# Bogolubov, Nikolai Nikolaevich

# BOGOLUBOV, NIKOLAI NIKOLAEVICH

(*b*. Nizhny Novgorod, Russia, 21 [old style 8] August 1909; *d*. Moscow, Russia, 13 February 1992),

*mathematics, theoretical physics*.

Bogolubov (the name can also be spelled Bogolyubov or Bogoliubov) was a prominent Russian and Ukrainian mathematician and mathematical physicist, one of the founders of non-linear mechanics and the quantum theory of many-body systems. Bogolubov also developed fundamental mathematical methods in kinetic theory, quantum statistics, quantum field theory, and the theories of superfluidity and superconductivity.

**In Lieu of Education.** Bogolubov descended from a family line of Russian Orthodox priests. His father, Nikolai Mikhailovich, taught philosophy and theology at a seminary in Nizhny Novgorod, and later at Kiev University; his mother, Olga Nikolayevna, gave music lessons. Nikolai was the oldest of the family’s three sons, all of whom eventually became prominent scientists. The boy’s childhood coincided with the turbulent period of the Russian revolution. He was mainly self-taught and received little formal education, other than a certificate from a seven-year secondary school in a Ukrainian village, where the family survived during the years of the Civil War, 1919–1921, and where his father served as a parish priest after his university chair of theology was closed by revolutionary authorities. During those village years, family friends and relatives aroused Nikolai’s interest in mathematics and noticed his exceptional aptitude for the subject.

In 1921 the Bogolubov family returned to Kiev, where the father accepted another parish. With the help of old faculty acquaintances, he obtained permission for his thirteen-year-old son to start attending an advanced university seminar in mathematics. There Nikolai’s talents came to the attention of a senior mathematician, Nikolai Mitrofanovich Krylov (1879-1955), who held the chair of mathematical physics at the recently organized Ukrainian Academy of Sciences. Krylov took the boy under his informal patronage and tutelage, and also offered him room and board at his house in 1925, after the rest of the Bogolubov family left Kiev and returned to Nizhny Novgorod following the father’s new church appointment. The same year Krylov obtained for Nikolai status at the Academy of Sciences as an *aspirant*, which was a junior academic position similar to that of graduate student. Because of Bogolubov’s “phenomenal talents” and advanced knowledge of mathematics, an exception was made to his minor age and lack of university courses. The Soviet educational system, in general, allowed much flexibility in the early post-revolutionary period: It was undergoing many different reforms and often permitted young students to skip certain formal stages and degrees in their scientific education.

At the age of fifteen Bogolubov published his first research paper. Many of his early works were authored together with Krylov in the latter’s fields of specialization: variational calculus, differential equations, approximate solutions. Sometimes they published their important results only in Ukrainian, sometimes in Russian, English, or French. Some papers appeared in established international journals with wide circulation, others in small and rare local publications. In 1930 the Bologna Academy of Sciences recognized one of Bogolubov’s early accomplishments with its special prize. Earlier that year the twenty-one-year-old completed his graduate studies and received from the Ukrainian Academy of Sciences the degree of Doctor of Mathematics. He continued working under Krylov’s supervision as research associate at the Academy.

**Non-linear Mechanics.** Starting around 1932, Bogolubov and Krylov extended their joint research from well established branches of mathematics to a new and practically unexplored area. In the course of the crash industrialization campaign in the Soviet Union, scientists came under increased pressure to turn their research work toward practical goals and produce immediately applicable results. Krylov, a mining engineer by initial education, was well prepared for the challenge, which led him and Bogolubov to a new class of mathematical problems, the theory of non-linear oscillations.

Some non-linear differential equations had previously been studied by Henri Poincaré in celestial mechanics, in the case of conservative systems. Krylov engaged in collaboration with the Institute of the Mechanics of Buildings and some other industrial research sites in Ukraine concerned with construction of power stations and aviation. In all these unrelated branches of industry he and Bogolubov encountered phenomena and practical problems that involved non-linear oscillations in essentially non-conservative systems. In the practically unlimited range of important cases, the oscillation process was close enough to the linear, harmonic one, whereas the non-linear part could be regarded as a small perturbation, resulting in the dependence of the period of oscillation upon the amplitude.

