$D$, the expectation values of thermal observables in $\boldsymbol{\Theta}_x^{(N)}$ evaluated in $\omega$ and in $\langle\,.\,\rangle_{\boldsymbol{\beta}(x)}$ coincide; in other words, with respect to these thermal observables at $x$, $\omega$ looks just like the thermal equilibrium state $\langle\,.\,\rangle_{\boldsymbol{\beta}(x)}$. Note that $\boldsymbol{\beta}(x)$ can vary with $x$, so an LTE state can have a different temperature and a different “equilibrium rest frame”, given by the direction of $\boldsymbol{\beta}(x)$, at each $x$.\
[**(C)**]{} With increasing order $N$, the sets $\boldsymbol{\Theta}^{(N)}_x$ become larger; so the higher the order $N$ for which the LTE condition is fulfilled, the more can the state $\omega$ be regarded as coinciding with a thermal state at $x$. In this way, the maximum order $N$ for which (\[lte-s\]) is fulfilled provides a measure of the deviation from local thermal equilibrium (and similarly, for the condition (\[lte-m\])).\
[**(D)**]{} Condition (\[lte-s\]) demands that, on $\boldsymbol{\Theta}_x^{(N)}$, $\omega$ coincides with a thermal equilibrium state at sharp temperature, and sharp thermal rest-frame. Condition (\[lte-m\]) is less restrictive, demanding only that $\omega$ coincides with a mixture of thermal equilibrium states, described by the probability measure $\varrho_x$, on the thermal observables $\boldsymbol{\Theta}_x$. Of course, (\[lte-s\]) is a special case of (\[lte-m\]).\
[**(E)**]{} Clearly, each global thermal equilibrium state, or 