed in terms of a particular solution of Eq. (67), Carrying out the transformation from ψp(ξ p ) → Z(x) we obtain the following expression for the thermal conductivity, where The particular solution for Z (−) is obtained as an expansion, in the complete set of orthonormal eigenfunctions, {φ n (x)}, of the homogeneous differential equation, where α n is the eigenvalue associated with the eigenfunction, φ n (x). The coefficients, {A n }, are determined from Eqs.
(67), (68) and (73) and the orthogonality condition, In particular, where is the overlap of the n th eigenfunction with the driving term in the transport equation. Also, we set λ (−) ≡ λ κ above and hereafter. The same term appears in the kernel of the heat current. Thus, the thermal conductivity, in particular, S κ (T ), is determined by the weighted sum over the eigenvalue spectrum,
The evaluation of the overlap integrals leads to the solution for the thermal conductivity (Eq. (2)) obtained by Brooker and Sykes, 11,37 and independently by Jensen et al. 20 , with transport time τ κ = τ in S ∞ κ , where which depends on the angular average of the scattering amplitude via λ κ and is independent of temperature. Thus, the thermal conductivity diverges as κ ∝ 1/T as T → 0 because the number of thermal excitations, the heat capacity and the number of final states for binary collisions are all vanishing as T . Note that λ κ = W (1 + 2 cos θ) / W is a measure of the relative importance of forward vs. backscattering, and is restricted to the domain, −1 < λ κ < 3. The resulting spectral sum, S ∞ κ , is finite since λ κ < 3. However, at any fixed temperature κ increases dramatically for quasiparticle scattering that is predominantly in the forward direction, i.e. for λ κ → 3. Note also that S ∞ κ → 5 6π 2 1 1− 1 3 λκ in this limit.

I. Exact Solution
Here we extend the exact solution for pure 3 He 11,20 to that of 3 He in aerogel described by the two-channel collision integral for binary quasiparticle collisions and quasiparticleaerogel scattering. Bennett and Rice extended the analysis of Refs. 11,20 to collisional scattering of s-and d-electrons combined with electron-impurity scattering. 10 Their results for the electrical and thermal conductivity are expressed as a sum over weighted integrals of products of Gegenbauer polynomials. Our analysis also starts from a two-channel extension of the integral equation of Refs. 11,20, i.e. Eq. (60). The results presented below are a closed form analytic solution to the linearized Boltzmann equation and thermal conductivity, and an exact perturbation theory result for the inelastic corrections to the elastic limit which is used to develop a very accurate approximate solution for the thermal conductivity that is valid for all temperatures (above T c ), pressures and aerogel densities, and is fast and easy to evaluate.
Elastic scattering by the aerogel modifies the form of the eigenfunctions for the nonequilibrium distribution function, and leads to an eigenvalue spectrum that varies strongly with temperature. The key parameter is the structure constant (γ) in Eqs. (62) and (63) in the differential equation for the distribution function. The temperature dependence is conveniently exhibited by scaling Eq. (63) in terms of the cross-over temperature, T ⋆ , defined in Eq. (5), The eigenfunctions of Eq. (73) for any γ > 1 must be bounded on the interval [−1, +1]. The singular points at x = ±1 have indicial equations with one physically allowed solution in the neighborhood of the singular point; in particular, since γ > 1 we select the physical solutions which must behave as Thus, we extract the behavior near the singular points and express where g n (x) is analytic on the domain [−1, +1], and governed by the differential equation, Analytic solutions on the finite domain can be represented as a Taylor expansion about x = 0, The differential equation determines the recurrence formula for the coefficients, Thus, the solutions break up into even and odd parity solutions depending on the coefficients G 0 and G 1 . For even parity solutions, we set G 0 = 0 and