Electrical effects in superfluid helium. I. Thermoelectric effect in einstein's capacitor
The Einstein's ideas about the thermodynamical fluctuational nature of some electrical phenomena and the difference
 of electrical potentials U in a capacitor at temperature T were proposed in 1906-1907. On base of these ideas
 we explain the experimental results, which were rec...
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| Date: | 2014 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2014
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| Cite this: | Electrical effects in superfluid helium. I. Thermoelectric effect in einstein's capacitor / D.O. Ledenyov, V.O. Ledenyov, O.P. Ledenyov // Вопросы атомной науки и техники. — 2014. — № 1. — С. 170-179. — Бібліогр.: 38 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860059447669293056 |
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| author | Ledenyov, D.O. Ledenyov, V.O. Ledenyov, O.P. |
| author_facet | Ledenyov, D.O. Ledenyov, V.O. Ledenyov, O.P. |
| citation_txt | Electrical effects in superfluid helium. I. Thermoelectric effect in einstein's capacitor / D.O. Ledenyov, V.O. Ledenyov, O.P. Ledenyov // Вопросы атомной науки и техники. — 2014. — № 1. — С. 170-179. — Бібліогр.: 38 назв. — англ. |
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| container_title | Вопросы атомной науки и техники |
| description | The Einstein's ideas about the thermodynamical fluctuational nature of some electrical phenomena and the difference
of electrical potentials U in a capacitor at temperature T were proposed in 1906-1907. On base of these ideas
we explain the experimental results, which were recently observed under the action of the second sound standing
wave in the electrical capacitors which take placed in the superfluid 4
He and in the torsional mechanical resonator.
The Einstein's approach, based on the interrelation of thermal, mechanical and electrical fluctuations, allows to obtain
the quantitative results, coinciding with the experimental datas for the correlations of the alternate low temperatures
difference in second sound wave and alternate electric potentials difference between the capacitor plates in superfluid
helium at as well as in the rotating mechanical oscillator.
Идеи Эйнштейна о термодинамической флуктуационной природе некоторых электрических явлений и
разнице электрических потенциалов U на обкладках электрического конденсатора при температуре T были
изложены еще в 1906-1907 годах. На основании этих представлений мы объясняем экспериментальные ре-
зультаты, которые недавно были получены под действием волны второго звука в электрических конденсаторах, находившихся при низких температурах в сверхтекучем гелии и в торсионном механическом резона-
торе. Подход, базирующийся на взаимосвязи термических, механических и электрических флуктуаций, по-
зволяет получить количественные результаты, совпадающие с экспериментальными данными по корреля-
ции малой переменной температурной разности в волне второго звука и переменной разности электрических
потенциалов между обкладками конденсатора как в сверхтекучем гелии, так и в торсионном механическом
резонаторе.
Ідеї Ейнштейна про термодинамічну флуктуаційну природу деяких електричних явищ і різницю електричних потенціалів U на обкладках електричного конденсатора при температурі T було ще в 1906-1907 роках. На основі цих пропозицій ми пояснюємо експериментальні результати, які недавно було отримано під
дією хвилі другого звуку в електричних конденсаторах, що знаходились при низьких температурах у надплинному гелії та в торсіонному механічному резонаторі. Підхід, що базується на взаємодії термічних, ме-
ханічних та електричних флуктуацій, дозволяє отримати чисельні результати, співпадаючі з експеримента-
льними даними за кореляцією малої змінної температурної різниці в хвилі другого звуку і змінної різниці
між обкладками конденсатора як у надплинному гелії, так і в торсіонному механічному резонаторі.
|
| first_indexed | 2025-12-07T17:03:14Z |
| format | Article |
| fulltext |
170 ISSN 1562-6016. ВАНТ. 2014. №1(89)
ELECTRICAL EFFECTS IN SUPERFLUID HELIUM.
I. THERMOELECTRIC EFFECT IN EINSTEIN'S CAPACITOR
D.O. Ledenyov, V.O. Ledenyov, O.P. Ledenyov
National Science Center «Kharkov Institute of Physics and Technology»,
Kharkov, Ukraine
The Einstein's ideas about the thermodynamical fluctuational nature of some electrical phenomena and the dif-
ference of electrical potentials U in a capacitor at temperature T were proposed in 1906-1907. On base of these ideas
we explain the experimental results, which were recently observed under the action of the second sound standing
wave in the electrical capacitors which take placed in the superfluid 4He and in the torsional mechanical resonator.
The Einstein's approach, based on the interrelation of thermal, mechanical and electrical fluctuations, allows to ob-
tain the quantitative results, coinciding with the experimental datas for the correlations of the alternate low tempera-
tures difference in second sound wave and alternate electric potentials difference between the capacitor plates in su-
perfluid helium at as well as in the rotating mechanical oscillator.
PACS: 05.40.-a; 67.40.Pm
INTRODUCTION
In 1906, Einsteinin his research paper on the Brown-
ian motion [1], firstly showed that there is an intercon-
nection between the thermodynamical fluctuations and
the electric effects in the linear electric circuit. In the
agreement with the Einstein’s conclusions, the electric
charge q(t) , which transfers through the cross section
of closed circuit conductor, is defined by the tempera-
ture T, closed circuit resistance R and observation time t
variables in the expression 2 2 /Bq t k T R= , where Bk is
the Boltzmann constant, the upper emphasis designates
the averaged value. The expression for the current is
2 2 /BI k T Rt= , and the equation for the difference of
electrical potentials is 2 2 /BU R k T t= in this electrical
circuit. In 1907, Einstein [2] showed that the electric
potentials difference U at plates of an electric capacitor
with the capacitance C must appear because of fluctua-
tional effects in surrounding medium at temperature T.
He proposed that the electric capacitor can be consid-
ered as a Brownian particle, which has one degree of
freedom, connected with its electric subsystem and in
the agreement with the Einstein’s calculations the fluc-
tuations for U can be observed in an electric capacitor.
According to the equipartition theorem for any degree
of freedom, it appears that the electric energy stored by
an electric capacitor is equal to the thermal energy as in
equation 2 2 2BCU k T= .
