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Article
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Physics
Dongfang Modified Equations of Molecular Dynamics
X. D. Dongfang
Orient Research Base of Mathematics and Physics,
Wutong Mountain National Forest Park, Shenzhen, China
The unitary principle test of the pressure equation of ideal gas gives a negative conclusion, so I
systematically revise the basic equation of molecular dynamics. The specific logic and conclusions
are as following. The classical molecular dynamics theory established the wrong physical model
of uniform motion of molecules under the action of an equivalent constant force, and the classical
equations of ideal gas pressure and temperature derived from this model that violates the principle
of mechanics is all incorrect. A variety of physical models of molecular interaction in accordance
with the mechanical principle is established, and the correct equation of ideal gas pressure is derived
consistently. It is proved that the pressure of an ideal gas is equal to the molecular energy per unit
volume, and the thermodynamic temperature of an ideal gas is equal to the quotient of molecular
average kinetic energy and Boltzmann constant. Various inferences of different models are consistent,
so they comply with the unitary principle. Finally, I introduce the problem of the definite solution of
the gas molecular rate distribution function that meets the limit condition of the speed of light, and
put forward experimental suggestions to verify the theoretical gas temperature correction equation.
Keywords: Ideal Gas; Average molecular kinetic energy; Molecular velocity distribution; Basic
laws of molecular dynamics; Dongfang formulas.
PACS number(s): 02.70.Ns—Molecular dynamics and particle methods; 31.15.Qg—Molecular
dynamics and other numerical methods; 05.20.Gg—Classical ensemble theory; 05.45.2a—
thermo dynamics, and nonlinear dynamical systems; 05.90.+m—Other topics in statistical physics,
thermo dynamics, and nonlinear dynamical systems; 52.65.Yy—Molecular dynamics methods; 82.60.-
s—Chemical thermodynamics
1 Introduction
Classical molecular kinetic theory
[1, 2, 5]
holds that the
thermodynamic temperature of an ideal gas is direct-
ly proportional to the average kinetic energy of a large
number of molecules. This qualitative conclusion is true.
However, it is not correct for Classical molecular kinetic
theory to establish the physical model of uniform mo-
tion of molecules under a constant force in the process
of deriving quantitative equations. Then, are the basic
equations of molecular dynamics such as pressure equa-
tion and temperature equation, which are the basic laws
of ideal gas derived from the incorrect model, correct?
Revealing mistakes of classical theories will always en-
counter great resistance, but the wrong theories will in-
evitably encounter logical difficulties. Different metrics
can be selected to describe natural laws. The inference of
physical theory must conform to some basic principles,
one of which is the Dongfang unitary principle
[3, 4]
: There
is a definite transformation relationship between differ-
ent metrics describing the natural law, and the natural
law itself does not change due to the selection of differ-
ent metrics. When the mathematical expression of natu-
ral laws under different metrics is transformed into one
metric, the result must be the same as the inherent form
under this metric, 1=1, meaning the transformation is
unitary. The unitary principle is a universal principle of
logical self consistency test, conforming to the unitary
principle is the prerequisite of theory.
It should be pointed out that the theory conforming
to the unitary principle may not be correct. But the
theory violating the unitary principle must not be cor-
rect. The pressure equation of ideal gas is tested by the
unitary principle. It is found that the classical molecu-
lar kinetic theory only chooses the momentum theorem
as one of the metrics for inferring the pressure equa-
tion of an ideal gas for calculation, and its quantitative
conclusion is not consistent with the calculation result
of the kinetic energy theorem which is another metric
for inferring the pressure equation of the ideal gas, and
does not meet the unitary principle. The reason is that
the classical molecular kinetic theory has established an
incorrect physical model of uniform motion of molecules
under the action of an equivalent constant force. The re-
lationship between the pressure of an ideal gas and the
average molecular velocity and the relationship between
the temperature of an ideal gas and the thermodynamic
temperature need to be corrected.
According to the unitary principle, various physical
models that conform to the mechanical principle for the
Citation: Dongfang, X. D. Dongfang Modified Equations of Molecular Dynamics. Mathematics & Nature 1 , 202104 (2021).
2 X. D. Dongfang Dongfang Modified Equations of Molecular Dynamics
interaction of ideal gas molecules are studied here, in-
cluding the equivalent constant force interaction model
of the frequent collision b etween single molecule gas and
the container wall and the pressure mo del of the uniform
circular motion of single molecule gas. Under the action
of an equivalent constant force, the motion of molecules
can only be equivalent to motion with constant acceler-
ation. This is one of the basic principles of mechanics.
