Int. J. Heat Mass Transfer.

Vol. 6, pp. 79-89.

ALEXANDERTO CELEBRATE

Printed in Great Britain.

Pergamon Press 1963.

SAVVICHTHE OCCASIONPREDVODITELEVOF NIS SEVENTIETH

BIRTHDAYA. V. LUIKOVInstitute of Energetics of the B.S.S.R., Academy of Sciences,Minsk, B.S.S.B., U.S.S.R.
(Received 14 April 1962)

ALEXANDERSAVVICH PREDVODITELEV,
Corresponding Member of the U.S.S.R. Academy of
Sciences, was born in a peasant family in thevillage of Bukrino in Ryazan province. After
finishing secondary school he entered the
Physicomathematical Faculty of the Moscow
University in 1912. On graduating at the
University he remained to prepare for professorship. In 1919 he passed the master’s ex~nations
and began lecturing on separate topics of optics
and molecular physics and working as a seniorassistant in the Physical Laboratory at the
University.
At the same time, Academician A. P. Lazarev
invited him to work at the Research Instituteof Physics and Biophysics, and in 1920 he wasinvited to teach physics at the Moscow Higher
Technical College.
Predvoditelev was the first man in the U.S.S.R.
to specialize in thermal physics, which he did in
1929. In 1930, he was chosen as professor andappointed as Chief of the Department of Thermal
Physics at the Moscow University. He was also
acknowledged as a member of the ResearchInstitute of Physics at this University. In thesame year Predvoditelev organized a thermophysical laboratory at this Institute.
Predvoditelev is an outstanding scientist who
has carried out a set of profound experimental
and theoretical investigations in various branchesof physics, mechanics, thermal physics, hydrodynamics and physical gas dynamics. Results of
his investigations are presented in more than 140
scientific works.
We shall dwell at length on Predvoditelev’s
investigations into the solution of problems on
gaseous, liquid and solid states of a substance,
wave processes and hydrodynamics, chemical

physics and combustion
theoretical physics.

theory as well as on

INVESTIGATIONS INTO GASEOUS LIQUID AND
SOLID STATES OF A SUBSTANCE [I-14]

In some investigations Predvoditelev suggested
an original model of a liquid structure whichallowed him to determine a number of laws which
gained widespread experimental confirmation.
He considers a liquid to be a quasi-crystal
arranged out of tetrahedrons in the same manneras that used by Thomson who made a model ofa lightwave carrier out of the same geometricalfigures. McCullough used this model as a basis
for writing down the prototype of the Maxwell
equations.
According to Predvoditelev this model isextremely convenient in elucidating those properties of a liquid which arise from the collective
interaction of its molecules. Molecules, or entire
complexes of molecules, may be elements of aquasi-crystal structure.
In the course of time, the quasi-crystal
structure conditioned by collective interactionsof molecules suffers complex periodical changes,
i.e. the elements composing it are not in constantand indispensable relation with all the otherelements which compose a liquid quasi-crystal.
A certain proportion of the liquid molecules isin a wandering state and subjected to statisticalgas laws. In colliding with the main elements ofa quasi-crystal, these wandering molecules
knock out the first ones and replace them. Sucha type of molecular collision with molecules
subjected to casual wanderings differs in no way
from those molecular collisions which areobserved in gas systems. But these collisions also
maintain a vibrational state on a quasi-crystal.
79

80

A. V. LUIKOV

Thus, in the model described, heat motion isofcomposed of bothcasual molecular motion andblack-body acoustic radiation of finite amplitude.
If the described state of a substance is subjected to an adiabatic effect, then there will be
kept invariable some combinationof the following quantities:
energy density of heat motion,
frequency of black-bodyacoustic radiation, thenumber of molecules per volume and thermalmolecular
velocity. These invariable
combinations of the quantities
mentioned
give the socalled adiabatic invariants.
Using
adiabatic
invariants,
Predvoditelev
proved in I948 that for liquids the following law
should hold :
K

CPP
-~--

,A4

Jc1
P,

-f(T).

(1)

Here K is the heat conductivity
coefficient of a
substance ; C’, is the specific heat of a substanceat constant pressure; p is the substance density:
A4 is the molecularweight and f(T) is someuniversal
temperaturefunction.
Thus,
this
mathematicalgroup of physical quantities does
not depend on the nature of a substance andvaries in the same way with temperature
for allthe substances.
It should be noted that many
years ago the German investigator
Weber cameacross this combination
purely empirically.
Fora long time the formula given above was referred
to as the Weber law with the only difference thatthe functionf(T)was considered
constant.
This was an error which made the Weber formula
inapplicablewhen it was comparedwith theaccumulating
experimental material. When Predvoditelev
showed that the function f(T) was
not a constant quantity, the above formula wasrevived.
In connexion
with the above-mentioned
considerationof the nature of heat motion inliquids,
Predvoditelev
undertook
the task ofdetermining
the relation
between this motion
and thermodynamic
temperature.
It goes without saying that from the viewpoint expounded
the local energy density of heat motionofliquids cannot be the same everywhere. It is ofa “hilly”
character,the “hillocks”
travelling
arbitrarily
over a space. To Predvoditelev’s
mind, the thermodynamic
temperature
is connected, only throughcertain thermodynamic

relations,with the average time or a\eragr:
statistical value of energy density of heat motionof a medium. Having submitted the deviationsof the nominal temperaturefrom the thermodynamic one to the laws of random wandering,
it appeared to be convenient
to estimate thequantity of the above heat “hillocks” by meansof these deviations.
Having dealt with the deviations given above
with the help of the Smolukhovsky-MarkovKolmogorov equation, Predvoditelev establishedanoth.er important
law regarding the change inthe heat conduction coefficient with temperature.
This law may be written as follows:
K z BP” :(.

