What is the idea behind coarse-graining?

Question

I don't think I fully understand the idea behind coarse-graining. I will elaborate. I was reading some lecture notes on statistical field theory and the text begins with some previous analyses on the $d$-dimensional Ising model. Then, the author uses the process of coarse-graining to get a more generalized version of this model, where the order parameter (magnetization, in case of the Ising model) becomes a continuous function $m(\vec{x}) \in [-1,1]$ of the position $\vec{x}\in \mathbb{R}^{d}$. This is what bothers me.

I am convinced that some systems cannot be explained through such simplistic models such as the Ising model, and it is important to consider order parameters depending on $\vec{x}$, so we can study nonhomogeneous systems too. What I don't seem to understand is why to coarse-grain an Ising (or some other simple) model to get this generalization. Landau's theory is full of hypothesis and approximations, so it would be natural to me if one said "ok, let us assume that Landau's theory generalizes to systems with more general order parameters (such as those depending on $\vec{x}$ or something else) and let's move on to the Landau-Ginzburg theory without further justifications." This would be just another postulate in the theory , in my point of view. To coarse-grain a system, in my understanding, sounds like trying to "deduce" or "justify" the generalization by turning it into something almost systematic. But (I guess) not every model in Landau-Ginzburg theory comes from coarse-graining some simpler model. Besides, what does one want when one uses coarse-graining in models like the Ising model? Aren't the previous analyses of (say) the Ising model enough so we really need to coarse-grain? Aren't we changing the essence of the model when we do that? Or, in case of Ising model, it is just a matter of didatic motivation? Furthermore, why turning it into something systematic and not just another postulate about the generalization of simpler models?

Answer 1

The crucial point is that, for a particular ("critical") temperature, the Ising model is (approximately) scale invariant. That is to say, the effect successive coarse-graining leads to a partition function with the same low-energy physics as the initial one. This procedure also works if one is close to the critical temperature, where the system is no longer invariant under a coarse-graining (or "renormalization group") transformation, but one can estimate how it behaves on long length scales by the procedure. (More concretely, for small deviations from the critical temperature, there is only a single large length scale which is relevant to the physics of the system, and a coarse graining procedure leads to a transformation on this length scale.) Essentially any time we are performing this renormalization group procedure, we are implicitly assuming that our system is close to being scale-invariant.

Therefore, if one is interested in the near-critical Ising model, the Landau-Ginzburg theory one obtains by coarse-graining the model should (and does) have the same low-energy physics and scaling properties as the Ising model, so one is justified in working with that instead. But if one is far from the critical point, then you're correct that a Landau-Ginzburg theory is not necessarily a good/useful description.

Answer 2

A couple points to add onto the other answer:

it is important to consider order parameters depending on $\vec{x}$ , so we can study nonhomogeneous systems too.

It's important to stress that the spatial dependence of the order parameter is not (necessarily) used to model nonhomogeneous systems. The Ising model is a perfect example: it's completely homogeneous everywhere, but you need to allow for spatial dependence of your order parameter to capture spatial fluctuations in your system. For example, if you want to ask the question: "how correlated are these two spins a distance $r$ away?", it is not enough to employ the zeroth level mean field theory where the whole system is uniform.

To coarse-grain a system, in my understanding, sounds like trying to "deduce" or "justify" the generalization by turning it into something almost systematic.

In general, you rarely (read: never) course grain by actually performing some sort of systematic averaging. It's a nice picture to have in the back of your mind, so that you know you're still describing the same system when you go from the microscopic model to the course-grained model. But in all but the most uninteresting systems, actually performing the course-graining step is neither possible nor necessary: to obtain the universal behavior, it is sufficient to write down the most general Landau-Ginsburg field theory which captures all of the symmetries of the microscopic model.

Besides, what does one want when one uses coarse-graining in models like the Ising model? Aren't the previous analyses of (say) the Ising model enough so we really need to coarse-grain?

There are two points to this. First, while the 1D/2D Ising models are "special" enough to admit exact solutions, most models do not. For sufficiently complicated models, extracting the critical behavior from the microscopic model is difficult or impossible. In that sense, it is purely didactic to course-grain the Ising model. But perhaps more importantly, a key insight gained from this course-graining picture is that different microscopic models have the same course-grained Landau-Ginsburg theory. You could consider an Ising model with just nearest-neighbor interactions, or you could include next neighbor interactions, or you could include 10th-next nearest neighbor interactions, or something even crazier. But these all course-grain to the same Landau-Ginsburg theory, which gives you the hint that all these models behave identically near the critical point.

Aren't we changing the essence of the model when we do that?

There are of course some things you lose in the process -- if we retained full knowledge of the microscopic model after course-graining, then it clearly couldn't be any more powerful than using the microscopic model itself. But the key point is that the long wavelength physics is identical, which means that course-graining captures the same physics in the vicinity of the critical point.