Crossover behavior of the thermal conductance and Kramers’ transition rate theory

Kramers’ theory frames chemical reaction rates in solution as reactants overcoming a barrier in the presence of friction and noise. For weak coupling to the solution, the reaction rate is limited by the rate at which the solution can restore equilibrium after a subset of reactants have surmounted the barrier to become products. For strong coupling, there are always sufficiently energetic reactants. However, the solution returns many of the intermediate states back to the reactants before the product fully forms. Here, we demonstrate that the thermal conductance displays an analogous physical response to the friction and noise that drive the heat current through a material or structure. A crossover behavior emerges where the thermal reservoirs dominate the conductance at the extremes and only in the intermediate region are the intrinsic properties of the lattice manifest. Not only does this shed new light on Kramers’ classic turnover problem, this result is significant for the design of devices for thermal management and other applications, as well as the proper simulation of transport at the nanoscale.

Using these dimensionless parameters, the exact expression for the heat current through the lattice can be written as where C is a dimensionless function of five dimensionless arguments. This very general expression can be simplified by (i) taking N → ∞ (see inset in Fig. 2 of main article) and by (ii) exploiting the fact that the heat current in such a model is proportional to ∆T [see Eq. (17) and the related discussion], then the function C cannot depend on T L and T R . These considerations result in where C is now a dimensionless function of just three dimensionless arguments. The expression for heat conductance of the 1D uniform harmonic model considered in this work can then be written in the limit of N → ∞ as The Langevin equations of motion for a lattice of harmonic oscillators, given by the Hamiltonian are a system of first-order linear differential equationṡ x n = p n /m n , K nm x m + γ n p n /m n = η n (t), where the mass can potentially be site-dependent in order to, e.g., examine mass disorder as in Disordered Harmonic Lattice discussed in main article. The matrix K contains the interaction potentials and [M] nm = δ nm m n is a diagonal matrix of the masses. These equations of motion can be expressed as where |q ⇒ The non-symmetric matrix G is given by where Γ is a diagonal matrix of the couplings to the Langevin reservoirs (and tr|Γ| = 2Γ, where Γ is the cumulative friction coefficient).
The formal solution of Eq. (7) is An important class of observables can be expressed as an average of a quadratic form of |q(t) over the statistical ensemble. For example, the heat current flowing from site n to site n + 1 can be written as a statistical average of An observable of this class can be represented by an operator O so that Averaging over the statistical ensemble and using Eq. 15 from Methods section of main article for the auto-correlation function of the random forces, one obtains where F is a diagonal matrix with its only non-zero elements given by F N +n,N +n = 2γ n k B T n , n = 1, 2, ..., N.
The spectral decomposition of the particular class of non-symmetric operators G above necessarily includes left and right eigenvectors, i.e., G = k λ k |k R k L |, where the components of right (column) and left (row) eigenvectors are denoted by [|k R Substituting these spectral decompositions into Eq. (13), setting t 0 → −∞ to find the steady-state value of the observable, and integrating, we obtain This equation allows one to draw very general conclusions regarding the dependence of the observable O on temperature without the need to explicitly specify G and O. Since operators O (e.g., heat current) and G do not typically depend on temperature, and the operator F in Eq. (14) is strictly linear with respect to temperature, O is then a linear form of T 1 , T 2 , ..., T N . Specifically, since in the harmonic lattice case we examine there is only two independent temperatures, T L and T R , O is necessary a linear form of T L and T R , i.e., where α L and α R are coefficients that depend only on specific parameters of the system and nature of operator O, but not temperature. For example, since the heat current has to vanish at thermodynamic equilibrium, we must have α L = −α R and, therefore, That is, the heat conductance does not depend on temperatures T L or T R at all. These considerations do not lead to the same conclusion in the case of anharmonic lattices, as illustrated in Fig. 5(b).

III. HEAT CURRENT IN 1D HARMONIC LATTICES
To use Eq. (15) to evaluate the heat current in the lattice of harmonic oscillators, one needs to construct operators G and O. Below we explicitly construct these matrices and derive an expression for the heat current.
The system under consideration is a lattice of harmonic oscillators with only nearest neighbor coupling, which yields a matrix K in Eq. (5) that is sparse. For an oscillator with index n somewhere inside the lattice (i.e., 1 < n < N ), the corresponding n th row of the matrix has only three non-zero components (K n,n−1 , K n,n , K n,n+1 ) = (−K, D+2K, −K), where K and D are parameters in the Hamiltonian. The end sites, n = 1 and n = N , are assumed to be coupled to "hard walls", i.e., to implicit oscillators with indices n = 0 and n = N + 1, respectively, whose coordinates are kept zero. This results in only two non-zero matrix elements in the very first and the very last rows of matrix K: According to Eq. (11), the operator J n corresponding to the heat current from site n to site n + 1 has only a single non-zero element Substituting this expression, together with Eq. (14) and the eigenvectors of G and G † , into Eq. (15) gives