The pressure equation deduced from different models is
consistent, and the conclusion conforms to the unitary
principle. The correct form of the ideal gas pressure e-
quation can be determined by using the idea of squeeze
theorem, and then the basic molecular dynamics equa-
tions such as the temperature and internal energy of the
ideal gas can be modified. The results show that the
average rate of ideal gas molecules is equal to the maxi-
mum probability rate, the pressure of ideal gas is equal
to the total kinetic energy of gas molecules in unit vol-
ume, and the product of thermodynamic temperature of
ideal gas and Boltzmann is equal to the average kinet-
ic energy of molecules. These modified conclusions are
called Dongfang formula of basic laws of molecular dy-
namics. Based on the Dongfang temperature formula,
I discussed the constant volume and constant pressure
specific heat capacity of gas, and proposed a new equa-
tion to explain the experimental measured value of gas
specific heat capacity.
In addition, considering that the speed of light is the
maximum speed of matter, and the upper limit of tem-
perature depends on the upper limit of the average ve-
locity of molecules, I put forward a definite problem
for determining the solution of distribution of molecu-
lar sp eeds with the condition of the light speed limit.
However, there are mathematical difficulties in the solu-
tion of this definite solution problem. At present, there
is no quantitative answer.
2 Classical equations of ideal gas violate u-
nitary principle
It is generally accepted that the pressure of a gas is
caused by frequent collisions between molecules. The
pressure of an ideal gas can be calculated either based
on a single molecule model or based on a multi-molecule
model. According to the unitary principle, the single
molecule model and the multi molecule model are two
metrics for calculating the ideal gas pressure, and the
conclusions of the two mo dels should be same.
Investigating an ideal gas consisted of the same
molecules, in which the mass of the molecule is m, the
average speed is ¯v, and the mean square value of the
velocity is v
2
. It is assumed that the average distance
between gas molecules or the average distance between
the gas molecules that is impacting the container wall
and the container wall is l, and l is just the free path of
the molecules. If the whole process of molecular collision
is equivalent to constant force action, the molecule’s mo-
tion is equivalent to a uniform decrease in velocity from
¯v to 0, whereas it increases uniformly from 0 to ¯v in the
opposite direction. That time is a collision periods. The
force a molecule marked i on the container wall and the
average reaction force on the molecule are equal, which
is denoted by F
i
.
The standard theory selects the multi particle mod-
el, first calculates the collision periods ∆t = 2l/¯v
iz
by a
component motion of uniform motion, such as the mo-
tion in oz direction. Then it uses the component form of
momentum theorem
(−F
i
∆t) =
(−2m
i
¯v
iz
) as one
of the metrics to calculate the equivalent constant force
F =
F
i
acting on the collision cross section A. So
(−F
i
l) =
−m
i
¯v
2
iz
. This result was wrongly writ-
ten as
(−F
i
l) =
−m
i
v
2
iz
in the textbook because
of the traditional interpretation of the average kinetic en-
ergy of the molecules, which actually violates the conclu-
sion of Maxwell’s velocity distribution of v
2
̸= ¯v
2
. The
average volume of a gas corresponding to a free path is
V =
Al, and the pressure is p = F /A, so
F
i
l = pV .
As a result, the product of pressure p and volume V is
pV =
mv
2
iz
. Classical theory makes further ener-
gy equipartition hypothesis mv
2
x
/2 = mv
2
y
/2 = mv
2
z
/2,
which gives v
2
ix
= v
2
iy
= v
2
iz
, so v
2
iz
= v
2
i
/3. This also
needs to be understood as v
2
i
/3 = v
i
2
/3, we can get the
classic pressure equation pV =
mv
2
i
/3. This equation
is combined with the experimental law pV = NkT
[6, 7]
of ideal gas to give the classical expressions of absolute
temperature and average kinetic energy of ideal gas,
kT =
1
3
mv
2
, ¯ε
k
=
1
2
mv
2
=
3
2
kT (1)
where k is the Boltzmann constant
[8]
, v
2
is the average
of the square of the velocity, T is the thermodynamic
temperature of gas, and ¯ε
k
is the average translational
kinetic energy of molecules.
