4.3.2. DynMaxEnt to Fourier Heat Conduction

The simplest possible dependence of entropy on the extra field **w** is quadratic (keeping in mind that *S* has to be a concave and even with respect to time reversal functional),

$$S(\rho, \mathbf{m}, \varepsilon, \mathbf{w}) = \int d\mathbf{r} \, s \left(\rho\_\prime \varepsilon - \frac{\mathbf{m}^2}{2\rho} - \frac{1}{2}a\mathbf{w}^2\right) \,. \tag{62}$$

Consequently, entropy is highest (for given fields *ρ*, **u** and *e*) at **w** = 0, which is the MaxEnt estimate **w**˜ . Plugging this value into Equations (61) leads to

$$\frac{\partial \rho}{\partial t} = -\partial\_k (\rho m\_k^\dagger)\_\prime \tag{63a}$$

$$\frac{\partial m\_{\rm i}}{\partial t} = -\partial\_{\dot{j}}(m\_{\rm i}m\_{\dot{j}}^{\dagger}) - \rho \partial\_{\dot{i}}\rho^{\dagger} - m\_{\dot{j}}\partial\_{\dot{i}}m\_{\dot{j}}^{\dagger} - s\partial\_{\dot{i}}s^{\dagger},\tag{63b}$$

$$\frac{\partial \mathbf{s}}{\partial t} = -\partial\_k \left( s m\_k^\dagger + w\_k^\dagger \right) + \frac{1}{\tau (s^\dagger)^2} (\mathbf{w}^\dagger)^2,\tag{63c}$$

$$0 = -\partial\_k s^\dagger - \frac{1}{\tau s^\dagger} w\_k^\dagger \,. \tag{63d}$$

Equation (63d) has the solution

$$\mathbf{\ddot{w}}^{\dagger} = -\tau \mathbf{s}^{\dagger} \nabla \mathbf{s}^{\dagger}.\tag{64}$$

Plugging this value into the rest of Equations (63), we obtain

$$\frac{\partial \rho}{\partial t} = -\partial\_i (\rho E\_{m\_i}),\tag{65a}$$

$$\frac{\partial m\_i}{\partial t} = -\partial\_j (m\_i E\_{m\_j}) - \rho \partial\_i E\_\rho - m\_j \partial\_i E\_{m\_j} - s \partial\_i E\_{s\prime} \tag{65b}$$

$$\frac{\partial \mathbf{s}}{\partial t} = -\partial\_k \left( s E\_{m\_k} - \tau E\_\delta \partial\_k E\_\delta \right) + \tau \left( \nabla E\_\delta \right)^2,\tag{65c}$$

which are Euler equations with Fourier heat conduction. Indeed, denoting local temperature *Es* as *T*, the entropy flux is

$$\mathbf{J}^{(s)} = -\boldsymbol{\tau} \boldsymbol{T} \boldsymbol{\nabla} \boldsymbol{T} = \frac{-\lambda \boldsymbol{\nabla} \boldsymbol{T}}{T} \, \tag{66}$$

where *<sup>λ</sup>* <sup>=</sup> *<sup>T</sup>*2*<sup>τ</sup>* is the heat conductivity and **<sup>q</sup>** <sup>=</sup> <sup>−</sup>*λ*∇*<sup>T</sup>* is the heat flux.

One can also seek higher order corrections by means of the infinite DynMaxEnt chain. The corrected value of **w** implied by Equation (63d) is

$$\mathbf{\dot{w}}^{(2)} = -\frac{\tau}{\alpha} T \nabla T,\tag{67}$$

which leads to energy

$$E = \int d\mathbf{r} \left(\frac{\mathbf{m}^2}{2\rho} + \varepsilon(\rho, s) + \frac{1}{2}\tau\mathfrak{a}\varepsilon\_s^2(\nabla\varepsilon\_s)^2\right). \tag{68}$$

This is a weakly non-local energy implied by the second-order DynMaxEnt correction.

In summary, Fourier's law, which tells us that heat flows from a hotter body to a colder body, can be derived by the dynamic MaxEnt reduction from the coupled dynamics of phonons and fluid mechanics. The only irreversibility on the higher level of description is present in the evolution equation for **w**. After the reduction, this irreversibility leads to irreversible terms in the equation for entropy (irreversible entropy flux and entropy production). Note that this reduction is again compatible with the asymptotic expansion carried out in [33].