IV. HEAT CURRENT IN THE SMALL γ REGIME
The most convenient way to analyze the small γ dynamics of the 1D lattice is to perform a perturbation expansion of eigenvalues and eigenvectors of operator G with respect to γ. G is at most linear with respect to γ and can be represented as G = G 0 + γG 1 , where G 0,1 are independent of γ. The zeroth-order eigenvalues and eigenvectors of G are then the eigenvalues and eigenvectors of G 0 , The solutions -normal modes of lattice vibrations with no friction -can be found in the standard way, first by scaling ("mass-weighting") by S = diag(M 1/2 , M −1/2 ), and then diagonalizing the mass-weighted coupling M −1/2 KM −1/2 via an orthogonal transformation T. The normal modes of the lattice are where u k n = M −1/2 T nk are the real-valued polarization vectors of the normal modes and n, k = 1, . . . , N enumerate the lattice sites and modes, respectively.
The eigenvalues and right eigenvectors (hence denoted with R below) of G 0 can be readily constructed from these and they come in pairs. The unnormalized "positive frequency" solutions are for k = 1, . . . , N and n = 1, . . . , N . The unnormalized "negative frequency" solutions are for k = 1, . . . , N and n = 1, . . . , N . The unnormalized "negative frequency" solutions are v 0,L N +n,k ∝ u k n , for k = N + 1, . . . , 2N , n = 1, . . . , N , and the same "extension" for k > N . The left and right eigenvectors clearly obey the right orthogonality relationship, but for each k have norm ∓2iω k for "positive" and "negative" frequencies, respectively.
The first-order perturbative correction to the eigenvalues of G 0 can be found in the usual manner, i.e., Using this expression, we can now expand the current. To do so, we will make use of Eq. (15). The matrix element k L |F|l L is linear with respect to γ at small γ, i.e., putting in the normalized zeroth order states k 0 L |F|l 0 L = n 2γ n k B T n u k n u l n / √ 4ω k ω l . Thus, we can focus on the quantity and show that it is zeroth order with respect to γ. We separate the summation in Eq. (15) into the off-diagonal (k = l) and diagonal (k = l) contributions. Assuming no degeneracy, the former is nonzero and well-behaved when zeroth-order eigenvalues and eigenvectors are used since the denominator in Eq. (15) does not vanish at k = l. The off-diagonal contribution is thus proportional to γ at small γ.
The diagonal contribution has a vanishing denominator as γ approaches zero, i.e., λ 0 k is pure imaginary which gives λ k + λ * k = 0 + n γ n (u k n ) 2 + O(γ 2 ). Performing the whole sum, however, we can pair the "positive" and "negative" frequencies: where the last factor is the same for both "positive" and "negative" frequencies. The term in parenthesis gives Thus, the sum over "positive" and "negative" frequencies gives a contribution that is of order γ. Therefore, the current for small γ is J ∝ γ.

V. HEAT CURRENT IN THE LARGE γ REGIME
The current in the large γ regime can also be calculated perturbatively in powers of 1/γ. We consider only single sites at each end connected to Langevin reservoirs, as any additional sites in the extended reservoir are decoupled from the lattice by higher orders in 1/γ. Now consider the matrix γG 1 + G 0 , where G 0 is the perturbation. The expansion is complicated by the fact that the "bare" matrix is highly degenerate, with 2N − 2 zero eigenvalues and 2 non-zero eigenvalues. The latter two are γ/m 1 and γ/m N , which may have an (unimportant) degeneracy, and we can take two of the zeroth order right (left) eigenvectors to be |N + 1 and |2N ( N + 1| and 2N |).
The degenerate space is spanned by the states |n with n = 1, . . . , N and |N + m with m = 2, . . . , (N − 1). The degeneracy is lifted in the normal way: Let P 0 project onto this subspace and diagonalize P 0 G 0 P 0 . This matrix has 2N − 4 eigenvectors with non-zero eigenvalues. The right eigenvectors with non-zero eigenvalues have the same form as Eqs. (23) and (24) except found from the lattice sites 2, . . . , (N − 1) only. The other two eigenvectors have a zero eigenvalue. Their degeneracy is not broken until the next order (1/γ), and requires diagonalizing the eigenvectors outside this subspace will be order 1/γ. As with the small γ regime, we need to group the different contributions into diagonal and off-diagonal (and then further distinguish between off-diagonal contributions between these three groups). We will break down the contributions to Eq. (15) in terms of each of the factors l R |J|k R , k L |F|l L , and λ k + λ * l .
For eigenvectors in B 0 , we get: O(1) (this is the same calculation as Eq. (30) as the states are eigenstates of G 0 on the internal lattice); O(1/γ) because although F is order γ, the contribution of |N + 1 and |2N to the eigenvector are order 1/γ; O(1/γ) because the eigenvalue has a real part that is order 1/γ (the imaginary part is order 1 but this cancels in λ k + λ * l ). Thus, the total contribution from a eigenvector in B 0 is order 1. However, just like in Eq. (30) for the small γ regime, this contribution always has a paired contribution from the negative (positive) frequency mode, which cancels this order 1 contribution. This gives an overall contribution at most of order 1/γ. For the off-diagonal terms, we give the contributions to each of the three factors for all the different possibilities in Table I. The highest order contributions from the off diagonal are order 1/γ. Thus, the sum over all contributions gives zero for the zeroth order term and the leading term is order 1/γ. We note that for both the small γ and large γ regimes, these well-defined perturbation expansions demonstrate that the the zeroth order terms are zero, a fact which is obvious for the small γ regime (i.e., there would be no driving force for the thermal current) but not the large γ regime. Moreover, due to the presence of the different temperature reservoirs at each end, the first order expressions are non-zero always unless the lattice has a broken link.
Coupled with the results of the anharmonic lattice, the perturbative expressions give evidence that the presence of these regimes are universal except in pathological cases.