In [2], the corresponding calculations were com-
pleted and it was shown that the electric potentials dif-
ference U can be measured with the use of experimental
measurement setup at appropriate selection of capaci-
tance С at room temperature T. On that time, the dis-
cussed experiment [2] was not done, because the Ein-
stein’s first research paper on the thermodynamic elec-
trophysics [2] was not appropriately noticed among his
other revolutionary research ideas. A few years later, de
Haas-Lorentz [5] developed these theoretical represen-
tations for other elements of linear electric circuits and
showed that, in the fluctuational case, the inductance L
will store the energy as in equation 2 2 2BLI k T= . In
all the considered cases, the thermodynamic fluctuations
can do both to create the electrical currents as well as to
scatter the electrical currents. The heating energy, which
is transforming at this process during time τ, is equal to
2 4 BU R k Tτ = (in this case U=IR , I is the electrical
current, τ is the time, τ <<R/L). The difference of ob-
tained result in the two times in the given expression in
comparison with the expression in [1] is connected with
the fact that the characteristic time periods for the linear
magnitude U and the quadratic magnitude U 2 are dif-
ferent in the two times. In 1928, Nyquist [6] considered
the spectrum of possible frequencies of electrical oscil-
lations in the linear RLC circuit, and found an intercon-
nection of the time τ with the frequencies band span
Δω = 2π /τ , in which the registration of signal is per-
formed. Nyquist [6] also pointed to the possible quan-
tum case, which can be realized at the following condi-
tion ħω >kBT, where ω is the circular frequency of
electric signal in linear circuit. In 1951, Callen and Wel-
ton [7] obtained the quantum fluctuation – dissipation
formula with consideration of contribution of zero oscil-
lations and found an important relation
( )2 / 2 BU R cth k Tω ωω ω= h h (also see [8-11]), where
ħ=h/2π , h is the Planck constant. As far as we know,
the experiment for linear circuit with a capacitor, pro-
posed by Einstein [2], was not completed in the XX
century. At the same time, the fluctuational electrical
noises in the resistors were researched widely. The re-
sults obtained in the fluctuation – dissipation theory for
the linear circuits were confirmed in both the classic
case as well as the quantum case, serving as one of the
possible methods for research of the electromagnetic
fluctuations and the Van der Waal’s forces [12, 13].
In 2004, Rybalko [3] measured the alternating dif-
ference of electrical potentials U' between the plates of
an electrical capacitor, placed in the superfluid helium
(helium II) at T<Tλ (Tλ=2.172K is the temperature of
superfluid transition). The wave of second sound with
the frequency of ω2 was propagating in helium II, and
was accompanied by the oscillations of temperatures
difference T' . The amplitude of oscillations of tempera-
ture was equal to T'≈ (10− 2…10− 4)K and, as it was
discovered in [3], the alternating difference of electrical
potentials with the magnitude U'≈ (10− 7…10− 9)V
was generated at the same frequency of ω2. The relation
ISSN 1562-6016. ВАНТ. 2014. №1(89) 171
of amplitudes satisfied the condition U'/T'≈kB/2e ,
where e is the electrical charge of an electron.
In 2005, Rybalko and Rubets [4] placed a capacitor
in the torsional mechanical oscillator with the helium II,
which oscillated with the frequency of Ω. In this case,
the alternating electrical signal U' with the approxi-
mately same amplitude at frequency of 2Ω was regis-
tered. Authors [3, 4] proposed that the potentials U' are
due to some effects, connected with the electrical po-
larization of helium II, however the possible physical
mechanism of polarization was not known yet.
Presently, there are more than ten well known theo-
retical researches [14-26], devoted to the interpretation
of experimental data in [3, 4]. In these research papers,
the propositions about possibility of existence some new
effects, including the inertial effects [16], which, in their
authors opinion, might lead to the electric activity and
polarization of superfluid helium in conditions of low
temperature experiments in [3, 4]. However, the critical
analysis of theoretical researches in [21, 22] demon-
strated that the proposed models forecast the magnitude
of oscillations of potentials difference U' in
10 5…10 6 times less than the amplitude observed in
[3, 4]. It is also not explained: why does the magnitude
of oscillations of potentials difference U' not depend on
the temperature T in [3], when all the properties of he-
lium II change in the researched range of temperatures
significantly.
We will show up below that, in our opinion, some
of the experimental results [3, 4], related to the thermoe-
lectric fluctuational effect, appear in a capacitor due to
the effect, considered by Einstein [2], when there is an
alternating difference of temperatures T ′ between the
plates of a capacitor. Let us take to the attention the fact
that the conditions (ħω2 , ħΩ )<<kBT are true for the
second sound wave and rotational oscillations in [3, 4],
therefore these effects belong to the classic case of fluc-
tuational processes, hence it is enough to use the repre-
sentations in [2] for their theoretical description.
FLUCTUATIONAL THERMOELECTRIC
EFFECT IN A CAPACITOR
As it is known, any physical body with the tempera-
ture T, is a source of electromagnetic radiation. The
spectrum of radiation depends on the body temperature
only, if the body and radiation are in the state of thermal
equilibrium. Following the Einstein [2], let us consider
the case, when the characteristic radiation frequency ω
satisfies the classic condition ħω<kBT . We will believe
that the quasi-statistical electric and magnetic fields are
dependent on the thermal fluctuations of electrical
charges. The state of physical system is characterized by
some values of its physical parameters 1 2, ,... nξ ξ ξ , de-
fining the thermodynamic state [2, 27]. In general case,
these parameters can characterize the microscopic state
as well as the macroscopic state of physical body. In the
microscopic case, when the physical body can be repre-
sented by a group of microscopic particles n, the value
of physical parameter iξ will characterize the particle i.
In the macroscopic case, the value of physical parameter
iξ will characterize all the physical body with different
states of freedom with indexes i. In the equilibrium
state, these parameters have some average values
1 2, ,... nξ ξ ξ . The interconnection between the entropy of
physical system 1 2( , ,... )nS ξ ξ ξ and the statistical weight
1 2( , ,... )nW ξ ξ ξ of a corresponding state of system is de-
fined by the Boltzmann expression
1 2 1 2( , ,..., ) ln ( , ,..., )n B nS k Wξ ξ ξ ξ ξ ξ= [27] . The prob-
ability of system’s presence in a particular state, charac-
terized by the parameter ξ i , is equal to
( ) ( ) ( )
i
i i i i i iP W W
ξ
ξ ξ ξ= ∑ . In the equilibrium state,
0i i iξ ξ ξ= = , and the entropy reaches its maximum val-
ue 0 10 20 0 0 10 20 0( , ,..., ) ln ( , ,..., )n B nS k Wξ ξ ξ ξ ξ ξ= . The
thermodynamic fluctuations lead to the deviation of
given parameters from their average values. The prob-
ability P that the parameter ξi has its values in the range
ξi0+dξi is defined by the completed work dA TdS= − ,
which is equal to the change of free energy in thermo-
dynamic system. At the change from ξ i 0 to ξ i , the
work will be equal to A TdS= −∫ , or
0 0( ) ln( )BA T S S k T P P= − − = . The probability of sys-
tem presence in this state is 0 exp( )BP P A k T= − . Let
us suppose that the deviations ξ i−ξi0, appearing at the
action of random fluctuations, are with the alternating
signs and small values, therefore A can be decomposed
in the Taylor series, which begins with the second order
term. Then, we will get 2
0( ) ...i iA b ξ ξ= − + , where b is
the constant. The probability of presence of system in
the state ξ=ξ i will be
2
0 0( ) exp[ ( ) ]i i i BP P b k Tξ ξ ξ= − − . (1)
We will obtain the probability magnitude P0 inte-
grating P(ξi) over ξ i (−∞, +∞). These limits can be set,
because of the convergence of an integer, which will be
equal to the one
( )
2
0 0
1/2
0
( ) exp[ ( ) ]
1.
