JEMEA Journal Vol.6 (2022.12)

学術論文
マイクロ波照射下での反応速度定数に対するエネルギー変動の影響
Effect of Energy Fluctuation on Reaction Speed Constantunder Microwave Irradiation
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: Langevin equation, Epi-thermal distribution, Correlated noise,
Abstract
There is a research field called a fluctuating system. In systems such as Brownian motion, the particles of interest
exchange kinetic energy from white noise and perform a random walk. The Langevin equation handles these events andtakes up fine particle friction and microwave irradiation to confirm the operation of the system. In this paper, we
incorporate microwave irradiation into the Maxwell-Boltzmann distribution and confirm that its non-thermal energy
exhibits the so-called "microwave effect". As a sample case using correlated noise, two-dimensional Ising model is
proposed as a specific example of Monte Carlo simulation. Results will vary depending on the strength of the noisecorrelation. In this way, the noise-to-noise correlation has a strong effect on physical quantities.
1.

Introduction
Methodologies have been paid attention for tracking
non-equilibrium thermodynamic fluctuations in thematerials. Specific examples include substances such as
clusters and colloids, and molecular motors.[1] It has been
experimentally confirmed that fine particles such as pollen
show Brownian motion, and it is theoretically known thatwhite noise shows a random walk accompanied by theLangevin equation. The situation is further complicatedby the input of energy into the system by microwaves.
Friction occurs when clusters and colloids move, white
noise transmission due to collisions with other particles,
and microwave irradiation. They conflict with each otherto determine the behavior of the system.
In this paper, we will consider how kinetic energy canbe introduced into the reaction speed. We will discuss thereaction speed of the system, taking into account the threequantities of microwave, white noise, and friction.
In general, the energy loss due to friction and the white
noise received from other particles compete with eachother to determine the movement of the particles. For

example, it is reported that the reaction time of
microwaves is 4 hours and the time required for a specificreaction by heat transfer is 40 hours,[2] which is exactly
the effect of microwaves. Here, noise with a timecorrelation is proposed as a random number. This is used
to examine the noise dependence of the reaction speed.
And then, time-correlated noise is taken as an example
of statistical mechanics. It is to be important the last resultof this paper.
As a system, we will consider liquid H2O as an exampleof microwave active material, and the Ising model can beconsidered as an application using correlated noise.
2. Formulation
Since we are considering liquid H2O as a system, it isheated by microwaves and a phase transition from liquid
to gas occurs. To give a broader example, chemical
reactions can be discussed in the same way. As a model,
the system surrounded by a heat bath with a constanttemperature and volume is treated.

34

JEMEA Journal Vol.6 (2022.12)

there is no time evolution of the system. Since 𝜆𝜆⃗ fluctuates
as a function of time, Eq. (1) becomes a function of time.
This is the reason why Eq. (1) includes noise andmicrowave.
Considering the kinetic energy, energy conservation
formula is as follows,𝑟𝑟
𝑊𝑊𝜇𝜇 = ∫𝑟𝑟 𝑓𝑓 𝑑𝑑𝑑𝑑 ∙ �−𝛾𝛾𝑟𝑟⃗ + 𝜆𝜆⃗ + 𝑃𝑃�⃗ × 𝐸𝐸�⃗ �(2)〈𝜆𝜆𝑖𝑖 (𝑡𝑡)𝜆𝜆𝑗𝑗 (𝑡𝑡′)〉 = 𝛿𝛿𝑖𝑖𝑖𝑖 Δ𝑡𝑡,𝑡𝑡 ′

(3)

𝑖𝑖

Here, 𝑟𝑟𝑖𝑖 , 𝑟𝑟𝑓𝑓 mean initial and final positions, respectively.
White noise 𝜆𝜆⃗ =(𝜆𝜆1 , 𝜆𝜆2 , 𝜆𝜆3 ,) in general,

Fig. 1 Schematic diagram of the epi-thermal distributionof continuously irradiated microwave. The middle line isa cross section of the microwave at a point in time 𝑡𝑡. The
distribution function does not depend on time because thesystem is surrounded by a heat bath.

