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That temperature of a black body (or any other body like an ideal gas in a container) is probabilistic in nature. Temperature represents the most probable energy distribution of particles.

Suppose a black body is at $50^\circ$C. Then there is a chance (no matter how small) that the temperature might get recorded as $0^\circ$C or $100^\circ$C, etc.

I want to know whether it can be known mathematically, what are the chances of recording the temperature of a black body as $50.01^\circ$C. I am using the numerical temperature as an example just to make things clear, we can generalise it for temperature T and deviation $\Delta$T.

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    $\begingroup$ "Temperature of a system is probabilistic in nature.Temperature represents the most probable energy distribution of particles".... can you backup this statement with relevant references. $\endgroup$
    – Qwerty
    Commented Jan 28 at 17:36
  • $\begingroup$ I’m voting to close this question because it has a wrong concept on what is the definition of temperature. $\endgroup$
    – anna v
    Commented Jan 28 at 18:01
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    $\begingroup$ @annav I would say that this question is rather interesting. The question correctly notes that the result of a measurement of temperature is not a "sure number" but indeed should fluctuate. I am interested to know how that fluctuation is calculated. $\endgroup$
    – DanielSank
    Commented Jan 28 at 18:34
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    $\begingroup$ Please explain how the temperature is being measured? $\endgroup$
    – ProfRob
    Commented Jan 28 at 18:39
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    $\begingroup$ It depends on the thermodynamic ensemble. $T$ is given exactly, without any deviations, in the thermodynamic limit or in the canonical ensemble ($T=$constant). $\endgroup$
    – Mauricio
    Commented Jan 28 at 20:06

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Since you mentioned an ideal gas, let's take that example. To measure its temperature, bring it into contact with a thermometer. For simplicity, suppose this thermometer is also an ideal gas, one held at constant pressure. In the thermometer, then, we can use $PV = nRT$ to infer the temperature based on its volume, which could be measured with a ruler. One would measure an erroneously high (low) temperature if statistical fluctuations drove an unusually high (low) amount of energy into the thermometer. Seen in this light, it seems clear that the probability of being off by $.01^{\circ}$C depends on the size of your thermometer and on the duration over which you make your measurement.

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