However, if we use the kinetic energy theorem as an-
other metric to calculate, according to the classical the-
ory, the molecular collision is equivalent to the constant
force model, then there is the equation
(−F
i
) dl =
0 − m
i
v
2
iz
/2
. Where
F
i
dl =
F
i
l. If the hy-
pothesis of equipqrtition of energy mv
2
x
/2 = mv
2
y
/2 =
mv
2
z
/2 is established, the product of the gas pressure p
and the volume V should be pV = Nmv
2
/6. Comb-
ing this formula with ideal gas experimental law of
pV = N kT gives the relationships that are complete-
ly different from (1),
kT =
1
6
mv
2
, ¯ε
k
=
1
2
mv
2
= 3kT (2)
Obviously, the two classical corollaries of momentum
theorem and kinetic energy theorem based on the as-
sumption of equipartition of energy are contradictory,
Mathematics & Nature (2021) Vol. 1 3
that is, the reasoning metho d of pressure equation and
temperature equation of ideal gas in classical thermody-
namic theory does not conform to the unitary principle,
and the reasoning is naturally incorrect. The momen-
tum theorem and kinetic energy theorem are two parallel
inferences of Newton’s law of motion, which constitute
two metrics of logical self consistency test
[9]
. Their de-
ductions must be consistent. The two conflicting deduc-
tions reveal the error of classical theory about concept
interpretation and calculation. There are many reason-
s why the classical derivation of ideal gas pressure e-
quation is wrong and difficult to find. For example, in
the calculation process, the classical theory expresses the
momentum theorem
F
i
dt =
2m
i
v
iz
remove the in-
tegral symbol, or use the molecular velocity distribution
function f (v) and the volume microelement dV = Adt
to calculate the molecular number N = f (v) dV , so
the intermolecular collision is equivalent to the constant
force, and the molecular motion is misunderstood as the
uniform motion, and the error of the collision time ∆t is
concealed. On the other hand, the unequal relationship
¯v
2
̸= v
2
between ¯v
2
=
N
0
vf (v) dN /N
2
= 8kT /πm
and v
2
=
N
0
v
2
dN
N = 3kT /m of classical theory from
Maxwell’s velocity distribution theory has not attracted
the attention.
Figure 1 The schematic diagram of the equivalent constant
force action of molecular collisions. When the molecular col-
lision is equivalent to the constant force, the whole process
of each completion of the collision in the free path can only
b e equivalent to the motion with constant acceleration, the
collision time is not the false inference ∆t = 2l/¯v
iz
of the s-
tandard course, but the correct inference ∆t = 2l/¯u = 4l/¯v
iz
,
where ¯u = ¯v
iz
/2.
The pressure is the result that the molecular colli-
sion is equivalent to the impact of the collision time
constant force. Since the interaction of the intermolec-
ular collisions is equivalent to the constant force, the
motion of the molecule can only be equivalent to a u-
niform linear motion. Figure 1 clearly shows that the
action time should be ∆t =
dt = 4l/¯v
iz
instead of
∆t = 2l/¯v
ix
calculated by classical thermodynamic the-
ory. This is because the velocity of a molecule i de-
creases from ¯v
iz
to 0 under the equivalent constant force
and increases from 0 to ¯v
iz
, with an average velocity of
¯u
iz
= (¯v
iz
+ 0)/2 = ¯v
iz
/2. and the time to return in
the free path l is ∆t = 2l/¯u
iz
= 4l/¯v
iz
. Using the com-
ponent form
(−F
i
∆t) =
(−2m
i
¯v
iz
) of momentum
theorem, the equivalent constant force F =
F
i
= P A
acting on the collision cross section A is further calculat-
ed, and the result should be
(−F
i
4l) =
(−2m
i
¯v
iz
),
so
(−F
i
l) =
−m
i
¯v
2
iz
2
. This is the same as the
result calculated by the direct kinetic energy theorem,
which implies the essential difference that ¯ε = m
i
¯v
2
iz
2
is the expression defined, while m
i
¯v
2
iz
2 is the result
of calculation.
(−F
i
l) =P Al = P V . Combing with
the so called energy equipartition hypothesis m
i
¯v
2
iz
2 =
m
i
¯v
2
i
6 gives the product of the pressure p and the vol-
ume V , which is also pV = Nm¯v
2
i
6, and then uses
the experimental law of ideal gas pV = NkT to de-
rive the formula (2). The conclusion that this derivation
does not contain incorrect concepts is obviously also de-
formed, which is due to the use of the energy partition
hypothesis which can not be proved.
To sum up, the model of uniform motion under the
equivalent constant force is not reasonable, and the cal-
culated collision time is wrong. The deduction process
and conclusion of the classical theory are not correct,
and the inference of any correct model based on the as-
sumption of the so-called equipartition theorem of kinet-
ic energy can not be accepted. The relationship between
temperature and molecular average kinetic energy is one
of the basic equations of the theory of thermodynamics.
Although it is not very influential in engineering, it has a
wide range of influence on the theory. The pressure and
temperature equations of ideal gas need to be corrected.
We need to find the inference that is consistent with the
scientific logic.
3 Dongfang pressure formula of ideal gas
Molecules are moving irregularly, so the velocity
changes continuously. Since the pressure of the gas gen-
erated in collisions between molecules, under equivalent
constant force the pressure from a frontal collision or
an oblique collision should be same. Why? A molecule
bounces-back after the frontal collision and occupy the o-
riginal channel, blocking other molecules collide with the
container. The molecule will fly in the other direction
if take an oblique collision, and the other molecules will
fill in the original channel to collide with the container
again. It is speculated that the frontal collision with low
collision frequency will pro duce a large force, and the
oblique collision with high collision frequency will pro-
duce a small force. Under the condition of that the same
average velocity of the same like molecules, the force per
unit area produced by the two collision models should
be same. As a matter of fact, a linear motion collision
under equivalent constant force can only be equivalent
4 X. D. Dongfang Dongfang Modified Equations of Molecular Dynamics
to a frontal collision. Otherwise, the molecules will de-
scribe a curved path in space, and the classical theory
on the pressure of an ideal gas is ineffective. Collisions
between molecules have an equivalent interaction space.