Krylov and Bogolubov developed methods of asymptotic integration for large classes of corresponding nonlinear differential equations with a small parameter, extending the methods of perturbation theory onto nonconservative systems in general. They studied invariant manifolds in phase space and developed methods for direct computation and approximations of periodic solutions. The results were important not only for fundamental mathematics, but also for solving practical problems in various fields of engineering. Their method of describing function (quasi-linearization), in particular, proved essential for the new field of non-linear control engineering developed since World War II.

At about the same time, starting 1930, and for analogous reasons, another group of Soviet researchers around Leonid I. Mandelstam and Aleksandr A. Andronov in Moscow and Nizhny Novgorod attacked similar problems with non-linear equations and oscillations arising from the tasks of radio engineering and communications. Combined, their work and the work of the Kiev group around Krylov and Bogolubov established a new research community and a subdiscipline within mathematical sciences, which they called non-linear mechanics. Krylov and Bogolubov summarized their main contributions to the field in a 1937 book, *Introduction to Non-Linear Mechanics*, translated into English by Solomon Lefschetz in 1943.

**Recognition amidst Troubled Times.** During the height of the Soviet anti-religious campaign around 1930, Bogolubov’s father was imprisoned for almost three years. Bogolubov reportedly sought advice of the Russian Church leader and would-be patriarch Metropolitan Sergius and requested an appointment with the head of the Soviet state police, Vyacheslav S. Menzhinsky, which eventually won his father’s release from prison. By accounts of those who knew him well, Bogolubov held deep and sincere religious beliefs throughout his entire life. At the same time, he also operated perfectly smoothly within the avowedly and often militantly atheistic Soviet polity, so that his absolute political loyalty could never be doubted. How exactly he justified to himself the apparent contradiction and managed the situation is hard to answer, because he kept his thoughts on the matter largely to himself.

His academic rise, in any case, was fast and steady. From 1936 onward, while holding a research position at the Ukrainian Academy, Bogolubov also taught as professor at Kiev University. He traveled abroad to conferences in France and Belgium in the years preceding World War II, when the Soviet Union already severely restricted its citizens’ foreign contacts and granted permission for foreign travel only on very rare exceptions. Bogolubov married Yevgeniya Aleksandrovna Pirashkova in 1937 and in 1939 was elected to the Ukrainian Academy of Sciences as a corresponding member. After the 1939 Soviet annexation of Bukowina, the formerly Austrian part of western Ukraine that had belonged to Romania since World War I, he was commissioned to visit the University of Chernivtsi to help reorganize and modernize the university curriculum along the Soviet pattern.

Fleeing Hitler’s 1941 attack on the Soviet Union and the occupation of Ukraine by German troops, most academic institutions were evacuated from Kiev to the east. In 1941**–** 1943 Krylov and Bogolubov worked and lived in poor conditions in Ufa near the Ural mountains. They applied their computational methods to new problems in war production, in particular to non-linear resonance in aviation engines. They also responded to criticism by some mathematicians regarding the difficult problem of divergence of approximate solutions by upgrading nonlinear mechanics to a much higher level of generality. Their related investigations extended into the general abstract theory of dynamical systems, in particular the qualitative analysis of equations in non-linear mechanics, introduction of ergodic sets, and the Krylov-Bogolubov theorem proving the existence of invariant measure.