12)

Here B is some constantdependingon thenature of a substance.
The above formula isvalid only for liquids, but Predvoditelev extended
it to the whole range of materials, from liquid togaseous. In this case Predvoditelev’s
formula isof the following form:
K z Kt +- B/j’ 3.

I.3)

Here Kt should be calculated
accordingtoordinary formulae of the gas-kinetic theory fora rarefied gas.
Besides the heat conduction
coefficient. theviscosity of a liquid is of vital importanceas acharacteristicof its state. Predvoditelev
considers that the solutionof this problem
isconnected
with those equations
which should
describe the motion of a medium creating a
black-body acoustic field. Predvoditelev followed
Debye’s procedure for solving the problem ofheat capacity of solids. As is known. Debye
submitted the heat motion in solids to ordinary
equations of elasticity theory. having restricted
the number of degrees of freedom of the elastic
continuum
by the cube of the atoms in a cubic
centimetre.
This courageous
attempt made by
Debye proved to be correct and gave goodresults, and is still unquestioned
to-day.
Predvoditelev’sview of the nature of’ heatmotion of liquids is close to Debye’s. He tried.
therefore,
to use hydrodynamics
equations.
corrected
for non-ideal
homogeneity.
for thedescriptionof the heat motion in liquids. This
attempt
gave good results. He succeeded inALEXANDERSAVVICH

establishing the relation between the heat conduction coefficient and the viscosity coefficient.
This relation has the following form:

Here E and b are constant. When a specificvolume is very large, the multiplier in roundbrackets becomes unity and the formula converts
exactly into the Maxwell formula for gases. The
value of the constant b is close to that of the
known Van der Waals constant but not equal toit.
Comparing the latter two formulae andeliminating the heat conduction coefficient,
Predvoditelev established the following law for
viscosity:--.- d77= (v _ b)+3’
Here A is a constant dependent on the nature ofa substance. He also generalized this latter
formula for liquids and gases. The above
generalization leads to such a relation:A
7 = qt + (u _ b)$3’
should be calculated by the formulae which
are given by the gas-kinetic theory for rarefied
gases.
To give a more complete idea of Predvoditelev’s scientific activity it is important tonote that all the relations he obtained have beenverified by thorough experiments. His pupils
N. B. Vargaftik, L. P. Filippov, Abas-Zade etal. confirmed Predvoditelev’s laws on various
materials and over wide temperature ranges.
Later, on the basis of his general views on
the nature of heat-medium motion, Predvoditelev succeeded in framing, in a novel way, thequestion of how to find a state equation forcompressed gases and liquids. He proves thatin dense gases and liquids the statistical indications, which are taken as components of heatvelocities, cannot be independent as in the Maxwell-Boltzmann theory which, in fact, determines
the type of the Maxwell distribution function. He
considers that the statistical indications correlate
amongst themselves. Limiting himself to the
rlt

F

81

PREDVODITELEVcase of isotropic correlation, i.e. considering thecorrelation coefficients rrj for each pair ofcomponents of heat velocities to be equal,
Predvoditelev obtains the following general
type of the state equation:
pv = RT#(r)-$E,(7)Here the value +E is the virial of internal forces;
the function #(r) is universal and has the following form:$(r)=(1 - r)V+ r12(1+ 2rj2--.

(1 + 3r)2(1 - 3r)3/2(8)

The correlation coefficient r proved to be proportional to density, the proportionality factor
being dependent on the nature of the substanceand practically unchanged with temperature.
Predvoditelev successfully verified his state
equation on a very extensive virtual material.
In his works Predvoditelev lucidly explained
numerous experiments of Mikhelson and his
co-workers on the measurement of the compressibility factor for gases over a wide range oftemperatures and pressures (up to 5000 atm).
Predvoditelev’s views on the critical state areinteresting. Proceeding from the fact that, on
approaching the state line between the liquidand gas, the correlation coefficients for a liquid
and its vapour should become equal, and considering the internal energy of each system to bea linear density function, he derives an equation
for the state lines between a liquid and vapour
in thermodynamic equilibrium. The line of aliquid is of the form:,,=,[l+A(T)Jrsj].

(9)

The line of a vapour is of the form:

Only at large distances from a critical point does
the density pS differ slightly from the criticaldensity per.
As to the quantities A(T) and CL,they aredefinitely related to the difference between aliquid and its vapour as well as to parameters ofinternal energy. The verification of the formulae

82

A. v.

LUIKOVobtainedfrom numerous
experimental
data
showed that for each liquid the quantities A(T)and p remain constant up to 0.2 per cent. Only
for transition
from one liquid to another does
the divergence reach 5 per cent. We may consider that these values are of a universal character and are equal to:A(T) = 2.73
As may be
Cailletet-Matias
relation, viz :and1~ = 1.05.

easily seen,rule follows
P.f -t-2

PO

the well-known
from the above

= ps.

K

(‘,Y/ (I:. n’)*:!

(17‘

R

96

,L ‘L
f 6’i

(iii

From this formula it is seen that only for hightemperatures,
when atomic heat capacity reaches
its limit equal to 6, does the right-hand side ofthis equality become the Lorentz
expression,
VIZ..

