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I have always taken the definition of temperature to be the kinetic energy Statistical mechanics definition of temperature as the average kinetic energy.

However I have been reading a paper where the dynamics is an atom chain in contact with heat reservoirs at the boundary. In this case the system has a non-equilibrium stationary state. In this paper the temperature (at atom $i$) is then defined as the variance of the momentum of atom $i$ under the non-equilibrium stationary state.

Why now has the definition of temperature changed from the mean squared of the momentum to the variance?

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    $\begingroup$ it would be useful to reference the paper $\endgroup$
    – jim
    Commented Feb 6 at 11:13
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    $\begingroup$ The point of the question you've linked to is that temperature is not always equal to the average kinetic energy. $\endgroup$
    – Michael Seifert
    Commented Feb 6 at 12:31

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First off, for gas etc. the mean momentum is zero so the mean of the square is the same as the variance. Secondly, the notion of temperature in equilibrium is a more general notion. It corresponds to the variance of momentum because of equipartition and the fact that kinetic energy is quadratic.

For out of equilibrium systems there is no general definition of temperature. Therefore by analogy, such a proxy is reasonable. But without any more information about the article, it’s hard to motivate their choice.

Hope this helps.

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