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I understand entropy is called the arrow of time because it is the only physical property which always increases in an isolated system (based on 2nd law of thermodynamics).

On the other hand, entropy is the degree of disorder of a system. When we talk about order/disorder, we are talking about a group of particles and their state relative to each other.

Does it mean for a single particle which is isolated (far away in vacuum space) time is still? Is it rational to suppose a particle in such state?

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You can certainly imagine such a Universe with one lonely structureless and cold particle, interacting with nothing and living in some eternal and unchanging spacetime. Of course such a Universe it is nothing like our own. My answer would be that since nothing ever changes in this Universe, there is not really an operational notion of time.

We can imagine this is a massive quantum particle living in Minkowski space; then there is at least a frequency associated with the Compton wavelength of the particle (in its rest frame), which would put a physical clock in the Universe. But there would be no arrow of time

Such a Universe is so weird that it borders on not being mainstream physics. But I can see its value as a thought experiment as in your question to clarify that, indeed, to measure time there need to be something changing, and to have an arrow of time there needs to be irreversibility -- in our Universe, this emerges through the second law because entropy was lower in the early Universe, and (on average) increases toward the future.

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    $\begingroup$ Good answer ! A single-particle universe has only state so only one value for entropy so it is simultaneously in its state of minimum and maximum entropy i.e. it has its Big Bang and its Heat Death at the same "time". $\endgroup$
    – gandalf61
    Jul 4 at 6:45
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    $\begingroup$ Ok. But you should be careful not to give the OP the impression that time is a consequence of "the arrow of time" (by what he seems to mean the 2nd principle of thermodynamics), which seems to be what he believes and is wrong. $\endgroup$ Jul 4 at 8:08
  • $\begingroup$ @Yvan Velenik could you please elaborate what you mean by time is not a consequence of entropy increase? Is time a separate entity which is just related to entropy? $\endgroup$
    – Ebi
    Jul 4 at 8:24
  • $\begingroup$ @YvanVelenik The perceptual past must be in the direction of decreasing entropy in order to allow the formation of reliable memories and records. In this sense, the perceptual arrow of time and the thermodynamic arrow of time are interlinked. $\endgroup$
    – gandalf61
    Jul 4 at 11:54
  • $\begingroup$ @gandalf61 Possibly. However, time exists even if it is not perceived. I did not understand his question as being about perceived time, but actual time. $\endgroup$ Jul 4 at 12:55

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