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Conflicting definitions of temperature?

  1. Oct 20, 2009 #1
    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?
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  3. Oct 20, 2009 #2


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    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.
  4. Oct 20, 2009 #3

    Andy Resnick

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    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.
  5. Oct 20, 2009 #4


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    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.
  6. Oct 21, 2009 #5
    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 [itex]g(E)\propto E^a[/itex] the temperature is incidently proportional to the energy.

    Hmm, that's a kinda content-less statement :wink:
    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.
  7. Oct 21, 2009 #6

    Andy Resnick

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    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.
  8. Oct 21, 2009 #7
    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
  9. Oct 21, 2009 #8

    Andy Resnick

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    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.
  10. Oct 21, 2009 #9


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    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).
  11. Oct 21, 2009 #10
    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.
  12. Oct 22, 2009 #11

    Andy Resnick

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    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.
  13. Oct 22, 2009 #12
    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.
  14. Oct 22, 2009 #13


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

    [itex]\Omega[/itex] = 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 [itex]S = k \log \Omega[/itex] (the famous equation which is engraved on Boltzmann's gravestone).

    Then define temperature via

    [tex]\frac{1}{T} = {\left( \frac {\partial S}{\partial U} \right)}_{V,N}[/tex]
  15. Oct 22, 2009 #14


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    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.
  16. Oct 22, 2009 #15

    Andy Resnick

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    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.
  17. Oct 22, 2009 #16
    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
    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.
  18. Oct 22, 2009 #17
    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.
  19. Oct 22, 2009 #18

    Andy Resnick

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    This is a science discussion; my opinion is irrelevant.

    Thanks for the reference.
  20. Oct 22, 2009 #19
    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.
  21. Oct 22, 2009 #20


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    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
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