By Cleo Xu ’26
In the summer of 2024 and continuing into the school year of 2024-2025, Conor Scheidt ‘25 worked on ways to model the diffusion and interaction of molecules with mentor Todd Gingrich. Traditionally, this is done by considering molarity—the moles of molecules per liter of volume. However, Scheidt realized that sometimes interesting chemical reactions are “averaged” out using this method, which results in a smooth curve. To avoid getting an averaged function, he considered analyzing individual molecules, or more specifically, the different states in which a given number of molecules can be. This method is not restricted to just microscopic particles, the generalization can also be used in modeling the number within a species in biology or even the cells within our body.
To undo the “average” though, is not as easy as it may seem. There are pre-existing methods, which allow one to run a number of simulations graphing the change of mass/molarity with respect to time and then obtaining a distribution. However, most methods are time-consuming and require a large amount of computing power. The difficulty lies in the number of simulations one must run in order to obtain a precise distribution. Scheidt pointed out that instead of brute-forcing to generate a graph, it is possible to reverse-engineer the process and to think about what such a distribution fundamentally represents—a probability graph.
To start, Scheidt assumed that this probability graph takes the form of a wave (Psi) and by borrowing concepts from quantum wave mechanics, one could conclude that the change of Psi with respect to time equals Psi being operated by a Hamiltonian. Usually, the Hamiltonian is introduced in a more physics-related context, namely Shrödinger’s equation, where it takes into account both the kinetic energy and potential energy of the system. The equation can be rearranged to obtain the relation: Psi(t)=e^(Ht)psi(0).
Consequently, the problem of modeling is broken into two parts: first, finding a good expression for H and second, finding the wave function Psi. In order to understand the expression Schedit used for H, let’s first explore the basic concepts of a lattice. A lattice is a structure that gives one information–a number, vector, or tensor–in a quantized space and time. Unlike a field, such as the gravitational field in physics, a lattice is not continuous and has discrete values at each point—hence why it is referred to as “quantized.” Now, assume we have two regions, A and B, and we have some molecules floating inside them. The molecules can move in any direction they desire, but practically, Scheidt only needed to consider two different movements of the molecule: cases where the molecules remain in the same region they started with fall and situations where the molecule cross regions (the latter is called “diffusion”). Interestingly, due to the matrix property of H, one can directly add up the H for diffusion and the H for interactions within the lattice region.
The Psi function is slightly more difficult to obtain. When one has a huge number of species to consider, where each species can have up to 100 states, the calculations become unreasonable to render—even for a computer. Scheidt decided to use a tensor network to simplify the process. Scheidt reasoned that all the states can be expressed in Matrix Product State (MPS). The advantage of considering the data in terms of a tensor network is that now it is possible to use matrix decomposition to create three matrices from the original matrix: one with horizontal rows, one with vertical columns, and one with diagonal elements. The diagonal matrix takes into account how the two other matrices interact with each other, which is where the simplification comes in. Scheidt was able to reduce the number of elements he needed to compute by setting certain parameters and crossing out the ones that did not meet the requirement without any loss of information.
After one has both H and Psi, the wave function is almost built except for the time component (the goal is not to model a diffusion at a specific point in time, but rather obtain a general trend of how the diffusions change over time). The way to introduce the time component is through an algorithm called TDVP—time-dependent variational principle. TDVP essentially “wiggles” a little part of the MPS. Now the entirety of the function is assembled: dPsi/dt = e^(Ht) * Psi.
There are still many difficulties this comparatively more approachable method faces, such as the fact that the term (Ht) is raised to the power of (e), rendering the calculation, despite the shortcuts and clever mathematical maneuvers, still a challenge. Another aspect is dealing with the “dimensions” of the matrix approach. Careful consideration of any limitations this method may pose is also needed. Scheidt continues to address these issues and work on this project at Andover.
For more information, read: https://journals.aps.org/prx/pdf/10.1103/PhysRevX.13.041006
Read more articles like this in our Fall 2024 Issue!
