Temperature is fundamentally tied to how entropy and energy relate to one another in a system.
For instance, see this post for some kooky stuff that can happen. In most “normal” systems, though, temperature is relatable to the internal energy (both potential and kinetic), and in many of those systems, the kinetic energy is relatable to the speeds of the constituents (but not linearly). But none of this is universally true or definitional.
This one is more straightforward, up to a point. Climbing the temperature ladder, you’ll pass a few sign posts marking the next stage of breakdown, listed here in round numbers for energy units (electron-volts) and temperature units (kelvin), with Boltzmann’s constant relating the two.
- 1 eV or 104 K: breakdown of atoms → formation of plasma
- 1 MeV or 1010 K: breakdown of nuclei into neutrons and protons
- 150 MeV or 1012 K: breakdown of nucleons → formation of quark-gluon plasma
- 150 GeV or 1015 K: electroweak phase transition → the Higgs field changes dramatically, and (thus) so too does almost everything familiar about particles and their interactions.
- [10 TeV or 1017 K: highest energies probed experimentally at any level of detail]
- ??? TeV or ??? K: new physics?
For the last item, there are many reasons why we expect something new to happen at some energy scale, and many are looking for evidence of this next interesting point. This isn’t only done by cranking the experimental energies up directly but also through indirect information gained through high-precision measurements of lower-energy phenomena whose behaviors are influenced by the very existence of higher-energy processes.
(Aside: it is thought that the early universe included a so-called “inflation” period with temperatures in the 1025 K ballpark. There are experiments looking for echoes of inflation in the cosmic microwave background. If found, this would be an indirect probe of obviously much-higher-energy physics than the above list reaches.)