generate the solutions from the recurrence relation: Similarly, for the odd-parity solutions we start from G 1 = 0 and find In either case Thus, the series solution diverges at |x| = 1 unless the expansion truncates at a finite value of m. This restricts the physical solutions for g n (x) to a set of polynomials, and an eigenvalue spectrum determined by the condition: Expressing the eigenvalue as 2α n = ǫ n (ǫ n + 1), we obtain ǫ n = γ + n , n = 0, 2, 4, . . . (n = 1, 3, 5, . . .) , for even (odd) parity solutions. The corresponding eigenfunctions are with the summation over even (odd) integers for even (odd) parity eigenfunctions. The coefficients can be expressed in terms of Gamma functions. In particular, for the odd-parity eigenfunctions, which are relevant for computing the thermal conductivity, is fixed by the normalization of φ n (x), and B(x, y) = Γ(x)Γ(y)/Γ(x + y) is the Beta function. 2 We can now evaluate the spectral sum in Eq. (77) to obtain an exact solution for the thermal conductivity. In particular, where Although Eq. (96) provides us with an exact, closed form solution for the thermal conductivity over the full temperature and pressure range, T c ≤ T ≪ T f , the sums defining S κ (T ) involve ratios of Gamma functions. Thus, care must be taken in evaluating these functions even for moderate values of their arguments. This is particularly true in the low-temperature limit, T ≪ T ⋆ , since the scaling parameter, γ becomes large. However, the limit T ≪ T ⋆ can also be evaluated using perturbation theory.

J. Perturbation Theory
At tempertures T ≪ T ⋆ inelastic quasiparticle collisions are relatively infrequent compared to elastic collisions off the aerogel. Thus, the inelastic collision integral in Eq. (34) is of order and we can formally expand the integral equation and the deviation from local equilibrium in the small parameter δ, The perturbation expansion through first order becomes, where L(ξ p1 ,p 1 ) represents the driving term on the left side of Eq. (34). For heat transport the zeroth-order solution of Eq. (100) is simply the distribution in the elastic scattering limit, and now provides the driving term for the first order correction in Eq. (101). This equation has the same integral kernel as that of Eq. (100) and so we can express the first-order correction in terms of an inelastic correction to the scattering time, where τ 1 is the first-order correction to the mean scattering time, τ el . This distribution function gives the first order elastic collision integral, The solution of the Eq. (101) can then be expressed in terms of τ 1 . The analysis of the inelastic collision integral, evalu-ated with the zeroth order nonequilibrium distribution function, δn pi , leads to where I 1 = I(ξ 1 )/T 2 , with I(ξ 1 ) given by Eq. 54, and with K(ξ) defined by Eq. 53. The resulting first-order correction for the collision time reduces to which vanishes in the "ballistic limit" for inelastic collisions, λ κ → 3.
The first order correction to the thermal conductivity is calculated by evaluating Eq. (21) with the first-order correction, δn After the integration over τ 1 (t), and scaling to the elastic limit for the thermal conductivity given in Eq. (33), we obtain,

III. RESULTS
Theoretical models for the quasiparticle collision probability, W (θ, φ), for pure 3 He have been proposed by a number of authors. 13,24,26,33 We use an extended version of the sp model introduced by Dy and Pethick 13 , described as the spd model in Sec. V. In Fig. 2

A. Results for 3 He-aerogel
Theoretical results for heat transport in 3 He-aerogel based on the two-channel solution for the thermal conductivity are shown in Fig. 3. In addition to the mfp describing the aerogel, the input data for bulk 3 He used to generate these results density (n), effective mass (m ⋆ ), Fermi velocity (v f ) and the Fermi liquid parameters (F s,a l ) for l ≤ 2, all of which are taken from the database provided in Ref. 17 and 19. The Fermi-liquid parameters are used to construct the inelastic scattering rate using the spd model as described in the Appendix (Sec. V).
