JEMEA Journal Vol.6 (2022.12)

学術論文
強い電磁場下での熱外放射分布によるマイクロ波効果の仮説
A Hypothesis of Microwave Effects Induced by An Epi-thermal
Radiations Distributions under Intense
平井隆司 1*, 中谷 伸 2, 佐藤元泰 2, 永田和宏 3
Takashi Hirai1*, Shin Nakatani2, Motoyasu Sato2, Kazuhiro Nagata3
1. 中部大学ミュオン理工学センター 〒487-8501, 愛知県春日井市松本町 1 200,
2. 中部大学大学院工学研究科 〒487-8501, 愛知県春日井市松本町 1200,
3. 東京工業大学名誉教授 〒224-0006, 神奈川県横浜市都筑区荏田東 3-1-13-401,
1.

Center for Science and Technology, Chubu University, 1200, Matsumoto-cho, Kasugai City, Aichi, 487-8501,
Japan
2. Graduate School of Engineering Studies, Chubu University, 1200, Matsumoto-cho, Kasugai City, Aichi, 487-8501,
Japan
3. Prof. Emeritus, Tokyo Institute of Technology, 3-1-13-401, Eda-higashi, Tsuzuki-ku Yokohama-shi, Kanagawa,
224-0006, Japancorresponding author*, e-mail address: heisuke2009@gmail.com
Keywords: Epi-thermal distribution, Diffusive equation, Non-thermal effect
Abstract
In the handling of microwave heating, it is useful to consider the same type of equations, the heat conduction equationand the diffusion equation. Due to the difference in the diffusion coefficients of each equation, there is a time delay
between the energy applied by the microwave and the thermal energy. From this delay, microwave energy is stored nonthermally, which directly breaks the bonds of the crystal (it means phase transition) before it becomes thermal energy.
Also, non-thermal energy is treated as acoustic phonon in the crystal. Therefore, we propose to treat phonon as an epithermal distribution function with an extended Bose-Einstein distribution, and in fact the application of microwaves formsa monochromatic peak in the distribution function. Its peak decays over time and becomes thermal energy. Such a process
is one of the indicators of how to understand the phenomenon of microwave heating.
1. Introduction
Microwave heating is described in the form of theinteraction of microwaves with charged and mass
substances. The presence of a finite mass causes a phaselag in response to microwave, thus increasing the phasecomponent. Since this is an irreversible process,microwave energy is absorbed by the system and does not
return.[1][2]Since microwaves have a monochromatic frequency,
the response of microwaves is described by the MaxwellBoltzmann distribution function when considering a
multi-particle system with a finite temperature. By
considering dumping here, it is known that there is a way
to record a monochromatic peak due to microwave in the
distribution function.[3] However, when considering solid,
it is not appropriate to describe the behavior of the systemwith the Maxwell-Boltzmann distribution function.
In such situations, it is possible and useful to define
lattice vibration (phonon).[4] If microwave is consideredas photon, photon and phonon are connected by electric
distortion in the single crystal that constitutes thesubstance, and interact to exchange energy. In this case,the Bose-Einstein distribution function can be usedinstead of the Maxwell-Boltzmann distribution function todescribe the behavior of the system for microwave. The
Bose-Einstein distribution function can directly bring in
the energy of microwave and describe the response of thesystem without taking the procedure of converting theenergy obtained by microwave into the momentum ofparticles with effective mass. Also, since phonon have a
Bose-Einstein distribution, it is important that the sameenergy can be accumulated.
When a substance is irradiated with microwave, a nonthermal energy field called phonon and thermal energy
can be considered. This is treated as separate equations inthis paper, and it is assumed that phonon is firstaccumulated as a non-thermal energy field, and that1