(11)

It is interesting to note that the known Hirschfelder equation, for greatly compressed gases andhigh temperatures,
follows as a specific case of
the Predvoditelev
state equation.
The state equation obtained by Predvoditelev
also enables, with the help of the known thermodynamic equality, calculation of that part of theinternal energy of a substance which is dependent on density. Such calculations are in a good
agreementwith experiment.
Many other investigators who tried to solve the problem of a
substancestate equation
did not succeed inreaching such a good agreement.
While studying mainly properties of gases andliquids, Predvoditelev
paid attention
to solids
as well. Here, he also obtained some interesting
results.
Proceedingfrom wave concepts
ofmolecular
heat transfer
analogous
to thosewhich he developed for liquids, he obtained avery simple formula for the heat conductivityof dielectrics, viz. :K .E .
1: J[pn;z)n]{y’Cp(h -i- 2pL)l-i- 2Y’h)ally conducting bodies from just the same viewpoint, Predvoditelev
discovered an interesting
generalizationof the Vierdermann-Frantz
law.
He found :

;.

(12)

Here C is the heat capacity of a substance;
C’N
is the molecular heat capacity of a substance ;p is the density; Xand p are the Lame coefficients ;R is the gas constant;d is the average distance
between atoms ; /3 and n are integers dependent
on the crystal structure of a substance. In his
works Predvoditelev
shows how to determine
these quantities.
Analysing molecular heat transfer in electric-

INVESTIGATIONSINTO WAVE PROCESSESHYDRODYNAMICS

AND[IS-221

Predvoditelev
was interested in this branch ot
science for a long period of time. He concentrated
mainlyon finding
the conditions
under which a wave surface might exist. Proceeding from the famous discussion between Frenkel
and Poisson and comparing it with the investigations of Hugonio and Hadamard,
Predvoditelcv
developed an extremely plausible general theoryof processes propagating
as a front. He established a strict definitionof wave surfaces. towhich all the integral surfaces should be attributed if they may be considered as a geometriclocation ofcharacteristics determined by Monge.
In wave processes,
such surfaces separatearegion of rest from motion or one motion fromanother. The existence of such a region requires
the observanceof special compatibility
conditions. In science these conditions were found inlwo
forms:
Frenkel-Poisson
and Hugonio
Hadamard.
Predvoditelev
showed that these
two formulae were always revealed in thosecases when a closed system of differential equations permits the following transformation
withrespect to the variables:,$ ---=J+, J’. Z) ; x:1.

(1%lntegral surfaces, corresponding
to a wavefrom.
are easily singled out withthe help of such atransformation.
Using the general determinationofthe compatibility
conditions given by Hugonio,
this transformation
facilitates the derivation of
the known Hadamard
identities and kinematic

ALEXANDERSAVVXCH

conditions. In the course of solving problems on
the propagation of acoustic waves in dispersive
media, Predvoditelev demonstrated the significance of these conditions. It may be said thatthe method worked out by Predvoditelev rarely
encounters obstacles. Any system of closed
differential equations allowing of wave solutions
is treated according to Predvoditelev~s method
so that the dispersion formulae may be easily
obtained in their general form without resortingto the integration of a system of equations. Thus,
Predvoditelev solved a great variety of concrete
problems in the field of physical acoustics. In
some of his ex~rimental works he exposed incorrect inte~~etations of the results obtained,
and explained their true nature.
Recently Predvoditelev succeeded in eliminating one defect in the Hadamard algorithm.
The Hadamard identity and kinematic conditions are applicable only in those cases when awave surface in its motion does not suffer
deformation. This means that they cannot be
used to solve problems on reflection, refraction
and diffraction of waves, i.e. in those processes
when a front deliberately transforms;
forexample, the front of an incident ray transforms
into that of a reflected ray.
Hadamard derived his identities and kinematic conditions proceeding from the postulate,the essence of which lies in the fact that it ispossible to make an entire operation of differentiation on a wave surface. In the case of thedefo~ation of a wave surface, it is obvious thatsuch an operation cannot be fulfilled. Substituting the operation mentioned for the variation operation, Predvoditelev succeeded ingeneralizing the Hadamard identity and kinematic conditions. The generalized conditions intheir application proved to be a powerf~ tool.
With the help of these conditions Predvoditelev
made a complete investigation of the reflection
and refraction of the so-called shock. He showedthat the Mach reflection and refraction was a
striking expression of the Hugonio compatability
conditions and was the most general law ofreflection and refraction. The conditions found
for the transformation of a wave front in the
Mach reflection and refraction are those fromwhich normal reflection and refraction may
easily be reduced as limitina cases. The Mach

PREDVODITELEV83

reflection, as is known, had not been properly
explained until the works of ~edvoditelev.
Predvoditelev paid much attention to wave
processes in which the transformation of a
substance took place. Using this general View on
the nature of wave processes, he solved a numberof probiems on the front propagation of combustion and detonation. Using combined processes as an example in such cases when heatpropagates in a chemically reacting medium,i.e. heat transfer and kinetics of substancetransformation proceed simultaneously, Predvoditelev demonstrated the conditions thatshould be observed in the fo~ationof atravelhng front. In this respect his works possess
an independent character as compared withthose now accepted in science. The majority ofscientists solve this problem as a boundary one.
In this direction Predvoditelev’s ideas areclosely connected with Mikhelson’s who did not,however, develop his views.
As a further step, Predvoditelev also explained
the nature of detonation by the use of similarreasoning. Not only did he demonstrate that a
jet flow might co-exist with shock rupture, as
is most frequently observed, but he also showedthat in some cases a jet-helical flow might coexist with a shock rupture and the surface of ashock rupture would rotate around some axis,
as is observed in the case of spin detonation.
Predvoditelev also used his own approach to
substantiate hy~od~amics
equations for a
viscous liquid from the viewpoint of the molecular kinetic theory. As is known, the aim here is
to use the solutions of the Boltzmann integral
differential equation of the first, second andthird approximations, depending on the character of a problem. The soiutions so found define
more exactly ordinary aerodyna~cs equations
by adding higher derivatives to them. This
introduces a new problem if they are used
specifically, viz. the introduction of the higher
derivatives entails an increase in the number ofboundary conditions. It is very difficult topredict these conditions for a given case. Thus
it would seem that one difficulty gives place
only to another, and there is no visible advantage. Predvoditelev’s attention was drawn
to one peculiarity which is usually observed on
application of the usual method, From the

84

A.