The calculation of the pressure by using the component
wise of momentum theorem repeats the equivalent inter-
action space. So the equation of gas pressure from this
algorithm is not accurate.
Figure 2 Model of single molecule of uniform circular motion
Collisions between gas molecules are complex. If
a molecule continuous to oblique impact with other
molecules, the average effect of this movement can be
equivalent to a lo cal uniform circular motion. As shown
in Figure 2, it is assumed that a molecule is tied to a
cylinder with the same diameter and height. The vol-
ume of the average space occupied by the molecule is
V
0
= πr
2
h, where r = d/2 and h = d. This is also the
size of the equivalent interaction space. Use ¯v to express
the average velocity of the equivalent uniform circular
motion of the molecules. Its average centripetal force is
F
xy
= m¯v
2
r. This force is provided by the equivalent
cylinder side wall. According to Newton’s third law, the
size of the average force on the side of the cylinder is
equivalent to this value, and the pressure takes the form
p
xy
= F
xy
/(πdh). Consequently, the calculated pressure
formula of the ideal gas based on the model of a single
molecule in a circular motion is
p
xy
V
0
=
1
2
m¯v
2
(3)
The results prompted us to reflect on new conclusions:
the average kinetic energy of an ideal gas is described
by the square value of the average velocity, and the gas
pressure is equal to the average molecular kinetic ener-
gy per unit volume rather than three times the average
kinetic energy of molecules.
All molecules remain random thermal motion. It usu-
ally establishes the elastic collision model to calculate
the ideal gas pressure. It is assumed that the aver-
age distance between molecules is d, and each molecule
takes up space of average volume V
0
= d
3
. The process
of a molecule and other molecules with frontal collision
can be equivalently regarded as that the molecule under
the action of the average constant force
¯
F does a round
straight-line movement with piecewise constant acceler-
ation. On average, in each collision periods it completes
a collision between any two molecules or one molecule
and the container. In this course, molecular velocity is
from ¯v to 0, then reversely from 0 to ¯v. The equiva-
lent beeline path length of the molecular back-and-forth
movement is 2d. It is obtained for the equivalent col-
lision time ∆t = 2d/(¯v/2). In the vertical direction of
movement, the equivalent constant force
¯
F per unit area
of the collision cross section is the pressure, p =
¯
F
d
2
.
As shown in Figure 3, the applications of the momen-
tum theorem
∆t
0
¯
F dt = m¯v − (−m¯v) and the kinetic
energy theorem
d
0
−
¯
F dy = 0 − m¯v
2
2 give the same
conclusion
¯
F = m¯v
2
2d. Apparently, the average ve-
locity of a molecule meets the relation ¯v · ¯v = ¯v
2
, and
the average molecular kinetic energy should be expressed
as E
k
= m¯v
2
2 rather than the classical expression
E
k
= mv
2
2. Consequently, the calculated pressure
formula of the ideal gas based on the model of a single
molecule in a frontal collision is
pV
0
=
1
2
m¯v
2
(4)
Figure 3 Model of frontal collision of single molecule
More generally, no matter which direction motion of
molecules, it has
d
0
F · dr = 0 − m¯v
2
2 in any col-
lision periods. For the equivalent constant force F,
d
0
F · d r = −
¯
F d = −pV
0
, combining them gives the
equation (4), showing that the above correction of the
pressure equation of an ideal gas is reliable.
Gas is composed of a large number of molecules in
thermal motion. Ideal gas molecules of statistical sense
are the identical particles. Thermal motion of the
molecule is disorderly, but even if we consider the dis-
ordered movement as an orderly movement, the average
molecular velocity will not change, so the pressure and
temperature of gas also remain unchanged. As shown in
Mathematics & Nature (2021) Vol. 1 5
Fgi.3, consider an ideal gas sealed in a cube container,
which the total molecular number is N. It is assumed
that in one collision periods there are 1/6 molecules
make directional collision in the six directions respec-
tively. The molecules of each direction in directional
collisions occupy 1/6 total volume of space of the gas,
so NV
0
= V . The average collision forces between the
molecules are equal, so are the average areas of the col-
lision cross sections. According to the equation (4), It
can be written as the following form
[10]
,
p
+x
NV
0
6
=
N
6
1
2
m¯v
2
, p
−x
NV
0
6
=
N
6
1
2
m¯v
2
p
+y
NV
0
6
=
N
6
1
2
m¯v
2
, p
−y
NV
0
6
=
N
6
1
2
m¯v
2
p
+z
NV
0
6
=
N
6
1
2
m¯v
2
, p
−z
NV
0
6
=
N
6
1
2
m¯v
2
There are the same gas pressures everywhere, p
+x
=
p
−x
= p, p
+y
= p
−y
= p and p
+z
= p
−z
= p. Com-
bining the six equations gives pNV
0
= Nm¯v
2
2, and it
is easy to obtain pNV
0
= Nm¯v
2
2. So, the ideal gas
pressure calculated by the model of large numbers of
molecules with ordered motion in a regular hexahedron
container takes the form
pV =
1
2
Nm¯v
2
(5)
It is now clear that the pressure of an ideal gas is de-
termined by the average molecular kinetic energy and
the average molecular space. Imaging that all molecules
in the container are at rest, and their volume is neg-
ligible. After adiabatic expansion with an equal pres-
sure, the molecular volume increase from 0 to V. Gas
does work while expanding, and the work in this course
is W =
V
pdV = pV . On the other hand, according
to the theorem of kinetic energy, this work is equal to
the total kinetic energy obtained by the gas molecules,
W = Nm¯v
2
2. Combining the above two equations
reads the above equation.