JEMEA Journal Vol.6 (2022.12)

energy diffuses into thermal energy after a short but finite
transition time. It is important to envision this transitive
process, and such claims have not existed in previous
studies.
In addition, it is experimentally reported that aphysicochemical reaction that cannot be explained byordinary thermal energy occurs in the microwave process,which is called "microwave non-thermal effect".[5]-[11]In this paper, we treat the issue as same type two
equations and assume that phonons are stored primary asa non-thermal energy field and that energy diffuses into
thermal energy after a short but finite transition time. It is
important to envision this transitive process, and suchclaims did not exist in previous studies. Section 2 shows
the system and model, and Section 3 shows the derivationand solution of the theoretical formula, and the entropy
generation process of the phonon field. Section 4
introduces computational examples and discusses theapplication of theory to latent heat processes in phasetransition.
Here, methane hydrate is applied as an object of
microwave heating. Methane hydrate is a microwavesensitive substance, but the ice produced by the solidsolid-phase transition is a microwave-insensitive
substance. Therefore, microwave acts only on the material
before the phase transition. This allows for more generalphase transitions and similar discussions during chemicalchanges.

On the other hand, when the system is continuously
irradiated with microwaves, the microwaves continue tobe incident from 𝑡𝑡 = 0 , so the theoretical andexperimental image is straight forward to understand.
We examine the epi-thermal distribution function of thesystem containing microwave. For that purpose, it isimportant to deal with two equations, the heat conduction
equation and the diffusion equation. These are similar
equations, but the heat conduction equation deals with the
diffusion of thermal energy in a substance, and the
diffusion equation deals with the diffusion of energy space.
These are independent and have different meanings. The
difference between these two equations is the magnitudeof the diffusion coefficients. For the diffusion coefficient
𝐷𝐷𝑃𝑃 of the diffusion equation, the diffusion coefficient 𝐷𝐷𝑇𝑇of the heat conduction equation can be tentatively set as
𝐷𝐷𝑇𝑇 = 10𝐷𝐷𝑃𝑃 . As we will see later, this assumption has been
confirmed to be reasonable. These equations withdifferent diffusion coefficient confirm the timedependence of microwaves. This assumption that the
diffusion coefficients are different will be very importantin future discussions. Here, in general, 𝐷𝐷𝑃𝑃 and 𝐷𝐷𝑇𝑇
depend on the temperature, but since the temperaturechange is small, both are easily set to constants.
The heat conduction equation and diffusion equation are
defined below.
𝜕𝜕𝜕𝜕(𝑟𝑟, 𝑡𝑡)(1)= 𝐷𝐷𝑇𝑇 𝛻𝛻 2𝑇𝑇(𝑟𝑟, 𝑡𝑡) ,𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕(𝑟𝑟, 𝑡𝑡)(2)= 𝐷𝐷𝑃𝑃 𝛻𝛻 2 𝜙𝜙(𝑟𝑟, 𝑡𝑡) .
𝜕𝜕𝜕𝜕The following analytical solutions are derived for thethermal energy 𝑄𝑄𝑇𝑇 (𝑡𝑡) and the mechanical vibrationenergy 𝑄𝑄𝑃𝑃 (𝑡𝑡) for the microwave pulse incident.
(3)
𝑄𝑄𝑇𝑇 (𝑡𝑡) = 𝐶𝐶𝑇𝑇0 [1 − 𝑒𝑒𝑒𝑒𝑒𝑒 (−𝑡𝑡⁄𝜏𝜏 𝑇𝑇 )]𝑒𝑒𝑒𝑒𝑒𝑒 (𝑖𝑖𝑖𝑖𝑖𝑖) ,(4)
𝑄𝑄𝑃𝑃 (𝑡𝑡) = 𝜙𝜙0 𝑒𝑒𝑒𝑒𝑒𝑒 (−𝑡𝑡⁄𝜏𝜏𝑃𝑃 )𝑒𝑒𝑒𝑒𝑒𝑒(𝑖𝑖𝑖𝑖𝑖𝑖) .
Here, the quantities are defined as 𝑄𝑄𝑇𝑇 (𝑡𝑡) = 𝐶𝐶𝐶𝐶(𝑡𝑡) and
𝑄𝑄𝑃𝑃 (𝑡𝑡) = 𝜙𝜙(𝑟𝑟, 𝑡𝑡), where 𝐶𝐶 is heat capacity. The quantity𝜏𝜏𝑃𝑃 = 1⁄𝐷𝐷𝑃𝑃 𝑘𝑘 2 is calculated by the velocity of the acoustic
phonon, and we can set 𝜏𝜏𝑃𝑃 = 10𝜏𝜏 𝑇𝑇 as described above. As
we will see later, microwave energy creates electric
distortion in the crystal, which is stored as phonon.
Then, the thermal energy due to continuous irradiation of
microwave and the energy of mechanical vibration due tocontinuous irradiation are calculated by the following
formulas;𝑡𝑡
𝑄𝑄 (𝑡𝑡′)^𝑇𝑇 (𝑡𝑡) = � 𝑑𝑑𝑑𝑑′ 𝑇𝑇
(5)
,
𝑄𝑄
𝜏𝜏 𝑇𝑇
𝑡𝑡 ′=0𝑡𝑡
𝑄𝑄 (𝑡𝑡′)^𝑃𝑃 (𝑡𝑡) = � 𝑑𝑑𝑑𝑑′ 𝑃𝑃
(6)
.
𝑄𝑄
𝜏𝜏𝑃𝑃
𝑡𝑡 ′=0Here, the epi-thermal distribution function is calculatedby thermal relaxation including monochromatic
microwave energy. As the distribution function, the BoseEinstein distribution function can be considered because
the energy of the system is stored in phonon.
Since there is a monochromatic energy inflow in this
distribution function, the distribution function has a peakat the point corresponding to the microwave. This peak issuppressed as a function of time, and the distribution