V. LUIKOV

molecular
kinetic
viewpoint,classical
aerodynamics equations (according to the way they
are obtained) are always valid if apparent transfer velocities of colliding molecules are mutuallyequal, i.e. if gradientsof apparent
velocities
along the mean free path of the molecules are
zero. In some cases this may not happen in aflow of rarefied gas. Consequently,
the classicalequations become inapplicable.
Using the concept of a physically infinitely
small volume, Predvoditelev
showed that on
going over from a moving discrete medium toan equivalent
flow in a continuum
it wasnecessaryto change the dimensionsof thephysically infinite small volume when passing
from point to point. By introducing
a specialparameter similar to volumetric and tangential
viscosity, one may take into account this peculiarity of the transition from a discrete medium tothe continuum
and write generalized
aerodynamics
equations
without
introducing
thehigher derivatives.
Consideringexperimental
investigations
on acoustic dispersion in rarefied
gases, Predvoditelev
demonstrated
the validityof the considerations
given.
INVESTIGATIONS
AND

INTO

COMBUSTIONCHEMICAL
THEORYPHYSICS[23-49]Predvoditelev
began
his scientificcareerinvestigating
Rayleigh light dispersion in turbid
media with particles 11~.and less. In this work heshowed that the general solution for the dispersed light intensity proposed, by the known
Germantheorist
Mi in the form of series
expansions
on spherical harmonics,
might bereplacedwith adequate
accuracyby electromagnetic radiation of dipoles and quadrupoles.
The concisenessof the formulae
obtainedconsiderably
facilitatedcalculationsof thedispersed light intensity in turbid media.
A study was undertakenusing dispersed lightfor the concentration
determinationof one ofthe componentsof photochemical
polymerization of anthracene
in organic solvents whichchanges into insoluble dianthracene
under theinfluence of ultra-violet light. This method was
extremely
convenient
for investigatingthekinetics of this particular
photochemical
conversion.
Predvoditelev
devotedmany
experimental

investigations
to photochemical
phenomenaoforganic dyes. mainly of the aniline type. He
concentrated
his attention
in the main on theeffect of film-dye thickness on the velocity ofmaterial transformations.
The film thickness 01’
a dye-stuff seriously influenced the rate of diccoloration.
and evident
pcriodicityof theprocess was found, with a period increasing \vith
the film-dye thickness.
The periodicityof theprocess
vanished
when
the film thickness
approached
I :A.
In order to explain the nature of the proces\
described, Predvoditelev investigated the photoelectric propertiesof films of dyes, since hc
considered
the photoelectriceffect to be theprimary event in any photochemical
reaction.
The investigations
confirmed
his idea. The
intensity of photoelectric
current was actuallyfound to vary periodicallywith the film thickness. The periodicityof the phenomenon
alsovanish.ed when the film thickness
approached
I g. At one time these investigations
roused greatinterest when it became necessary to explain tht:
nature of the photoelectriceffect.
Since dye-stuffs belong to the class 01 doelectric media, Predvoditelev
began experimcntai
investigationsof photoelectric
propertiesoi
dielectric crystals, the crystals being chosen froma class of complex inorganic compounds.
These
compounds
were chosen mainly bccausc, in :L
spatial structureof a substance,
metal atomx
capable of loosing electrons are surroundedby
a ahcll of molecules (water. ammonia) which can
decclcrate
electronemissivity.
It was consequently possible to evaluate this deceleration
by means of the magnitudeof photoelectric
current and thus to analyse the nature of it
molecular
field between a metal atom and thestructural formations
round i2.
At the same time Predvoditelcv
studied III?
kinetics of disintegrationof complex inorganic
compounds
in vacuum, mainly dehydrationofcrystalline hydrates, using interesting
methods.
In these experiments
the kinetics of disintegration was studied by the decrease in weight of thesubstance investigatedon a specially designed
microbalanceand by recording
the rcacticc
force of the escaping molecules on a torsion
balance. Both instruments
wcrc placed into thesame vessel with the gas evacuated.
Such aALEXANDERSAVVICHsimultaneous investigation of the dehydration
process allowed ~edvoditelev to establish an
important law. Kinetic energy of the escaping
molecules proved to be exactly equal to the heatof reaction effect calculated per molecule.
Numerous complex compounds confirmed thisresult. Thus, Predvoditelev found a new noncalorimetric method for measuring heat offormationof chemical compounds.
These
measurements are of interest not only in thisrespect, but they also explained the role of theMaxwell distribution in the flow of molecules
which form the reactive force applied to bladesof the torsion balance.
In these experiments Predvoditelev
also
accomplished the easy determination of the
progressive separation of water of crystallization; that in itself is of great interest.
Spatial water distribution in crystalline hydrates is frequently in the form of shells andthese break down progressively. In his experiments, Predvoditelev investigated the kinetics ofsuch series disintegration in detail and measured
separately the heat of formation of each structural shell, an operation which could hardly bedone by calorimetric experiments.
In connexion with; the solution of someengineering problems, Predvoditelev and his
pupils conducted a set of interesting investigations on chemical transformation of a substancein electric discharges at atmospheric pressure.
Reactions were considered in a corona dischargeof direct and ~gh-frequency current as well as
in flame discharges on installations which generated an electromagnetic wave of about 4 m.
In these investigations, resonant transformation of a substance was discovered, and it was
thus proved that under suitable conditions thehigh-frequency discharge was more effective
than discharges obtained from constant electric
voltage.
He and his pupils thoroughly investigatedconversion of a sulphurous gas into sulphuric
acid, air nitrogen into nitric acid, and acetylene
into butadiene, etc.
Predvoditelev carried out extensive investigations into heterogeneous and homogeneous
combustion of solid, liquid and gaseous fuels,
For combustion of solid fuels he evolved an
entire theory of a process in which he managed