By two kinds of single-molecule limit models, using
the momentum theorem and the theorem of kinetic en-
ergy respectively to calculate the pressure of an ideal gas
reads the equations (3) and (4) that are the same, show-
ing the self-consistency of the correction logic, which ac-
cords with the unitary principle. Let N express the total
number of total number of molecules of an ideal gas, the
gas volume of the ideal gas is V = NV
0
. Use the e-
quation (3) or the (4) can also export the equation (5).
Equation (5) is called the Dongfang pressure formula of
ideal gas, which shows that the pressure of an ideal gas
is equal to the total kinetic energy of molecules in unit
volume.
4 Dongfang Temperature formula of ideal
gas
The internal energy of the ideal gas is equal to the
product of the gas pressure and the gas volume. Since
the potential energy between any ideal gas molecules is
zero, its total kinetic energy is the product of the total
gas molecules number and the kinetic energy of the gas
molecules. Therefore, the corrected gas pressure equa-
tion (5) of the ideal gas satisfies the law of conservation
of energy, it is only another representation of the en-
ergy of an ideal gas. Equation (5) combines with the
ideal gas law
[11, 15]
pV = NkT to get immediately the
temperature equation of ideal gas,
kT =
1
2
m¯v
2
= ¯ε
k
(6)
The results indicated that the thermodynamic temper-
ature of an ideal gas is equivalent to the quotient of the
average translational kinetic energy and Boltzmann con-
stant. Equation (6) is called the Dongfang Temperature
formula of ideal gas. Using the equation (6), the re-
lationship between the internal energy of ideal gas and
the physical quantities such as the average kinetic ener-
gy or thermodynamic temperature of the molecule
[16-19]
is revised to
U =
1
2
Nm¯v
2
= P V = nRT (7)
where n is the number of moles of an ideal gas. These
modified consequences satisfy the laws of conservation
of momentum and energy. In principle, the relevant
theories
[12, 20-24]
established on the basis of the classical
ideal gas pressure, temperature and internal energy need
to be corrected accordingly. We are convinced that the
revised conclusion is in goo d agreement with the experi-
mental observations under the condition that the effects
of various objective factors are fully taken into account.
Ideal gas is a simplified model of the actual gas that
neglects the potential energy of intermolecular interac-
tion. It is an infinitely thin gas from the macroscopic
point of view and follows the ideal gas equation of state.
However, from the perspective of energy transformation
and conservation, the ideal gas molecules are the sin-
gle particles with kinetic energy but no binding energy.
Even the extremely thin monatomic gas cannot form an
ideal gas, because the single atom is a multi particle sys-
tem composed of nuclear and extra nuclear electrons. In
the effective understanding of the microscopic structure
of matter, only a large number of neutrons which are s-
tored in a container can form an ideal gas in a real sense,
but in fact no such ideal gas exists.