is defined like this. The white noise defined here has acharacteristic time determined by the frequency factor.
That is, white noise has a value of Δ𝑡𝑡,𝑡𝑡′ = 1 at 𝑡𝑡 − 𝑡𝑡′ =0.1 [𝑚𝑚𝑚𝑚 ], and otherwise Δ𝑡𝑡,𝑡𝑡′ = 0. On the other hand,
δ𝑖𝑖𝑖𝑖 = 1 for 𝑖𝑖 = 𝑗𝑗 , and 𝛿𝛿𝑖𝑖𝑖𝑖 = 0 for 𝑖𝑖 ≠j. So, time factor
Δ𝑡𝑡,𝑡𝑡′ is unusual definition of noise. But this definition isimportant in this study.
As the certain fine particles, molecules andpolarizations generally follow the Maxwell-Boltzmann
distribution function, but for the introduction of
microwave monochromatic energy, one must consider
that there will always be a peak in the MaxwellBoltzmann distribution function. This is called the epithermal distribution.
As an example, let's define and explain a distributionfunction that takes two positions.
We introduce the multiplication type distributionfunction.[3]In this system, microwave is continuously applied. At
this point 𝑡𝑡 , the epi-thermal distribution function isconsidered clipped, as shown in Fig. 1. This epi-thermal
distribution function loses its peak with the passage oftime due to friction between particles, and eventually
becomes heated and changes to a normal distributionfunction. However, since the system is in contact with theheat bath, the shape of the distribution function does notchange at continuous time. The time scale taken up in Fig.
1 is seconds, but the time scale until the microwave falls
into heat is 𝜇𝜇𝜇𝜇 . Therefore, local temporal heating by
microwaves is not observed on the scale of seconds, andit looks the same as normal heating such as electricheating.
The Langevin equation is introduced as equation of
motion containing a fine particle with existing of random
work. In the case of one particle surrounded by many
particles, it can be written as follows.
𝑚𝑚

𝑑𝑑𝑣𝑣�⃗𝑑𝑑𝑑𝑑= −𝛾𝛾𝑟𝑟⃗ + 𝜆𝜆⃗ + 𝑃𝑃�⃗ × 𝐸𝐸�⃗(4)𝑓𝑓𝑀𝑀 (𝑣𝑣) = 𝑓𝑓0 (𝑣𝑣)𝑔𝑔𝑀𝑀 (𝑣𝑣)
𝑓𝑓0 (𝑣𝑣) = �𝑚𝑚

2𝜋𝜋𝑘𝑘𝐵𝐵 𝑇𝑇

𝑒𝑒𝑒𝑒𝑒𝑒 �−

𝑔𝑔𝑀𝑀 (𝑣𝑣) = 𝑒𝑒𝑒𝑒𝑒𝑒 �𝛿𝛿𝑣𝑣,𝑣𝑣𝜇𝜇

(1)

𝑣𝑣̅ =

Here, 𝛾𝛾 is the coefficient of friction, 𝜆𝜆⃗ is white noise, 𝑃𝑃�⃗ isthe polarization coefficient, respectively. The quantity 𝐸𝐸�⃗is electric field of single mode microwave, and theconcrete formula is 𝐸𝐸�⃗ (𝑥𝑥, 𝑡𝑡) = 𝐸𝐸�⃗0 𝑒𝑒𝑒𝑒𝑒𝑒(𝑖𝑖𝑖𝑖𝑖𝑖 − 𝑖𝑖𝑖𝑖𝑖𝑖) . The
sample size (few 𝑐𝑐𝑐𝑐3 ) of this study is much smaller than
1/𝑘𝑘 (𝑘𝑘~10−6 order), so there is no need to think about
spatial changes. Further mentioning the time change, since
the frequency is GHz, it is an oscillation on a scale on theorder of 10−9 [s]. The order is different by 103 because
the time scale considered is 10−6 [s]. Therefore, thevibration of the electric field takes an approximation thatconsiders only the large modulation.
If 𝜆𝜆⃗ is removed from Eq. (1), it becomes the equation of
motion of the polarization element in 𝐸𝐸�⃗ . When the averagevelocity is calculated from the Maxwell-Boltzmann
distribution, the kinetic energy is consumed by friction, sothat the average velocity does not depend on time, and∞∫0 𝑣𝑣𝑓𝑓𝑀𝑀 (𝑣𝑣)𝑑𝑑𝑑𝑑∞∫0 𝑓𝑓0 (𝑣𝑣)𝑑𝑑𝑑𝑑

