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All real things have temperature. Temperature can be measured in various ways. We have reached to a great precision in measuring change in time and space , however I am not sure to what extent change in temperature can be measured? I searched the Internet but found no relevant results.

My question is : With what maximum precision a change in temperature can be measured ? Is there any limit to the measurement of precision in change of temperature ?

PS : I am getting suggestions that my question is probably a duplicate but here is the difference: I am not asking the minimum temperature attainable. I am asking with what maximum precision a change in temperature can be measured. For example - if I take two glasses of water ,or anything suitable ,under same constraints and heat them simultaneously and I stop heating one of the glass few seconds or minutes or hours before the second glass then there will be temperature difference. Now , How sensitive are our measuring instruments to measure the change in temperature? If I heat one glass for 30 minutes and second for 31 minutes or 30.5 minutes then say first glass will show temperature T and other glass will show T plus delta T, now , how fine the delta T can be ?

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    $\begingroup$ Does this answer your question? Absolute zero: how close can we go? $\endgroup$
    – Roger V.
    Oct 30, 2023 at 9:50
  • $\begingroup$ As to the question in the body of the post, what magnitude of change of temperature do you want to detect? A difference of 0.001K or of 10K? Around room temperature or cryogenic? $\endgroup$
    – my2cts
    Oct 30, 2023 at 10:11
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    $\begingroup$ @RogerV. The question asks what is the smallest measurable temperature difference, not the smallest measurable temperature. $\endgroup$ Oct 30, 2023 at 10:28
  • $\begingroup$ There you go - World's most sensitive thermometer made from light $\endgroup$
    – Farcher
    Oct 30, 2023 at 12:43
  • $\begingroup$ This question asks about the best temperature precision, not best temperature difference. Cold atoms can probably measure temperature differences at the nK level but this is on clouds that are at sub uK temperature, so not mind blowing precision. $\endgroup$
    – Jagerber48
    Oct 30, 2023 at 16:53

4 Answers 4

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For a bulk substance under relatively normal conditions as you've asked about in your edited question, the "regular" standard is approximately a millikelvin (equal to 0.001 C) using platinum RTDs (resistive temperature devices, here is an example). I've also found this MIT PhD thesis (scroll down to get to the PDF, which also contains a summary of other high-precision temperature measurement techniques which you can read) which uses a laser interferometer to measure the change in level of a classical liquid thermometer and claims sub-microkelvin resolution. However, the problem with this is that it does not have sub-microkelvin accuracy - it can measure changes in temperature very precisely, but not absolute temperatures. In principle you could calibrate it to a known temperature, but it is very hard to get a temperature standard good to better than a millikelvin. The triple point of water (the temperature at which solid, liquid, and gas coexist) is only known to a precision of 0.1 mK.

So, in summary: the absolute temperature of an everyday substance can be measured to 0.1 mK at the very best, in practice more like 1 mK with significant effort. The change in temperature can however be measured with microkelvin resolution, if sufficiently isolated from the outside environment.

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  • $\begingroup$ Along these lines, pit-viper cells detect tiny but rapid changes in temperature, down to 3mK. However, they are insensitive to the much larger ~15K nighttime poikilotherm body temp range. $\endgroup$ Nov 1, 2023 at 7:39
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Lately, there's a lot of buzz around a research field called quantum thermometry. The goal here is to estimate the temperature of atomic systems close to absolute zero. This means finding out the exact temperature of super cold gases, electrons in superconductors, quantum dots, and a whole lot of other cool systems. At these tiny scales, quantum effects play a big role in how precisely we can measure. So, one of the things we're trying to figure out is the basic limits for temperature measurements in these quantum systems.

When we assume unbiased estimators, which means our temperature guesses average out to $T$, there's a cool rule that sets a limit to how accurate our guesses can be. This limit is called the quantum Cramér-Rao bound. This bound describes the minimum possible variance of an unbiased estimator for the temperature - basically, no matter how fancy our measurement tools get, we can't do better than this bound when trying to guess the temperature of quantum systems.

\begin{equation} \Delta T^2 \geq \frac{1}{n\mathcal{F}(T)}, \end{equation} where $n$ is how many times we repeat the measurement. The more times we measure, the better our estimate usually gets. The function $\mathcal{F}(T)$ is the quantum Fisher information. It's a measure of how much information we can squeeze out of our quantum system about the temperature. So, the right side of the equation gives us the best (smallest) variance achievable given the number of measurements and the informativeness of our system.

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    $\begingroup$ "and a whole lot of other cool systems" - nice! $\endgroup$ Oct 30, 2023 at 15:34
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    $\begingroup$ @MarkoGulin I didn't resist the joke ahahah I'm happy that someone spot it. $\endgroup$
    – Alex
    Oct 30, 2023 at 15:36
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    $\begingroup$ It's a nice word play, well done. And for the answer as well :) $\endgroup$ Oct 30, 2023 at 15:37
  • $\begingroup$ Is there some practical bound or estimate on the quantum Fisher information for temperature measurement? $\endgroup$ Oct 30, 2023 at 16:41
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You have a nice answer about an engineering limit (millikelvin-ish) with a link to some primary literature which I have not read. But that's an engineering answer. A more interesting answer is whether there is a fundamental physical limit on measurable temperature differences.

Those of us who live in the macroscopic world tend to consider temperature as a continuous phenomenon. But our best understanding of temperature, for more than a century, is as a statistical phenomenon, relating changes in the internal energy $U$ of a system to its changes in its entropy $S$:

$$ \frac{\partial S}{\partial U} = \frac 1T $$

If you are unfamiliar with this fundamental definition, there are several minus signs and one-overs that you have to account for in order to recover the usual statement that heat moves from hot to cold in order to increase entropy. But the statistical nature comes from the definition of the entropy, $S = k\ln \Omega$, in terms of the number $\Omega$ of microscopic states with the same internal energy $U$, and the fact that the internal energy $U$ is itself quantized.

This statistical nature means in practice that there is no such thing as an extended object at a "constant temperature" which can be measured to infinite precision. You have comments which refer to temperature gradients and temperature drifts. But I'm talking about a temperature version of shot noise, where adjacent small regions in a bulk material have different temperatures just because energy and entropy come in lumps.

The size of the intrinsic temperature fluctuations will be larger for smaller subsystems and for lower temperatures. Some quantitative examples would be interesting to add, but I'll have to return later to do that (if there is interest).

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    $\begingroup$ This answer seems like the beginning of one that I hoped may be the eventual accepted answer.. 🙂 $\endgroup$ Nov 1, 2023 at 6:12
  • $\begingroup$ @LamarLatrell I suspect it overlaps with the answer about "quantum thermometry," but I don't know the literature. $\endgroup$
    – rob
    Nov 1, 2023 at 16:49
  • $\begingroup$ This makes me wonder if LIGO can do some super-sensitive measurements of the temperatures of their mirrors, since they are sensitive to their thermal fluctuations. $\endgroup$
    – Rococo
    Nov 27, 2023 at 2:37
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Temperature is not a fundamental property, but a kind of average of the energy of a number of particles. Take for example a volume of gas. Within the gas there may be small temperature variations due to statistical fluctuations. The average of a larger amount of gas can be measured with greater precision than for a small amount of gas, because the statistical variation is greater for a smaller number of particles. This statistical nature of the property temperature, makes it difficult, if not impossible, to answer your question in an absolute way.

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