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The Planck and maximum temperature

In the Planck temperature scale, $0$ is absolute zero, $1$ is the Planck temperature, and every other temperature is a decimal of it. This maximum temperature is believed to be $1.416833(85)\times 10^{32}$ Kelvin, and at temperatures above it, the laws of physics just cease to exist.

It states that max. Possible temperature is plancks Temperature but i have read that temperature of negative kelvin is hottest temperature and is hotter than infinite temperature means also hotter than plancks Temperature but how its possible

(From What happens as you approach/cross the Planck temperature?)

I expect it's impossible to cross the Planck temperature, just like it's impossible to cross absolute zero or the speed of light.

At the Planck temperature, you start producing miniature Planck-mass black holes, which are the hottest black holes that can exist. If you try to put more energy in the system, you would get larger black holes, which are cooler, and they would start absorbing stuff and cooling things down.

If it's not possible, then how can we talk about infinite temperature?

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    $\begingroup$ Possible duplicate of Why is there no absolute maximum temperature? $\endgroup$ – John Rennie Aug 2 '18 at 7:28
  • $\begingroup$ I don’t believe that the link provided answers OP’s question as answers there focus mainly on special relativity but do not focus on the quantum effects that may limit to a Planck temperature. $\endgroup$ – Tausif Hossain Aug 2 '18 at 7:49
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    $\begingroup$ Possible duplicate of What happens as you approach/cross the Planck temperature? $\endgroup$ – Tausif Hossain Aug 2 '18 at 7:52
  • $\begingroup$ Hi, welcome to Physics SE! Please don't paste pictures of text. Typed-out, formatted text is index-able by search engines, and shows up better on different devices' displays. For formulae, try MathJax instead. I've edited it here. $\endgroup$ – user191954 Aug 2 '18 at 10:20
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    $\begingroup$ That question is not a duplicate of any of the suggestions: it's explicitly asking about how to reconcile infinite-temperature claims with the Planck temperature. $\endgroup$ – stafusa Aug 3 '18 at 7:05
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The point is that a system cannot be heated above the Planck temperature, which does not necessarily preclude systems from existing at a higher temperature (effectively $\infty$, here), just like the impossibility to accelerate a particle to $c$ doesn't mean there aren't particles speeding at $c$.

One could also say that the apparent paradox arises from using a naive concept of temperature, when a more precise one is necessary.

The answers you quote already clarify the role of the Planck temperature as an upper limit. Negative absolute temperatures exist in particular systems (typically with population inversion) and are better understood in a statistical mechanics context - see, e.g., Physical significance of negative temperature and How to make physical sense of negative temperatures. Why negative temperatures are effectively infinite (with regard to heat exchange) is well explained in the answers to: Prove that negative absolute temperatures are actually hotter than positive absolute temperatures.

Also the Wikipedia entry on negative temperatures is useful:

Temperature is loosely interpreted as the average kinetic energy of the system's particles. The existence of negative temperature, let alone negative temperature representing "hotter" systems than positive temperature, would seem paradoxical in this interpretation. The paradox is resolved by considering the more rigorous definition of thermodynamic temperature as the tradeoff between energy and entropy contained in the system [...] Systems with a positive temperature will increase in entropy as one adds energy to the system, while systems with a negative temperature will decrease in entropy as one adds energy to the system.

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With a temperature of $T$, an energy-$E$ state's occupation probability per particle is proportional to $\exp -E/k_B T$. For finite $T>0$, increasing $T$ makes the high-energy states more likely to be occupied, increasing the mean energy. At $T=\infty$ the occupation probability is $E$-independent, but for $T<0$ the high-$E$ states are more likely to be occupied, giving an even higher mean energy. This is called population inversion; it's an important part of how lasers work. The hottest temperature is therefore actually $-\infty$, for which only the highest-energy state is occupied. (You might want to work out the mean energy as a function of $T$ for a $2$-state non-degenerate system to get a feel for all this.) It can be easier to think about this using $\beta:=1/(k_B T)$, the thermodynamic beta.

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