𝑊𝑊𝜇𝜇

𝑘𝑘𝐵𝐵 𝑇𝑇=�

𝑚𝑚𝑣𝑣 22𝜋𝜋𝑘𝑘𝐵𝐵 𝑇𝑇

�

2𝑘𝑘𝐵𝐵 𝑇𝑇
𝜋𝜋𝜋𝜋

(5)

�

𝑒𝑒𝑒𝑒𝑒𝑒 �(6)𝑊𝑊𝜇𝜇𝑘𝑘𝐵𝐵 𝑇𝑇�(7)

Here, Eq. (4) is the epi-thermal distribution function, Eq.
(5) is the usual Maxwell-Boltzmann distribution function,
Eq. (6) is the epi-thermal component of distributionfunction, Eq. (7) is the average velocity obtained from Eqs.
(4)-(6), respectively. And 𝛿𝛿𝑣𝑣,𝑣𝑣𝜇𝜇 , 𝑊𝑊𝜇𝜇 are Kronecker delta
and kinetic energy caused by noise. It should be noted thatthe exp term in Eq. (7) is multiplied by the usual velocity
distribution. The initial M means multiplication andindicates the monochromatic appearance of the
microwave immediately after it is turned on.
Using the expression epi-thermal distribution, thermalenergy and non-thermal energy are mixed on the samegraph. The peak structure is non-thermal energy, and it
takes 𝜇𝜇𝜇𝜇 to change to thermal energy.
We introduce an added type distribution function.[4]35JEMEA Journal Vol.6 (2022.12)

𝑓𝑓𝐴𝐴 (𝑣𝑣) = 𝑓𝑓0 (𝑣𝑣) + 𝑓𝑓0 (𝑣𝑣)�𝑣𝑣 − 𝑣𝑣𝜇𝜇 �𝑔𝑔𝐴𝐴 (𝑣𝑣)

𝑔𝑔𝐴𝐴 (𝑣𝑣) = �𝑣𝑣̅ =𝑊𝑊𝜇𝜇 =

𝑚𝑚(𝑣𝑣−𝑣𝑣𝜇𝜇 )2𝑒𝑒𝑒𝑒𝑒𝑒 �−

2𝜋𝜋𝑊𝑊𝜇𝜇𝑣𝑣∫0 𝑐𝑐 𝑣𝑣𝑓𝑓𝐴𝐴 (𝑣𝑣)𝑑𝑑𝑑𝑑
𝑣𝑣∫0 𝑐𝑐 𝑓𝑓0 (𝑣𝑣)𝑑𝑑𝑑𝑑
2
𝑣𝑣𝜇𝜇
𝑚𝑚
𝑣𝑣𝑐𝑐2

2𝑘𝑘𝐵𝐵 𝑇𝑇√𝜋𝜋

�

2

+

≅�
4

𝑚𝑚(𝑣𝑣−𝑣𝑣𝜇𝜇 )22𝑘𝑘𝐵𝐵 𝑇𝑇
𝜋𝜋𝜋𝜋

2𝑊𝑊𝜇𝜇

�

�1 + 𝑊𝑊𝜇𝜇 �
2𝑣𝑣𝑐𝑐

� 𝑒𝑒𝑒𝑒𝑒𝑒 �

𝑣𝑣𝜇𝜇

− 1�

value indicating the influence of these.

