A.A. Vlasov, and N.N. Bogolyubov, the Forerunners of Quantum Electrodynamics
All written below is the child of my brain, although the meetings and discussions of these two great physicists actually took place, and these discussions resulted in the substantiation of the famous Vlasov equation in the seminal work by N.N. Bogolyubov “Dynamic Methods in the Statistical Physics”...
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irk-123456789-132982010-11-05T12:02:35Z A.A. Vlasov, and N.N. Bogolyubov, the Forerunners of Quantum Electrodynamics Rukhadze, A.A. Хроніка, бібліографічна інформація, персоналії All written below is the child of my brain, although the meetings and discussions of these two great physicists actually took place, and these discussions resulted in the substantiation of the famous Vlasov equation in the seminal work by N.N. Bogolyubov “Dynamic Methods in the Statistical Physics” published in 1946. This work, together with Vlasov’s one, not only proved the statistical physics of many-particle systems with Coulomb interaction, but also laid the corner stone of the method of electrodynamic perturbation theory, which brought about the creation of quantum electrodynamics at the end of the 1940s. Все, що написано – плiд моєї фантазiї, хоча зустрiчi та дискусiї цих двох великих фiзикiв дiйсно мали мiсце. Цi дискусiї привели до обґрунтування знаменитого рiвняння Власова у видатнiй роботi М.М. Боголюбова “Динамiчнi методи в статистичнiй фiзицi”, яку було опублiковано в 1946 роцi. Ця робота разом iз роботою Власова не тiльки обґрунтувала статистичну фiзику багаточастинкових систем iз кулонiвською взаємодiєю, але й заклала основу методу електродинамiчної теорiї збурень, яка в кiнцi 40-х рокiв привела до створення квантової електродинамiки. 2010 Article A.A. Vlasov, and N.N. Bogolyubov, the Forerunners of Quantum Electrodynamics / A.A. Rukhadze // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 151-153. — англ. 2071-0194 PACS 52,57K http://dspace.nbuv.gov.ua/handle/123456789/13298 en Відділення фізики і астрономії НАН України |
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Хроніка, бібліографічна інформація, персоналії Хроніка, бібліографічна інформація, персоналії |
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Хроніка, бібліографічна інформація, персоналії Хроніка, бібліографічна інформація, персоналії Rukhadze, A.A. A.A. Vlasov, and N.N. Bogolyubov, the Forerunners of Quantum Electrodynamics |
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All written below is the child of my brain, although the meetings and discussions of these two great physicists actually took place, and these discussions resulted in the substantiation of the famous Vlasov equation in the seminal work by N.N. Bogolyubov “Dynamic Methods in the Statistical Physics” published in 1946. This work, together with Vlasov’s one, not only proved the statistical physics of many-particle systems with Coulomb interaction, but also laid the corner stone of the method of electrodynamic perturbation theory, which brought about the creation of quantum electrodynamics at the end of the 1940s. |
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Rukhadze, A.A. |
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Rukhadze, A.A. |
| title |
A.A. Vlasov, and N.N. Bogolyubov, the Forerunners of Quantum Electrodynamics |
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A.A. Vlasov, and N.N. Bogolyubov, the Forerunners of Quantum Electrodynamics |
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A.A. Vlasov, and N.N. Bogolyubov, the Forerunners of Quantum Electrodynamics |
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A.A. Vlasov, and N.N. Bogolyubov, the Forerunners of Quantum Electrodynamics |
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A.A. Vlasov, and N.N. Bogolyubov, the Forerunners of Quantum Electrodynamics |
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a.a. vlasov, and n.n. bogolyubov, the forerunners of quantum electrodynamics |
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Відділення фізики і астрономії НАН України |
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Хроніка, бібліографічна інформація, персоналії |
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A.A. Vlasov, and N.N. Bogolyubov, the Forerunners of Quantum Electrodynamics / A.A. Rukhadze // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 151-153. — англ. |
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A.A. VLASOV AND N.N. BOGOLYUBOV, THE FORERUNNERS OF QUANTUM ELECTRODYNAMICS
A.A. VLASOV AND N.N. BOGOLYUBOV,
THE FORERUNNERS OF QUANTUM ELECTRODYNAMICS
A.A. RUKHADZE
A.M. Prokhorov General Physics Institute, Russian Academy of Sciences
(38, Vavilov Str., Moscow, Russia)
PACS 52,57K
c©2010
All written below is the child of my brain, although the meetings
and discussions of these two great physicists actually took place,
and these discussions resulted in the substantiation of the famous
Vlasov equation in the seminal work by N.N. Bogolyubov “Dy-
namic Methods in the Statistical Physics” published in 1946. This
work, together with Vlasov’s one, not only proved the statistical
physics of many-particle systems with Coulomb interaction, but
also laid the corner stone of the method of electrodynamic per-
turbation theory, which brought about the creation of quantum
electrodynamics at the end of the 1940s.
