### How a physicist implemented a computer algorithm that changed statistics for every field.

What happens inside ice when it melts? How do atoms arrange themselves into sparkling facets on a crystal? Where does the delicate, lacy structure of a snowflake come from?

These are hard questions to answer, because each of these systems involves so many different tiny atoms - there are a whooping 10^23 molecules just in a gram of water! There are so many of them that it’s not a simple matter of putting them under a microscope and watching them as they crystallize or melt. Instead, we have to get creative to understand the microscopic motion of the matter that makes up our world. One tool we have is a computer. Since there are too many atoms to understand all of their interactions, we need to come up with clever ways to model how atoms zoom around. These models are called simulations, and the sets of steps we need to achieve them are called algorithms.

One of the first people to make these kinds of simulations possible was a physicist named Arianna Wright Rosenbluth. The key idea behind the simulations she programmed is that instead of keeping track of all the ways atoms can move, we mostly only need the ones that are most likely. This makes the problem dramatically easier for the computer. The algorithm she wrote and programmed in the 1950s is very efficient and still hugely important to science today. In fact, Dr. Rosenbluth was instrumental in changing the way we think about probability, statistics, and how we know what we know.

In [Arianna Wright Rosenbluth's] effort to answer the simple question of how solids melt into liquids, she developed one of the most important computer algorithms of all time.

## A sharp mind

Arianna Wright Rosenbluth was born in Texas in 1927. As a young woman, she loved reading, especially books like The Wizard of Oz series, and fencing. She fenced competitively throughout college, won duels against both women and men, and even qualified for the Olympics twice.

Arianna was also a brilliant student. She graduated from college when she was only 18 years old, with honors in both physics and math. After college, she moved to Boston to continue her education. She got a master’s from Radcliffe College, the women’s college associated with Harvard University, and later a PhD in physics from Harvard. At the time, there weren’t many women in her department. Arianna was even told by one professor she wanted to work with that he wouldn’t accept any women into his lab. But she kept learning and doing research, and at the age of 22, she became the 5th woman ever to receive a doctorate in physics from Harvard.

## MANIAC at work

When Dr. Rosenbluth was starting her scientific career, there was one topic in physics that loomed larger than any other - how to harness the power of atoms. Just a few years prior, the Los Alamos Laboratory had developed the world’s first atomic bombs, weapons that would be used in deadly attacks against the cities of Hiroshima and Nagasaki. But that was unfortunately just the beginning of a race to develop more and more deadly weapons exploiting this atomic power.

Arianna joined Los Alamos as a staff scientist in 1951, along with her husband, Marshall Rosenbluth. At the time, the laboratory was focused on developing new types of weapons based on fusion–merging two atoms together into one–instead of fission–splitting an atom in two. Arianna Rosenbluth did some work verifying calculations for fusion-based bombs, but soon started focusing on a new instrument that was being built at the lab: a computer called MANIAC I (Mathematical Analyzer Numerical Integrator and Automatic Computer Model I).

MANIAC I was what was called a “fast computing machine.” It was massive, weighing over 1000 pounds, and incredibly temperamental. Any directions given to the machine had to be translated onto punch cards, which were sheets of paper with holes punched into it in a sequence that the computer could understand. Dr. Rosenbluth was one of the few people at Los Alamos who knew how to code and use the machine. But what’s more, she also was a physicist who could understand and develop new algorithms for the machine to implement. The combination of these skills would prove really important for her next project.

## An algorithm for best guesses

Arianna and a team of other scientists, including her husband Marshall, Nicholas Metropolis, and Augusta and Edward Teller, wanted to figure out how to calculate what’s called an equation of state. This is an equation that describes how different variables we can measure–like temperature, pressure, or volume–depend on each other in complex systems of many particles all moving around. These kinds of equations are crucial for a field of science called statistical mechanics. We can think of statistical mechanics as the connection between things that are very tiny and behavior that we can see. It describes how a lot of small things all moving around randomly can come together to act in a predictable way.