PREDVODITELEV85

to elucidate completely the role of natural andforced diffusion of a gas maintaining combustionof solid fuel. His theoretical propositions were
well confirmed by numerous experiments whichwere conducted by his pupils over a long periodof time.
He also gave the theory of bedded fuelcombustion. A furnace with liquid slag disposal
was designed on the basis of this theory.
For homogeneous combustion, Predvoditelev
considered a problem on inflammation of agaseous fuel in laminar and turbulent flows, He
worked out the theory of limiting concentrationand that of flame propagation mentioned above.
Latterly he and his pupils were working on thetheory of two-phase combustion. In this direction the most interesting results were obtainedin his laboratory and experiments’ of great
precision were carried out. Conditions ofignition, formation and motion of a combustion
front in two-phase gas mixtures were foundout.
A feature of Predvoditelev’s work on homogeneous and two-phase combustion is his search
for the principles whereby chemical kinetics andthe laws of heat propagation could be combined.
He proves that in the presence of chemical
processes isothermal surfaces satisfying the heat
conduction equation acquire the character ofwave surfaces. This follows from the fact thateach point of an isothermal surface may beconsidered as a source of the origin of a chemical
process. Regarding the process mentioned,
Predvoditelev’s views bear a strong resemblance
to the theory of the origin of secondary waves
in optics (the Huygens principle). The above
circumstances allowed him to develop a special
mathematical apparatus which made it possible
to obtain a finite result to practical or theoreticalproblems, by means of comparatively simple
methods.
INVESTIGATIONS INTO THEORETICAL PHYSICS

143451
Predvoditelev’s investigations on theoreticalphysics are noted for their great originality. One
group of these investigations is connected withthe application of the Thomson-Joukovksi
method for evaluating the stability of mechanicalmotions in atomic systems. Thus, in a number

86

A. V. LUIKOV

of works, Predvoditelev
reveals
the purely
geometric character of “jumps” of electrons inatoms from one orbit to another. He proves that“jumps”
are only convenient
as a model forplotting
a calculation
algorithm.
This model
actually results from the complex motion ofelectrons over a surface of double curvature.
Proceeding from a unified model of the atomic
structure, such consideration
allowed Predvoditelev to establish a relation between its serial
radiation and the Planck radiation.
As is known, in modern statistical physics,
statisticalsystems are based on the so-called
quasi-ergodic
hypothesis
which is still being
discussed even now since Maxwell’s original
work
on this problem.
Superimposingonstatistical systems stability conditions of motionof their elements, Predvoditelev
attempted
toexclude this hypothesis. The detailed analysis ofstatistical systems from these positions permitted
the author to divide all the systems into twoclasses. To one class belong statistical systemsof Maxwell-Boltzmann
and Gibbs, and, to theother, all the statistical systems united by thegeneralized Schriidinger equation. This generalized equation
unites the form of motionofstatistical elements,
according to the laws of
randomwandering,with the mode of theirmechanical motions, described by the Hamilton
characteristic functions. In this case, the generalized Schriidinger equation is characterized
by aspecial modulus
of statistical
distribution.
If
this modulus is purely imaginary, we may obtaina quantum system. If this modulus is real, thenwe may describe such molecularsystems inwhich statisticalsystems appear in the formtaking correlation into account. In the course of
its life such a system may degenerate either intoa purely mechanicalor into a purely statisticalsystem, for example, into the system of Gibbs
orMaxwell-Boltzmann.
In these
works,
Predvoditelev
also explained
what conditions
should be observed
so that a non-quantum
system could transform
into a quantum enc.
In general, statistical systems may fluctuate.
Adhering to his main suppositions, Predvoditelev
also made an attempt to explain this aspect.
Predvoditelev
did not confine himself to theanalysis of systems in which statistical elements
1.
possessed constant mass, but he also analysedsuch statistical systems in which mass depended
on velocity, for example, according to the lawsof the theory of relativity. In this case he showedthat the Dirac statistical systems were not justspecial cases, but lay within the framework of
the generalized
SchrBdinger
equationwhich
might be transformed
into four equations of thefirst order when actuallysolvingproblems.
Besides the indispensable
compatibility
conditions discoveredby Hugonio-Hadamardforwave processes were observed
a remarkablepeculiarity of such a concept.
In one of his works, Predvoditelev proves thatthe Hugonio-Hadamard
compatability
conditions, expressed in the form of the Hadamard
identity and kinematic conditions,are a prototype of an operator
methodof quantum
mechanics.
In addition, according to the very characterof the Hugonio-Hadamard
compatability
conditions, they must contain a mathematical
formof the Heisenberg uncertainty
principle. In oneof his works, Predvoditelev proves this assertion.
Thus, the mathematicalform of the Heisenberg
uncertainty
principle is not confined to quantum
mechanics,
but is in fact a special form of‘of specific qualitiesof all wave
expression
processes.
Applying features of stability of motion tostatisticalsystems
accordingto Thomson
Joukovski,
Predvoditelev
found a more convenient
expression
for motion
throughafunction.
This expression had the same significance with respect to disturbed motions as the
Hamiltoncharacteristic
functions
had with
respect to undisturbed
motions.
By means of a number of particular examples
taken from atomic physics, Predvoditelev showed
the usefulnessof his type of equations
foi
disturbed motions.
To characterize Predvoditelev’s
wide interestsin the field of theoretical physics, we may mention his work devoted to the theory of elementary
particles. He considers a space filled with a force
field in which there exist closed discontinuity
surfaces, and he shows that such surfaces willthemselves behave like material bodies possessing properties
corresponding
either to Bose’s
statistics or to Fermi-Diracstatistics.
If these particles are brought to decay. weALEXANDERSAVVICH