The experimental laws of gas are all derived from the
actual gas under certain conditions. The specific heat
capacity of the actual gas is related to the law of the
temperature of the gas, but it is not the only connec-
tion. This is because the heat absorbed by the actual
6 X. D. Dongfang Dongfang Modified Equations of Molecular Dynamics
gas molecules not only increases the total kinetic en-
ergy of the molecules, but is also absorbed by atoms
or molecules to increase the energy of the atoms in the
molecules and molecules. Ignoring the heat absorption
of the container wall molecules, the energy relation of
the constant volume process is,
Q =
(∆ε
i
+ ∆E
molecule
+ ∆E
atom
) (8)
Among them, Q is the heat absorbed by the gas with
the total number of N,
ε
i
= N ¯ε = N kT , E
molecule
the energy of the molecule, and E
atom
the energy of the
atom. The principle formula for calculating the average
specific heat capacity c
v
=
1
N
Q
∆T
of each molecule in
the process of the fixed volume is,
c
v
= k +
1
N
∆E
molecule
+ ∆E
atom
∆T
(9)
The specific heat capacity of any gas has its own wider
distribution range
[25-27]
. The molecules of a diatomic or
polyatomic gas will absorb more energy than the single
atom. According to the experimental measurements of
the constant volume ratio of the gas to the heat, the
monatomic gas c
v
≈ 3k/2, the diatomic gas c
v
≈ 5k/2,
the polyatomic gas c
v
≈ 7k/2. In the process of con-
stant pressure, the volume of gas expands, the gas is
doing work to the outside world, and the surface area
of the solid container increases, and the number of solid
molecules that absorb heat increases. Ignoring the heat
absorption of the container wall molecules, the energy
relation of the constant pressure process is,
Q =
(∆ε
i
+ ∆E
molecule
+ ∆E
atom
) + W (10)
Where W is the work done by the gas in the process of
constant pressure. In this way, From this, the principle
formula for calculating the average specific heat capacity
c
p
=
1
N
Q
∆T
of each molecule during constant pressure
is,
c
p
= k +
1
N
∆E
molecule
+ ∆E
atom
∆T
+
W
∆T
(11)
Considering the influence of the container, when the
gas molecules absorb heat and thus the temperature
changes, the molecules in the container wall absorb en-
ergy and the temperature changes synchronously, which
is also related to the material of the container. Consid-
er the transition of the atomic energy level, the specific
heat capacity c
v
of constant volume and the specific heat
capacity c
p
of constant pressure cannot be constant at
all times. Therefore, it is very difficult to theoretically
derive the precise formula for the specific heat capacity
of gas.
However, the experimental law of gases is not directly
related to the energy of atoms or molecules.Therefore,
the Dongfang Temperature formula of ideal gas based
on the correct physical model and the correct reasoning
of the ideal gas is generally applicable to the thin real
gas.
5 Problem for determining solution of ve-
locity distribution function
The equation (6) is consistent with a conclusion of s-
tatistical physics. According to Maxwell velocity distri-
bution law, if the temperature of an ideal gas is T , the
number of molecules with the speed v makes up percent-
age f (v) of the total number of gas molecules
[5]
, where,
f (v) = 4π
m
2πkT
3/2
v
2
e
−
mv
2
2kT
(12)
where m is the particle mass, and kT is the product
of Boltzmann’s constant and thermodynamic tempera-
ture. The corresp onding speed of the extreme point of
the above distribution function, df (v)/dv|
v=v
p
= 0, is
the most probable speed.
v
p
=
2kT
m
(13)
This is just the equation (6). It follows that the average
molecular speed is equal to the most probable speed.
Our modified ideal gas temperature is consistent with
the conclusion of statistical physics.
However, Maxwell velocity distribution law predicting
the statistical average values conceals a difficulty that
the roots of physical quantities with the high orders are
different. The corresponding calculations violate the ba-
sic mathematical rules of
|a|
2
= |a|. For example,
⟨v⟩ =
1
N
N
0
vdN =
∞
0
vf(v)dv =
8kT
πm
and
⟨v
2
⟩ =
∞
0
v
2
f(v)dv
1/2
=
3kT
m
It was obvious that
⟨v
2
⟩ ̸= ⟨v⟩, the logic is not self-
consistent! Classical theory defined these two calcula-
tions, respectively, as the statistically average velocity
and root-mean-square speed
[28-30]
, but ignored the na-
ture why the calculations are very different. In fact,
these results have no effects of experimental observa-
tions, and the physical quantities that have the effect of
experimental observation are the average velocity or the
most probable speed. A theory based on the relation-
ship
|a|
2
̸= |a| is only a formal theory. An inference
of the formal theory is often similar to but deviate from
the true law. Investigating the p ower radiated from a
black body in terms of its temperature. According to
the Stefan–Boltzmann law
[31-33]
,
j
∗
=
2π
5
k
4
15c
2
h
3
T
4
∝ ¯v
8
(14)
Mathematics & Nature (2021) Vol. 1 7
Where j
∗
is the black-body radiant exitance, h is the
Planck’s constant, k is the Boltzmann constant, and c is
the speed of light in a vacuum. Now that the term ¯v
8
appears in the Stefan–Boltzmann law, then according to
the Maxwell velocity distribution law, we can define the
generally statistical averages to compare their roots,
n
⟨v
n
⟩ =
1
N
N
0
v
n
dN
1/n
Apparently, different values of n lead to the results vary
widely. These formal consequences have no physical
meaning. But giving them different definitions or as-
sumptions respectively would be representative of the
logical difficulties.
