Written by: Simon Nirenberg ‘28
Edited by: Parsa Lajmiri ‘26
In an increasingly interconnected world, Monte Carlo methods have emerged as a powerful example of how disparate fields—from quantum chemistry to economics—can converge through shared mathematical techniques.
The Birth of Monte Carlo: A New Kind of Estimation
The year was 1946. Physicist Stanisław Ulam was working at the Los Alamos National Laboratory on nuclear fission technologies when he encountered a perplexing roadblock: calculating the paths of neutrons as they diffused through atomic bomb cores. Despite repeated attempts, no exact equation could capture the probabilistic nature of neutron diffusion. During an illness that sidelined him from work, Ulam passed the time by playing solitaire. As he played, he began to wonder how he might be able to make predictions for the course of the game:
“After spending a lot of time trying [combinatorics], I wondered whether a more practical method … might not be to lay it out … one hundred times and simply [count] the number of successful plays.”
Rather than solving for an exact equation to predict outcomes, Ulam realized that he could simulate many individual paths and construct a probability distribution to use instead. It quickly struck Ulam that the same approach could be applied to his neutron problem [1]. He shared this insight with his colleague, John von Neumann, and the two got to work formalizing the method, which they dubbed "Monte Carlo" after the famous Monaco casino. Unlike casino games, however, Monte Carlo would turn out to be more than profitable in the long run.
The Fundamentals of Monte Carlo:
Modern Monte Carlo methods retain the fundamental simplicity that Ulam recognized in his approach to solitaire. Instead of calculating expectation values and outcomes using complicated, exact mathematics, Monte Carlo methods take large numbers of random samples and use their aggregate behavior to approximate properties of the distributions that underlie them—the more samples taken, the higher the accuracy of the predictions. This makes Monte Carlo an attractive method for solving complicated integrals, differential equations, and other complex systems driven by many hard-to-measure variables [2]. A basic application of Monte Carlo to an integration problem is shown in Figure 1.
Expanding Horizons: From Physics to Finance
In its early years, Monte Carlo remained a niche method, primarily limited to complex problems in particle motion and atomic physics. However, in 1964, economist David Hertz introduced Monte Carlo methods to finance, using them to predict market and portfolio risks [3].
Hertz’s reasoning was strikingly similar to Ulam’s. In theory, financial quantities, like stock prices, should be exactly predictable—but in practice, doing so would require intimate knowledge of nearly everything about the world at the time of calculation. Before Hertz, financial analysts typically relied on historical trends, which could only account for one potential future outcome. Hertz reimagined stock prices as undergoing a random walk—a process in which a particle decides to move up or down at each moment based on a mix of random chance and the conditions of the moment just before it. By adding terms for trends and volatility, Hertz was able to run Monte Carlo simulations for thousands of potential future stock price trajectories, generating a probability distribution for various financial outcomes [3].
Though financial Monte Carlo methods have grown more sophisticated since Hertz’s time, the core idea remains the same: leverage random sampling to explore possible market conditions without relying solely on previous data [2].
Unifying Fields: Quantum Monte Carlo and Beyond
Today, Monte Carlo methods have expanded far beyond physics and finance. In particular, Quantum Monte Carlo (QMC) has grown to immense prevalence in computational chemistry and condensed matter physics due to its relatively efficient and accurate approach to the quantum many-body problem, which describes the difficulty of modeling energetics and dynamics in quantum systems of 3 or more particles [4]. This challenge is fundamental to a wide array of modern fields, including computational pharmaceutical development, crystal structure prediction, and materials engineering.
Monte Carlo has also seen widespread use in ecology, where researchers have applied it to model population dynamics and species interactions under uncertainty [5]—a subject becoming increasingly urgent given the looming impacts of climate change.
Conclusion: Embracing Uncertainty
Time and again, mathematical methods have demonstrated their ability to unite seemingly disparate fields. However, while many such methods, like calculus, rely on exactness and continuity, Monte Carlo uniquely thrives on randomness, welcoming the unpredictability of real-world systems rather than shying away from it. In a culture that often prioritizes organization and structure, the effectiveness of Monte Carlo serves as a powerful reminder that embracing randomness can lead to valuable insights and innovations. Ultimately, it is through recognizing the value of both order and disorder that we can navigate the complex nature of both science and everyday life.
References:
Eckhardt R. Stan Ulam, John von Neumann, and the Monte Carlo method. Los Alamos Science. 1987;15:131-7.
Kenton W. Monte Carlo Simulation: What It Is, How It Works, History, 4 Key Steps. Investopedia. Updated 2024 Jun 27. Available from: https://www.investopedia.com
Hertz DB. Risk analysis in capital investment. Harvard Business Review. 1964;42:95-106.
Mareschal M. The early years of quantum Monte Carlo (1): the ground state. EPJ H. 2021 Dec;46(1):11.
Link WA, Cam E, Nichols JD, Cooch EG. Of Bugs and Birds: Markov Chain Monte Carlo for Hierarchical Modeling in Wildlife Research. The Journal of Wildlife Management. 2002 Apr;66(2):277.