may determine their absolute mass from thevalue of a statistical distribution modulus.
Further the author proves that such particles
will possess a whole spectrum of masses. The
formulae he found, finish up with an extremely
great variety of elementary particles. Among
this variety there figure proton nuclei, helium,
etc. With the help of the relations found,
Predvoditelev also determines the dependence
between isotropic number T and mass number
A. This relation is of the following form:
T = 0.0136 A312.
The above formula completely embraces the
Harkinson empirical relation which is asfollows :0.0073 A5’3
-T = 1 + 0.0073 A213’
Predvoditelev published this work in the formof a discussion.
Recently Predvoditelev has been studying thegeneral properties of a physical space byanalysing the entire scientific material accumulated by geometricians and physicists for many
centuries. He has come to the conclusion that
our cognition is dual. For example, if we
assume rectilinear and uniform motion as abasis of mechanics, considering this as a modelwith which all other motions are compared,
then, as Newton showed in his origins, we shallhave varied motions of the Euclidean physicalspace. If, on the other hand, we consider all the
motions of material bodies uniform and,proceeding in geodesic lines, we may then end
up with a variety of physical spaces instead of
motions. The outstanding Norwegian mathematician Sophus Lie showed that both constructions were equivalent. It cannot be therefore
asserted which one is correct. As anotherexample, at first sight, well-known experimentson the p-decay of cobalt lead to a concept thatour physical space has no minor symmetry.
However, a number of investigators showedthat it was possible to increase the number ofphysical properties of a material particle(neutrino) in a similar manner and the space
properties will remain in their familiar form.
According to the duality principle, it is possible
either to increase the number of properties of aphysical space by preserving the number of the

PREDVODITELEV

87

found properties of material particles, or toincrease the number of properties of a material
particle by preserving the number of foundproperties of a physical space. Both views arecorrect, and our cognition can never escapefrom the interpretation of the described natural
phenomenon in two ways. Such is the interaction between a concious person and the world
around. The number of examples of the dualistic
principle may be increased many times. Predvoditelev wrote a small monograph on thisproblem.
Predvoditelev’s r8e as an educator of scientificpersonnel is great. The growth of the scientificpersonnel of Predvoditelev’s school took place
under the conditions of a live interaction
between theory and practice. His skill in assisting
the development of the creative initiative andself-dependence of scientific workers contributed
a great deal to the general growth of scientificinstitutions. In this man, we have a talented
educator of scientific specialists, who has
trained more than 110 skilled scientific workers
among whom there are twenty-five doctors andprofessors.
It is necessary to note Predvoditelev’s great
work in the field of methodology and the historyof physics, in throwing light on the role ofRussian scientists in the development of scienceand in the organization and planning of scientificresearch work. Predvoditelev took the lead in
the publication of Emmys on the Developmento~~~y~jc~ in Rosin. He is an author of some
papers in this collection.
Between 1937 and 1946, Predvoditelev was
director of the Research Physics Institute andDean of the Physical Faculty at the Moscow
State University.
Predvoditelev fulfilled extensive work as
Chairman of the University Scientific Technical
Board. He was responsible for getting firstclass equipment for the new University buildings.
Highly praising Predvoditelev’s scientific,
teaching and public work, the Soviet Government decorated him with the Order of Lenin,
four Orders of the Red Banner of Labour, theOrder of the Red Star, and Medals for Valorous
Labour in the Great Patriotic War and forDefence of Moscow.
Predvoditelev is an ardent patriot of the88

A. V. LUIKOVMotherland
and gives all his strengthandknowledge to the developmentof Soviet science.
Let us wish him great success in his fruitful work
and in his varied scientific and academic activity.
MAIN SCIENTIFIC WORKS OFPREDVODITELEV
1. On stability of mechanical motions in atomic physics(0 prochnosti mekhanicheskikh dvizhenii v atomoi
fizike), Z/z. Eksp. i Tear. Fiz. 4, vyp. 1, 43-59 (1934).
2. On the theory of liquid viscosity and molecularassociation (K teorii vyazkosti zhidkostei i assotsiatsiya molekul),Zh. Eksp. i Teor. Fiz. 3. vyp. 3. 217.
229 (1933).