As the war fortunes reversed, Bogolubov returned from the evacuation, at first to Moscow in 1943, and in 1944 to the recently liberated Kiev. As the Dean of the Department of Mathematics and Mechanics, he handled the difficult tasks of the postwar and post-occupation reconstruction of Kiev University. Bogolubov applied his new methods of perturbation theory also to the problems of statistical mechanics. Already in 1939, together with Krylov, he published a paper on the Fokker-Planck equation and the emergence of stochastic regularities in dynamical systems, followed by a series of investigations on the theory of stochastic differential equations and chains. Bogolubov’s idea of the hierarchy of times in non-stationary statistical physics, in particular, became crucially important for the subsequent development of the statistical theory of irreversible processes. In the classic 1946 monograph, *Problems of Dynamical Theory in Statistical Physics*, Bogolubov developed the method of chains of equations for the distribution functions in both equilibrium and non-equilibrium statistical mechanics and derived kinetic equations for systems with various types of interactions (short-range, and long-range Coulomb forces). The book laid foundations for the subsequent development of non-equilibrium statistical mechanics. In 1947 Bogolubov was elected to the USSR Academy of Sciences as corresponding member, and the following year the Ukrainian Academy promoted him to full member.

**Theoretical Physics and Many-Body Systems.** In the postwar years Bogolubov’s research style and interests changed. Up until 1941 he published mostly in collaboration with Krylov, a teacher and colleague many years his senior. After the war, Krylov’s advanced age and increasingly poor eyesight hampered his research activities. Bogolubov, now in his mid-thirties, became a widely recognized scholar leading independent new programs of investigations often in collaboration with junior students and research associates. Together with Yuri Alekseyevich Mitropolsky, Bogolubov continued developing non-linear mechanics and asymptotic methods of mathematical physics, including the method of quick convergence, but more and more of his time was devoted to theoretical physics, where he used his superior powers as a mathematician to help attack some of the most central and difficult problems in the field.

One such problem, generally considered insoluble in both classical and quantum theory, concerns the treatment of many-particle systems with interactions. Even when the forces and the laws of movement governing individual particles are known exactly, the situation when several such particles mutually and simultaneously influence one another cannot be resolved mathematically. Only when such interactions could be considered severely limited, such as in the model of ideal gas where particles are rare, classical physics was able to develop extremely powerful mathematical descriptions and methods. Such descriptions could also be modified to the case of quantum—but still non-interacting—particles, in the model of an ideal quantum gas. While enormously successful in their proper domain, ideal-gas descriptions were hard to apply realistically in the cases of many real-life systems when constituent particles are packed together closely and strongly interact with each other, such as in liquids, solids, and condensed matter in general.

One promising strategy of circumventing insurmountable mathematical difficulties had been initiated by Soviet theorists Yakov Il’ich Frenkel and Lev Davidovich Landau. They suggested treating particles in condensed matter collectivistically rather than individually and proposed models of so-called “collectivized particles” or “elementary excitations,” which were the quanta of collective movement of many constituent particles at once. With the help of such assumptions, Landau in 1941 explained the strange property of superfluidity in liquid helium. Despite successful applications, some basic ideas behind the collectivist approach were intuitive rather than rigorous, and in the absence of solid proof, aroused controversy. In 1947 Bogolubov refuted such doubts by demonstrating mathematically how collective excitations, which he called “quasiparticles,” arise naturally in the model of an ideal gas, when one adds a weak interaction between its particles. The solution could be found if the interaction was sufficiently small, but already in this case it reproduced the main features of the collective excitation model and helped to justify the general approach.

Bogolubov’s paper proved enormously influential internationally, as it established the quasiparticle concept on a solid mathematical foundation and essentially transformed it into a universally accepted method, the basic tool of contemporary quantum physics of condensed matter. In the case of a non-ideal Bose gas, his microscopic theory of 1947 justified and somewhat corrected Landau’s more phenomenological theory of superfluidity. In the case of a non-ideal Fermi gas, Bogolubov with collaborators in 1958 developed some earlier ideas of Herbert Fröhlich and Leon N. Cooper into a consistent quantum theory of superconductivity. Their theory appeared practically simultaneously and independently of a similar microscopic explanation of superconductivity achieved by a different method by John Bardeen in collaboration with Cooper and J. Robert Schrieffer (the BCS theory of superconductivity). The same year Bogolubov also applied the quasiparticle approach to the treatment of atomic nucleus predicting nuclear superfluidity.

