set of all future-directed timelike Minkowski vectors. It will be assumed that the map is smooth if $D$ is open. Then we say that [*$\omega$ is a local thermal equilibrium state of order $N$ with mixed temperature distribution $\varrho$*]{} (for short, $\omega$ is a $[D,\varrho,N]$-LTE state) if $$\begin{aligned}
     \label{lte-m}
     \langle \eth_{\mu_1 \cdots \mu_n} :\Phi_0^2:(x) \rangle_\omega & =
      \langle \eth_{\mu_1 \cdots \mu_n} :\Phi_0^2:(0) \rangle_{\varrho_x}\end{aligned}$$ holds for all $x \in D$ and $0 \le n \le N$, where $$\begin{aligned}
     \label{average}
    \langle \eth_{\mu_1 \cdots \mu_n} :\Phi_0^2:(0) \rangle_{\varrho_x} & =
    \int_{V_+} \langle \eth_{\mu_1 \cdots \mu_n} :\Phi_0^2:(0) \rangle_{\boldsymbol{\beta}'} \,d{\varrho_x}
    (\boldsymbol{\beta}')\end{aligned}$$

In this definition, $\Phi_0$ can more generally also be taken as the linear scalar field with a finite mass different from zero. Let us discuss a couple of features of this definition and some results related to it.\
[**(A)**]{} The set of local thermal observables used for testing thermal properties of $\omega$ in (\[lte-s\]) and (\[lte-m\]) is $\boldsymbol{\Theta}_x^{(N)}$, formed by the Wick-square and all of its balanced derivatives up to order $N$. One can generalize the condition to unlimited order of balanced derivatives, using as thermal observables the set $\boldsymbol{\Theta}_x^{(\infty)} = \bigcup_{N = 1}^{\infty}
\boldsymbol{\Theta}_x^{(N)}$.\
[**(B)**]{} The condition (\[lte-s\]) demands that at each $x$ in 