(8)
(9)

𝑤𝑤𝜇𝜇 = � 𝑃𝑃(𝜆𝜆𝑗𝑗 )𝑒𝑒𝑒𝑒𝑒𝑒 �−
𝑗𝑗

(10)

− ∑𝑗𝑗 ∫𝑟𝑟 𝑓𝑓 𝑑𝑑𝑟𝑟𝑗𝑗 ∙ �−𝛾𝛾𝑟𝑟𝑗𝑗 + 𝑃𝑃�⃗𝑗𝑗 × 𝐸𝐸�⃗ �(11)

2𝑘𝑘𝐵𝐵 𝑇𝑇𝜋𝜋𝜋𝜋

̀ = 9𝑚𝑚𝑒𝑒 3 𝑣𝑣𝜇𝜇2 /8√𝜋𝜋 .
𝑊𝑊

�1 +

̀
𝑊𝑊

𝑘𝑘𝐵𝐵 𝑇𝑇

(12)

�

̀ is
with
It can be seen that 𝑊𝑊
introduced in the form of addition to the usual distributionfunction. Whether this multiplication type distribution in
Eqs. (4)-(7) or addition type distribution Eqs. (8)-(12)becomes important because multiplication type shows
initial state and addition type shows medium state of timescale, respectively. From multiplication to addition, thereaction speed changes shape over time.
Eqs. (4) and (8) are not normalized in the form ofincluding microwave peaks. This is because the peak is
not thermal energy. The microwave peak of the epithermal Maxwell-Boltzmann distribution in Eq. (4) islisted in the same figure, but is a separate physical quantity.
In Eq. (8), the Maxwell-Boltzmann distribution and themicrowave peak are partially mixed, and the situation isdifferent from Eq. (8). After a few 𝜇𝜇𝜇𝜇, the non-thermal
peak turns completely thermal energy due to friction withother particles. Eq. (4) shows the state immediately after
the microwave is injected, and Eq. (8) shows the state inwhich non-thermal energy accelerates particles slower
than the peak. This means the passage of time whenmicrowave is applied.
The reaction speeds 𝜅𝜅𝑀𝑀 , 𝜅𝜅𝐴𝐴 are obtained for the case of
the definition formula Eq. (4) to be multiplied by the
distribution function and the case of the definition formulaEq. (8) to be added. [4]𝜅𝜅𝑀𝑀 =𝜅𝜅𝐴𝐴 =

𝑞𝑞 ‡ 𝑘𝑘𝐵𝐵 𝑇𝑇

𝑒𝑒𝑒𝑒𝑒𝑒 �−𝑞𝑞𝑎𝑎 𝑞𝑞𝑏𝑏 ℎ̀
𝑞𝑞 ‡ 𝑘𝑘𝐵𝐵 𝑇𝑇+𝑊𝑊
𝑞𝑞𝑎𝑎 𝑞𝑞𝑏𝑏ℎ𝐸𝐸 ∗ −𝑊𝑊𝜇𝜇�𝑘𝑘𝐵𝐵 𝑇𝑇𝐸𝐸 ∗𝑒𝑒𝑒𝑒𝑒𝑒 �−

𝑘𝑘𝐵𝐵 𝑇𝑇∑∞𝑘𝑘 ∑𝑗𝑗 𝑃𝑃�𝜆𝜆𝑗𝑗,𝑘𝑘 �𝑒𝑒𝑒𝑒𝑒𝑒 �−(15)

𝑡𝑡−𝑡𝑡𝑗𝑗,𝑘𝑘𝜏𝜏Θ�𝑡𝑡𝑗𝑗,𝑘𝑘 �� = 0(16)

where 𝑘𝑘 is an index of fine particle number of interest.
This is the numerical calculation result. The important
thing is that the average of random numbers will be zero.
The Eq. (16) means the kinetic energy of the system ispreserved. Therefore, the problem of the many fineparticles system is competition of friction and the energy
given by microwave independent of one fine particle's
noise.
Now let's change the topic from H2O to magnetization.
The partition function can be defined by using timecorrelated noise.
𝑍𝑍 = ∑𝑗𝑗,𝑘𝑘 𝑒𝑒𝑒𝑒𝑒𝑒 �−𝑊𝑊𝑗𝑗,𝑘𝑘
𝑘𝑘𝐵𝐵 𝑇𝑇�𝑊𝑊𝑗𝑗,𝑘𝑘 = 𝑊𝑊0 𝑃𝑃�𝜆𝜆𝑗𝑗,𝑘𝑘 �𝑒𝑒𝑒𝑒𝑒𝑒 �−〈𝑋𝑋〉 = ∑𝑗𝑗,𝑘𝑘