i i i i B i
B
P d P b k T d
P k T b
ξ ξ ξ ξ ξ
π
+∞ +∞
−∞ −∞
= − − =
= ⋅ =
∫ ∫ (2)
It follows from the above that ( )1/ 2
0 BP b k Tπ= .
Taking to the consideration that the equilibrium
electric and magnetic fields are defined by the thermo-
dynamic state of the body, which is characterized by its
temperature, then, in the classic case, the intensities of
the electric field E and the magnetic field H have to be
among the values of parameters ξi [27]. The deviation of
average energy from equilibrium value may be written
as
2 2
0 0 0 0
1 1( ) ( )
2 2E HA V E E V H Hεε μμ= − + − , (3)
where E is the intensity of electric field, H is the inten-
sity of the magnetic field, E0 and H0 are the equilibrium
magnitudes of electric and magnetic field, ε is the rela-
tive electric permittivity, ε0 is the free space permittiv-
ity, μ is the relative magnetic permeability, μ0 is the free
space magnetic permeability, VE is the volume, in
which the electric field is concentrated in the capacitor
with capacitance C and VH is the volume, in which the
172 ISSN 1562-6016. ВАНТ. 2014. №1(89)
magnetic field is concentrated in the induction L. The
expressions for the electric and magnetic energies are
similar and quadratic by the fields in both cases. Let us
note that there are the resonance phenomena in the LC
electrical circuit. The fluctuational oscillations of the
electric and magnetic fields have the maximum ampli-
tudes near to the resonance frequency ω0 = (LC)−1/2 in
the LC electrical circuit, while the fluctuational oscilla-
tions of the electric and magnetic fields have a wide
spectrum similar to the white noise in an electrical ca-
pacitor. We will analyze the response of the linear RC
electric circuit and limit our research by the considera-
tion of the case, directly related to the experiment [3]
and only connected with the electric field in the capaci-
tor C.
The average work A , which must be done by the
system in order to change the intensity of electric field
from E0 to E, can be represented as the Gaussian inte-
gral
2 2
0 0 0 0
1 1( ) ( ) ( ) .
2 2E EA V E E V E E P E dEεε εε
+∞
−∞
= − = −∫ (4)
Here, the probability is
2
0 0( ) exp[ ( ) ]E BP E P b E E k T= − − , where
( )1/2
0E BP b k Tπ= , b>0 and 0 2Eb V εε= . The integer,
which defines the average work, can be transformed as
1/2 ( )Bk T
A I ϕ
π
= , where 2 2 1/2( ) exp( ) 2I dϕ ϕ ϕ ϕ π
+∞
−∞
= − =∫
and ( ) ( )1/2
0 02EV E Eϕ εε= − (the integer I(ϕ) is shown
in [28]). The final result 2BA k T= is equal to the
thermal energy, related to the one degree of freedom of
macroscopic body. This is explained by the fact that the
part of thermodynamic system, which is connected with
electric field, can have the only one predetermined ori-
entation of direction of vector E in every point of space
in a capacitor. In a plane capacitor, the vector E is di-
rected toward the shortest direction n0 between the two
plates, and the task is effectively reduced to the one di-
mensional case. In particular case, it is possible to as-
sume that there is no the external electric field, E0=0 .
At homogenous temperature, the two orientations of
vector E=±En0 have equal probabilities, hence the
time dependent fluctuating electric field E(t) is random
with alternating plus-minus sign. The electric field aver-
age magnitude is ( ) ( )
2 2
1 1
0
t t
t t
E t E t dt dt= =∫ ∫ , if the time
interval ( t 2− t 1 )>>RC , where R is some equivalent re-
sistance of circuit with an electrical capacitor in Fig. 1.
The characteristic time of correlation τ=RC corre-
sponds to the time of attenuation of fluctuation in a cir-
cuit of a capacitor. Let us note that the resistance R has
the temperature T, and the fluctuational electric current
appears in it in agreement with the fluctuation-
dissipation theorem. The fact that the resistance R
doesn’t represent a main source of the fluctuations of
potentials difference, but defines the time of relaxation
of the fluctuations (see below), is a main distinctive fea-
ture of such simple linear scheme. As it was shown
above, the average quadratic magnitude of the electric
field
2 2
2
1 1
( ) ( )
t t
t
t t
E E t E t dt dt= ∫ ∫ is not equal to the nil.
The full energy of electric field in a capacitor with vol-
ume VE can be expressed by its macroscopic capaci-
tance C as 2 2
0
1 1
2 2EV E CUεε = , where the potentials
difference is
0
( )
d
U E x dx= ∫ , and the capacitance of plate
capacitor only depends on the geometric dimensions
0 plC S dεε= , where d is the distance between the
plates, and Sp l is the square of plate of a capacitor.
From this, we come to the expression obtained by Ein-
stein
21 1
2 2 BCU k T= . (5)
Fig. 1. Electrical circuit with capacitor С and
conditional resistance R; difference of temperatures
between plates is T'~T0 sin(ω2 t)
This expression doesn’t depend on the resistance R
in the electric circuit, which connects the plates of a ca-
pacitor, that is the considered case, when the frequency
of thermal fluctuations is significantly more than 1/RC.
It is possible to write 1 1
2 2 BqU k T= , where q CU= is
the average fluctuating electric charge of a capacitor,
and 2U U= is the average quadratic value of poten-
tials difference. The charge of capacitor in fixed time
moment времени q(t) consists of the discrete number
of electrons, while its average quadratic value CUN
e
=
can be fractional as a result of time averaging (e is the
charge of an electron). The stored electric energy is
1 1
2 2 BNeU k T= , and it is proportional to the tempera-
ture. If T and C are constants the average quadratic dif-
ference of potentials decreases at increase of a number
of fluctuating electrons
1/2 1 Bk TkTU
C eN
⎛ ⎞= =⎜ ⎟
⎝ ⎠
(6)
and reaches its maximum at N=1 . The electric energy,
related to the one fluctuating electron, is Bk T
e U
N
= .
The full energy, expressed by the charge and capaci-
tance, is equal to
2 2 21 1 1
2 2 2 B
q N e k T
C C
= = . (7)
ISSN 1562-6016. ВАНТ. 2014. №1(89) 173
The same expression between the number of fluctu-
ating electrons and the temperature 2N T∝ is true in
the metal in the degenerative electron Fermi-system at
the temperature T, which is much less than the tempera-
ture of degeneration (see [9], §113). We didn’t take to
the account the Fermi statistics of conduction electrons.
The matter is that the electric fields of conduction elec-
trons in the metal are exactly compensated by the
charges of metal nucleuses at the distances r≈a0 , where
a 0 is the distance between the nucleuses; and the condi-
tion of electro-neutrality is true in the metal. The trans-
lation invariance of metal’s crystal grating allows hav-
ing the free transfer of conduction electrons in the met-
al. In this case, the Fermi statistics has to be taken to the
account in view of the big density of states and degen-
eration of electronic system. The electrons, which are
situated near the Fermi surface in the momentum space,
mainly contribute to the energy and charge transfer. The
thermoelectric coefficient in metal at low temperature in
a few kelvins is very small: α∝kT/εF 10− 4 [9], εF is
Fermi energy. In our case, the electrons, which create
both the charge at plates of a capacitor and the fluctuat-
ing electric field of a capacitor, can be considered as the
classic particles, because their surface density at plates
of a capacitor is not big, hence their Fermi energy εF is
very small. The characteristic de Broglie wavelength of
the electrons will be significantly bigger in comparison
with the distances between the nucleuses; and the dis-
tance between the energy levels is small in comparison
with the temperature Δε kBT . The electric field, cre-
ated by the electrons, isn’t shielded by the charges of
nucleuses at distances between the atoms in the metal,
and propagates at the macroscopic distance d>>a 0 in
the space between the plates of a capacitor. In the ex-
periment [3], a general number of the electrons wasn’t
big indeed (see below). The calculation of energy of this
electric field was completed by the authors above.