Since the molecular speed cannot reach the speed of
light, it can only describe roughly the gas molecules ve-
locity distribution by Maxwell speed distribution func-
tion of depending on the normalized conditions that the
speed limit is infinity
[34-36]
. Formally, using the rela-
tivistic energy equation E
2
= p
2
c
2
+ m
2
0
c
4
can rewrite
the Maxwell velocity distribution functions
[37-41]
, but we
cannot strictly prove the result of rewriting. In fact,
the relativistic normalization coefficient written in the
speed of light
[42]
can not be expressed by the elemen-
tary function, so it is difficult to obtain the accurate
rate distribution function. It usually considers that a
velocity distribution function gives the probability, per
unit speed, of finding the particle with a speed near v.
We found a velocity distribution function can also be
understood as describing the probability of a molecule
reaches the speed v, and the corresponding results are
not affected. An accurate gas molecule velocity distri-
bution function should be determined by the following
boundary value problem,
ψ (v = 0) = 0, ψ (v = c) = 0
c
0
ψ (v) dv = 1,
dψ (v)
dv
v =¯v
= 0
¯v =
c
0
v
n
ψ (v) dv
1/n
, n = 1, 2, · · ·
(15)
We do not know yet whether this molecular velocity dis-
tribution function exists, and whether the iconic factor
of relativity is included in the solution. However, it can
be sure that the Maxwell velocity distribution function
must be an approximation of the solution of this bound-
ary value problem.
6 Comments and conclusions
The Dongfang Temperature formula of ideal gas was
discovered when studying the com quantum law
[?]
of stel-
lar radiation temperature, which promoted the normal-
ization principle test of the basic equation of molecular
dynamics and found its errors, leading to the genera-
tion of systematic modified equations. According to the
unitary principle of logical self consistency test, all the
problems and conclusions proposed in this paper are easy
to understand. In short, establishing various reasonable
physical models for calculating the pressure of an ideal
gas and modifying the basic equations of pressure and
temperature of an ideal gas, the conclusions obtained
are consistent and therefore cannot be overturned. The
modified equation is named as the Dongfang formula of
the basic laws of molecular dynamics to distinguish it
from the incorrect equation of classical molecular dy-
namics.
Computations that violate the common sense of
physics in physics standard courses are usually difficult
to be discovered directly, and correcting such calcula-
tions is not easy to be accepted. The main reason is
that some simple but trivial causal relationships have
not attracted the attention of researchers, and main-
stream physicists’ excessive pursuit of personal fame and
wealth has led to hegemony and sophistry that have
been shrouded in the physical world. Even if wrong in-
ferences and even wrong theories are found, they have
been permanently strangled. But we believe that future
scientists will distinguish between correct and wrong in-
ferences and make the right choices. Since the average
force produced by a gas molecular collision is equivalent
to a constant force, the motion of a gas molecule must
be equivalent to a motion with constant acceleration;
otherwise all the inferences are contrary to the law of
conservation of energy. Various equivalent motions in
the process of molecular collision constitute a variety of
model metrics. Momentum theorem and kinetic energy
theorem constitute two kinds of computational metric-
s, which belong to sub metrics. It is only necessary to
investigate the equivalent molecular motion process in
one of collision periods, and calculate the pressure, tem-
perature and internal energy of the ideal gas
[28-30]
in-
dependently in various metrics. The calculation results
of the same physical quantity must be consistent. This
is the specific application of the principle of normaliza-
tion. The physical process and conclusion following the
unitary principle are the most reliable.
The ideal gas pressure is equal to the kinetic ener-
gy of the molecule in a unit volume, and the product
of the thermodynamic temperature of the ideal gas and
the Boltzmann constant is equal to the average kinetic
energy of the molecule. In the laboratory, the number
of molecules, gas pressure and gas temperature of the
same kind of gas in an adiabatic sealed container can
be measured. If the average speed of the same kind of
gas molecules at the same temperature is measured, the
Dongfang Temperature formula of ideal gas can be ver-
ified experimentally. Therefore, the measurement of the
average rate of gas molecules at a certain temperature
is a meaningful basic experiment.
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Guided Reading
For decades, I have systematically and carefully s-
tudied the basic theories of physics and found that not
all physical inferences are correct. When studying the
generalized wave equation and its exact solution, I found
that the classical equation of the ideal gas temperature
was wrong, and found the cause of the error. In this pa-
per, the modified equation of the ideal gas temperature
is given, and the relevant experimental explanations are
Mathematics & Nature (2021) Vol. 1 9
reviewed. It is a very cautious thing to revise the ba-
sic theorem of classical theory. The popularization of
physics and mathematical logic in this article makes me
rack my brains. In the past decade, it has been misread
by commentators of academic journals. Ignoring mali-
cious slander and strangulation, of course, I thank all
those who read this article. The concise logical structure
of this paper is listed here. I hope this article will not
cause misunderstanding again.
• Calculation errors in standard textbooks
1. The classical molecular kinetic theory establish-
es the equivalent constant-force model to describe
molecular collisions, this is not a problem.