At the beginning of the 1940s, after the famous work
by A.A. Vlasov had been published (Zh. Eksp. Teor.
Fiz., 1938) – most likely, after N.N. Bogolyubov had
studied it – the latter became a frequent visitor from
Kyiv to Moscow and spent a lot of time at the Fac-
ulty of Physics of the Moscow State University, in hot
discussions with A.A. Vlasov. N.N. Bogolyubov was in-
terested in the essence and substantiation of the Vlasov
kinetic equation – or, as it is conventionally referred to,
the equation with a self-consistent field. In fact, one
may assert that A.A. Vlasov ingeniously foresaw this
equation, and his arguments in its favor did not satisfy
N.N. Bogolyubov, except for that it is an explanation
of a large cycle of experiments carried out by I. Lang-
muir and his collaborators. Really, before A.A. Vlasov,
only the Boltzmann kinetic equation – written down as
early as at the end of the 19-th century and not rec-
ognized by the scientific community, in particular, by
H. Poincaré, for a long time – was known in the kinetic
theory of gases. H. Poincaré perfectly understood the
meaning of a small parameter introduced by L. Boltz-
mann,
a0n
1/3
0 � 1, (1)
where a0 is the size of a gas atom or molecule, and n0
is the atomic concentration (the number of particles in
a unit volume). This condition which is referred to as
the ideal-gas one says that the average distance between
atoms in the gas is much longer than the atomic size, so
that the atoms spend much more time in a free (ther-
mal) flight, being engaged in collision processes for only
a small time fraction; or, to put it in other words,
τcm ∼ a0/vT � n
−1/3
0 /vT ∼ 1, (2)
where vT =
√
T/m is the average thermal velocity of an
atom, m is its mass, and T is the gas temperature mea-
sured in energy units. At the same time, H. Poincaré
could not understand altogether how an equation that
describes the dissipation can be obtained for a Hamil-
tonian system that conserves energy. It seems that
N.N. Bogolyubov had an answer to H. Poincaré’s ques-
tion even before his discussions with A.A. Vlasov started.
At the beginning of the 1940s, he had already been de-
veloping a time hierarchy of correlation functions for a
gas of short-range interacting particles (the famous Bo-
golyubov chains), and he understood that, under condi-
tion (1) – or, equivalently, condition (2) – this chain of
equations can be cut to obtain a closed finite system of
equations. In this case, in the zeroth-order approxima-
tion with respect to parameter (1), the Liouville equa-
tion, which describes a gas of noninteracting particles
(the ideal gas), appears. In the first approximation with
respect to parameter (1), only the pair correlation func-
tions are taken into account, with ternary and higher-
order correlations being rejected. As a result, a kinetic
equation is obtained, which describes a gas, where only
pair collisions between particles are made allowance for,
and which is known as the Boltzmann equation.
But what can we do with a gas with the Coulomb in-
teraction between particles? Such a gas, with the bless-
ing of I. Langmuir in 1929, was called plasma. I. Lang-
muir not only gave a name to the ionized gas (the over-
whelming number of its particles are in the charged
state), but also carried out its fundamental experimen-
tal and theoretical researches, which won him the No-
bel Prize in 1932. The high award evidenced the im-
portance of the object under investigation, i.e. plasma.
Plasma is too widespread in the nature; it is a light-
ning, atmospheric and laboratory discharges in gases and
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 151
A.A. RUKHADZE
solids, the ionosphere of the Earth, an interplanetary
gas, stars, nebulae, and, at last, plasma in solids (metals
and semiconductors). Not accidentally did D.A. Frank-
Kamenetskii, who wrote one of the first textbooks on
plasma physics, call plasma the fourth aggregate state
of the substance. Meanwhile, a simple hydrodynamical
model of plasma proposed by I. Langmuir for the ex-
planation of his experiments, although having a bright
success in some cases, was totally out of place in the
others.
L.D. Landau was the first who understood the neces-
sity of plasma description making use of the kinetic equa-
tion. In 1937 (Zh. Eksp. Teor. Fiz., 1937), he paid
attention that the ideal-gas condition (1) is not suitable
in the plasma case, since the typical radius of interac-
tion between particles in plasma, the Debye radius, is
much larger than the average interparticle distance, i.e.
the inverse inequality, as compared with inequality (1),
takes place:
rDn
1/3 ≈
(
T
e2n1/3
)1/2
� 1. (3)
Here, e is the particle charge, and T is the plasma tem-
perature. However, it is condition (2) that says that the
average potential energy of interaction between charged
particles is much lower than their average kinetic (ther-
mal) energy, i.e.
η =
e2n1/3
T
� 1, (4)
which is an equivalent to inequality (3). Just this con-
dition was introduced by L.D. Landau as the ideal-gas
criterion for plasma.