Now, let us consider the thermoelectric effect for
the fluctuational electrons, appearing in the case, when
the plates of a capacitor have some different tempera-
tures. In the experiment [3], the plates of a capacitor
were in He II at certain equilibrium temperature T , and
the wave of second sound generated the alternate differ-
ence of temperature T ′ ( t )=T ′ 0exp(iω2 t ) between the
plates (T ′ ( t)<<T) . We will assume that one of plates
of a capacitor is at constant temperature T, and the tem-
perature of second plate is T+T ′ ( t ) . The change of
thermal and electric energies of electrons at variation of
plate temperature has to be taken to the account in view
of the presence of a gradient of temperature dT ′ /dr be-
tween the plates of a capacitor. In the linear response
theory [27], each of N electrons of electric fluctuation
takes the additional energy, which is equal to 1
2 Bk T ′ .
This is accompanied by an appearance of corresponding
addition to the potentials difference U ′ . Herewith, the
change of a full number of fluctuating electrons N at the
action of small difference of temperatures T ′ is also
small N ′<<N .
Fig. 2. Conditional chart of fluctuational depend-
ence of potentials difference U(t) between plates of ca-
pacitor on time t at some constant temperature T (a)
and on frequency (b), where ωT is the Brownian fre-
quency of potential difference in capacitor and ω2 is the
frequency of second sound
Let us consider some conditional dependence of po-
tentials difference U(t) between the plates of a capacitor
on the time in Fig. 1,a and on frequency dependence in
Fig. 1,b. The following expressions to characterize the
function U(t) can be written, if the temperature T on the
plates of a capacitor is same:
2( ) 0, ( ) 0U t U t= ≠ , (8)
2 2( ) ( ) 0U t U t+ −= ≠ , (9)
( )2 2 2( ) ( ) ( ) BCU t C U t U t k T+ −= + = , (10)
2 2( ) ( ) 2BCU t CU t k T+ −= = , (11)
2 ( ) 2BCU t qU k T+ += = , (12)
where ( )1/2
2U U= and / 2q CU Ne+= = , where N is
the average number of fluctuating electrons at the both
plates in a capacitor.
Let us assume that there are the upper and lower
plates in a conditional condensator (see Fig. 1). We
define the potentials difference as U+, when the
electrons, creating the charge, are at the upper plate in a
capacitor. We assign the potentials difference as U−,
when the electrons, creating the charge, are at the lower
plate in a capacitor. Moreover, we assume that the
temperature is changing in time and is equal to T+T'( t )
at the upper plate of a capacitor; and the temperature is
constant and is equal to T at the lower plate of a
capacitor. The oscillating system in [3] can be divided
by two parts: 1) the mechanical part, which is connected
with the fluid He II, and 2) the eletrical part, which is
consisted of a capacitor in the fluid He II, and its
electrical circuit, including the voltmeter with the input
resistance R. As it is shown above, the oscillations in-
duced by the wave of second sound in the helium nor-
mal and superfluid subsystem are accompanied by the
oscillations in the electrical part, which is in the ther-
modynamic equilibrium with the mechanical subsystem.
These oscillations are accompanied by the appearance
174 ISSN 1562-6016. ВАНТ. 2014. №1(89)
of the potentials difference and currents, flowing by the
circuit and depending on its resistance R. In general
case, the correlation function for the voltage U in the
linear RC circuit depends on the time t as
( )1 2 1 2( ) ( ) expBU t U t k T C t t RC< >= ⎡− − ⎤⎣ ⎦ .
Let us assume that, in the discussed experiments, the
time constant of linear circuit RC satisfies the condition
| t 1 - t 2 |<<RC , where | t 1 - t 2 |≈2π /ω2 in [3], and
| t 1 - t 2 |≈2π /Ω in [4]. This allows us to simplify the con-
sideration and assume that the exponential term in the
correlation function is close to the one. Let us think that
the heat capacity of metal walls in a capacitor and their
near surface areas, where the electric charge is accumu-
lated in a capacitor at low temperatures, is small in
comparison with the heat capacity of helium fluid, and
is not dependent on the thermal state of the system,
which is defined by the fluid only. We don’t count the
possible Kapitza temperature jump on the boundary of
metal, because the amplitude of oscillations of tempera-
ture T' is significantly smaller than the equilibrium
temperature T.
In an agreement with the above conditions, it is
possible to write an expression for the full thermal and
electric energies of a capacitor with the plates at
different temperatures, taking to the consideration that
T' T
( )
( )
2 2( ) ( ) ( )
1 1
2 2 2B B B
C U t U t CU t
Nk T T N k T k T
+ + −′+ +
⎛ ⎞′ ′ ′= + + + +⎜ ⎟
⎝ ⎠
(13)
and for the (+) plate of a capacitor under action of T' ,
will be equal to
( ) ( )2 1( ) ( )
2 2B B
NC U t U t k T T N k T+ +
⎛ ⎞′ ′ ′ ′+ = + + +⎜ ⎟
⎝ ⎠
. (14)
The second term in the right part of equations is
connected with the change of a number of electrons at
an action of the temperature T' . Now, we have to take to
the consideration the fact that its frequency is
significantly smaller than the frequency of thermal
fluctuations, hence it can be considered as an analogue
of quasistationar effect. In this case, a number of
fluctuating electrons changes by N' and the energy of
each electron changes by kT' /2. Let us open the first
quadratic term in the left side of equation (14), and
considering the difference of times of averaging τ τ′
for U+ and U+′ , let us leave the unaveraging ( )U t+′ in
the second term. Thus, we will get
( )
2 22 ( )
1 ( ) ( ) ( ) ( ).
2 2B B B
CU CU U t CU
Nk T T t k T t N t k T t
+ + + +′ ′+ +
′ ′ ′ ′= + + +
(15)
In the left and right parts of the equation, the second
order terms, which have the dashes, are small, and they
can be neglected. Then, we will obtain the expression
for the first order terms with U+′ and ( )N t′ :
1 12 ( ) ( )
2 2 2B B B
N NCU U t k T k T k T t+ +
+⎛ ⎞′ ′ ′ ′= + = ⎜ ⎟
⎝ ⎠
(16)
or
1( ) ( )
2 B
NNeU t k T t+
+⎛ ⎞′ ′= ⎜ ⎟
⎝ ⎠
. (17)
Let us note that, in the standard theory of
measurements [29], the term at left part of equation
(16), is assumed to be equal to the nill, that is true, if the
time of averaging is significantly more than the
characteristic time of change T ′ . However, in [3] and
[4], the technique of synchronous detection was used,
and theis contribution was registered at second sound
frequency of ω2 , which is characteristic for T ′ , that is
why it was not averaged and is different from the nill.