3. Viscosity of liquids and gases from the viewpoint ofcyclic motions (Vyazkost’ zhidkostei i gazov s tochki
zreniya tsiklicheskikh dvizhenii). Zh. Eksp. i Tear.
Fiz. 3, vyp. 3, 230-236 (1933).
4. Some invariant quantities in the theory of heat conduction and liquid viscosity (0 nekotorykh invariantnykh kolichestvakh
v teorii teploprovodnostiivyazkosti zhidkostei), Zh. Fiz. Khirn. 22, No. 3, 339.
348 (1948).

5. Some considerations on the operator method of wave
mechanics (Nekotorye soobrazheniya ob operatornom metode volnovoi mekhaniki), Vesfnik Mod.
Univ. No. 2, 57-65 (1949).
6. On molecular heat transfer in liquids (0 molekulydrnom teploobmene
v zhidkostyakh),
Dokl. Akatl.
Nauk

SSSR,

Fiz. Khim. 72, No. 2, 323-326(1950).

7. On the coefficient of heat conduction, viscosity of liquids and compressed gases (0 koeffitsiente teploprovodnosti i vyazkosti zhidkostei i szhatykh gazov). Sb.
statei, posvyashchennyi
pamyati akademika P. P.
Lazareva, Izd. Akad. Nank SSSR, 84-l 12 (1958).
8. On the character of heat motion in liquids 1~111(0 kharaktere teplovogo dvizheniya v zhidkostyakh
I-III), Znzh. Fiz. Zh. 4, No. 6, 3--l 2 ( 1961) : 4, No. 7,
4-10 (1961); 4, No. 8, 3-10 (1961).
9. iSber die Abhlngigkeit der Fliissigkeitsdichtenvander Temueratur. Zeit. ”fiir Phvsik Bd. 36. Hf. 7. 557
562 (192ii).
IO. Zur Frage der Abhlngigkeit
der Dichtc van derTemperatur, Zeit. fiiv Physik Bd. 40, Hf. 6. 474-416(1926).
11. Das Maxwellsche Relaxationsgesetz
und die innere
Reibung der Fliissigkeiten, Zeit. fiir Physik t&i. 49,
Hf. 3/4, 279-294 (1928).
12. The possibility of considering gas systems as calculated sets with correlation attributes. Papers 1, 2, 3,
4, 5 and 6 (0 vozmozhnosti rassmotreniya gazovykh
sistem, kak schetnykh mnozhestv s korreliruyushchimi priznakami. Stat’i 1, 2, 3, 4, 5 i 6). In press.
13. Heat conduction of electrical conductors (0 teploprovodnosti elektroprovodyashchikhtel), Z/r. Eskp.
i Teor. Fiz. 4, vyp. 8, 838-854 (1934).
14. Heat conduction of solid heat insulators (0 teploprovodnosti tverdykh teplovykh izolyatorov), Zh.
Eskp. i Teor. Fiz. 4, vyp. 8, 813-837 (1934).
1.5. Molecular kinetic substantiation of hydrodynamics

equations(0 molekulyarno-kineticheskom
obo+novanii uravnenii gidrodinamiki), 120. Akurl. Nuuk
SSSR, Otdd. Tekh. Nouk No. 4, 545-560 (194,6, Theoretical examination of vibratory movement oj
the flame front in closed vessels. Trudy 7 Mezhdunarodnogo simpoziutna po gorrniyu i vrryvam,
21 l--214, London (1958).
17. On general properties of discontinuity procc\~> andacoustic phenomena in solid non-uniform bodies(Ob obshchikh svoistvakh razryvnykh protscasov iakusticheskikh yavleniyakh v tverdykh neodnorodnykh telakh). Applicationsof ultra-acoustics
tosubstance investigation.
Proceedingsof the 4th
conference, 27.-45, Moscow ( 1957).
non-uniformities
in the thcl)rl I>!
IX. Hydrodynamic
combustion and explosions (0 gidrodinamichesklkh
ncodnorodnostyakh
v teorii gorcniya i vzryvov).
Gas dynamics and combustion physics. Izd. :\kad,Nauk SSSR, 5--44, Moscow I1950).
IO. On conditions of regular morirln ~11‘rtrong bhoch
ruptures and detonation iOb usloviyakh regulyarnogo
dvirheniya sil’nykh udarnykh rarryvcn i detonatsii)
Physical gas dynamics. Izd. Akad. Nauk SSSR. 5 -13.
Moscow (1961).
20. On dispersion of transvcrsc and longitudinal wavch
in liquids in relaxation. Part I (K voprosu o dispcrbii
poperechnykh i prodol’nykh voln v relaksiruyushchikh zhidkostyakh. Chast’ I), b’ertnik Mo.yk. l/m?.
Series Ill, No. 3, IO--l8 (1961).
21 On dispersion of acoustic wave\ in relaxation media.
Part II (0 dispersii akusticheskikh voln v rclaksiruyushchikh sredakh. Chast’ I I ). Vmfnik Work. (lair.
Series Ill, No. 4, 44-52 (1961).
22 Motion of a combustion zone as a hydrodynamic
non-uniformity(Dvizhenie
zany goreniya
kakgidrodinamichcskoi
neodnorodnosti).
Sb. trudov,
posvyashchennyi yubileyu G. M. Krzhizhanovskogo.
Izd. Akad. Nauk SSSR. 793-816 (1959).
23. ijbcr den Einfluss des Kristallwassers auf den photoclektrischen EfTekt in den Kristallhydraten
I., Zc~ic.
f?ir f’hvsik Rd. Hf. I, 60.-76 (I 927).
24. (jber &n Einfluss des Kristallwassers a~~f den Photucfl‘ekt in Kristallhydraterr II ._%eif. .fiir P/ly.cik Bd. 44.
Iif. 3. 207-215 (1927).
der H&)2s. Uber die absoluten Geschwindigkeiten
Molekiile, welche bei der Dehydratation VOII Kristallhydraten herausfliegen, Zeir. flir F’hJxik Bd. 54, Hi
1,‘2, IS--l60 (1929).
36. On one possible method ol’ measuring the heap offormation of chemical compounds capable of tli+
integrating in vacuum (Oh odnom vozmozhnvm
metode iamereniya teploty obraLovaniya khimicheckikh
soedinenii,
mogushchikhraspadat’syaxvakuume), Z/I. Pvikl. Fi:. 5, 101-I 14 (1928).
of dehydration heat in vacuum
27. ‘The determination
on a torsion balance (K voprosu ob clpredelenii
teploty degidrdtatsii\ vakuumc na krutil’nykh
vesakh), Z/1. frik/. Fiz. 6, vyp. 3-4, 96-103 ( 1929).
28. On the theory of the Bunsen flame (K teorii plameni
Bunzena), Zh. Tekh. Fiz. 5, vyp. 8. 1484-1487 (1935).
29. On the theory of gas reactions in a high frequency