But his next step – namely, he followed L. Boltzmann
and wrote a Boltzmann equation (a Liouville equation
that makes allowance for pair collisions) for plasma as a
gas of charged particles – was, strictly speaking, incor-
rect. As a by-product, he masterfully overcame the well-
known Coulomb divergence, by introducing the well-
known Coulomb logarithm (in essence, the logarithm of
the inverse gas parameter η (see Eq. (4)) when writing
down the final Coulomb collision integral.
Exactly in a year, A.A. Vlasov published his famous
equation with a self-consistent field in the article cited
above and substantiated it making use of literally the
same words as L.D. Landau did. In particular, the in-
teraction sphere had to include plenty of particles, i.e.
condition (3) had to be satisfied. However, the follow-
ing words were quite different. Following A.A. Vlasov’s
speculations, if it is so, every particle interacts, in the
first approximation, with all other particles simulta-
neously or, in other words, with the electromagnetic
field created by all particles in the plasma. As a re-
sult, we obtain, in the first approximation, a system
of equations that consists of the Liouville kinetic equa-
tion, in which the Lorentz force is presented, and the
Maxwell equations describing the fields under the action
of the Lorentz force. The field sources in the Maxwell
equations are the densities of charges and currents cre-
ated by all charged particles in the plasma. Together,
those equations comprise the Vlasov–Maxwell system
of equations, or the equations with a self-consistent
field.
Well, what can we do next? How would the equations
look like, if the next approximation order is taken into
account? This issue excited N.N. Bogolyubov, and it
was a subject of his hot discussions with A.A. Vlasov,
at the beginning of the 1940s, in a university audi-
ence on Mokhovaya Street in Moscow, where N.N. Bo-
golyubov came several times from Kyiv. The hot dis-
cussions between N.N. Bogolyubov and A.A. Vlasov re-
sulted in the monography by N.N. Bogolyubov men-
tioned above. In this monography, he applied the
quantum-mechanical electrodynamic method to the sta-
tistical physics problems for the first time. N.N. Bo-
golyubov proceeded from a Hamiltonian consisting of
a sum of Hamiltonians of free particles and the field,
and a Hamiltonian of interaction between them (but not
more). By applying the perturbation theory method
(the expansion into a power series of ε2) to his famous
chain of equations for correlation functions, N.N. Bo-
golyubov obtained – as the first approximation in ε2 –
the Vlasov kinetic equation and – in the next approx-
imation (with an accuracy to ε4) – the Vlasov equa-
tion with the Landau collision integral. Thus, a con-
secutive derivation of kinetic equations for gases by
N.N. Bogolyubov had been completed. Surprisingly, this
method is known as the Bogolyubov–Bhatnagar–Gross–
Krook (BBGK) one, although the works of the oth-
ers appeared independently and, as the saying is, some
later!
It was much later, when the quantum electrodynamics
had already been created, that R. Balescu, using the
Feynman diagram method, showed the following:
1) if only a vertex diagram is taken into consideration
(the particle radiates or absorbs the field), the Vlasov
equation is obtained;
2) if both a vertex and an exchange diagram are taken
into consideration (one particle radiates the field, and
another particle absorbs it), the Vlasov equation with
the Landau collision integral is obtained;
152 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
A.A. VLASOV AND N.N. BOGOLYUBOV, THE FORERUNNERS OF QUANTUM ELECTRODYNAMICS
3) if all crossed exchange diagrams are summed up (the
ladder approximation), the Vlasov equation with the
Lenard–Balescu collision integral (which takes a plasma
polarization into account due to the particle interaction)
is obtained.
Hence, there is a good reason to refer to the sys-
tem of equations that describes the plasma kinet-
ics as the Vlasov–Landau–Bogolyubov–Maxwell sys-
tem, in the ascending order of contribution made
by each of the participants into the physics as a
whole!
Received 00.00.09.
Translated from Ukrainian by A.I. Voitenko
А.О. ВЛАСОВ I М.М. БОГОЛЮБОВ –
ПОПЕРЕДНИКИ КВАНТОВОЇ ЕЛЕКТРОДИНАМIКИ
А.А. Рухадзе
Р е з ю м е
Все, що написано – плiд моєї фантазiї, хоча зустрiчi та дискусiї
цих двох великих фiзикiв дiйсно мали мiсце. Цi дискусiї приве-
ли до обґрунтування знаменитого рiвняння Власова у видатнiй
роботi М.М. Боголюбова “Динамiчнi методи в статистичнiй фi-
зицi”, яку було опублiковано в 1946 роцi. Ця робота разом iз
роботою Власова не тiльки обґрунтувала статистичну фiзику
багаточастинкових систем iз кулонiвською взаємодiєю, але й
заклала основу методу електродинамiчної теорiї збурень, яка
в кiнцi 40-х рокiв привела до створення квантової електроди-
намiки.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 153
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