**Quantum Field Theory and the “Bogolubov School.”.** Like many leading Soviet physicists of the immediate postwar period, Bogolubov was invited to participate in the atomic project and in 1948 joined a classified research group at the Institute for Chemical Physics in Moscow. He never fully abandoned connections to Kiev and Ukraine, but started teaching part-time at Moscow University, spending increasing amounts of time in the capital, for several years maintaining two residences, and eventually relocating to Moscow. Although Bogolubov did not play a central role in the development of atomic weapons, in 1950–1953 he lived and worked in a secret location, later known as Arzamas-16, where the Soviet analog of the Manhattan Project’s Los Alamos laboratory operated (the town’s original historic name, Sarov, was restored to it only during the post-Soviet era). After the successful test of the first hydrogen bomb in 1953, Bogolubov returned to civilian research. That same year he became a full member of the Soviet Academy of Sciences, the highest rank in the country’s academic hierarchy.

From 1950 on Bogolubov’s interests extended to quantum field theory, where he could apply successfully his already developed mathematical treatments of many-body systems, as well as new methods. He founded his approach on Heisenberg’s S-matrix formalism with the goal of achieving a more rigorous and consistent representation of renormalization procedures. On the basis of the “Bogolubov microcausality condition,” in 1956 he provided a rigorous proof of dispersion relations, the line of research that eventually led Bogolubov to the so-called axiomatic formulation of quantum field theory. Together with Dmitry V. Shirkov, he also proposed in 1955 the renormalization group method, an extremely useful tool in practical calculations in quantum electrodynamics. In 1961–1963 Bogolubov suggested the mathematical idea of quasi-averages in statistical physics, which proved instrumental for the development of the general concept of broken symmetry and the modern theory of phase transitions. The 1964–1965 papers by Bogolubov and coauthors analyzed the symmetry properties of quark models in strong interactions and introduced a new quantum number for quarks, which subsequently became known as quark’s color.

As one of the country’s top-ranked scientists, Bogolubov accepted an increasing number of administrative appointments at various academic and research institutions. From 1949 on he directed the theoretical physics department at the V.A. Steklov Mathematical Institute of the USSR Academy of Sciences, and from 1953 headed the chair of theoretical physics at Moscow State University. In 1956 Bogolubov organized the theoretical physics laboratory at the Joint Institute for Nuclear Research in Dubna near Moscow and led it until 1965, when he was elected director of the entire institute. In 1966 he successfully lobbied for the creation of the Institute of Theoretical Physics at the Ukrainian Academy of Sciences in Kiev and became its first director. In all this academic entrepreneurship, Bogolubov proved extremely capable of working and manipulating the bureaucratic labyrinths of the Soviet political and administrative system. But he never became a Communist Party member, which in the late Soviet period was typically expected of an administrator at such high rank. The expanding number of institutions he and his associates controlled served as the seats for the flourishing “Bogolubov school of theoretical physics.”

The clustering of the academic community into “research schools,” clan-like groups around top-ranked scientists, became a common social phenomenon in late Soviet science. Scientific schools cultivated high esprit de corps and educated and nurtured their members from the late college years onward. They maintained distinctive research programs and styles, and a researcher often remained associated with a chosen school for his or her entire academic career. Typically scientific schools did not mix, and their members tended to work together in the same or friendly institutions. Soviet scientists often perceived the existence of such schools as a natural and necessary feature of science itself, universal and beneficial for the very progress of knowledge. Administrative leaders of Soviet science were expected to establish and maintain their own schools in the institutions they headed, and as their associates matured, if an opportunity availed itself, branch out and occupy or establish new institutes and laboratories. During the late Soviet period in the discipline of theoretical physics, two scientific schools were particularly successful in pursuing such a strategy—one associated with Bogolubov, the other with Landau. They respected each other’s accomplishments and often profited from each other’s ideas, but also developed a strong sense of institutional rivalry and competition for resources, positions, national and international reputation. To a significant degree, the school identities and rivalries survived the death of their founders and continued well into the post-Soviet era.

