Time and entropy

We conclude by noting that the irreversible increase in entropy is something that emerges only when considering macroscopic systems. In fact, if we look at a system from a microscopic point of view its properties are perfectly invariant under time reversal. More formally, if ${\displaystyle (\mathbb {Q} ,\mathbb {P} )}$ is the representative point of our system in phase space we know that solving Hamilton's equations for the Hamiltonian ${\displaystyle {\mathcal {H}}}$ of the system gives us the motion ${\displaystyle (\mathbb {Q} (t),\mathbb {P} (t))}$ of this point; however if ${\displaystyle (\mathbb {Q} (t),\mathbb {P} (t))}$ is a solution of Hamilton's equations so is ${\displaystyle (\mathbb {Q} (-t),\mathbb {P} (-t))}$[1]: in other words if we have a solution of Hamilton's equations and we reverse time we obtain another valid solution and so another physically possible evolution for the system[2].

However, this time-reversal invariance does not hold any more if we consider macroscopic systems. For example, consider the two following processes:

• the equilibration of a cold and a hot body into two bodies with the same temperature
• the adiabatic expansion of a gas: a box is divided by a wall into two halves of volume ${\displaystyle V}$, one completely empty and one containing a gas of ${\displaystyle N}$ particles and energy ${\displaystyle E}$; opening a hole in the wall the gas will spontaneously fill all the volume of the box. This is clearly an irreversible process so the entropy of the system increases (while the energy of the gas, and thus its temperature, does not change)[3]

It is clear that they will never occur spontaneously in reverse, and that in order to do so we must do some kind of work on the system: two bodies in thermal contact at the same temperature will never spontaneously undergo a transformation at the end of which they will have two different temperatures (we must use some mechanical work to transfer heat), or a gas contained in a box will never fill on its own only one half of the volume (we must compress it)[4].

Therefore, entropy seen as a "measure" of irreversibility is what gives us the definition of the arrow of time, namely it allows us to distinguish past from future. We will cover the relationship between entropy and time in more detail in Entropy and the arrow of time.

1. This is true in all physical cases. One can always find "pathological" Hamiltonians for which this is not true, but generally they have no physical significance; of course we are thinking about classical systems, otherwise we can easily find systems which are not time-reversal invariant (as known, the laws of particle physics are invariant only under ${\displaystyle CPT}$).
2. "Visually", we can think to watch a video representing the motion of the particles of our system: if we rewind it we obtain another possible microscopic evolution of the system.
3. We can also see this from the microcanonical definition of entropy. In fact the phase space volume of the system in the two cases is:
${\displaystyle \Omega (V)=\int _{V}d\mathbb {Q} \int _{\mathbb {R} ^{3N}}d\mathbb {P} \delta (E-{\mathcal {H}})\quad \qquad \Omega (2V)=\int _{2V}d\mathbb {Q} \int _{\mathbb {R} ^{3N}}d\mathbb {P} \delta (E-{\mathcal {H}})}$
and since the integrand is always positive and the domain of integration in configuration space is larger after the adiabatic expansion, we indeed have ${\displaystyle \Omega (2V)>\Omega (V)}$ and so the entropy increases in the process.
4. "Visually", as before, this means that if we are watching a video of the evolution of a macroscopic system we can always tell if the video is being rewinded or not.