From this, we will obtain the final result
( ) ( 1) 10
( ) 2 2 2
B BU t k kN
T t N e e N
α +′ + ⎛ ⎞= = − = − − ⎜ ⎟′ ⎝ ⎠
. (18)
The equation (18) corresponds to the thermoelectric
coefficient α for the electrons, which take part in the
fluctuation process at presence of the changing
temperatures difference T'( t)between the plates in a
capacitor.
This result quantitatively corresponds to the magni-
tude of α, obtained in the experiment in [3]. It also fol-
lows from the expression (18) that the negative elec-
trons charge is accumulated at heated plate in a capaci-
tor at increase of temperature (T ′>0 ) . In the experiment
[3], the plate with the high temperature was negatively
charged in a capacitor.
Thus, the creation of small enough oscillations of
temperatures difference T ′ω2∝T ′ 0 s in(ω2 t ) by the
wave of second sound in the fluid helium II has to be
accompanied by an appearance of the additional in-
phase potentials difference U ′ ω2=αT ′ 0 s in(ω2 t ) ,
where α =−kB/2 |e|≈ 4.3126 ⋅10−5 V/K. The equilib-
rium temperature T is not included in (18), hence the
results don’t depend on the magnitude of equilibrium
temperature T, as it was observed in [3]. In the re-
searched case [3], the alternate electrical field reached
the magnitude E ′ 0∼ (10− 4…10− 6) V/m in a capaci-
tor. The helium atoms didn’t contribute a notable
change to the obtained result, because the polarizational
effects for the helium II atoms are proportional to the
square of intensity of electric field as in the case of all
the inert gases with filled electron shells.
ELECTRICAL POTENTIALS DIFFERENCE
OSCILLATIONS IN CAPACITOR IN
MECHANICAL TORSIONAL RESONATOR
Let us analyze the results of experiment [4], in
which the oscillations of electric potentials difference
U ′ ( t ) at the electrodes of cylindrical electric capacitor
(Fig. 3), connected with the mechanical torsional reso-
nator, were observed in the range of temperatures from
1.4 K to Tλ=2.172 K.
The first plate of a capacitor includes a fixed elec-
trode (1); the second plate of a capacitor consists of the
cylindrical surfaces (2) and (4) with the covers (3) and
ISSN 1562-6016. ВАНТ. 2014. №1(89) 175
(5), involved in the rotational movement. The maximum
amplitude of electric potentials difference was
U ′≈ 1.5 ⋅10−7 V at temperature T≈ 2 K [4]. The he-
lium II as the liquid or as the saturated or not saturated
film, covering the inside surface of a resonator, was
placed in a resonator. In an agreement with the equation
(18), the oscillations of the electric potentials difference
at such a scale can be originated by the oscillations of
the temperatures difference T ′∼ 10− 3K in a given ca-
pacitor. Considering the dependence of relative concen-
tration of superfluid component on the temperature, we
can find that, at the temperature T=2 K , the magnitude
is ρ s /ρ ≈ 0 .3, where ρ s is the density of superfluid
component, and ρ is the full density. In agreement with
the data in [30], at the above temperature T, the change
of density of superfluid component ρ′s, appearing at the
action by oscillations of given temperatures difference
T ′ , doesn’t exceed the magnitude 34 10Sρ ρ −′ ≈ ⋅ .
Let us consider the case, when there is a saturated
film of superfluid helium inside the resonator. As it is
known [30], the thickness of saturated film is
∼ 25…30 nm, and it contains the 100…120 layers of
the helium atoms approximately, and it covers all the
inside surface of a resonator. The thickness of film
slightly decreases in the upper part of a resonator, but
this effect has no principal influence in this case, hence
we don’t consider it. It is visible that the change of den-
sity ρ′ is commensurable with the mass of approxi-
mately 0 .4…0.5 single layer of helium at the above
described oscillations of temperature. The flow of small
amount of the superfluid component from one part of a
resonator to another part will result in the change of dis-
tribution of temperatures in a resonator at adiabatic con-
ditions due to the above considered mechanism, result-
ing in the observed effect [4].
Let us define the physical mechanism and forces,
which have the influences on the dynamics of the nor-
mal and superfluid components of the helium II, and
their transfer during the rotation of a resonator in the
considered research task. In [4], the resonator conducted
the rotational oscillations ϕ ( t )=ϕ 0s in(Ω t ) , where ϕ 0
is the amplitude of angular oscillations, expressed in the
radians, Ω is the angular frequency. The velocity of ro-
tation of resonator’s walls is
υϕ (r)=rdϕ( t ) /d t=rϕ 0Ω cos(Ω t ) , where r is the ra-
dius of given point of rotating surface. At the surface (4)
in Fig. 3, the velocity of rotation had a maximum value
[4]. It is possible to assume that, in view of the big vis-
cosity and small thickness of the superfluid film, the
normal helium component is fully drag by the rotating
resonator’s walls, taking part in the rotational movement
with the velocity υ n ,ϕ (r , t ) , which is equal to the veloc-
ity of walls rotation υϕ (r) . The centrifugal force
Fc=ρ n (υ n ,ϕ )2 /r , acting on the normal component, can
be regarded as small enough in the considered case, as-
suming that the radial velocity of normal component is
υn,r ≈ 0. The superfluid component is not involved in the
rotational movement of resonator’s walls with the ve-
locity smaller than the critical velocity of vortices gen-
eration, that is υ s ,ϕ = 0. Let us note that, during the os-
cillations of a resonator, the superfluid component can
make the oscillating movements with some radial veloc-
ity υ s , r ≠ 0, and it can flow from the regions with the
small radius r, that is from the surface (2) and from the
central regions of covers (3) and (5) to the side surface
(4), which has the biggest radius R, and then flowing
backward. Let us clarify the cause of oscillating move-
ment. In some sense, the oscillations of such a type are
similar to the oscillations between the two connected
vessels, when the helium II levels inside the vessels are
close to the equilibrium [31], and the vessels are con-
nected by the thin constriction or when the interflow of
the helium II between the vessels is done by the means
of the superfluid helium film at presence of outer pres-
sure.
The system of hydrodynamic equations for the he-
lium II was firstly proposed by Landau, and then devel-
oped by Khalatnikov (see [32-34]). Let us write the
equation of movement for the superfluid component at
coordinate r
22
,( )
0
2 2
n n ss s
s s rt
ϕρ υ υυ υ
ρ ρ μ
ρ
⎛ ⎞−∂
+ ∇ + − =⎜ ⎟⎜ ⎟∂ ⎝ ⎠
, (19)
where μ is the chemical potential. Let us assume that the
rotational oscillations are generated with the small an-
gular amplitude and with the small linear velocity υϕ ,
which is less then the critical velocity of vortex genera-
tion. In the initial moment, the superfluid component
will be in the state of quiescence with the velocity
υ s=0, and by selecting the phase of oscillations, when
∇μ=0 , the equation (19) can be written
, 2
,2
s r n
s s r nt ϕ
υ ρ
ρ ρ υ
ρ
∂ ⎛ ⎞
= ∇ ⎜ ⎟∂ ⎝ ⎠
, (20)
where υn
2=[rϕ 0Ω cos(Ω t )] 2 .
