What is the true meaning of temperature? Is it global or local?

[Hiero]

Temperature is often imagined as a scalar field varying over (physical) space. Yet temperature seems to be generally defined as the (reciprocal) rate of change (with fixed V,N) of the entropy of a system w.r.t. the energy of that system; a seemingly global property. Is this definition only for equilibrium? Is temperature meaningful in non-equilibrium dynamics?

Also, is it true that, unlike say energy, temperature is not defined for any micro-state? Temperature is instead a property of an ensemble (average) of systems not a single system. Further, temperature also comes from the entropy which in itself depends on our knowledge and which control parameters we consider (if I understand right) so then temperature also depends on that?

Im just trying to understand the modern statistical perspective of temperature and entropy (especially the limits of them in non equilibrium) so anything anyone thinks is relevant or insightful is welcomed!

Edit:
I have to go but I’m also wondering about, if it is a global property, and we divide the system into subsystems, how small is temperature meaningful? I would ask if it’s meaningful down to the Planck box except temperature exists over real space not phase space so isn’t the Planck scale meaningless?

 

[Chestermiller]

Do you think the room that you are in is at thermodynamic equilibrium? If you have a thermometer and its reading varies from one point in the room to another, what do you think the thermometer is measuring at each point?

 

[Hiero]

Do you think the room that you are in is at thermodynamic equilibrium?

Well that depends what room we’re talking about. I suppose a room truly in equilibrium must have uniform temperature.

Lets say we have a room with a heater on one side and an open window on the other side (or alternatively a box of gas with a hot wall and an opposite cold wall) so that just as much energy flows in (from the heater) as flows out (the window). Then we don’t have equilibrium because there is a flow of energy, yet we can imagine a steady state where temperature is constant in time but not uniform over space. (What would you call this? Steady states seem like a weaker form of equilibrium.)

If you have a thermometer and its reading varies from one point in the room to another, what do you think the thermometer is measuring at each point?

Assuming the temperature is not changing at some point and we leave the thermometer there long enough to equilibrate then it would be measuring temperature at that point.

That’s the intuitive picture of temperature; a scalar field over space with possibly different values at each point. My main two confusions are:
1) the mathematical definition seems to imply that it as a global property of a system (so then the only way to make it local would be to take smaller and smaller subsystems around a point).
2) is the temperature even meaningful in the case of non-equilibrium? (We can’t wait for the thermometer to equilibrate if the temperature is changing in time.)

 

[Chestermiller]

Suppose that it is a very rapid-response thermocouple. Are you saying that the instantaneous value that it is measuring is meaningless and unpredictable?

 

[Hiero]

Suppose that it is a very rapid-response thermocouple. Are you saying that the instantaneous value that it is measuring is meaningless and unpredictable?

Well there must be some equilibration time, even if its rapid. If the time scales over which it equilibrates are longer than the time scales over which the temperature truly varies, then yes it seems meaningless to measure it.

That’s not to say the temperature itself is undefined, but just that we couldn’t measure it experimentally in that case.

Anyhow I’m more curious about the theoretical meaning of temperature. For instance is $T=\frac{\partial E}{\partial S}$ the general definition, or only at equilibrium? (If it is general,) E and S are the energy and entropy of the system, hence temperature seems to be a property of the system as a whole, which aligns with the use of T as a (macro-)state variable.

 

[jbriggs444]

Well there must be some equilibration time, even if its rapid. If the time scales over which it equilibrates are longer than the time scales over which the temperature truly varies, then yes it seems meaningless to measure it.

Nicely stated.

That’s not to say the temperature itself is undefined, but just that we couldn’t measure it experimentally in that case.

I would say that this makes temperature itself undefined in such a situation. Or at least fuzzy.

 

[phinds]

I would say that this makes temperature itself undefined in such a situation. Or at least fuzzy.

I think "imprecise" would be more precise.

 

[Omega0]

Well there must be some equilibration time, even if its rapid. If the time scales over which it equilibrates are longer than the time scales over which the temperature truly varies, then yes it seems meaningless to measure it.

Thats it! Thermodynamics is something which has perhaps 3, 4, 5 or say 100 degrees of freedom. The simple reason is that is no fun for anyone to deal with 100000000000000000000000 and the good news is, that we seem to can break it down to some basic things we can measure. This is how thermodynamics began to exist. It was always defined as something to break down the degrees of freedom to something to make it more understandable.

$T=\frac{\partial E}{\partial S}$ the general definition, or only at equilibrium? (If it is general,) E and S are the energy and entropy of the system, hence temperature seems to be a property of the system as a whole, which aligns with the use of T as a (macro-)state variable.

