# Conflicting definitions of temperature?

1. Oct 20, 2009

I thought that temperature is a measure of energy density, which means that at the vacuum energy has a minuscule temperature above absolute zero. However, I read at http://www.newton.dep.anl.gov/newton/askasci/1993/physics/PHY59.HTM that "At absolute zero, all motion does not cease,..." which would seem to contradict the idea of absolute zero as a state of zero energy density which is attainable with a probability approaching zero. So, is the definition of "temperature proportional to energy density" flawed?

2. Oct 20, 2009

### Born2bwire

Yes, temperature is related to the average kinetic energy of the system. Any system, classically, that is at absolute zero will have no kinetic energy, and thus no movement. However, it can still have a potential energy. Heck, even the vacuum field at 0 K has a very dense energy density that is divergent with frequency.

I'm not sure about quantum mechanics though. I would still feel that at 0 K there is no movement. However, even in vacuum there are still quantum field fluctuations. For example, charged particles can couple with the field fluctuations and this has real effects in quantum electrodynamics. But I am not sure if we can say that this will cause true movement of a system brought to 0 K. That would seem to require energy being taken out of the vacuum field to do work which as far as I know is not known to be possible, at least as a constant dynamic. Casimir force for example can draw objects closer but it will eventually hit a static point. Well, somebody with a far greater understanding of statistical physics could correct me here.

3. Oct 20, 2009

### Andy Resnick

Temperature is not a measure of the total energy density, temperature in thermodynamics is analogous to 'mass' in mechanics. Just as we say "mass is the amount of material", we can say "temperature is how hot an object is". Trying to say much more than that generally leads to either highly restrictive uses of the quantity (such as a mechanical basis for temperature), or curious nonphysical temperatures (such as occurs in two-state systems during population inversion).

In order to define temeprature sensibly, one needs a more general definition than is supplied by ideal-gas definitions (e.g, the temperature is a measure of how fast the atoms are moving). Defining the temperature of a body, for example, requires the body be in equilibrium.

4. Oct 20, 2009

### f95toli

First of all, Yes, there is still zero-point motion at 0K. This is because in QM the temperature is usually "defined" as a parameter of a bath of harmonic oscillators; and even when you set "T" in these equations to zero things move.
There is a section on this is Gardiner's book on open quantum systems (I don't remember the title).

And as already been stated: There is no good all-encompassing definition of "Temperature". The concept is used in many situations where the "classical" meaning of the world does not apply.
At very low temperatures the word is VERY ambiguous and you basically have to look at the exact circumstances of a given experiment to understand what the "T" in the equations actually refers to.

5. Oct 21, 2009

### Gerenuk

Temperature is in general not defined by the average energy. If you look at the basis of statistical mechanics, then you find that temperature is defined so that the Boltzmann distribution gives you the correct mean energy.
$$E=\frac{\sum_c E_ce^{-E_c/k_BT}}{\sum_c e^{-E_c/k_BT}}$$
where E is the energy you measure in the system and the sum is over all possible configurations c.

Only for the special case where $g(E)\propto E^a$ the temperature is incidently proportional to the energy.

Hmm, that's a kinda content-less statement
But there is a general definition for general systems. I can't remember how it goes. Do you know?

That would be the case for my definition of temperature, but I don't see a problem with that. It's only non-physical if you believes temperature should be the average energy and therefore positive :uhh: That's why that definition is general.
Actually for system not in equilibirium there wouldn't be a temperature defined. But surely someone has generalized my definition of temperature so that it encompasses all distribution and converges the the normal definition for the Boltzmann distribution.

6. Oct 21, 2009

### Andy Resnick

I'm not sure why you say that. You didn't seem to mind "mass is a measure of how much stuff there is". In both cases, a physical property is defined in terms of a mathematical objects: a scalar quantity that also allows for ordering (T1 >T2, for example). The statement also allows for numerous other quantitative treatments: changes in temperature, for example. It really is the most fundamental statement possible.

There's nothing inherently wrong with non-physical mathematical solutions; I am simply saying that the *physical* basis of physics must be primary to the *mathematical language* of physics. Otherwise, based on what criteria do we exclude solutions as non-physical?

