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I see that temperature is defined always in equilibrium. But systems which are not in equilibrium with their environment. How is temperature defined in these cases?

Humans for example, have a body temperature, though they are not in equilibrium with their environment. How is temperature defined in this case?

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We can only talk about temperature in a nonequilibrium system when such a system is locally in thermal equilibrium.

A nonequilibrium system does not have one specific temperature, as it is not in equilibrium (as you point out). We can however, define a temperature at every point, provided that locally the system will be in equilibrium. We can in that case, put a thermometer at that point and as soon as the thermometer comes in equilibrium with our system at that point, we can measure it's temperature there.

As Alireza points out, in the example of a human (not being in thermal equilibrium with its environment) we can still talk about the local temperature of the human body (which inside the body will be higher than at the surface of the skin). Using a thermometer we can locally measure the body temperature at the point of contact. Typically we are interested in the 'core temperature' of the body, this is why we try to put the thermometer as 'deep' inside the body as possible (e.g. in an armpit or ear).

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  • $\begingroup$ There is a large class of non-equilibrium systems for which the local equilibrium hypothesis breaks down. For such systems defining a local temperature does not make sense. Yet there are various proposed definitions of temperature in systems far from equilibrium. $\endgroup$ – Yiteng Apr 25 '17 at 11:59
  • $\begingroup$ @JgL "We can however, define a temperature at every point, provided that locally the system will be in equilibrium" there's some sort of "metric function"? $\endgroup$ – M.N.Raia Jun 30 '18 at 15:16
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You need to acquire a different perspective here. What we measure as e.g. human body temperature is the result of human body being in thermal equilibrium with the thermometer used. So again it is an equilibrium situation.

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  • $\begingroup$ I think the question is more about the conceptual issue of 'defining temperature in a nonequilibrium system' than being concerned with the example. (I edited the question to make a clear distinction between what I think was the original question and the example he gave). $\endgroup$ – JgL Mar 16 '17 at 18:58
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What the other answers are missing is something that needs to be made explicit -- when the system is not in equilibrium, there is not a single temperature. In other words, for a non-equilibrium system, there can be multiple temperatures.

For instance -- if a system has the translational, rotational, vibrational and electronic modes all excited but not in equilibrium, there can be 4 temperatures. One for translation, one for rotation, one for vibration, and one for electronic. As the system reaches an equilibrium (through colliding molecules), the 4 temperatures approach one another and eventually all reach the same, equilibrium value. Assuming the system reaches equilibrium.

In the example of the human body, this is also true. The body has one temperature, the thermometer has another. Two temperatures are correct for the system. Over time, the body cools (very slightly -- not enough to be noticed really) and the thermometer heats up. Eventually, the system reaches an equilibrium and you can read the temperature.

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In many non-equilibrium systems, the local hypothesis still applies and we can indeed define temperature locally as JgL pointed out. Much of this work was done by Prigogine and others decades ago. However, there is a large class of systems currently under study, for which this hypothesis does not apply.

Defining temperature in such systems is trickier, but there are various proposals for extending results from equilibrium thermodynamics. Most of these are based on the well-known measures of temperature that work in equilibrium or well-understood non-equilibrium situations (some of which perhaps more fundamental than others), such as:

  1. In equilibrium thermodynamics, $$ T^{-1} = \left( \frac{\partial S}{\partial U} \right). $$
  2. Equipartition theorem (kinetic theory): $$ \langle \frac{1}{2} m v^2 \rangle = \frac{3}{2} N k_B T. $$
  3. Relation to fluctuations, e.g. Einstein relation for a Brownian particle $$ \frac{D}{\mu} = k_B T. $$

It turns out that some of these properties no longer apply out of equilibrium and need to be modified. Definition such as entropy need to modified and become local quantities as far as such definitions make sense. Hence the first definition might still hold locally in certain situations.

More importantly, the equipartition principle doesn't hold anymore in general. It is broken for harmonic oscillators coupled to some external force, or for an ideal gas under a heat flux. As a result, one could define separate temperatures for each degree of freedom like tpg2114 pointed out. As a consequence, any thermometer measuring only one or a few degrees of freedom cannot establish an absolute temperature of the system even locally. Two systems which have the same reading might not be in equilibrium with each other when brought together. Hence the zeroth law of thermodynamics in its original form does not apply out of equilibrium.

Review paper: J. Casas-Vázquez and D. Jou, Temperature in non-equilibrium states: a review of open problems and current proposals

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Temperatures can also be defined for systems in a steady state, and probably a living human body is in a steady state. It maintains a constant temperature higher than the environment by producing more heat than what it radiates.

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