Dr. Rosenbluth and the other scientists wanted to see if this new computer could help them understand the messy system of lots of atoms moving around. To simplify matters, they pretended that their atoms were instead small circles. They wanted to see if they could model those “atoms” undergoing an important physical process called a phase transition, like melting from a solid to a liquid. But that simulation demanded a lot of memory and computing power, more than the rudimentary MANIAC computer had. Instead, they needed a clever trick to solve these equations without simulating the motion of every single particle.

The team ended up using a method called a Monte Carlo algorithm to solve this problem using randomness. Monte Carlo is a famous casino, where games are played based on the roll of a die or the random motion of a ball. While Monte Carlo algorithms do use probability to solve problems quickly, not all guesses are equal – the algorithm succeeds by prioritizing the best guesses. To use these methods for atomic problems, Dr. Rosenbluth and the team at Los Alamos simulated a bunch of configurations of their circular “atoms.” Although they used random sampling to help them pick these configurations, the snapshots they took of their atoms weren’t entirely determined by chance. Instead, they looked more at configurations that have a lower energy and are more likely to exist. Their algorithm goes like this:

Simulate a random snapshot of the particles and calculate its energy.

If the energy is lower than the previous state, accept that snapshot as a “step” in the simulation.

If the energy is higher, randomly decide to accept or reject this snapshot, but weigh that acceptance probability based on how low the energy is.

This penalizes the more rare, high-energy states and focuses the simulation on the most likely situation, allowing simulations of lots of particles to be run much more quickly and efficiently than was ever possible before.

Arianna Rosenbluth wrote this algorithm into computer code and figured out how to make it work. That implementation became known as the Metropolis algorithm, after the first author listed on the paper, but Dr. Rosenbluth’s contributions to the project are so important that some have proposed renaming it to the Rosenbluth algorithm. The algorithm is part of a larger family of algorithms called MCMC methods, or Markov chain Monte Carlo algorithms. Now that we have even more powerful computers to use these algorithms, they have become incredibly important statistical tools. Even though they were originally written to model atoms, MCMC methods are now used for unrelated problems. They’re used in almost every field of science, from astronomy to biology to data science.

## An Exceptional Legacy

Arianna Wright Rosenbluth’s career in physics was unfortunately a very short one. She left the field after having her first child, since it was expected at the time that women would remain in the home and focus on family life. But even though she was only a professional scientist for about a decade, the effect that she had was massive. In her effort to answer the simple question of how solids melt into liquids, she developed one of the most important computer algorithms of all time. The legacy of her research has changed the fundamental way scientists think about probability and inferring the truth from data. To date, more than 50,000 scientific papers have cited the influence of her work on the Metropolis Monte Carlo algorithm – an algorithm we may someday remember as the Rosenbluth Monte Carlo algorithm.

Written by Caroline Martin

Edited by Katie Fraser and Yanting Teng

Activities by Taylor Contreras

Video by Caroline Martin and Madelyn Leembruggen

Sources

## Learn more about the methods Arianna used!

Estimate (20-30 min)

Probability and randomness can be a helpful tool in many ways. Lets estimate the area of shapes using a Monte-Carlo method. You will need a piece of paper and a marker.

Start with a square piece of paper, with a known area.

Draw any closed shape you would like inside the square.

Now random mark 100 dots on the circle.

Count the number of dots on the inside of the circle

The estimated area is then area of the square times the number of dots / 100

Play (10-20 min): When isolated, a particle will move randomly due to its kinetic energy. Check out this activity to see if you can predict the random walk of this particle to beat the game! You can think about how difficult a problem it would be to try and predict the random walk of many particles that can also bump and interact with each other, like Arianna and her team set out to solve.

Investigate (30-40 min): Arianna studied how to predict how many atoms will interact with each other under different conditions like temperature and pressure. An atom can move randomly, however, when affected by temperature, pressure, and the atoms around it, it will have some probability to move in any direction. This transition probability can be described by a Markov chain. Learn more about Markov chains here.