The cross-over from the high-temperature regime dominated by inelastic quasiparticle collisions to the low-temperature regime dominated by elastic scattering by the disordered medium occurs over a fairly broad temperature range for dilute aerogels with long mfp. The elastic regime below T ≃ 5 − 10 mK is well described by κ = κ el + O(T 3 ), with κ el given by Eq. (33). The pressure dependence of the slope of κ el (T ), while not visible in Fig. 3, is shown clearly in Fig.  4. Note that lim T →0 κ/T can provide a determination of the elastic mfp for the aerogel. Figure 4 for κ/T highlights the deviations in the thermal conductivity from the elastic limit limit even at temperatures of order a few milli-Kelvin. Similarly, in the high temperature limit the product, κT , approaches the bulk 3 He limit determined by inelastic scattering. Significant deviations from the pure 3 He limit are shown in Fig. 5 over a wide range of temperatures above T ⋆ . conductivity, i.e. κ = 7.7 mW/mK at T = 2.22 mK. Comparison of these two data points with the theoretical predictions for this pressure are shown in Fig. 6.
If the difference in the mfp with and without the 4 He is attributable to spin-exchange scattering of itinerant 3 He spins by the localized solid 3 He spins, 6,34 then we can estimate the contribution to the scattering rate from indirect spinexchange scattering to be, and thus a mean time for spin-exchange scattering of τ spin ≃ 0.15 µsec, i.e. several orders of magnitude longer than the mean time for elastic scattering off the aerogel strands, τ el ≃ ℓ3 He+ 4 He /v f ≃ 8.6 ns. For scattering off a random distribution of N s localized spins via a Kondo interaction, the Born approximation implies an additional contribution to the scattering rate, Thus,we estimate the indirect exchange interaction to be which is in agreement with the order of magnitude estimate for J ind inferred from the absence of a low-field A 1 − A 2 transition in 3 He-aerogel. 34

FIG. 6: Comparison κ/T vs. ln(T [mK]) with experiments. Data shown as
is from the Lancaster group for a 98% aerogel and a pressure of p ≃ 0 bar. 30 Data shown as (purple ) is from the Stanford group with two monolayers of 4 He coating (without 4 He) the silica strands at high pressure (p = 32.4 bar), also for 98% aerogel but grown in a different laboratory. Theshow limT →0 κ/T . The Lancaster group also reports results for the lowtemperature (i.e. T ≪ T ⋆ ) thermal conductivity of normal 3 He-aerogel at low pressures, for aerogels with porosities of 95% and 98%. 30 Results for ̺ = 98% reported in Ref. 30 yield a much smaller mfp, ℓ = 950Å, than the Stanford data, suggesting significant differences in aerogels of the same density prepared under different growth conditions. Note that the authors of Ref. 30 attribute the deviations from the theoretical curve onsetting near T ≃ 20 mK (ln(T [mK]) ≃ 3 ) to Kaptiza boundary conductance through the experimental cell walls.
Although these results provide estimates for the aerogel mfp they do not provide a test of the the theory. Measurements of the thermal conductivity over the full temperature and pressure range of normal 3 He-aerogel should provide a strong test of the two-channel theory based on homogeneous disorder since we have an exact solution for the thermal conductivity in this model. Conversely, if significant deviations from the theoretical predictions are observed they could indicate new physics associated scattering and transport of fermionic excitations in a correlated random medium.

C. Scaling Function
The exact solution for the thermal conductivity in the twochannel scattering theory for 3 He-aerogel can be expressed in terms of a scaling function. Normalizing κ by the thermal conductivity in the elastic limit, κ el , from Eq. (33) gives, Note that τ el /τ in ≡ (T /T ⋆ ) 2 , and that S κ (T ) calculated from Eq. (77) [Eq. (96) in Sec. II I] provides the exact scaling function, F (x, λ), since S κ (T ) depends only on x = T /T ⋆ and the scattering ratio, −1 < λ κ < 3. Thus, the test of the twochannel transport theory would be to demonstrate that the thermal conductivity of 3 He-aerogel obeys the scaling behavior over the full temperature and pressure range of the normal state, and a broad range of aerogel density and mfp.