2. Theoretical models and Experimental devices
As a model, we consider a closed system excluding
radiation containing microwaves as energy to enter andexit. In fact, this model can be realized as an experimental
system with heat insulating material and vacuum double
glazing. So, it can be said that this model is a feasiblemodel both theoretically and experimentally.
The methane hydrate produced in the laboratory is
several centimeters large 𝑙𝑙 , the wavelength of the
microwave 𝐿𝐿 is long enough ( 𝑙𝑙 ≪ 𝐿𝐿 ), and it can beconsidered that there is no spatial dependence of the
microwave. In addition, since microwave is directional
electromagnetic wave, when a single mode is considered
as a device, the direction of irradiating microwave is thex-axis, and the amplitude of the electric field is the y-axis.
In this paper, we consider the following two types of
microwave irradiation; (i) pulse incident and (ii)continuous incident. We will discuss by comparing these
two examples.
3. Formulation
The pulse incident of microwave needs to be an
extremely short time. Theoretically, the delta function canbe used, but experimentally it requires a finite time ofincidence, albeit for a short time. Specifically, thevibration time of microwave is about [ns], and as will
come out later, the time for mechanical vibrations of
microwaves to decay in the system is several [μs]. By
handling a certain amount of vibration, it can be said thatmicrowave is sufficiently on the order of delta functions.

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JEMEA Journal Vol.6 (2022.12)