Bogolubov’s ideas and methods spread to become classical, working, and indispensable tools in numerous branches of the exact sciences. He received almost every possible recognition—national as well as international— except the Nobel Prize, which somehow avoided him even though more than one of his accomplishments could have deserved it. By the time he retired from most of his administrative duties in 1989 at the age of eighty, the society around him was in a state of utter turmoil. Mikhail Gorbachev’s perestroika unleashed social forces that would ultimately destroy the Soviet Union and, along with it, much of the scientific empire, infrastructure, and research community Bogolubov so carefully created as a result of many laborious years. Scholars do not quite know how he perceived and lived with this new contradiction in life, but in his last public statement he welcomed the news of the restoration and reopening to believers of the church in Nizhny Novgorod where his father once served as a priest.

## BIBLIOGRAPHY

### WORKS BY BOGOLUBOV

With Nikolai M. Krylov. “Sur les equations de Focker-Planck déduites dans la théorie des perturbations à l’aide dune méthode basée sur les propriétés spectrales de l’hamiltonien perturbateur.” *Zapiski kafedri matematichnoi fiziki AN URSR*4 (1939): 5– 157.

With Nikolai M. Kryloff. *Introduction to Non-Linear Mechanics*, translated by Solomon Lefschetz. Princeton, NJ: Princeton University Press, 1943. Russian originals published in 1934 and 1937.

*Problems of a Dynamical Theory in Statistical Physics*. Cambridge, MA: Harvard University Press, 1959. Russian original, 1946.

“On the Theory of Superfluidity.” *Journal of Physics* 11 (1947): 23–32.

*Lectures on Quantum Statistics*. Edited by Lewis Klein and Solomon Glass. London: Macdonald, 1968. Ukrainian original, 1949.

With Yuri A. Mitropolsky. *Asymptotic Methods in the Theory of Non-Linear Oscillations*. New York: Gordon and Breach, 1961. Russian original, 1955.

With Dmitrii V. Shirkov. “Charge Renormalization Group in Quantum Field Theory.” *Nuovo Cimento* 3 (1956): 845–863.

With Dmitrii V. Shirkov. *Introduction to the Theory of Quantized Fields*, translated by G.M. Volkoff. New York: Interscience, 1959. Russian original, 1957.

With Vladimir V. Tolmachev and Dmitrii V. Shirkov. *A New Method in the Theory of Superconductivity*. New York: Consultants Bureau, 1959. Russian original, 1958.

With Anatoli A. Logunov and Ivan T. Todorov. *Introduction to Axiomatic Quantum Field Theory*. Reading, MA: W.A. Benjamin, 1975. Russian original, 1969.

*Selected Works*, 4 vols. Edited by A.M. Kurbatov. New York: Gordon and Breach, 1990–1995. Published in Russian in 3 volumes (Kiev, 1969–1971). A twelve-volume Russian edition of *Collected Works* is being prepared, with several volumes already published as of 2007.

### OTHER SOURCES

*Nikolai Nikolaevich Bogoliubov: Matematik, Mekhanik, Fizik*. Dubna, Russia: Ob’edinennyi institut iadernykh issledovanii, 1994. A volume of recollections and essays by Bogolubov’s students and coworkers.

*Nikolai Nikolaevich Bogoliubov*. Dubna, Russia: Ob’edinennyi institut iadernykh issledovanii, 1989; updated edition, 2004. Edited by Dmitrii V. Shirkov and Alexei N. Sisakian. The full bibliography, plus a biographical essay and a review of major works.

*Alexei Kojevnikov*

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