*2.4. Linear-Response Regime*

To compare with the much more studied linear-transport regime, we here present the relevant transport properties in this limit. Specifically, with small applied voltage and thermal bias, we can write the heat and charge current in the convenient matrix form [21],

$$
\begin{pmatrix} I \\ J \end{pmatrix} = \begin{pmatrix} G & L \\ M & K \end{pmatrix} \begin{pmatrix} V \\ \Delta T \end{pmatrix} \tag{16}
$$

where (only due to linear response!) *J* = *J*L = −*J*R, and the matrix elements are defined as

$$\mathbf{G} = \frac{e^2}{h} \mathcal{Z}\_0, \quad \mathbf{L} = -\frac{\mathcal{M}}{T\_0} = \frac{e}{h} \mathbf{k}\_\mathbf{B} \mathcal{Z}\_1, \quad \mathbf{K} = -\frac{1}{h} (k\_\mathbf{B}^2 T\_0) \mathcal{Z}\_2. \tag{17}$$

with

$$\mathcal{Z}\_{\rm n} = \int\_{-\infty}^{\infty} dE \, D(E) \left( \frac{E}{k\_{\rm B} T\_0} \right)^{\rm n} \left( -\frac{\partial f\_0(E)}{\partial E} \right). \tag{18}$$

Here *f*0(*E*) is the Fermi-Dirac distribution in Equation (4) with *Vα* = 0 and *Tα* = *T*0. In the same limit, the charge-current noise reduces to the equilibrium noise, given by *SI* = 2*k*B*T*0*G*, in accordance with the fluctuation-dissipation relation.

Another performance quantifier, which is often used in the linear response, is the figure of merit *ZT*. It is given by [5]

$$ZT = \frac{L^2}{GK - L^2 T\_0} T\_{0\prime} \tag{19}$$

in terms of the response coefficients given above.