To reiterate, AFAIK, there is no rational generalized defintion of temperature that holds for all physical systems. One may define "effective" temperatures, but these are not things we can measure with a thermometer.

Here's a practical example: for all practical purposes, we exist at constant temperature and pressure. Yet we exist in a state far from equilibrium- equilibrium for us means we are dead and decomposed. How can this be reconciled, other than simply stating "well, I can take my temperature with a thermometer so the temperature exists."? That's not a rational definition of temperature.

The same problem exists for simpler systems- sandpiles, a hard-sphere gas of bowling balls in zero-g conditions, etc. etc. Assigning a single, unique temperature to a hard-sphere gas is usually done in terms of the volume fractions (in order to correlate to phase transitions), but that does not correlate with the temperature of the (for example) bowling balls.

7. Oct 21, 2009

### Gerenuk

Oh, that last statement is also useless in a way. If I imagine I want to measure temperature, what would I do assuming all I know is to measure "hotness"? It's just a shift in definition - just as useful as saying "because god wanted it so".

I heard of definitions similar to "let's define two reference system with determined temperature", but I cannot recall exactly how they work.

I'm not sure what you mean. Can you please explain what you mean by "non-physical"? But you may not refer to violated laws that only follow for the special case when temperature is proportional to energy. In that case the preconditions are of course not satisfied.

I only know the temperature defined by basic statistical mechanics. To me it seems, the only reason why it's not applicable to all systems is, because someone invented a contradictory parameter and also called it "temperature".

Actually that's a very good point. One can think how much "huge number statistics" one needs and how homogeneous a medium has to be to define temperature. I think for this one can go back to the derivation of entropy and examine what happens for a small system.

One just says as an approximation the body is in a constant temperature state. There are deviation in details. They are small for thermodynamical purposes, but essential for us.

I heard of these examples, but I don't know the details. Why do they call it temperature in the first place? What are the conditions to justify calling a parameter temperature?

Last edited: Oct 21, 2009
8. Oct 21, 2009

### Andy Resnick

There's a lot here... I'll do my best:

No, it's not the same thing as saying 'god says so'. I'm talking about the foundations of physical theory- in order to have a theory, one must first formally define objects and concepts. 'hotness', like 'quantity', is a primitive concept- and 'mass' has no meaning without 'quantity'. It may seem trivial and silly, but saying 'there are measurable properties of things that are positive real numbers' is an important concept. Because positive real numbers are not physical objects, and there is no reason to assume that positive real numbers correlate with anything real.

You may be referring the 'the zeroth law', which is a way of defining temperature. The zeroth law is a definition of equilibrium, that's all.

Ok- why do you agree that negative temperatures are non-physical? We can define negative energies, why not temperatures?

Statistical mechanics is not the foundation of all of physics.

I think you missed my point. I am not in any way close to equilibrium, and neither are you. Yet we can both use a thermometer to measure our temperature. If temperature can only be defined for a body in equilibrium, how is it that we have a temperature of 98.6 F?.

Our deviation from equilibrium is not small! We are, by one measure (the concentration of ATP relative to ADP), orders of magnitude away from equilibrium.

Now *that's* a good question! I don't have an answer, other than if one writes dE/dS (or something like that), you get a parameter that acts like T.

9. Oct 21, 2009

### f95toli

There are not generally accepted criteria. I actually know some people who work in temperature metrology and not even they know. They basically stay avay from situations where there is any ambiguity. Which, btw, is why the latest international temperature scale(ITS-90) is only defined down to 650 mK. There have been attempts to extend it to lower temperatures but they haven't been successfull (you can buy sensors for lower temperatures that can be traced to NIST, but that is not an "offical" calibration).

10. Oct 21, 2009

### Gerenuk

OK, so you have a block of wood. How would you measure temperature then? Don't forget you are not given a magic device that measures "hotness". Saying temperature is hotness is just a shift of definition. What is hotness then?

No, there are guys around who define temperature with reference systems and for very general systems. A system that could be anything like a chess board with a cup of water on it. But I don't remember their very mathematical precise way. It probably has to do with ergodic theory or so.

I didn't write I don't agree with negative temperatures. I was asking you why you say that a temperature definition (probably mine) can be non-physical. Negative temperatures are in fact physical.

