*The science of collaboration*

**Name: **Tatyana Ehrenfest-Afanasyeva**Nationality:** Russian/Dutch**Lived: **1876-1964**Fields**: Mathematics, physics and education**Claim to fame:** Made major contributions to the foundations of statistical mechanics and statistical thermodynamics

Close to the turn of the 20th century, a young Russian student, Tatjana Afanasyeva (1876-1964), moved to Germany and met the physicist Paul Ehrenfest. He was to become her collaborator husband, and the two would embark on a career of fascinating scientific discovery.

### A meeting of minds

Tatyana Alexeyevna Afanasyeva was born in Kiev, then part of the Russian empire. When her father died, young Tatyana was raised by an uncle and aunt in St Petersburg. In the imperial capital, as in all Russian territory, women were not permitted to attend universities or other prestigious institutions. Instead, Tatyana completed her education at a women’s school where she was trained as a teacher and took some science-related courses.

In her mid-20s, she decided to leave Russia and continue her studies in Göttingen, Germany which, at that time, was a major centre for mathematics and physics. There, Afanasyeva met Austrian physicist and mathematician Paul Ehrenfest. Paul wondered why Tatyana did not come to the mathematics club meetings, and when he discovered that it was because women were not allowed to attend, he battled to get the rule changed. The friendship that developed between Tatyana and Paul led to their marriage in 1904. Over the years, the partnership enabled the scientific talents of both to fl ourish.

#### Faith no more

In 1907 the couple returned to St Petersburg where, under Russian law, a union between two people of different religious persuasions could not be sanctioned. As a Jew, Ehrenfest could not live with his wife, who was a Russian Orthodox, unless the two declared they were not affi liated to any religion. They both officially renounced their religions so that they could live in Tatyana‘s homeland. During that time Tatyana began to develop a new approach to teaching geometry and mathematics.

In 1912, Ehrenfest was given a professorship at the University of Leiden in the Netherlands, and the couple moved again. They might have gone from there to Prague, where Albert Einstein invited Ehrenfest to succeed him as Professor of Theoretical Physics, but at the time it was not possible for professors without religious faith to be appointed to universities in the Austro-Hungarian Empire. The two remained in the Netherlands throughout their career.

### Explaining the fundamentals

After the untimely death of physicist Ludwig Boltzmann in 1906, the editor of the *Encyclopaedia of Mathematical Sciences* asked Ehrenfest to write a review of statistical mechanics based on Boltzmann‘s work. Boltzmann had been one of the key supporters of atomic theory (a controversial topic at the time); the importance of his work may have been overlooked without the efforts of Tatyana and her husband. Together, the duo spent several years writing the piece and completed their classic review, The conceptual foundations of *the statistical approach in* *mechanics, in 1911*.

The Ehrenfest-Afanasyeva review detailed the work of the Austrian physicist and his school, explaining statistical thermodynamics and irreversibility, and highlighting some fundamental contributions to statistical mechanics. This field of physics uses probability to predict the behaviour of a complex system. Statistical mechanics paved the way for the development of mathematical models that are used today to understand the behaviour of complex systems.

The review added significant value to Boltzmann‘s original work, and was widely praised for its clear style. Tatyana and her husband explained the principles of statistical mechanics using well-chosen examples, and combined a logical analysis of the key theories with a full breakdown of the unresolved issues. The couple‘s ability to communicate these complex ideas so succinctly made the information more accessible to scientists, and contributed in no small part to advances in the field.

#### Influence on the home front and beyond

Ehrenfest-Afanasyeva had a unique and rigorous approach to examining fundamental issues, and was mainly interested in questions of entropy and chance. Knowledgeable, quick-thinking and with a critical mind, Tatyana was passionate about reforming mathematical pedagogy. She organised a series of workshops for teachers, and in 1924 published an elaborate didactic method for teaching geometry that received significant attention. Between 1926 and 1933, she set out a program for teaching geometry and mathematics in Russia.

The results of Tatyana‘s collaborative work with Paul were largely credited to her husband, whose contributions to science are well known. The couple had two sons, one of whom, Wassik, had Down‘s syndrome, and two daughters. Tragically, Paul ended his life in 1933, first shooting their son Wassik before turning the gun on himself.

Two of the children followed in their parents‘ scientific footsteps: their son Pavlik became a physicist and daughter Tatyana followed a career in mathematics. The younger Tatyana studied both in Leiden and Göttingen, and received her PhD from the University of Leiden in 1931; she is credited with having made significant contributions to mathematics.

### Scientific achievements

Tatyana Ehrenfest-Afanasyeva worked closely with her husband, Paul Ehrenfest, to write the famous review of the work of Austrian physicist Ludwig Boltzmann and his school. She also conducted widespread research on the science of teaching, randomness and entropy, and substantially influenced the work of her husband throughout his productive career. Ehrenfest-Afanasyeva’s contribution to science is documented in monographs and articles she published in Russian, Dutch and German. Her work in mathematical pedagogy, which was originally met with much criticism, laid the foundations for the world-renowned mathematics teaching reform that gained popularity during the 1970s.