Playwright Tom Stoppard has made the St. Petersburg paradox well known to the modern audience when his two characters in Rosencrantz and Guildenstern are Dead defy the laws of probability as Rosencrantz, for over 90 coin flips without a break, lands a coin on its head, thereby “winning” the coin. While “heads” is what gamblers aim for—after a series of tails in order to up the ante—the paradox is that reality differs from the expected outcome. In a recent report in Physical Review Letters, Jake Fontana of the US Naval Research Laboratory and Peter Palffy-Muhoray of Kent State University write, “Here, we demonstrate a physical realization of this paradox using tensile fracture.” Their choice of material is fiber, specifically polyester and polyamide. However, the researchers are not strictly concerned about materials failure as much as they are about leveraging materials science to show that this model can be useful in a broad range of fields—including weather forecasting or financial markets—where failure and reliability are essential.     

“Failure is a fascinating example of systems whose behavior is dominated by rare events,” Fontana tells MRS Bulletin. “What could be more intriguing than looking for manifestations of the most improbable things that are hiding out there?” 

The St. Petersburg paradox was introduced by Nicolas Bernoulli in 1713, and continues to be a reliable source for insights in decision theory. “The St. Petersburg paradox, … and its connection to failure proposed nearly 20 years ago, is confirmed experimentally in Fontana et al. They perform a challenging set of fiber strength experiments on a deceptively simple system, a one-dimensional fiber,” says Irene Beyerlein of the University of California, Santa Barbara, who was not involved in the study.    

The well-known Weibull analysis is currently used to model the probability of fracture; however, it is a theory of probability based on empirical evidence, according to Fontana and Palffy-Muhoray. Since the failure of materials is due to the rare occurrence of large defects, the researchers turned to the comparable probability theory of St. Petersburg paradox—which, in gambling terms, would be called the theory of “long shots.”

Considering that the force necessary to fracture the fiber is a linear function of the size of the defect in the fiber, then—the researchers surmise—the force needed to fracture the fiber “can be taken to depend linearly on the logarithm of the fiber length,” which is in alignment with St. Petersburg paradox.  

“Since larger and larger defects are less and less likely, increasing the sample size by a constant factor increases the (largest) defect size by a constant amount. This is why the dependence on sample size is logarithmic,” Fontana says.

The researchers plotted tensile strength for polyester and polyamide fibers at lengths from 1 mm to 1 km. They fit the datasets using both the St. Petersburg and Weibull models. They found that the former proved a closer match in agreement with experimental results.

However, as seen in the graph, both models are comparably accurate. What benefits the St. Petersburg model over Weibull, according to Fontana, is “simplicity—we revealed that materials failure can now be viewed as a game.”

Zdeněk Bažant of Northwestern University, who was not involved in the study, commented to MRS Bulletin that “the direct application of the St. Petersburg model to mechanical failure of fiber is too simplistic.” Bažant and two of his colleagues communicated to MRS Bulletin, “In the paper, the defect size is defined in the direction of the specimen length, but with no consideration of damage localization. The St. Petersburg model considers that the length of the damage zone varies with the specimen length, which contradicts the quasibrittle fracture mechanics, particularly the inevitability of damage localization. The length of the damage localization band is a materials property.”

Fontana says, “We provided a simple relationship between the ultimate strength of some materials and their size, demonstrating that defects in materials accumulate similarly to profits in the St. Petersburg game and the strength of materials depends on the logarithm of their size.”

“This validation could open the way for some applications in the study of failure mechanisms in high strength materials for load-bearing applications,” Beyerlein says.

Read the abstract in Physical Review Letters.