Let us consider the above written dependence of
υ n ,ϕ (Ω) and derive the final expression for the force,
acting on the superfluid component as
( ), 2 2
0 1 cos(2 )s r s n
s r t
t
υ ρ ρ
ρ ϕ
ρ
∂
= Ω + Ω
∂ (21)
Fig. 3. Torsional resonator with fixed metal
electrode (1) and oscillating cylindrical metal cavity
(2, 3, 4, 5) (after [3]). Arrows show directions
of superfluid helium component flow, when velocity
of rotation of a resonator is at maximum
176 ISSN 1562-6016. ВАНТ. 2014. №1(89)
The right part of the equations (20), (21) is the Ber-
noulli force which acts on the superfluid component,
transferring it to the part of a resonator, where the linear
velocity υ n ,ϕ has its maximum value. It is well known
fact that the Bernoulli force can not originate in the ex-
periments to research the Venturi effect in the helium II
in the pipe with constriction [35-36]. In agreement with
[36], in this case, the condition rotυ s=0, where the di-
rection υ s coincides with the direction of movement of
normal component υ n in the pipe, has to be true for the
superfluid component. This results in the equality of the
helium II levels, measured in the region of wide pipe
and in the region of constriction of wide pipe. In the re-
searched case, the appearing superfluid component ve-
locity υ s , r is directed along r and orthogonal to the
normal component velocity υ n ,ϕ in the equation (16).
Under these conditions, the Bernoulli effect has to orig-
inate, because the expression rotυ s=0 is true in view
of the existing orthogonality between the velocity vector
and the direction of circular contour bypassing. The in-
terflow of the superfluid component, which doesn’t
transfer the entropy, results in an appearance of the dif-
ference of temperatures T ′ between the resonator’s re-
gions with the small and big coordinates r. In the turn-
ing points of a resonator, the velocity of normal compo-
nent is υ n ,ϕ=0, and the Bernoulli force is equal to the
nil. The thermomechanical effect, which creates a con-
trary stream of the superfluid helium II and results in an
origination of the oscillation process, plays a main role
at the turning points of a resonator. The phase of oscilla-
tions of the helium II can be shifted in relation to the
resonator’s oscillations, because of the inertia of the
flow of mass, however this effect can be disregarded in
view of the small mass transfer. These forced oscilla-
tions of the temperatures difference and the flow of su-
perfluid component at the axe r can transform to the
fading oscillations, continuing for some time after the
moment of resonator’s rotation stopping, as observed in
[4].
Going from the calculations, the contribution by a
cylindrical capacitor with small radius with the elec-
trodes (1) and (2), and the covers of a big volume reso-
nator with the electrodes (1) and (3) and the electrodes
(1) and (5) to the full value of capacitance of a sectional
capacitor exceeds the capacitance of a capacitor with the
electrodes (1) and (4) in more than 5 times. Therefore,
namely the plates (2), (3) and (5) will create the main
electric response of a sectional capacitor. In the connec-
tion with the interflow of superfluid component from
these surfaces to the surface (4) during the oscillations,
all the plates will experience the increases of tempera-
tures periodically. Therefore, the plates will charge neg-
atively in an agreement with the equation (18), but the
central electrode (1) will charge positively, because it
will have the lower temperature, comparing to the other
plates. To account for the fluctuational contribution, let
us represent the variables as T=T0+T ′ and
υ n=υ n 0+υ′ , where the part with the index 0 corre-
sponds to the equilibrium value, but the second part cor-
responds to the value, which depends on the time and
originates as a result of the oscillations of a resonator.
Let us evaluate the amplitudes of temperature difference
T ′ , appearing between the central and distant regions of
a resonator.
Let us substitute dμ=−σdT+dP/ρ , where σ=S/ρ ,
in the equation (19):
2
0
2
,0 ,
( )
2
( ) 0.
2
s s s
s s s
s n
n n s
d
P T T
dt
ϕ
ρ ρ
ρ ρ σ
ρ
ρ ρ
υ υ
ρ
′+ ∇ + ∇ − ∇ +
− ∇ + − =
υ
υ
υ
(22)
Let us assume that the gradient of external pressure
∇P≈0 and also the relation is true υ s /υ n<<1 and
∇υ s
2≈0 , which are in an agreement with the conditions
of experiment. In the extremum points dυ s /d t=0 , and
we can derive the intercoupling expression between the
temperature and the velocity of normal component of
the helium II by integrating the two last terms in the
equation (22) over the r (see the similar solution for the
thin capillary in [32] §140). From the equation (22) we
will obtain
2
0 , 0( ) ( )
2
s n
n n s T Tϕ
ρ ρ
υ υ ρ σ
ρ
′+ = − + . (23)
In this case , 0n nϕυ υ<< where 0nυ i s the average
thermal velocity of the helium atoms, υ n ,ϕ is the velocity
of the normal component on the surface of rotation. For
the numerical evaluation of the effect’s magnitude, let
us write the expression (23) for a capacitor with small
diameter (the surface number (2), see Fig. 3) and a ca-
pacitor with big diameter (the surface number (4),
see Fig. 3), taking to the consideration the volumes V1
and V2 of films of fluid helium II, situated in every ca-
pacitor. Let us assume that, in the adiabatic case in the
corresponding phase of wave process, the superfluid
component of helium II, which flows from the small ca-
pacitor, results in an increase of temperature at the sur-
face (2) and in a small change in a kinetic term at left
part of equation (23), because of the small magnitude of
surface radius and small velocity of its rotation. This
superfluid helium, inflowing into a big capacitor at the
surface (4), leads to the small decrease of temperature in
view of its small volume in comparison with the volume
V2, situated inside it, but the term, connected with the
kinetic energy, is significantly increased, because of big
velocity of rotation of the surface. Let us assume that
the linear velocity of rotation doesn’t increase above the
critical value, hence the superfluid component is not
trapped by the oscillating resonator and υ s ,ϕ=0. Let us
sum up the equations (23) for the both considered ca-
pacitors, leaving the main terms, which have an influ-
ence on the change of temperature only.
Then, as in the case of the equation (16), we will ob-
tain
2 2
2 0 0 , , 1 0( 2 ) ( )
2
s n
n n n n sV V T Tϕ ϕ
ρ ρ
υ υ υ υ ρ σ
ρ
′+ + = − + , (24)
where υ n ,ϕ is the velocity of the normal component of
helium on the rotation surface (4) (see Fig. 3).