The exact solution for κ/κ el × (T /T ⋆ ) ≡ x F (x, λ κ ) is shown in Fig. (7). The calculation of spectral sum, S κ (T ), was carried out using arbitrary-precision floating point arithmetic in order to evaluate the ratios of the Gamma functions or large arguments that enter Eq. (96) with sufficient precision to obtain accurate results for the triple sum that defines S κ (T ).
In particular, the points labeled "exact" in Fig. (7) were obtained with the floating point precision set at 55 digits and each sum was cutoff after 30 terms were computed. One can obtain reasonably good results with a lower precision setting for the floating point arithmetic, but double precision on a 32-bit machine limits the accuracy of the results, particularly in the limit T < T ⋆ .
Also shown in Fig. (7) are calculations of the scaling function based on an approximate analytic formula that is numerically fast and easy to evaluate. The approximate scaling function is constructed from the the asymptotic limits for S κ (T ) for T ≫ T ⋆ and T → 0, as well as the leading order perturbative result for T ≪ T ⋆ , as described below.
The limiting behavior for the exact scaling function is known from the asymptotic limit, x ≫ 1, and perturbation theory about x = 0. In particular, where S ∞ κ is given by Eq. (80).
The most common approximate solution for multi-channel scattering is based Matthiessen's Rule, which in this context can be expressed as i.e. the total transport scattering rate is the sum of independent rates for purely elastic and purely bulk inelastic transport. The resulting expression for the the thermal conductivity, normalized to its value in the elastic limit, defines the approximate scaling function, F MR (x, λ κ ) given by, The MR scaling function deviates from the exact result of Eq. (115) for the leading order finite temperature correction. Curiously, the exact result for the leading order correction is equal to that obtained from F MR (x, λ κ ) by approximating S ∞ κ with just the first term of the sum in Eq. (80). This approximation is very good in the limit of nearly forward scattering. In this limit the inelastic channel leads to large thermal transport for T T ⋆ . As a result the MR scaling function gives a very good approximation to the exact scaling function in the limit of large λ κ for all x. This is shown clearly in Fig. (8). However, the MR scaling function deviates from the exact scaling function when backscattering in the inelastic channel is significant, i.e. for λ κ 1.0. These deviations are also clearly visible in Fig. (8).
We can try to improve on the MR scaling function by incorporating the exact perturbative result for F (x, λ κ ) for x ≪ 1. We construct an interpolation formula that connects the exact asymptotic limits. A simple extension of Matthiessen's interpolation formula is the two-parameter, rational polynomial function, which has the limiting forms, We then fix the coefficients from the exact asymptotic limits for F (x, λ κ ) in Eq. (115), Although this approximate scaling function works well for the x ≪ 1, it does a poor job in the intermediate and hightemperature region x 1 (green curves in Fig. 8), and is particularly poor for λ κ → 3. If we consider the leading order correction to the asymptotic limit x → ∞ we obtain For the polynomial approximate we obtain, while the MR scaling function gives Both approximate scaling functions give the correct sign for the leading order correction, however in the limit λ κ → 3, where we know the MR scaling function approaches the exact result, we see that C MR 4 is large and negative, whereas C poly 4 → −1. This discrepancy in F poly is traced to the contamination of the temperature region x > 1 by the exact solution that is valid for x ≪ 1.