𝑊𝑊𝑗𝑗.𝑘𝑘
𝑍𝑍𝑒𝑒𝑒𝑒𝑒𝑒 �−(17)
𝑡𝑡−𝑡𝑡𝑗𝑗,𝑘𝑘

𝑊𝑊𝑗𝑗,𝑘𝑘
𝑘𝑘𝐵𝐵 𝑇𝑇𝜏𝜏�Θ�𝑡𝑡𝑗𝑗,𝑘𝑘 ��

(18)
(19)

By using Eqs. (17)-(19) statistical mechanics quantities intime-correlated noise can be calculated. If we propose
some kind of fluctuating physical quantity, a distributionfunction by time-correlated noise is defined and thethermodynamic quantity 〈𝑋𝑋〉 can be calculated. Since thisstatistical mechanics method usually requires theaccumulation of many samples, it is useful to treat thenoise distribution function defined here in that the numberof samples is small.
As an application example of correlated noise, the Isingmodel can be considered, and it is generally known that itexhibits ferromagnetism. The formulation of Ising model
is,

(13)
�

𝑖𝑖

Here, 𝑤𝑤𝜇𝜇 = 𝑊𝑊𝜇𝜇 /ℎ is normalized kinetic energy, 𝑃𝑃(𝜆𝜆⃗𝑗𝑗 ) is a
Gaussian distribution probability as a function of 𝜆𝜆⃗𝑗𝑗 . Also,𝜏𝜏 is the relaxation time of the decay of kinetic energy, andΘ(𝑡𝑡𝑗𝑗 ) is the step function. As a result, Eq. (15) defines anequation that ticks 𝑗𝑗 over time at intervals of about
frequency factor 10 kHz. [5] This equation takes into
account friction loss, energy gain from the microwave
field, and noise.
Taking the average of the random numbers of many fineparticles in the first term of Eq. (15) gives the following
relationship.

Here, 𝑣𝑣𝜇𝜇 is defined by dumping point velocity. The cutoff
𝑣𝑣𝑐𝑐 is introduced to prevent the divergence of the integral.
The initial A means to add. Eq. (8) shows the following.
It means moving particles in a distribution with low
kinetic energy based on the velocity of 𝑣𝑣𝜇𝜇 to a region withhigh kinetic energy by microwave energy. In order tomake the cutoff easy to understand, letting, 𝑣𝑣𝑐𝑐 = 2𝑣𝑣𝜇𝜇 , Eqs.
(10), (11) become simple as follows.
𝑣𝑣̅ = �𝑟𝑟

𝑡𝑡 − 𝑡𝑡𝑗𝑗
Θ�𝑡𝑡𝑗𝑗 ��𝜏𝜏(14)Here, the quantities 𝑞𝑞 ‡ , 𝑞𝑞𝑎𝑎 and 𝑞𝑞𝑏𝑏 are partition functions
for phase (chemical) change, state a and state b,
respectively. It is noteworthy that Eqs. (13), (14) are in
perfect agreement with the results of Nagata &Kodama.[5]̀ have not
The velocity 𝑣𝑣𝜇𝜇 and energy 𝑊𝑊𝜇𝜇 , 𝑊𝑊
substantially discussed time dependence. Then, Eq. (15)is newly defined as a time-dependent kinetic energy as a