Stricly speaking we are not in equilibrium. But the material in the thermometer is and so it can show you a temperature value. As the thermometer only interacts with the kinetic motion of our molecules, which themselves are roughly in equilibirium, it is in equilibrium with the motion of our molecules only.

For this you have a different temperature from mine. As an approximate and to define the only reasonable temperature I use the kinetic motion of molecules only. Everything else doesn't permit the definition of temperature anyway.

I thought about that, but the problem is that entropy S is even less defined than temperature.

11. Oct 22, 2009

### Andy Resnick

Hopefully, you are starting to see that temperature cannot be measured unless you have a thermometer- which is defined as a device to measure some *physical property* of the system, and that satisfies certain properties, analogous to having a ruler or a clock- being able to compare different measurements,for example. Enunciating those properties is 'thermometry', and the foundations of thermometry are not compeletely understood as of now.

You keep saying this, but have not supplied a reference (I would like to read the article). In any case, ergodic theory does not cover glassy states, so I don't see how it can be a truely general definition.

Really? Kelvin would disagree with you.

Again, defining the temperature in terms of 'average kinetic energy' is too restrictive. It may be useful for elementary considerations, but it cannot constitute a foundation of thermometry.

12. Oct 22, 2009

### Tac-Tics

It's not flawed. It's just classical. It provides a good estimate for large systems. But it breaks down when you get to a certain extremely cold temperature due to the uncertainty principle.

Temperature is a measure of kinetic energy, and energy is the "dual" quantity of time. If you freeze a particle to 0K and measure the energy of a system exactly, then by the uncertainty principle, you have no idea when that measurement was valid.

13. Oct 22, 2009

### Staff: Mentor

In what way is entropy not well defined in statistical mechanics?

$\Omega$ = number of microstates of a system which comprise a given macrostate (specified by total energy U, volume V, number of molecules N, for e.g. a gas).

Entropy $S = k \log \Omega$ (the famous equation which is engraved on Boltzmann's gravestone).

Then define temperature via

$$\frac{1}{T} = {\left( \frac {\partial S}{\partial U} \right)}_{V,N}$$

14. Oct 22, 2009

### f95toli

The problem is that it is very hard to see how one would use these equations when dealing with e.g. the cooling of a single mode of a resonator.

15. Oct 22, 2009

### Andy Resnick

Right- another good example is the electromagnetic field. It's possible to define a temperature for a single configuration of the field- black body radiation. Any deviation from that, such as passing the light through a filter that removes only a narrow region of frequencies, results in non-thermal light that cannot be assigned a temperature.

16. Oct 22, 2009

### Gerenuk

Hmm, that's again a shift in definition only and not getting to the point. You still haven't specified how you want to measure temperature. Instead you rely on other people providing you a device called thermometer.
To illustrate what I mean by giving a specific system for measurement here is how I would measure temperature defined by my statmech equation above:
I observe the system and measure its total energy at different times. From this energy data I plot a histogram of the energy distribution and fit it to an exponential law. The exponent of the exponential law gives me the temperature. If it doesn't fit an exponential law, then the system is not in equilibirium and doesn't have a temperature.
This method is general enough to include the statmech and thermodynamics concept of temperature.

I keep saying that I do not recall how they did it. It was a lecture where I quickly noticed that it was to mathematical and abstract for me. But some part of the talk was based on
http://arxiv.org/abs/math-ph/0003028
That's sort of their method and temperature was defined similarly. Maybe one can find a paper search for these guys.

It doesn't matter if Kelvin disagrees. He probably used temperature for the thermodynamics of ideal gases only.
But anyway, please finally post you own opinion. Do you find negative temperatures unphysical and if so then why?

That's not what I wrote. My general definition is the statmech one. For the special case of the interaction between an ideal gas and and human body, the kinetic of the molecules plays a role only.

I wasn't clear enough. I meant, if you want to measure the temperature of a piece of wood, refering to theoretical equations about entropy makes the task only harder. Or just really try to imagine which step by step instructions you would try to follow to measure temperature. What would it be? What is a microstate? How do you count them?
But remember that some equation you might know only apply to an ideal gas and not to a block of wood.

17. Oct 22, 2009

### twofish-quant

Temperature is what a thermometer measures. Trying to define temperature is like trying to define length. For that matter how do you define "cup" or "spinach." Ultimately you end up pointing to things that are cups, things that aren't cups, and as long as we agree on what is a cup then we are good.