We might remedy this problem with a two-parameter interpolation that limits the contamination between x ≪ 1 and x ≫ 1. In particular, consider the approximate scaling function, For x ≪ 1, Note that there are only exponentially small corrections to the leading order result for x ≪ 1 coming from the terms that are fixed by the asymptotic solution for x ≫ 1. Similarly, for x ≫ 1, the term that is fixed by the exact solution for x ≪ 1 is now exponentially small and we obtain, Using these expansions and the exact leading order asymptotic limits we obtain This two-parameter interpolation formula yields a better approximation to the exact scaling function, particularly for λ κ 1. However, F exp under estimates the maximum in x F (x, λ κ ), and this deviation is enhanced as λ κ → 3, as is clear from Fig. (8). The basic result of this analysis is that the MR scaling function, F MR , is accurate in the limit of large λ κ , but deviates from exact scaling for λ κ 1. By contrast the two-parameter exponential scaling function, F exp , is accurate in limit λ κ < 1, but shows increasing errors from exact scaling in the cross-over region, x ∼ O(1), for 1 < λ κ < 3. This suggests that we combine these two scaling functions into a single scaling function by weighting the respective regions of accurate scaling, i.e.
where the weight function p(λ κ ) is chosen on the physical domain, −1 < λ κ < 3, to satisfy, p(−1) = 1, p(+3) = 0. Thus, the simplest weight functions which map the physical domain onto the interval [0, 1] are The quadratic weight function, i.e. s = 2, leads to remarkably good agreement with the exact scaling function for the entire domain of λ κ and reduced temperature, x = T /T ⋆ . This comparison is shown in Fig. (7). Note that the maximum deviation for any of the computed values is less than 0.6%, and careful examination shows that these small errors occur near the maxima of x F (x, λ κ ). Thus, the main result here is that Eqs. 118, 127, 132 and 133 provide numerically fast and accurate formulas for calculating the thermal conductivity over the full temperature and pressure range within the two-channel scattering theory for normal 3 He-aerogel.

D. Scaling for 3 He-aerogel
The analysis of the pressure dependence of the thermal conductivity of pure 3 He based on the spd scattering amplitude described in Sec. III and App. V implies that the thermal transport scattering parameter is nearly pressure independent, i.e. λ κ ≃ 1.3 for 5 bar p ≤ 34 bar with a smooth drop to λ κ ≃ 1.0 as pressures between 5 and 0 bar (see inset of Fig. 2).
Pressure independence of the scattering parameter, λ κ , implies that the thermal conductivity for all temperatures above the superfluid transition, all pressures and all elastic meanfree paths should collapse to a single scaling function when normalized to its value in the elastic scattering limit, i.e. lim T →0 κ = κ el given in Eq. (33). Thus, for 3 He-aerogel we expect that thermal conductivity for all T , p and ℓ to collapse to the narrow band of scaling functions shown in Fig.  9. A complete set of measurements of the thermal conductivity of 3 He-aerogel for all T , p and a wide range of aerogel mfp would provide a strong test of this theory, particularly the assumption of uncorrelated disorder described by a single mf p.

IV. SUMMARY
Liquid 3 He impregnated into silica aerogel is a model system for investigating the effects of quenched disorder on the properties of a strongly correlated Fermi liquid. In the normal Fermi liquid the transport of heat, mass and spin by fermionic excitations exhibits cross-over behavior from a high temperature regime, where inelastic scattering dom-inates, to a low temperature regime dominated by elastic scattering off the aerogel. The exact solution to the twochannel Boltzmann-Landau transport equation reported here provides quantitative predictions for heat transport in liquid 3 He-aerogel. An approximate solution derived from the asymptotic solutions and perturbation theory is accurate to less than 0.6 %. A key result of this work is the scaling function, F (T /T ⋆ , λ κ ), that describes the exact solution for the normalized thermal conductivity, κ/κ el , for all pressures, temperatures (above T c ) and aerogel density. A complete set of measurements of the thermal conductivity of 3 He-aerogel for all T , p and a wide range of aerogel mfp would provide a strong test of this theory, particularly the predicted scaling behavior based on two-channel scattering and the assumption of homogeneous disorder described by a single mf p. Conversely, systematic deviations from the predicted scaling function behavior should provide a quantitative measure of the role of fractal correlations associated with the structure of the aerogel. The limited data that is available already hints that two-channel scattering is insufficient and that spin-exchange scattering between itinerant 3 He spins and localized 3 He spins contributes to the low-temperature thermal conductivity. the total spin S = 0 and S = 1, which we label as the singlet (s) and triplet (t) amplitudes, Thus, we use a short-hand notation, T ↑↓ ≡ T ↑↓;↑↓ = T ↓↑;↓↑ (139) and express the spin-projection amplitudes in terms of the singlet and triplet amplitudes The T-matrix can then be expressed in terms of T t,s and the corresponding symmetric (triplet) and anti-symmetric (singlet) spin matrix elements, where Since there are only two independent amplitudes it is often useful to use the symmetric and anti-symmetric amplitudes defined as Inverting, we have The two sets of amplitudes, T s,a or T t,s , define different, but equivalent representations for the spin-dependent T matrix.