Let us believe that the value υ nϕ
2 is quadratically
small, and it can be disregarded. Equalizing the linear
T ′ and υ nϕ terms, we will find the dependence of am-
plitude of oscillations of the temperature T ′ in a small
ISSN 1562-6016. ВАНТ. 2014. №1(89) 177
capacitor on the velocity of rotation of surface υ nϕ in a
big capacitor (4) as
2 0 , 1n n nT V Vϕρ υ υ ρσ′ = − . (25)
Let us conduct the qualitative evaluation of ampli-
tude of oscillations of temperature, for example, at the
temperature T =2 K. In [4], the maximum linear veloc-
ity of rotation was υ n ,ϕ=7 ⋅10− 4m/s . At the given tem-
perature the thermal velocity for the helium atoms is
0nυ ≈ 100 m/s , the specific entropy of helium II is
σ ≈ 940 J /kg ⋅K, ρ n /ρ = 0.7 , and the relation be-
tween the volumes of the helium II films on the surfaces
(2) and (4) is V2/V1 ≈ 16.7 . Then, going from the
equation (25), we will obtain T ′≈ 1.3 ⋅10−3K. The os-
cillations of difference of electric potentials will be
mainly defined by the change of temperature in a ca-
pacitor with small radius (electrodes (1) and (2) ), and in
an agreement with the equations (18) and (25), we will
obtain
2 2 0 , 12
B
n n n
k
U V V
e ϕρ υ υ ρσΩ′ = . (26)
This result is in a good qualitative agreement with the
data, obtained in [4]. From the equation (21), it follows
that υ n ,ϕ∼rϕ 0Ω cos(Ω t )(1−s in 2(Ω t )) , and the max-
imum value of the electric potentials difference must be
observed at the maximum deflection angle of a resona-
tor, but not at the moment, when the maximum value of
acceleration is reached, as confirmed in [4].
Thus, the rotation of an oscillator in [4] results in
both an increase of the normal component with subse-
quent interflow of superfluid component of the helium
II at the Bernoulli force action on the resonator’s wall
(4) with the biggest radius of rotation as well as an ap-
pearance of the biggest difference of temperatures be-
tween the plates of a capacitor with the small radius of
rotation (1-2). In the experiments [3] and [4], the physi-
cal foundations of observed processes are based on the
thermoelectric effect, appearing for the Einstein's fluc-
tuating electrons at presence of the temperatures differ-
ence between the plates of a capacitor.
DISCUSSION
Going from the research by Einstein [1], the authors
developed the theory of thermoelectric effect for the
fluctuating electrons, allowing to explain the results of
experiments in [3] and [4], which are connected with the
thermal effects, accompanied by an appearance of the
small electric fields in a capacitor in the helium II. The
thermoelectric coefficient α seems to be equal to the
value, which corresponds to the value in the known
classic theory by Drude [37], and approximately in 10 4
times bigger, than it can be in the metals at such low
temperatures, where it is reduced on kBT/εF . Using the
expression (7), it is possible to evaluate a total number
of fluctuating electrons in the experiment [3], where the
capacitance of a capacitor was C ≈ 5 ⋅10− 1 3F [39]. We
obtain that the total number of fluctuating electrons is
23N ≈ at temperature T =2 K. At T = 1,4 K, the
total number of fluctuating electrons is 19N ≈ . In the
researched case, the average number of fluctuating elec-
trons is small, and they can be considered as an ensem-
ble of classic particles, without taking to the account the
Fermi statistics. The change of temperature fields in the
He II is directly connected with the transfer of the su-
perfluid and normal components of the He II, and ob-
served at the propagation of waves of second and third
sounds or at the interflow of superfluid films [31, 33].
The dielectric constant ε of the fluid helium II de-
pends on the temperature, and it can have an influence
on the capacitance magnitude of a capacitor. At the
change of temperature from Tλ to 1 K, the relative value
of change of density is approximately Δρ /ρ ≈ 10− 3 ,
therefore the change of Δε /ε has the same value. The
correction of coefficient α in this mechanism is near
10− 8 V/K, that is why it was not taken to the consid-
eration in this case. Undoubtedly, it is necessary to take
to the account this correction in the case of experiments
with the big magnitudes of electric fields.
In our opinion, the maximum U ′ at temperature of
around 2 K (see Fig. 3) in [4] is connected with the
maximum of amplitude of oscillations of heat flux
W ′=ρCT ′u 2 in the case of second sound propagation,
where C is the heat-capacity of the helium II. During the
propagation of superfluid film, the maximum W ′ is in
the same temperatures range, though the velocity of su-
perfluid film propagation is approximately one order of
magnitude less than the velocity of second sound propa-
gation.
We don’t provide the exact numerical analysis of
frequency spectrum in [3], because it significantly de-
pends on a number of conditions of experiment, which
are not provided in [3].
Let us draw attention to the fact that the clear dif-
ference of amplitudes U ′ at the rotation of a resonator
in different directions is visible on the dependence of
the amplitude of oscillations U ′ on the time, described
by the equation (21) (see Fig. 2) in [4]. This difference
of amplitudes U ′ can be explained by the varying ve-
locity of normal component of helium υ n ,ϕ , appearing
in view of the non-symmetric polishing of resonator’s
plates, but not because of the presence of some circulat-
ing superfluid He II stream, as it was assumed by the
researchers [4]. In the case of circulation, the time semi-
periods of oscillations, connected with the rotation of a
resonator in different directions, must not be equal,
however they are equal precisely in [4].
Let us pay attention to the fact that the generation
of vortices at the velocities above the critical velocity
results in a suppression of the electric effect [4], because
the magnitude of flowing superfluid helium stream, the
oscillations of concentrations difference of Helium II
components, the thermal and electric effects are signifi-
cantly limited, because both the superfluid component
of Helium as well as the normal component take part in
the rotational movement during the process of vortices
generation, which is characterized by the average value
expression: <rotυ s ,ϕ>≠ 0.
In this research, the we don’t consider the effect
[3], connected with the generation of oscillations of se-
cond sound, when the alternate difference of electric po-
tentials U ′ with the magnitude in 108…10 1 0 times
bigger, than the magnitude in this research, was applied
178 ISSN 1562-6016. ВАНТ. 2014. №1(89)
to the additional capacitor inside the second sound reso-
nator, reaching the electric field magnitude
E ≈ 1.67 ⋅104V/m. In this case, the electric field can
not be considered as small enough, and its influence on
the properties of superfluid Helium II have to be taken
to the account. The different research approach has to be
used to interpret this effect, which can be discussed in
our next research paper.
CONCLUSION
The theory of thermoelectric effect with the fluctua-
tional electrons at the plates of a capacitor in the super-
fluid Helium II with the oscillations of second sound
wave is created. The results of theoretical calculations
are in good agreement with the experimental data, ob-
tained at the research of electric signals in a second
sound resonator and in a rotational resonator in [3, 4].
The similar described thermoelectric effect can be real-
ized in the capacitors at the creation of temperatures dif-
ference between the plates at various temperatures. The
thermoelectric effect can be used in numerous meas-
urement systems in view of a big value of thermoelec-
tric coefficient. In the authors opinion, the capacitor can
represent an effective thermoelectric transducer of a
new type.
Authors express thanks to A.S. Rybalko [38] for the
discussion and elaboration on some details of experi-
ments in [3, 4].
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Статья поступила в редакцию 02.12.2013 г.