𝐻𝐻 = −𝐽𝐽 ∑〈𝑖𝑖,𝑗𝑗〉 𝑆𝑆𝑖𝑖 𝑆𝑆𝑗𝑗 − ℎ ∑𝑖𝑖 𝑆𝑆𝑖𝑖

(20)

The spin of Ising model is defined by 𝑆𝑆𝑖𝑖 = 1/2 for sample
number 𝑖𝑖 ∈ [0: 0.5] and 𝑆𝑆𝑖𝑖 = −1/2 for 𝑖𝑖 ∈ [0.5: 1] in the
prepared correlation noise [0:1] defined by first term of36

JEMEA Journal Vol.6 (2022.12)

Fig.2 Extended distribution function. (a) When the
distribution function is disturbed by multiplication. (b)When a disturbance is added to the distribution function
by addition.

Fig.3 Schematic diagram of reaction speed by Langevinequation. (a) When the distribution function is disturbed
by multiplication. (b) When a disturbance is added to the
distribution function by addition. The quantities W
oscillate with time.
Fig.4 Time-dependent kinetic energy. (a), (d) Friction =microwave irradiation (b), (e) Friction > microwave
irradiation (c), (f) Friction < microwave irradiation,
respectively. And then, (a), (b), (c) for one fine particleand (d), (e), (f) for average of 100 fine particles,
respectively.

Eq. (15). The quantities 𝐽𝐽 and ℎ is defined by interactionenergy between nearest neighbor spins and exchange
energy under the magnetic field, respectively.
3. Results and Discussion
The calculated results of Eqs. (4),(8) are shown in Fig. 2.
In Fig. 2(a), the microwave energy has a delta functional
type peak structure. On the other hand, in Fig. 2(b), thedumping structure can be seen. When a microwave isinput at 𝑡𝑡 = 0, the energy is applied to the system as adelta functional peak. Here, thermal energy (MaxwellBoltzmann distribution) and microwave energy (peakstructure) should be understood separately.
Schematic diagrams of the reaction speed are shown in
Fig. 3 plotting Eqs. (13), (14). In Fig. 3(a), the energy of
the activated complex 𝐸𝐸 ∗ is reduced by the quantity 𝑊𝑊𝜇𝜇 .
̀
On the other hand, Fig. 3(b) shows vibration of energy 𝑊𝑊
in state a. In both cases, the reaction speed is accelerated.
These Figs. 2(a) and (b) correspond to Figs. 3(a) and (b),respectively. The difference in the behavior of these peaks
is the passage of time. These corresponds to raising thelevel of basic energy, and means that phase transitions andchemical changes can occur more easily below thetransition temperature.
Here, by introducing white noise with a frequency factorof 10kHz to the reaction speed again, the effect of noise
can be introduced to the reaction speed. Since therelaxation time of microwaves is several 𝜇𝜇𝜇𝜇,[3] the time
resolution is much lower than the frequency factor
10kHz,[6] that is, 0.1ms. Therefore, the noise effect is
updated for each frequency factor and applied to thereaction speed. Here, the Fig. 4 shows the kinetic energy
defined by the Eq. (15). The noise is no longer white noise,
but time-dependent correlated noise defined by theexpression the first term of Eq. (15).
Fig. 4 shows three correlations of friction, microwave
irradiation, and correlated noise in the Eq. (15) as afunction of time. Figs. 4(a), (b) and (c) show the kinetic
energy of a single fine particle, and Figs. 4(d), (e) and (f)show the average kinetic energy of 100 fine particles. Thismeans kinetic energy is preserved at the system as in the
Eq. (16). The difference in 𝜏𝜏 is the strength of the internoise correlation. If 𝜏𝜏 is small, the noise correlation is
weak, and if 𝜏𝜏 is large, the noise correlation is strong.
Since 𝜏𝜏 is the scattering memory of the fine particles, and
it is generally said that the first memory is lost after threecollisions, 𝜏𝜏=3 is a reasonable number. In all discussions,when the microwave outweighs the friction, the kinetic
37

