Here I will only discuss how to compute the four point susceptibility using the known trajectory of many particles. In our work we focus on tracking the diffusive motion of individual colloids in the supercooled or glassy state. So this definition will apply to such situations. This definition will also apply to molecular dynamics or Monte Carlo simulations where the trajectories of the molecules are known. Other methods for measuring the four-point susceptibility include using light scattering and electric susceptibility data.

Let's start by assuming that we have N particles of known trajectory r(t), where r is the position of the particle at time t. The simplest definition of the four-point susceptibility χ₄ is the fluctuations in time of the percentage of particles that are mobile. To clarify this definition we need to define which particles are mobile and which are not. We define a particle to be mobile if its displacement over a time interval Δt is larger than the some threshold displacement ΔL.

So how do you choose Δt and ΔL? If there are any particular length scales and time scales for which you want to know the dynamic heterogeneity you can pick those values. A more common approach is to vary the two numbers until you find which pair of values maximizes χ₄ because this is the length scale and time scale where the dynamics are most heterogeneous.

Ok, back to the math. Once Δt is chosen, we can define a displacement

Δr_i(t) = r_i(t + Δt) - r_i(t),

where i indexes the ith particle at time t. Using the displacements and the chosen value of ΔL we can define which particles at time t are mobile. After labeling each particle mobile or immobile the percentage Q(t) of mobile particles can be computed. Note that for each time t there is a value of Q associated with that time. For instance, if there are 100 different moments in time where you can compute a displacement, then you should have 100 different values of Q.

Now comes the simple part. The four-point susceptibility is the product of N and the variance in Q(t) with time, where we write this as

χ₄ = N(⟨Q(t)²⟩ - ⟨Q(t)⟩²).

The reason we introduce the factor of N is because the variance in Q is inversely proportional to N. Therefore if we have 10 times as many particles the variance in Q decreases by a factor of 10, but the χ₄ remains constant.

There are other slight mathematical ways to calculate χ₄ by changing the definition of Q. We can write our previous definition of Q mathematically as

Q = ⟨Θ(Δr - ΔL)⟩_i,

where the average is carried out over all the particles at time t and Θ is a heaviside function. The way we think of Q is that it's an average of weights where we use the displacements to compute the weights. In the previous example we used the heaviside function to weight the displacements. However there are other ways we can weight the displacements. To give one example we could use an exponential such that

Q = ⟨exp(-Δr/ΔL)⟩_i.

Besides the two examples given so far, there are other ways the displacements can be weighted.

Finally, what does χ₄ tell us? In the first definition I provide, χ₄ is a measure of the average number of particles whose dynamics are correlated. This is important because in both molecular and colloidal glasses correlated dynamics are an essential feature that gives rise to the anomalous behavior of glassy materials. Quantifying how the correlated dynamics change with temperature for molecular glasses or with volume fraction for colloidal glasses provide insight on how the correlated dynamics change as the glass transition is approached. The four-point susceptibility is just one tool out of many for quantifying the correlated dynamics.

To see how we use χ₄ to study correlated dynamics in supercooled colloidal glasses read the article below. This article also discusses other methods we used to quantify the correlated dynamics.

• "Spatial and temporal dynamical heterogenities approaching the binary colloidal glass transition"
  T Narumi, SV Franklin, KW Desmond, M Tokuyama, & ER Weeks, to be published in Soft Matter
     Abstract / PDF