What is the difference between fine-grained and coarse-grained entropy?

Question

So I am not a native English speaker, but I understand the words "coarse" and "fine" grained. But I really don't know what to make of them in the context of entropy. I have encountered this many times in the Preskill notes, but now these terms also appears in one paper I have to read and I really do not understand it. Can someone explain? I'll give you the context of the paper:

We wish to define a measure of the “lifetime” of information stored in a thermalized system, K, such as a black hole stretched horizon or a thermal cavity. Suppose at time t = 0, the system K is in some pure state ψ which is a typical member of a thermal ensemble with energy E and temperature T. Furthermore, let N(E) be the number of possible states at energy E. To begin with, the system has zero fine grained entropy and coarse grained entropy equal to log N.

So what do they mean with this and how are these different? I just want an explanation for both terms actually.

Answer 1

In general, a fine-grained description of a system is a detailed description of its microscopic behaviour. A coarse-grained description is one in which some of this fine detail has been smoothed over [1].

Coarse-graining is at the core of the second law of thermodynamics, which states that the entropy of the universe is increasing. It is important to recognize that a critical property of a coarse-grained description is that it is “true” to the system, meaning that it is a reduction or simplification of the actual microscopic details. It involves integrating over component behaviour. A wonderful intuitive description can be found in this answer.

Wikipedia says:

The exactly evolving entropy, which does not increase, is known as fine-grained entropy. The blurred entropy is known as coarse-grained entropy. Leonard Susskind in this lecture analogizes this distinction to the notion of the volume of a fibrous ball of cotton: On one hand the volume of the fibres themselves is constant, but in another sense, there is a larger coarse-grained volume, corresponding to the outline of the ball.

Mathematically speaking, fine-grained entropy $s$ is defined as the functional $$s = I[\rho] = \int \rho \ln \rho \ \mathbb{d}q \ \mathbb{d}p$$ over the whole space. It is also known as Gibbs' entropy or information entropy.

By Liouville's theorem, $\dfrac{d\rho}{dt} = 0$ and we find that $s$ remains constant in time, $\dfrac{ds}{dt} = 0$

Following Gibbs’ original idea, a fine-grained distribution $\rho$ can be coarse-grained by performing a local average over each cell (or partition) in phase space. Coarse-grained entropy $\bar{s}$ is defined as the functional $$\bar{s} = I[\bar{\rho}] = \int \bar{\rho} \ln \bar{\rho} \ \mathbb{d}q \ \mathbb{d}p$$ where $$\bar\rho = \sum_{C} p_{\rho}(C)\rho_{C}(\mathbf{x})$$ with $\sum_{C}p_{\rho}(C) = 1$ and each $p_C$ gives the probability of the system in Boltzmann state $C$. The following relation holds too: $$ p_{\rho}(C) = \int_{C}\rho(\mathbf{x}) \mathbb{d}q \ \mathbb{d}p = \int_{C}\bar{\rho}(\mathbf{x}) \mathbb{d}q \ \mathbb{d}p = p_\bar{\rho}(C) $$

The macroscopic properties of a fine-grained distribution $\rho$ are completely encoded in its coarse-grained version $\bar{\rho}$. As was established by Gibbs (also can be seen as an application of Jensen inequality) we find $$s \leq \bar{s}$$ A stronger result is that if the measure of the phase space is infinite while the measure of each partitioned cell is bounded then $\text{max}(\bar{s}) = + \infty$ as proved here.

Answer 2

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

The answer of @Abhay Hegde is a good one. But I would like to complement his answer with an intuitive example. Let’s take a system of particles expanding in a room, initialised in the bottom left corner. As an observer, we can separate our room into different “boxes”, and then count the number of particle in each box. These boxed will give us a coarse grained representation of the system. Assuming that the highest entropy state would correspond to the case where all boxes contain an equal number of particle, we can compute the entropy as:

$$S = \sum N_i \log(N_i),$$ where $N_i$ is the number of particles in each box. Interrestingly, the computed entropy depends on the grid resolution used to define the boxes. As we increase the resolution, our knowledge of the system gradually increases, up to the point where we have perfect knowledge of all the microstates when the grid size is very small (128x128). In this situation, as the hidden information is always zero, the entropy stays constant. Thus, for a very high resolution, our knowledge about the system is no longer coarse-grained, but fine-grained. That's because there will never be more than one particle in the same box.