JEMEA Journal Vol.6 (2022.12)

follow the epi-thermal Maxwell-Boltzmann distribution.
Although the forms of reaction speed are different, there
is no substitute for the conclusion that the reaction speed
increases in both cases. The important point here is thatthe reaction rate is discussed below the transition
temperature. Energy exceeding the transition temperature
can be obtained by superimposing microwave energy on
thermal energy and adding noise energy.
Furthermore, as kinetic energy, energy loss due tofriction, energy acquisition by microwaves, and energy
transmitted from correlated noise during particle
movement have been concretely shown as a function oftime. These friction and microwave energy have showedthat the kinetic energy have competed as a function oftime. Here, when one fine particle has moved, it has beena well-known random walk. On the other hand, when the
motion of many particles has been averaged, themessiness disappears and only the friction energy loss due
to noise in the entire system. Energy loss has been caused
only by friction, and that to be heat by diffusion process.
The method of statistical mechanics for the 2D Isingmodel has been calculated as a Monte Carlo simulation.
Here, time-correlation has been introduced as noise. As aresult, it has become clear that the curvature of the spincomponent has depended on the magnitude of thecorrelation. In this way, when dealing with physicalquantities where correlation becomes important, it has
been predicted that this correlation will become important.
Thus, the effect of correlated noise determined by thefrequency factor is expressed as the Brownian motion of
one particle of interest. This effect defines a newcorrelated noise for each frequency factor with respect toreaction rate. This is different from the conventional
Monte Carlo simulation.

Fig.5 Spontaneous spin polarization at zero magnetic field
and induced spin polarization under magnetic field in
Ising model. The solid line indicates that there is nomagnetic field, and the dotted line indicates that themagnetic field is applied. The noise-to-noise correlation is
set to 𝜏𝜏 = 1, 5[𝑚𝑚𝑚𝑚].

energy rises as a function of time, and conversely, when
the friction outweighs the microwave, the kinetic energy
decays as a function of time.
In this calculation, the ergodic theorem that the timemean and the population mean are equal is used.
Eqs. (17)-(19) is applied to the 2D Ising model Eq. (20)with periodic boundary condition. The 100 sites of
10 × 10 are shown as one sheet, 50 sheets are prepared,
and the Monte Carlo simulation method uses the noise
represented by Eq. (15). It can be seen in Fig. 5 that the
spin polarization increases and shows a constant value at
low temperature region. This value rises when an external
magnetic field is introduced, but it shows a constant valueeven at zero magnetic field. This predicts the emergenceof a spontaneous magnetic field and meets therequirements of this model. The difference between 𝜏𝜏 = 1
and 5 is whether the inter-noise correlation is weak orstrong. The correlation of spins between cells in the Isingmodel is not included in the calculations by the usualMonte Carlo method. Due to the inter-noise correlation
defined in this paper, the phase transition temperature atwhich magnetization occurs changes depending on themagnitude of 𝜏𝜏.

References
1. A. W. C. Lau, D. Lacoste and K. Mallick, Phys. Rev.
Lett, vol 99, (2007), pp. 158102.
2. G. Zhang and S. K. Hanson, Chem. Commun, 49,(2013), pp8371-8373.
3. T. Hirai, S. Nakatani, M. Sato and K. Nagata, JEMEA,(2022), to be published.
4. S. Takayama, M. Sato, and J. Fukushima, PSP Book,
Erwann-Guenin, (2015) chapter 2.
5. K. Nagata and K. Kodama, ISIJ International, vol 62,
No.8 (2022), pp. 1607-1617.
6. K. Hara, M. Hayashi, M. Sato and K. Nagata, ISIJ, vol52, No. 12 (2012), pp. 2149-2157.

4.

Summary
In this paper, we have examined two ways of thinkingof the Langevin equation as an epi-thermal distributionfunction, and have looked at the interpretation of the tworeaction speeds. As a result, the reaction speed has
increased in both cases. Here, we have defined the system
Langevin equation as the fine particles of interest for the

Manuscript received:
Revised:Accepted:

38

Aug. 23, 2022
Nov. 15, 2022
Nov. 18, 2022