function converts the peak energy into thermal energy.
The epi-thermal Bose-Einstein distribution function isdefined as follows.
(7)
𝐹𝐹(𝐸𝐸, 𝑡𝑡) = 𝑓𝑓(𝐸𝐸)𝑔𝑔(𝐸𝐸, 𝑡𝑡) ,1(8),
𝑓𝑓(𝐸𝐸) =𝑒𝑒𝑒𝑒𝑒𝑒(𝐸𝐸 ⁄𝑘𝑘𝐵𝐵 𝑇𝑇) − 1
𝑄𝑄𝑃𝑃 (𝑡𝑡)(9)𝑔𝑔(𝐸𝐸, 𝑡𝑡) = 𝑒𝑒𝑒𝑒𝑒𝑒 �𝛿𝛿𝐸𝐸,𝜀𝜀�.
𝑘𝑘𝐵𝐵 𝑇𝑇
Here, the quantity 𝜀𝜀 = ℏ𝜔𝜔𝑀𝑀 with 𝜔𝜔𝑀𝑀 = 2.45 [GHz] isdefined.
As a specific thermodynamic quantity, entropy byphonon can be considered. This is given by the wellknown harmonic oscillator entropy.
ℏ𝜔𝜔
ℏ𝜔𝜔
ℏ𝜔𝜔
𝑐𝑐𝑐𝑐𝑐𝑐ℎ− 𝑙𝑙𝑙𝑙 �2𝑠𝑠𝑠𝑠𝑠𝑠ℎ�� . (10)𝑆𝑆(𝑡𝑡) = 𝑁𝑁𝑘𝑘𝐵𝐵 𝑇𝑇 �
2𝑘𝑘𝐵𝐵 𝑇𝑇
2𝑘𝑘𝐵𝐵 𝑇𝑇
2𝑘𝑘𝐵𝐵 𝑇𝑇
Here, 𝑇𝑇 = 𝑇𝑇(𝑡𝑡) is a function of time, and the entropy canbe calculated by substituting Eq. (3) or Eq. (5) into Eq.
(10).
4. Results
The energy, distribution function, and entropy are plottedfor (i) pulse incident and (ii) continuous incident of
microwave, respectively. Let’s show each case.
4.1. Pulse incident
Fig. 1(a), in which the microwave energy and thermalenergy incident on the system are plotted, is calculated by
Eqs. (3) and (4). It can be seen that 𝑄𝑄𝑇𝑇 (𝑡𝑡) takes longer tosaturate than 𝑄𝑄𝑃𝑃 (𝑡𝑡).
Distribution function Fig. 1(b) is calculated by Eqs. (4),(7), (8), (9). This distribution function has a peak at 2.45
[GHz] and decays over time. Looking at Fig. 1(b), thepeak actually disappears at about 3 [μs]. The distributionfunction absorbs the peak energy and the temperature rises
slightly. Specifically, in Fig. 1(b), the distributionfunction is drawn by defining T=1 [K] at t=0 [μs], butwhen t=0.05 [μs] temperature T=1.22 [K] and t= 0.3 [μs]temperature T=2.2 [K].
The phonon entropy shown in Fig. 1(c) is calculated by
Eqs. (3) and (10). Entropy can be seen to converge aroundt=3 [μs]. A certain amount of energy is applied to thesystem, and it is a stable time until it is completely stored
in the system as thermal energy. So, the entropy decays
with respect to the time required for diffusion.
4.2. Continuous incident
Fig. 2(a), which shows the energy of the system, iscalculated by the following formula by Eqs. (5) and (6).
^𝑝𝑝 (𝑡𝑡) starts from zero at the start of
The quantity 𝑄𝑄
microwave incident 𝑡𝑡 = 0 [μs] and becomes constant at^𝑇𝑇 (𝑡𝑡) rises exponentiallyabout 3 [μs], but the quantity 𝑄𝑄
at t =0[μs] to 3[μs], and almost linearly from t = 3[μs].
The distribution function shown in Fig. 2(b) is
calculated by Eqs. (6), (7), (8), and (9). Unlike pulse
incident, no peak occurs at 𝑡𝑡 = 0. The peak once rises,
falls again, and then gradually rises. The temperatures are
T=1[K] at t=0[μs], T=1.48[K] at t=0.15[μs], T=2.31 [K]
at t=0.25[μs], T=5.94[K] at t=0.75[μs], and T=18.66 at
t=1[μs]respectively.

Entropy is calculated by equations (5) and (10). Entropy
does not converge even at time t = 5 [μs].
In Fig. 3, the peak structure of Fig. 2(b) is described asa function of time. The increase or decrease of this value
is essentially different from that of pulse incident.
5. Discussion
First, we consider the microwave of the pulse incident.
In Fig. 1(a), microwave is first stored in the system asmechanical energy and absorbed in the system as nonthermal energy at about 0.3 [μs]. On the other hand, thethermal energy becomes stable at about 1 [μs].
In Fig. 1(b), the epi-thermal Bose-Einstein distributionfunction is taken up as the epi-thermal distributionfunction. Methane hydrate is formed as a guest moleculecontaining methane in the bond of H2O molecule, but evenif we try to think of H2O itself as a distribution function,
the molecule is trapped in the crystal structure, so theeffective mass is considered. However, it cannot be
handled by the Maxwell-Boltzmann distribution because
effective mass of H2O is not a good approximation
because H2O is fixed at the crystal and does not move