ЭЛЕКТРИЧЕСКИЕ ЭФФЕКТЫ В СВЕРХТЕКУЧЕМ ГЕЛИИ.
1. ТЕРМОЭЛЕКТРИЧЕСКИЙ ЭФФЕКТ В ЭЛЕКТРИЧЕСКОМ КОНДЕНСАТОРЕ
Д.О. Леденёв, В.О. Леденёв, О.П. Леденёв
Идеи Эйнштейна о термодинамической флуктуационной природе некоторых электрических явлений и
разнице электрических потенциалов U на обкладках электрического конденсатора при температуре T были
изложены еще в 1906-1907 годах. На основании этих представлений мы объясняем экспериментальные ре-
зультаты, которые недавно были получены под действием волны второго звука в электрических конденса-
торах, находившихся при низких температурах в сверхтекучем гелии и в торсионном механическом резона-
торе. Подход, базирующийся на взаимосвязи термических, механических и электрических флуктуаций, по-
зволяет получить количественные результаты, совпадающие с экспериментальными данными по корреля-
ции малой переменной температурной разности в волне второго звука и переменной разности электрических
потенциалов между обкладками конденсатора как в сверхтекучем гелии, так и в торсионном механическом
резонаторе.
ЕЛЕКТРИЧНІ ЕФЕКТИ В НАДПЛИННОМУ ГЕЛІЇ.
1. ТЕРМОЕЛЕКТРИЧНИЙ ЕФЕКТ В ЕЛЕКТРИЧНОМУ КОНДЕНСАТОРІ
Д.О. Леденьов, В.О. Леденьов, О.П. Леденьов
Ідеї Ейнштейна про термодинамічну флуктуаційну природу деяких електричних явищ і різницю елект-
ричних потенціалів U на обкладках електричного конденсатора при температурі T було ще в 1906-1907 ро-
ках. На основі цих пропозицій ми пояснюємо експериментальні результати, які недавно було отримано під
дією хвилі другого звуку в електричних конденсаторах, що знаходились при низьких температурах у над-
плинному гелії та в торсіонному механічному резонаторі. Підхід, що базується на взаємодії термічних, ме-
ханічних та електричних флуктуацій, дозволяє отримати чисельні результати, співпадаючі з експеримента-
льними даними за кореляцією малої змінної температурної різниці в хвилі другого звуку і змінної різниці
між обкладками конденсатора як у надплинному гелії, так і в торсіонному механічному резонаторі.
|
| id | nasplib_isofts_kiev_ua-123456789-79939 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T17:03:14Z |
| publishDate | 2014 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Ledenyov, D.O. Ledenyov, V.O. Ledenyov, O.P. 2015-04-09T08:25:07Z 2015-04-09T08:25:07Z 2014 Electrical effects in superfluid helium. I. Thermoelectric effect in einstein's capacitor / D.O. Ledenyov, V.O. Ledenyov, O.P. Ledenyov // Вопросы атомной науки и техники. — 2014. — № 1. — С. 170-179. — Бібліогр.: 38 назв. — англ. 1562-6016 PACS: 05.40.-a; 67.40.Pm https://nasplib.isofts.kiev.ua/handle/123456789/79939 The Einstein's ideas about the thermodynamical fluctuational nature of some electrical phenomena and the difference
 of electrical potentials U in a capacitor at temperature T were proposed in 1906-1907. On base of these ideas
 we explain the experimental results, which were recently observed under the action of the second sound standing
 wave in the electrical capacitors which take placed in the superfluid 4
 He and in the torsional mechanical resonator.
 The Einstein's approach, based on the interrelation of thermal, mechanical and electrical fluctuations, allows to obtain
 the quantitative results, coinciding with the experimental datas for the correlations of the alternate low temperatures
 difference in second sound wave and alternate electric potentials difference between the capacitor plates in superfluid
 helium at as well as in the rotating mechanical oscillator. Идеи Эйнштейна о термодинамической флуктуационной природе некоторых электрических явлений и
 разнице электрических потенциалов U на обкладках электрического конденсатора при температуре T были
 изложены еще в 1906-1907 годах. На основании этих представлений мы объясняем экспериментальные ре-
 зультаты, которые недавно были получены под действием волны второго звука в электрических конденсаторах, находившихся при низких температурах в сверхтекучем гелии и в торсионном механическом резона-
 торе. Подход, базирующийся на взаимосвязи термических, механических и электрических флуктуаций, по-
 зволяет получить количественные результаты, совпадающие с экспериментальными данными по корреля-
 ции малой переменной температурной разности в волне второго звука и переменной разности электрических
 потенциалов между обкладками конденсатора как в сверхтекучем гелии, так и в торсионном механическом
 резонаторе. Ідеї Ейнштейна про термодинамічну флуктуаційну природу деяких електричних явищ і різницю електричних потенціалів U на обкладках електричного конденсатора при температурі T було ще в 1906-1907 роках. На основі цих пропозицій ми пояснюємо експериментальні результати, які недавно було отримано під
 дією хвилі другого звуку в електричних конденсаторах, що знаходились при низьких температурах у надплинному гелії та в торсіонному механічному резонаторі. Підхід, що базується на взаємодії термічних, ме-
 ханічних та електричних флуктуацій, дозволяє отримати чисельні результати, співпадаючі з експеримента-
 льними даними за кореляцією малої змінної температурної різниці в хвилі другого звуку і змінної різниці
 між обкладками конденсатора як у надплинному гелії, так і в торсіонному механічному резонаторі. Authors express thanks to A.S. Rybalko [38] for the discussion and elaboration on some details of experiments in [3, 4]. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Физика и технология конструкционных материалов Electrical effects in superfluid helium. I. Thermoelectric effect in einstein's capacitor Электрические эффекты в сверхтекучем гелии. 1. Термоэлектрический эффект в электрическом конденсаторе Електричні ефекти в надплинному гелії. 1. Термоелектричний ефект в електричному конденсаторі Article published earlier |
| spellingShingle | Electrical effects in superfluid helium. I. Thermoelectric effect in einstein's capacitor Ledenyov, D.O. Ledenyov, V.O. Ledenyov, O.P. Физика и технология конструкционных материалов |
| title | Electrical effects in superfluid helium. I. Thermoelectric effect in einstein's capacitor |
| title_alt | Электрические эффекты в сверхтекучем гелии. 1. Термоэлектрический эффект в электрическом конденсаторе Електричні ефекти в надплинному гелії. 1. Термоелектричний ефект в електричному конденсаторі |
| title_full | Electrical effects in superfluid helium. I. Thermoelectric effect in einstein's capacitor |
| title_fullStr | Electrical effects in superfluid helium. I. Thermoelectric effect in einstein's capacitor |
| title_full_unstemmed | Electrical effects in superfluid helium. I. Thermoelectric effect in einstein's capacitor |
| title_short | Electrical effects in superfluid helium. I. Thermoelectric effect in einstein's capacitor |
| title_sort | electrical effects in superfluid helium. i. thermoelectric effect in einstein's capacitor |
| topic | Физика и технология конструкционных материалов |
| topic_facet | Физика и технология конструкционных материалов |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/79939 |
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