ALEXANDERSAVVICH

electric discharge (K teorii gazovykh reaktsii v
v~ok~h~totn~melektriche~om
razryade), Zh.
Fiz. Khim. 6., VYU.
__ 4.418427 (19351.
30. Combustion of a ‘carbon particle in a gas flow
(Gorenie ugol’noi chastitsy pri obtekanii ee potokomgaza), Z/z. Z’ekh. Fiz. 10, vyp. 16, 1311-1323 (1940).
31. On the theory of a process of burning-up of a carbon
channel (K -teorii -protsessa vygoraniya ugol’nogokanala), Z/z. TX,%. Ez. 11, vyp. 10. 893-901 (1941).
32. The effect of solid admixtures on the velocity offlame propagation in fuel gas mixtures (0 vhyanii
tverdykh primesei na skorost’ rasprostraneniya
plameni v goryuchikh gazovykh smesyakh), Z/r. Tekh.
Fir. 7, vyp. 18-19, 1801-1811 (1937).
33. Combustion of walls of a carbon channel with forced
oxygen diffusion (Gorenie stenok ugol’nogo kanala
uri vynuzhdennoi diffuzii kisloroda). Zh. Tekh. Fiz.
”io, vyp. 13, 1113-1120 (1940).
34. On the kinetics of chemical reactions of solid andgaseous components resulting in the formation ofcomplex compounds(K kinetike khimicheskikh
reaktsii tverdoi i gazoobraznoi komponent, vedushchikh k obrazovaniyu kompleksnykh
soedinenii),
Zh. Priki. Fix. 3, VYU.
__ 3._67-73 (1927).
35. Carbon combustion (Gorenie &e&da).
Izd. Akad.
Nauk SSSR, Moscow (1949).
36. On velocities of chemica1 reactions in turbulent flows(0 skorostyakh khimicheskikh reaktsii v potokakb),
Inzh. Fiz. Zh. Part I, 3, No. 11, 3-10 (1960); Part II,
4, No. 1, 14-21 (1961).
37. Flame propagation in two-phase mixtures. Proceeding of the seventh International Symposia
on combustion and explosions, London (1958).

PREDVODITELEV

89

38. On the gas formation with carbon combustion in abed (K voprosu o gazoob~vanii
pri gorenii ugiya
v sloe), Izu. Akad. Nauk SSSR 10,1329-l 340 ( 1947).
39. The Arrhenius laws in chemical kinetics of gas reactions (0 zakonakh Arreniusa v khimicheskoi
kinetike gazovykh reaktsii), Dokl. Akad. NaukSSSR. Kkim. 57. No. 7.685-688 11947).
40. The dependence’of exhaustion of a catalyst bed on
the velocity of gas motion (Zavisimost’ istoshcheniya
sloya katalizatora
ot skorosti dvizheniya gaza),
Dokl. Akad. Nauk SSSR, Fiz. Khim. 127, No. 3,
602-605 (1959).
of a
41. The dependence between the concentration
waste gas and time of adsorbent exhaustion (Zavisimost’ mezhdu kontsentratsiei otkhodyashchego gazai vremenem istoshcheniya adsorbenta), Doki, Akad.
Nauk SSSR, Fiz. Khim. 127, No. 4, 825-827 (1959).
state of chemical gas
42. On the quasi-equilib~um
systems (0 kvaziravnovesnom sostoyanii khimicheskikh gazovykh sistem). Vestnik Mosk. Univ. 10, 4357 (1947).
of an atom
43. On one geometrical interpretation
structure of hydrogen by Rutherford-Bohr (Ob odnoi
geometricheskoi interpretatsii stroeniya atoma vodoroda po Re~rfordu-Boru),
J. RMssiaa Fiz. Chem.
Sot., Physicai Part, 61, vyp. 5,4431t50 (1929).
44. Possible classification of statistical systems (Vozmozhnaya klassifikatsiya
statisticheskikh
sistem).
Vestnik Mosk. Univ. No. 7, 23-42 (1947).
45. Studies on space and time in modern science (Uchenieo prostranstve i vremeni v sovremennoi nauke).
Vestnik istorii i metodo~ogii estestueaayk~ nauk No. 3,
Izd. Mosk. Univ. (1962).