Fig.1 (a) thermal energy 𝑄𝑄𝑇𝑇 (𝑡𝑡) and pulse microwave
energy 𝑄𝑄𝑝𝑝 (𝑡𝑡). (b) epi-thermal Bose-Einstein distributionfunction. (c) entropy
plotted as a function oftemperature. 𝑇𝑇(0) is the temperature at t=0[μs].
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JEMEA Journal Vol.6 (2022.12)

freely. Thus, we focus on the phenomenon that ultrasonicwave is generated in the crystal from the electric distortion
generated when microwave is applied to the crystal andaccumulate as phonon, and we decided to handle the
distribution function of phonon with the extended BoseEinstein distribution.
In Fig.1(c), entropy approaches stable over time. Since
the amount of energy input to the system is constant, whenthe energy incident by microwave is completely converted
to thermal energy, it will not evolve any further. This is
the reason why the entropy converges to a constant value
in about several tens of [μs].
Then, we consider the microwave of the continuous
incident.
In Fig. 2(a), the microwave supply and the process of
conversion to heat are balanced, and the microwave
energy converges to a constant value. This occurs in about
3 [μs] time scale. Thermal energy always receives energy
from microwave and continues to be heated, so it rises

Fig.3 the peak structure as a function of time plotted at
Fig. 2(b).
almost linearly except for t =0[μs] to 0.3[μs].
In Fig. 2(b), since the microwave started to enter at
𝑡𝑡 = 0 [μs], the result that the peak in the microwave
distribution function starts to rise from zero is valid. The
behavior of the peak is complicated, but in the region
where 𝑡𝑡 is small (0 < 𝑡𝑡 < 2.5 [μs]), the peak rises once
because the mechanical energy input to the system heats
up due to the difference in relaxation time between^𝑝𝑝 (𝑡𝑡). This is because it is stored as non^𝑇𝑇 (𝑡𝑡) and 𝑄𝑄
𝑄𝑄
thermal energy in the system before it changes to. So,
this peak drops once at about 0.5 [μs]. After that, as aresult, the peak value rises on the distribution function.
Such peak behavior is shown in Fig.3 as a function oftime. This is the effect of the continuous exposure of
microwaves to the system, which is a major difference
from pulse injection.
In Fig. 2(c), the entropy does not become stable nomatter how much time passes. This is because the system
is constantly being supplied with energy by microwaves.
In other words, entropy generation continues at any time.
While microwave energy is transformed into thermalenergy and distributed, more microwave is supplied. So,
the entropy does not settle to a constant value. Since thesystem undergoes a gradual phase transition due tomicrowave, phonon is no longer a good physical quantity
over time. Thus, the phase transition proceeds before theentropy continues to rise to stable, the methane hydrate
collapses, and the method of handling this system exceeds
the range covered in this study.
When the problem of "microwave non-thermal effect" isconsidered, it is necessary to consider the energy path asmentioned above. (a) The system is irradiated withmicrowaves. (b) Microwave energy changes to electric
distortion. (c) Electric distortion induces phonon in the
single crystal. (d) The molecular structure of H2O isbroken by the accumulation of this phonon. If 𝐷𝐷𝑇𝑇 = 𝐷𝐷𝑃𝑃 ,the path from (a) to (d) does not need to be considered.
The input microwave energy is immediately relaxed andconverted into heat. However, in reality, 𝐷𝐷𝑇𝑇 = 𝐷𝐷𝑃𝑃 is notalways the case, so a mechanism for storing non-thermalenergy inside the crystal is indispensable. This is apositive reason why non-thermal energies such as electric
distortion and phonon must be considered.
In fact, when microwave energy falls below the^𝑇𝑇 (𝑡𝑡) and continuous
Fig.2 (a) thermal energy 𝑄𝑄^𝑝𝑝 (𝑡𝑡). (b) epi-thermal Bose-Einstein
microwave energy 𝑄𝑄
distribution function. (c) entropy plotted as a function oftemperature. 𝑇𝑇(0) is the temperature at t=0[μs].

