It is a simple fact the entropy in the exact meaning in dynamical system does not change over time if the system is measure-preserving and ergodic. However, it is often said that the coarse-grained entropy increases over the time. I was wondering to know is there any rigorous proof for this statement?

The coarse-grained entropy increases over the time.

  • $\begingroup$ I don't think there is a proof of the statement. Otherwise it wouldn't be called the second law. Furthermore, I think its not even true as it is written. For any closed system one can see that there is a time (Heisenberg time) when the state will return to itself. Thus the statement is only true at times smaller than Heisenberg time (which is very long indeed). $\endgroup$ Mar 6, 2018 at 9:14
  • $\begingroup$ closely related question: physics.stackexchange.com/q/202522/226902 $\endgroup$
    – Quillo
    Sep 21, 2022 at 15:21

3 Answers 3


The probability density $\rho(p(t),q(t))$ of an Hamiltonian system follows Liouville's equation:

$$\frac{d\rho}{dt}=0 \tag{1}\label{1}$$

The intuitive interpretation of \ref{1} is that the probability density evolves in phase space like an incompressible fluid: it can stretch and deform, but its volume in phase space remains constant (figure below, from Wikipedia, shows a 1D example).

enter image description here

This means that if you define the entropy $S$ as

$$S=-\int \rho \ln(\rho) \ dp dq$$

Then $S$ is a conserved quantity:

$$\frac {dS} {dt} = 0$$

The solution proposed by Gibbs is that $\rho$ is actually coarse grained. This means that there is a maximum amount of "stretching" $\rho$ can undergo: if it is stretched further, the volume it takes in phase space will increase. I tried to sketch a 1D example below: on the left you have a "fine-grained" $\rho$, which follows equation \ref{1}, while on the right you have a coarse-grained $\rho$, which takes more phase space volume as it evolves in time.

enter image description here

The problem with the coarse grained entropy is that it is not clear what is the mechanism that should led to a coarse graining of $\rho$. It could be that phase space itself is quantized, as the Heisenberg uncertainty principle seems to suggest. Or it could be, as suggested by Jaynes (E.T. Jaynes; Gibbs vs Boltzmann Entropies; American Journal of Physics,391,1965), that blurring of entropy simply reflects our loss of knowledge about the system. Blurring can be the mechanism that connects the reversible microscopic dynamics to the irreversible macroscopic dynamics, but at the moment there is no consensus in the scientific community about the exact nature of the blurring mechanism.


I love @valerio answer, but I will give it a shoot with a very intuitive explanation.

Coarse grained entropy is an observer dependent quantity which can be though as measuring the ignorance of the observer about the system (or equivalently, the amount of information that can be extracted from the system). For instance observing a box full of bouncing ideal gas particles, I realistically ignore the exact (micro)state of the system, but I can measure that the macroscopic properties are the thermodynamic quantities (P,T,V,etc.).

Suppose now that something happen to the box. If the process is reversible, new microstates will become possible, but an equal number will become forbidden by the constraints given by the measured thermodynamic quantities. If the process is not reversible, more microstates will become possible with respect to forbidden ones. In effect now there are more possible microstates compatible with (P,T,V,etc.), therefore more entropy.

As a concrete example, think about a box containing white particles on one side and black particles on the other side, with a physical divisor in the middle. The act of removing the divisor imply that (besides velocity, position, etc,) now I'm also ignorant about the color of the balls in a given volume of the box.

In summary, in absence of microscopic measurements our ignorance of the system will go up or at best stay constant.


Very interresting stuff ! I wrote an article summarizing several aspect about coarse grained entropy. https://aurelien-pelissier.medium.com/on-the-conservation-of-information-and-the-second-law-of-thermodynamics-f22c0645d8ec

@Valerio's answer is a good explanation about what's happening on an abstract level (i.e. the phase space get more and more "filaments" which induce a "blurring"). Regarding possible explanations behind this blurring mechanism, an attractive one was proposed by Lloyd in 1988. He hypothesised that quantum uncertainty, and the way it spreads as particles become increasingly entangled, could be the true source of the blurring mechanism. In his view, information becomes increasingly diffuse, but it never disappears completely. States in equilibrium are maximally entangled and the approach to equilibrium can be thought of in terms of the spread of entanglement. So, although entropy increases locally, the overall entropy of the universe ($S_{tot}$) stays constant.

In practice, this mean that the growth of the entropies of the parts (the coarse grained entropy that we observe) is canceled by the growing negative entanglement entropy. That entanglement entropy ($S_{en}$) has no measurable consequences and the entropy described by the second law ($S_{obs}$) is the one we can monitor: $$S_{tot} = S_{obs} + S_{en}$$ Note that while technically, entropy is never negative, here $S_{en}$ is the conditional entropy of the hidden states ($H$) given the observable states ($O$) , which is can take negative values for quantum systems. $$S(OH) = S(O) + S(H|O)$$

  • $\begingroup$ Hi! Is this the paper you are referring to? $\endgroup$
    – Urb
    Apr 18, 2022 at 8:29
  • $\begingroup$ Hi, I was refering to Lloyd PhD thesis, where he first proposed his theory. Lloyd, Seth. Black Holes, Demons and the Loss of Coherence: How complex systems get information, and what they do with it. Diss. Rockefeller University, 1988. (web.archive.org/web/20120607195224/http://meche.mit.edu/…) $\endgroup$ Apr 18, 2022 at 22:36

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