2. The classical molecular kinetic theory does not cal-
culate the collision periods between molecule and
container wall in detail, the molecular motion un-
der the equivalent constant force action is treated
as a uniform motion, and the collision half period-
s is understood to be ∆t/2 = l/¯v. These are all
wrong.
3. The classical molecular kinetic theory have use an
unfounded hypothesis, v
2
x
= v
2
y
= v
2
z
, which is mag-
nified to the equipartition theorem of energy, and
this hypothesis is actually not true;
4. On the basis of the ab ove unreasonable logic, clas-
sical molecular dynamic theory uses the momen-
tum theorem, one of the deductions of Newton’s
law of motion, to derive an incorrect pressure for-
mula pV =
mv
2
i
/3 and an incorrect temperature
formula kT = mv
2
/3;
5. However, if the kinetic energy theorem, another d-
eduction of Newton’s law of motion, is used, the
logical inference of the classical molecular dynamic
theory should be pV =
mv
2
i
/3 and kT = mv
2
/3.
6. The molecular collision half periods of the equiva-
lent constant force action model is actually ∆t/2 =
2l/v, If one would again quote the unfounded hy-
pothesis, v
2
x
= v
2
y
= v
2
z
, the inference of the mo-
mentum theorem would also be kT = mv
2
/3, this
conclusion is obviously not acceptable.
Therefore, even if the so-called the equipartition theo-
rem of energy based on the unfounded hypothesis v
2
x
=
v
2
y
= v
2
z
is recognized, the conclusion of the molecular
dynamic theory can only be kT = mv
2
/3 rather than
kT = mv
2
/3. But the hypothesis of v
2
x
= v
2
y
= v
2
z
is not
reasonable, and the two classical results are not correc-
t. I build a variety of models and calculate them from
different angles to get consistent results.
• Calculation correction and correct inference
1. In one collision periods, the article considers two ex-
treme models to describe the collision of molecules:
uniformly accelerated linear motion and uniform
circular motion;
2. For circular motion model, the article uses the cen-
tripetal force equation to calculate the ideal gas
pressure, the result is pV = Nm¯v
2
2, where N is
the total number of molecules and V is the vol-
ume of ideal gas. So, the temperature equation is
kT = m¯v
2
2, which called the Dongfang Tempera-
ture formula of ideal gas;
3. For the uniform acceleration linear motion model,
one can prove that the collision time of a molecule
in the free path is ∆t = 4l/¯v. The article uses
the momentum theorem and the kinetic theorem
to calculate the ideal gas pressure respectively, and
the results are the same equation, pV = Nm¯v
2
2,
where N is the total number of molecules. Thinking
about it further also reads the ideal gas tempera-
ture equation kT = m¯v
2
2;
4. Now that the conclusions of the two extreme mod-
els are the same, by the squeezing theorem, the
temperature of the ideal gas inevitably satisfies the
formula kT = m¯v
2
2. The actual motion of the
molecular collisions is between the uniform circu-
lar motion and the special model of the unifor-
m velocity linear motion. Therefore, the equation
kT = m¯v
2
2 is the correct inference of the relation-
ship between the ideal gas temperature and kinetic
energy;
5. The revised calculation explains that the physical
quantity which has physical significance is the aver-
age velocity ¯v rather than the average square speed
v
2
. The hypothesis v
2
x
= v
2
y
= v
2
z
of classical text-
book is unable to be proven, and it’s actually su-
perfluous.
One can also establish a variety of different physical
models to prove the correction formulas of basic laws
of molecular dynamics, thus realizing that some simple
concepts and calculations in physics may be misunder-
stood, and the errors of the conclusions derived from the
incorrect methods have been difficult to be found. Phys-
ical computation requires a universal test rule to avoid
errors and meaningless arguments.
Latest findings
Dongfang Angular Motion Law and Operator Equations
Quantum mechanics based on Planck hypothesis and statistical interpretation of wave function has achieved great success in describing the discrete law of micro motion. However, the idea of quantum mechanics has not been successfully used to describe the discrete law of macro motion, and the causality implied in the Planck hypothesis and the application scope of the basic principles of quantum mechanics have not been clarified. Here, I first introduce the angular motion law and its application, which seems to be of no special significance as a supplement to the very perfect classical mechanics, but plays an irreplaceable role in testing whether the core mathematical program of quantum mechanics of the operator evolution wave equation meets the unitary principle. Then, the operator evolution wave equations corresponding to the angular motion law are discussed, and the necessity of generalized optimization of differential equations are illustrated by the form of ordinary differential equations. Finally, the real wave equation which is superior to the Schrödinger equation in physical meaning but not necessarily the ultimate answer is briefly introduced. The implicit conclusion is that Hamiltonian can not be the only inevitable choice of constructing wave equation in quantum mechanics, and there is no causal relationship between operator evolution wave equation and quantized energy in bound state system, which indicates that whether the essence of quantum mechanics can be completely revealed is the key step to unify macro and micro quantized theory.