Also trying to fit things to a Boltzmann distribution won't work. For most systems you end up with quantum interchange effects and chemical potentials. Also systems that are not in equilibrium have well defined temperatures. Also if you define temperature in terms of energy distributions, then you have the not insignificant problem of trying to define "energy".

The other thing is that suppose I give you a system that doesn't follow Boltzmann's equations, but gives you a well defined temperature when I stick a thermometer in it. Then I just toss Boltzmann's equations because they are wrong.

18. Oct 22, 2009

### Andy Resnick

This is a science discussion; my opinion is irrelevant.

Thanks for the reference.

19. Oct 22, 2009

### Gerenuk

OK, so what is a thermometer then? You have to start going down to lower concepts like measuring energy or time at some point.

I reiterate my question: Which (hypothetical) procedure would you perform to measure temperature? You are sitting in a lab, but no-one has left a thermometer, so you have to build one. A simple gas thermometer won't be general enough though - at least my Boltzmann procedure can deal with more general cases.

Length is defined by the speed of light and a certain duration of a physical process. These are preconstructed by nature. Thermometers do not come from nature.
In fact eventually all explainations and devices probably should end up using the observables length and time only.

The statmech book says that chemical potentials are a direct consequence of the Boltzmann distribution if you apply it correctly.

I have not studied that topic yet. I suppose there exists temperature that coincides with my definition for equilibrium cases, but generalize for non-equilibirium also. Hope I learn that at some point.

They are not wrong. Your thermometer follows the Boltzmann equations and keep in mind that the temperature reading refers to the temperature of your thermometer no not directly the body you are probing!
Due to interactions with a non-equilibrium system the thermometer will acquire a certain equilibrium state for itself. In fact knowing the physical laws and applying the Boltzmann equation to the liquid in the thermometer will predict you the correct temperature.

OK, so what would you do? You did indeed made fair points where the Boltzmann definition might fail, but what is a better suggestion? You have to suggest something better that explains at least as much as the "Boltzmann temperature".

The "Boltzmann temperature" explains all of thermodynamics and all of undergrad statmech.

20. Oct 22, 2009

### f95toli

But again, there IS no "general" definition of temperature.
Most of the fixed points on the international temperature scale (ITS-90) are based on triple points, although the lowest points use the melting curve of He-3. Hence, these points are "classical".

However, no one -including the people who manage the ITS (I know some of them) claim that this this is more than a practical scale. The reason why it hasn't been extended to lower temperatures is because the concept of temperature is so ill defined.

I use a nuclear orientation(NO) thermometer in my lab to measure temperatures between 15mK and 200 mK. Some of the equipment(including the Co-60 source) I am using is actually "leftovers" from a project that aimed to extend the ITS to lower temperatures using NO thermometers. However, they never succeeded; mainly because the temperature that is measured by NO (essentially the phonon temperature of the Co) is not necessarily the temperature relevant in experiments (usually the electronic temperature, which can be hundreds of mK higher if the e-p scattering times are long or the system is noisy).
Hence, extending the ITS using this method wouldn't actually be of much use.

The fact that there are several relevant "temperatures" when working below 1K is something a lot of people do not appreciate, it is definitely something I've had to point out many times when writing referee reports. It is also a very common error in published papers.

Last edited: Oct 22, 2009
21. Oct 22, 2009

### Gerenuk

I was asking for a temperature that includes statmech and ideal gases and also extends to as much as possible. So basically the most general temperature possible to define - i.e. the best one can do.
I was also asking which other parameters exists that claim to be called temperature. Also I asked for criteria for a parameter to be called temperature. So basically I was asking why conflicting definitions have the right to be called temperature. They might be... but what physical equation or concept is the reason?
Of course I could call the height of water in a glass temperature. It even would have some of its properties. But why would I do that?

Try to forget all equipment you have and all the tables you are given! Imagine you are the first scientist to construct a thermometer. How would you gauge it?
I assume make use of common statmech (i.e. Boltzmann distribution and temperature as I stated) and hope that your physical model of the microscopic process you used for the statmech equation is correct (just as people do for gases). Only this way you can relate a macroscopic quantity to something which is called temperature and derives from the Boltzmann distribution.