The T s,a amplitudes are the amplitudes for the T-matrix expressed in terms of the direct "particle-hole" channel, 1 → 3 and 2 → 4, For quasiparticle scattering on the Fermi surface the scattering amplitudes, T t,s , reduce to functions of the directions of the quasiparticle momenta on the Fermi surface, T t,s (p 1 , p 2 ; p 3 , p 4 ) T t,s (p 1 ,p 2 ;p 3 ,p 4 ) .
Furthermore, rotational invariance implies that T t,s can be expressed in terms of the relative direction cosines, The fourth column of equalities follows from momentum conservation for |p i | = p f , The conservation law also implies that there are only two independent angles. We adopt Abrikosov and Khalatnikov's parametrization 4 in terms of the angle θ between the two incoming momenta, and φ, the angle between the planes defined by n =p 1 ×p 2 and n ′ =p 3 ×p 4 , Thus, T t,s (p 1 ,p 2 ;p 3 ,p 4 ) = T t,s (θ, φ).
The Pauli exclusion principle requires the T-matrix to be anti-symmetric under exchange of either the initial or the final state of the two fermions. Thus, the spin-singlet (triplet) amplitude is necessarily symmetric (anti-symmetric) under exchange of the initial or final momenta, or in terms of the scattering angle, Thus, we can formally expand the singlet (triplet) amplitudes as a sum over even (odd) functions of cos(mφ), Note that T t vanishes for φ = π/2 and φ = 3π/2. For these angles the momentum transfer in the direct and exchange channels is identical, in which case exchange symmetry requires the triplet amplitude to vanish identically.
Microscopic analysis of the two-particle propagator and its relation to the quasiparticle scattering amplitude leads to an identity between the scattering amplitude in the forward direction, and the Landau parameters, F s,a ℓ , that define the quasiparticle molecular fields. In terms of the symmetric and anti-symmetric amplitudes in the p-h channel, Landau's identity for the forward scattering amplitude is 23 , where A s,a ℓ = F s,a ℓ 1 + F s,a ℓ /(2ℓ + 1) .

s-p-d Scattering
Several microscopic and phenomenological theories have been proposed for the quasiparticle scattering amplitude in 3 He. 13,24,26,33 Here we adopt a slightly modified version of the model proposed by Dy and Pethick. 13 They proposed a minimal model for the scattering amplitude that obeys exchange anti-symmetry. In particular, if we assume the singlet and triplet scattering amplitudes are to a good approximation given by the m = 0 and m = 1 terms, we have T s ≃ A s (cos θ) and T t ≃ A t (cos θ) cos φ. In this case we can fix the expansion coefficients of A t,s (cos θ) in terms of the forward-scattering amplitudes, A s,a ℓ , and thus the Landau parameters, F s,a ℓ , 13 T t ≃ ℓ≥0 (A s ℓ + A a ℓ ) P ℓ (cos θ) cos φ .