If you are using terms of thermodynamics you should - sorry - better understand thermodynamics. Thermodynamis is a way to describe processes. To understand what a "system" is... that is very fundamental. It is a wrong starting point to thing about thermodynamics as a bunch of equations.
No equation or "law" in physics makes sense if there isn't a measurement which makes a physicist trust in the statement and which seems to be forgotten from time to time - which fits to the conditions.
Short: You have direct measurements and indirect measurements -if you want to measure. You have to be sure in which framework your equations are correct or have sense and then you have to decide which measurement is appropiate.
"Temperature" is something we physicists (at least me) found appropriate to descibe a local (scalar) feature if there are a bunch of things happening. Particle party.
The point is: If you want to measure something then it has to be always in an appropriate framework if it should prove your theory or at least work in your theory.
If the definition of the thing you would like to measure is out of your framework then you shouldn't measure it in your framework - and if you do, it can be very wrong.
As an example: To say "I measured a temperature of 0 degrees Kelvin" obviously makes no sense.

 

[Hiero]

I would say that this makes temperature itself undefined in such a situation. Or at least fuzzy.

Yes I would agree except for the possibility of discovering a better thermometer which works even faster. Why should the limits of temperature depend on technology?

"Temperature" is something we physicists (at least me) found appropriate to descibe a local (scalar) feature if there are a bunch of things happening.

By “a bunch of things happening” I assume you mean macroscopically (so we have non equilibrium)? Then what is the definition of temperature you use? The only definition I know intimately relates to entropy and energy according to dE=TdS

The point is: If you want to measure something then it has to be always in an appropriate framework if it should prove your theory or at least work in your theory.
If the definition of the thing you would like to measure is out of your framework then you shouldn't measure it in your framework - and if you do, it can be very wrong.

Youre right; we can’t fit non equilibrium dynamics into the framework of equilibrium assumptions. The fundamental basis of statistical mechanics is the tendency of chaotic systems to mix available phase volumes. This gives a justification of the ergodicity of equilibrium and the uniform probability of states in the micro-canonical (constant energy) ensemble, from which classical statistical mechanics can be derived. It’s crucial that systems are given enough time to “lose memory” of the trajectory of the system. That’s where the generality comes from. I suppose the only way to handle non-equilibrium is case by case.

I’m still a student I’m just interested in how far statistical mechanics has been developed away from equilibrium. More insight is welcome. Thank you all for the replies, I will continue to study the fundamentals.

 

[jbriggs444]

Yes I would agree except for the possibility of discovering a better thermometer which works even faster. Why should the limits of temperature depend on technology?

If a macroscopic region cannot equilibriate before it changes its "temperature" then a better measurement tool won't help. You may be able to put bounds on what the temperature would be if the region had one, but any value you do come up with for temperature will necessarily be suspect.

You can make the region smaller so that it equilibriates faster. But you eventually run into the problem that the region becomes too small to have a meaningful temperature.

 

[Omega0]

By “a bunch of things happening” I assume you mean macroscopically (so we have non equilibrium)? Then what is the definition of temperature you use? The only definition I know intimately relates to entropy and energy according to dE=TdS

... okay and now? How does this help you if you don't have the conditions which are needed for this equation? Thermodynamics needs local equilibria to work, this is one of the spices. The next one is we need a big big bunch of things being there. Without particles we have no thermometer and without a big bunch statistical mechanics makes no sense. In thermodynamics we speak about "control volumes". A control volume is never a mathematical point. We have a lot of things happening there but the trick is that we are not interested in the party going on there, we just want to see the results - messy house, unhappy parents, sleeping students in parents bedroom or whatever. All of those numbers, like the students in the bedroom the next day are part of the measurement process which is based on the idea that the party is over. It makes no sense to count the students the day before or while the party is going on if you are interested in numbers coming from the equilibrium (which is very likely over when the parents come home). The measurement always has to fit to your situation - and the value you will get is only useful if the measurement is appropriate in the context.

It’s crucial that systems are given enough time to “lose memory” of the trajectory of the system. That’s where the generality comes from. I suppose the only way to handle non-equilibrium is case by case.

Yes, and a system in statistical mechanics is obviously a bunch of particles.

I will continue to study the fundamentals.

I am still a student because I want to learn more and more, day by day. You are not alone.
What I want to give you as ad advice on your way is: Always check if the things you have in mind are describing the situation properly. If you apply an equation: Which is the picture of the nature you will use? Check double if it is valid in the context.
The more important: Check your appartment/house statistically at 9AM to find out what happened in the control volume if you haven't been part of the party.