But that's a completely different source of troubles. Of course if the thermometer doesn't interact the right way with the electronic system, then these two energy systems won't equilibrate and will have different temperatures.

All these methods rely on assumptions about the statmech equations of the thermometer. I mean again at some point someone must have gauged the macroscopic parameter to something that obeys the laws for temperature.
These can only be guessed within an approximation so essentially all these temperature might be wrong. Luckily the a low density gas behaves quite like an ideal gas for which it is easy to measure the temperature.

I think temperature is a cardinal scale, so you cannot arbitrarily scale/transform it, without disturbing some equations?

22. Oct 22, 2009

### Mute

Temperature is abstractly defined as the property of two connected systems that is constant when there is no net energy flow between them. From the statistical mechanical view point if I have two systems connected by prevented from exchanging energy and I then release the constraint that they can't exchange energy, then the two systems will eventually evolve to configurations where the net energy in both of them doesn't change in time, which defines our "equilibrium". The two systems need not have the same energy, but it turns out that

$$\left(\frac{\partial S_1}{\partial E_1}\right) = \left(\frac{\partial S_2}{\partial E_2}\right)$$

Whatever this is, it's equal between the two systems. We choose to call this thing 1/T, where we call T "Temperature":

$$\frac{1}{T} = \left(\frac{\partial S}{\partial E}\right)$$

So, in any system where there is a property that we can measure that somehow behaves like energy (it doesn't actually have to be an energy - it just has to behave somehow like one mathematically), the net flow of which between two systems doesn't change in time defines an equilibrium condition analogous to thermal equilibrium in ideal gases, eg. We may then define a temperature-like variable $\theta$ by

$$\frac{1}{\theta} = \left(\frac{\partial S}{\partial \mathcal E}\right),$$
where $\mathcal E$ is the energy-like variable. S is the entropy, of course, defined by

$$S = k\ln \Omega.$$
Here, k is NOT necessarily Boltzmann's constant. In thermodynamics the Boltzmann's constant value isn't really fundamental - it just fixes the units we measure temperature or energy in. Aside from this constant, the number of microstates of the system is, in principle, a well defined number that doesn't really care about whether or not the system we're studying is a physical thermodynamic system. For example, I can calculate the entropy of a deck of cards: the number of possible sequences of a standard deck of cards is $\Omega = 52!$, so $S = \log_2 (52!)$, where I chose k such that entropy is measured in bits. It's not a thermodynamic entropy, it's an "information" entropy. One could argue that statistical mechanics is simply information theory applied to systems which exchange energy, particles, etc.

So, this is in general our theoretical definition of temperature. In systems where we can define a temperature-like variable we expect it to play a role analogous to temperature in a physical system. So, while $\theta$ might not be a physical temperature, it might still be the variable we tune to induce a "phase transition" in our system of interest.

Rather than find the definition of temperature that covers the most cases, it is probably best to pick several definitions which overlap in certain regimes. The statistical mechanical definition is very nice, but it is perhaps not always practical. So, for practical purposes what we can do is find a system in which we can define some other definition of temperature, "A", that coincides with our stat-mech definition in some regime, and if we then want to measure temperature in a regime where our stat-mech definition is cumbersome but our other definition still works, we can specify yet another definition of temperature, "B", that coincides with our definition A in this region where our stat mech definition is impractical.

For example, in the classical limit, the stat mech definition coincides with the definition of temperature defined by relating "thermal energy" to the average kinetic energy of the ideal gas. If we then consider a block of wood, for which we might never hope to calculate the temperature from the stat mech defintion, we can measure it with our "average kinetic energy definition". We might then be able to find another quantity which matches the "average kinetic energy definition" in a regime where "average kinetic energy definition" is no longer equal to the stat mech definition, and take that as another definition of temperature, which might still work in a regime where the "average kinetic energy definition" fails. (This is conceptual like analytic continuation of complex functions, I suppose).

You can certainly scale it. All that really amounts to is changing the units I chose to measure temperature in. I can set $k_B = 1$ if I so desire; all that results in is me measuring temperature in joules instead of Kelvin. There's no fundamental difference.