The quasiparticle lifetime, τ in (T ) in Eq. 1, as well as the thermal transport time, τ κ (T ), due to binary quasiparticle collisions in pure 3 He are determined by angular averages of the spin-averaged transition probability, with λ κ ≡ W (1 + 2 cos θ) / W , where W = 1 4 W ↑↑ + 1 2 W ↑↓ and the angular average is defined in Eq. 51. Writing W ab = π 2 −1 N −2 fW ab , the transition probability can be expressed in terms of the dimensionless singlet and triplet scattering amplitudes, and the quasiparticle lifetime becomes, Note that for weighted averages ofW in which the weight function is even under exchange (i.e. φ → φ + π) the cross term in Eq. (171) vanishes. Similarly, for the thermal transport time we can write λ κ = Λ κ / W where Λ κ = (1 + 2 cos θ)W .
In the spd model the Fermi-surface average of the rate becomes, We evaluate this rate in terms of Legendre expansion of the forward-scattering amplitudes. For either singlet or triplet channel, The angular average of A 2 is given by where Similarly, for the angular averages of the form, with These coefficients are listed in Table I for ℓ, ℓ ′ ≤ 2.
The input for our calculations of the transport properties of bulk 3 He as well as 3 He-aerogel are the Fermi-liquid parameters. The measured values of these parameters are collected in Ref. 19, and are also available online. 17 The Landau interaction parameters, F s 0 , F s 1 , and F a 0 are accurately known from measurements of the heat capacity, first-sound velocity and magnetic susceptibility of pure normal 3 He, while determinations of F s 2 and F a 1 have also been obtained from  I: Coefficients defining the angular averages of W (C ℓℓ ′ ) and W (1 + 2 cos θ) (L ℓℓ ′ ) in the spd scattering model. measurements of the zero sound velocity and spin-wave resonance for normal 3 He, respectively. However, these parameters are not as accurately determined. Less is known about the magnitude and pressure dependence of the ℓ = 2 contribution to the exchange interaction, F a 2 , 14 and much less is known quantitatively about the Landau interaction parameters corresponding to harmonics ℓ > 2, although evidence of interactions in higher order scattering channels is suggested by the observation of high frequency pair exciton 32 modes in superfluid 3 He-B. 12 The scattering model we use throughout is defined by Eqs. (166) and (167) with the added assumption that we truncate the expansion, i.e. set A s,a ℓ = 0 for ℓ ≥ 3. This approximation is reasonable if the contributions to the Fermi-surface averages of the scattering rate fall off sufficiently rapidly with increasing ℓ > 2.
The spd model with the Fermi liquid data for ℓ ≤ 2 as input qualitatively describes the decrease in the transport time, τ κ T 2 , with increasing pressure (shown in Fig. 2) and is within 25% of the experimental values for τ κ T 2 over the full pressure range. However, the comparison clearly shows that the spd model, or the accuracy of the known Fermi liquid data is inadequate, or both. The most problematic aspect of the spd model as it stands is that the FSSR is badly violated, when evaluated with A s,a ℓ = 0 for ℓ ≥ 3. In particular the largest violation in the FSSR, is at low pressures, which is also where the discrepancy (refer to Fig. 2) between theory (solid line) and experiment (red diamonds) is greatest. This is a significant violation of the Pauli exclusion principle, and is an indication that either the determinations of F a 1,2 are inaccurate, that there is significant weight in the interaction channels for ℓ ≥ 3, or both.
Respecting the Pauli exclusion principle, by enforcing the FSSR, is likely more important than knowing precisely the distribution of higher angular momentum channels that account for the missing weight in Eq. 180. Thus, we enforce the FSSR by fixing the least known material parameter in the spd model, i.e. we replace The importance of enforcing the FSSR appears to be born out by the improvement between theory (black dots) and experiment shown in Fig. 2. For pressures below p ≤ 25 bar the agreement is nearly perfect. Thus, the deviations between theory and experiment at higher pressures likely reflects real limitations of the spd model, i.e. there is scattering that reduces heat transport that is outside the spd scattering model.