4JEMEA Journal Vol.6 (2022.12)

microwaves is shown. This argument holds true not only
for methane hydrate, but also for phase transitions andchemical changes that differ in microwaves, as in the caseof the reducing action of iron.

hydrogen bond energy of H2O, which is about 0.2 [eV], if
microwave is considered as photon, it is necessary to store
about 100,000 times photon energies. This is the reason
why 𝐷𝐷𝑇𝑇 = 10𝐷𝐷𝑃𝑃 is defined by changing the size of 𝐷𝐷𝑃𝑃
and 𝐷𝐷𝑇𝑇 .
The time required for the accumulation of about 30,000
times photons is calculated to be about 2.5 [μs], which isconsistent with the time scale in which the mechanicalvibration energy generated by microwaves becomes
thermal energy. As a result, it is shown to be on the same
scale as the discussion so far, demonstrating the validityof this paper.
As an example of the discussion on chemical changes,
the reaction FeO + CO → Fe + CO2 is taken as an
example, which is a reduction reaction at 800 [℃], istaken as an example.[12] The binding energy is an orderof magnitude different from that of hydrogen bonds, so
set it to 1 [eV]. Then, using the same calculation as formethane hydrate, about 600,000 times photon
accumulations are required to break the bond. According
to the same calculation as before, about 25 [μs] isrequired for photons to accumulate. Since the frequency
factor of reaction of this reaction is 7.7 [kHz] and it
required for photon accumulation reciprocal is 13 [μs], itis the same order as the time.
Therefore, it can be said that this methodology is generaland that the contents discussed here are also useful forother systems that react with microwave.

References
1. S. Nakatani, M. Sato, T. Hirai, and K. Nagata,
Irreversible Energy Flow and Entropy Production by
Microwaves, JEMEA Journal Vov. 5 (2021.12) pp5357.
2. S. Nakatani, M. Sato, T. Hirai, and K. Nagata,
Uncompensated Transformations by Localized
Selective Absorptions of Microwave in
Inhomogeneous Media, JEMEA Journal Vol.5(2021.12) pp49-52.
3. A.C. Metaxas, ‘’Industrial Microwave Heating’’,Peter Peregrinus Ltd, London U.K.(1983), reprinted
with minor corrections (1988), in Chapter 3
Theoretical aspects of volumetic heating, pp 89-97.
4. John H. Booske, Reid F. Cooper and Ian Dobson,
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Manuscript received:
Mar. 30, 2022
Revised:May 9, 2022
Accepted:May 18, 2022

6. Summary
The non-thermal effect on irradiating methane hydratewith microwave has been discussed. The heat conduction
equation and the diffusion equation have been taken up,
and discussions have been carried out from the standpointthat their diffusion coefficients are different. Two types of
microwave injection methods have been dealt with in thissystem. That is, (i) pulse incident and (ii) continuous
incident. In (i) and (ii), the behaviors of microwave energy
and thermal energy have showed completely different
behaviors, respectively. The Epi-thermal distributionfunction is also different, and the appearance of peaks
depends on the assumptions of (i) and (ii). It is important
to be aware of this difference in actual experiments.
In contrast to the physical phenomenon of breaking
hydrogen bonds with microwave energy, the current
situation is that it is necessary to accumulate microwaves
as photon about 30,000 times in terms of photon energy.
On the other hand, when 𝐷𝐷𝑃𝑃 ≠ 𝐷𝐷𝑇𝑇 , it is indispensable toconsider a mechanism for storing non-thermal energy
inside the crystal, and this is a situation where it isnecessary to go through a non-thermal process called
electric distortion and ultrasonic phonon.
In reality, the input microwave does not completely heatup inside the system, but it is necessary to consider theenergy loss.[13] It is necessary to consider this energy
when comparing it with the experiment.
In this paper, as an example, the phenomenon that
occurs when methane hydrate is irradiated with5