Negative temperature is certainly a physical thing, if you interpret it properly (and, depending on how you are defining "temperature" in that statement!). Via wikipedia, "a system with a truly negative temperature is not colder than absolute zero; in fact, temperatures colder than absolute zero are impossible. Rather, a system with a negative temperature is hotter than any system with a positive temperature (in the sense that if a negative-temperature system and a positive-temperature system come in contact, heat will flow from the negative- to the positive-temperature system)."

Negative temperatures as defined in the "average kinetic energy" sense are impossible. As you yourself say, it is too restrictive. From the stat mech definition, however, negative temperatures are allowed, as you seem to be aware. So, it's certainly a physical thing from the stat mech viewpoint. (the wikipedia article: http://en.wikipedia.org/wiki/Negative_temperature)

Not in any way close to equilibrium with respect to what? Equilibrium is not a property of a system on its own, it's a property of a system with respect to another system. In the case relevant to defining a body temperature, that system is the environment, and we are most certainly in an energetic equilibrium with that: the amount of energy we are radiating away must be equal to the amount of energy we're absorbing from the environment - that is, the net flux of energy between our bodies and the environment is constant. (Of course, there are times when we are not in such an equilibrium, but there are periods of time where things are in a steady state and this applies). You might argue that temperature is still ill-defined because energy isn't constant, but we're in a steady state, and there is a generalization to steady state processes. In this case, "temperature" is defined by

$$\frac{1}{T} = \left(\frac{\partial s}{\partial u}\right),$$
where u is the energy density and s is the entropy density. See the wikipedia article on the Onsager reciprocal relations for more info: http://en.wikipedia.org/wiki/Onsager_reciprocal_relations

Last edited: Oct 22, 2009
23. Oct 22, 2009

### Andy Resnick

I would claim you are confusing the object being studied with the model. Statistical mechanics is a model of some physical phenomena, it is not the phenomena itself.

Lots of equations used in physics have solutions that do not correspond to physical reality- we take the principal square root of the kinetic energy when calculating the velocity, because negative speeds are nonphysical.

With respect to the chemical reactions that serve to keep us alive rather than dead. The ratio [ATP][Pi]/[ADP] is 10^8 higher than equilibrium conditions (ATP = adenosine triphosphate, ADP = adenosine diphosphate, Pi = phosphate), and this excess free energy is how we derive useful work from hydrolysis of ATP.

I used to think the Onsager relations were interesting. I read a nice rebuttal by Truesdell, and now I see they are a simple linearization of irreversible thermodynamics, and so do not apply to systems of interest (to me). 10^8 is much larger than 1.

24. Oct 23, 2009

### Mute

The point of the models is to be able to make physical predictions (or at least gain a qualitative understanding of the phenomenon). A negative temperature is a physical prediction if you interpret it properly. Whether or not any systems with negative temperatures are known is another matter - for one thing it relates back to the discussion on how to measure temperature: we certainly couldn't measure a negative absolute temperature with an ideal gas thermometer, but that's a problem with our thermometer, not necessarily our notion of temperature. A negative temperature has a well defined interpretation in the model; solutions in problems that give negative velocities typically do not.

Okay, but that's irrelevant to defining a body temperature. We define the body temperature with respect to an energetic equilibrium between our bodies (as a whole system) and the environment, and such an equilibrium exists. We don't need to worry about the ratio [ATP][Pi]/[ADP] in defining such a temperature.

Sure, they apply to steady state systems where a notion of local equilibrium is definable. As far as defining a body temperature goes, that should be sufficient. They certainly are not sufficient for systems far from equilibrium.

25. Oct 23, 2009

### Gerenuk

So my thought was: Would the transformed quantity $T^*=T^2$ also be a valid temperature or are there more constraints for something to be called temperature?

What bothered me long time ago, when I didn't know about statmech: If you are only able to measure macroscopic observables, then how can you ever deduce entropy without knowing about temperature first? That also means you cannot use this definition to define temperature?

I believe if you are fond enough to assume approximations for the system, then the statmech temperature immediately includes all "thermal average energy" definitions. The thermal average definition follows from $g(E)\propto E^\alpha$ which you probably could approximate to the relevant energy system in a block of wood.

So basically statmech temperature will never contradict the thermal dynamical average one. And if you want to use the latter, just claim that the correct precondition for statmech is fullfilled.