CITE THIS NOTEBOOK: Study on the average speed of particles from a particle swarm by Tao Guo. Wolfram Community OCT 26 2022.
ORIGINAL ARTICLE: Guo, T. Study on the average speed of particles from a particle swarm derived from a stationary particle swarm. NATURE, Scientific Reports, 11, 13290 (2021). https://doi.org/10.1038/s41598-021-92402-w
ORIGINAL ARTICLE: Guo, T. Study on the average speed of particles from a particle swarm derived from a stationary particle swarm. NATURE, Scientific Reports, 11, 13290 (2021). https://doi.org/10.1038/s41598-021-92402-w
Article abstract: It has been more than 100 years since the advent of special relativity, but the reasons behind the related phenomena are still unknown. This article aims to inspire people to think about such problems. With the help of Mathematica software, I have proven the following problem by means of statistics: In 3-dimensional Euclidean space, for point particles whose speeds are c and whose directions are uniformly distributed in space (assuming these particles’ reference system is , if their average velocity is 0), when some particles (assuming their reference system is ), as a particle swarm, move in a certain direction with a group speed u (i.e., the norm of the average velocity) relative to , their (or the sub-particle swarm’s) average speed relative to is slower than that of particles (or the same scale sub-particle swarm) in relative to . The degree of slowing depends on the speed u of and accords with the quantitative relationship described by the Lorentz factor -. Base on this conclusion, I have deduced the speed distribution of particles in when observing from .
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0
R
u
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0
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u
R
0
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0
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u
c
2
c
2
u
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u
R
0
It has been more than 100 years since the advent of special relativity, but the reasons behind the related phenomena are still unknown. This article aims to inspire people to think about such problems.
Model: There are countless point particles with certain masses in an infinite 3-dimensional space. Their speed is c, the motion directions of each particle are evenly distributed in a 3-dimensional space, and there is no interaction between these particles. Our research object is a subset of such particles, which has a fixed average speed u relative to the total particles.
Definition: Particle swarm composed of k particles is called k-particle. Individual particle is called 1-particle. The set of the total particles is called stationary frame, and the subset with average speed u is called moving frame .
Definition: Particle swarm composed of k particles is called k-particle. Individual particle is called 1-particle. The set of the total particles is called stationary frame
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u
Suppose that the standard deviation of the projection of the 1-particle velocity onto each equivalent coordinate axis is σ. Then, the velocity of a k-particle is the velocity of the overall center of mass of the k particles, which is the average of the velocity vectors of all these 1-particles. The standard deviation of the projections of the k-particle velocities onto each equivalent coordinate axis is . Accordingly, the projections onto each coordinate axis (approximate) follow NormalDistribution[0,]. As a result, the speeds of k-particles follow the MaxwellDistribution[].
σ
k
σ
k
σ
k
The average speed of k-particles is
In[]:=
v
σ
k
Out[]=
2σ
2
π
k
For the k-particles in different frames ( and ) and with different standard deviations ( and ), the ratio of their average speed /=
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u
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0
σ
u
σ
0
v
u
v
0
In[]:=
2
2
π
σ
u
k
2
2
π
σ
0
k
Out[]=
σ
u
σ
0
Therefore, the ratio of to is the ratio between the average speeds of k-particles in and .
σ
u
σ
0
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u
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0
A 3-dimensional Cartesian (rectangular) coordinate system Oxyz is established in .
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The coordinates of a random vector with norm c can be expressed as x = c Cos[θ] Sin[ArcCos[η]], y = c Sin[θ] Sin[ArcCos[η]] and z = c η, where θ UniformDistribution[{-π,π}] and η UniformDistribution[{-1,1}]
( https://mathworld.wolfram.com/SpherePointPicking.html ). Here we only take the coordinates on the x-axis. The density function can be expressed as:
( https://mathworld.wolfram.com/SpherePointPicking.html ). Here we only take the coordinates on the x-axis. The density function can be expressed as:
In[]:=
=TransformedDistribution[cCos[θ]Sin[ArcCos[η]],{θUniformDistribution[{-π,π}],ηUniformDistribution[{-1,1}]}]
Out[]=
TransformedDistributionc
1-
cos(x.1),x.1UniformDistribution[{-π,π}],x.2UniformDistribution[{-1,1}]2
x.2
Let the direction of the velocity of be parallel to the z-axis in the direction of increasing z. Then, the mean value of the velocity component of the particles in along the z-axis must be u.
Under the assumptions that all particles in are represented by vectors with their starting points at the origin of the coordinate system and that the point (0, 0, u) is taken as the dividing point of the z-axis, the vectors in can be separated into two groups: the components of the vectors above this dividing point and the components of the vectors below it.
These vectors randomly enter from with equal probability. Therefore, the distribution of the vectors in can be thought of as a mixed distribution of the vector distribution of the components above the dividing point and the vector distribution of the components below the dividing point.
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u
Under the assumptions that all particles in
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These vectors randomly enter
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The component distribution of the vectors whose components are above (0, 0, u) on the x-axis is denoted by , and its probability density is written as
1
In[]:=
1
u
c
Out[]=
TransformedDistributionc,1
1-
cos(x.1),x.1UniformDistribution[{-π,π}],x.2UniformDistribution2
x.2
u
c
The component distribution of the vectors whose components are below (0, 0, u) on the x-axis is denoted by , and its probability density is written as
2
In[]:=
2
u
c
Out[]=
TransformedDistributionc
1-
cos(x.1),x.1UniformDistribution[{-π,π}],x.2UniformDistribution-1,2
x.2
u
c
The component distribution of the vectors whose components are above (0, 0, u) on the z-axis is denoted by , and its probability density is written as
3
In[]:=
3
Out[]=
UniformDistribution[{max(-c,u),c}]
The component distribution of the vectors whose components are below (0, 0, u) on the z-axis is denoted by , and its probability density is written as
4
In[]:=
4
Out[]=
UniformDistribution[{-c,min(c,u)}]
The mixture distribution of and is
3
4
In[]:=
34=MixtureDistribution[{w,1-w},{,}]
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4
Out[]=
MixtureDistribution[{w,1-w},{UniformDistribution[{max(-c,u),c}],UniformDistribution[{-c,min(c,u)}]}]
The mean value of the mixture distribution 34 is
In[]:=
Simplify[Mean[34],Assumptions0<u<c]
Out[]=
1
2
When the mean value of the components on the z-axis of this mixed distribution is u, the mixture weights w can be determined.
Letthemeanvalueexpressionbe(c(2w-1)+u)=u,thenfindtheweightw
1
2
In[]:=
Reduce(c(2w-1)+u)u,w
1
2
Out[]=
(u0∧c0)∨c≠0∧w
c+u
2c
Then, the mixed distribution 12 consisting of and can be calculated in accordance with this weight w.
1
2
In[]:=
w=;12=MixtureDistribution[{w,1-w},{,}]
c+u
2c
1
2
Out[]=
MixtureDistribution,1-,TransformedDistributionc,1,TransformedDistributionc
c+u
2c
c+u
2c
1-
cos(x.1),x.1UniformDistribution[{-π,π}],x.2UniformDistribution2
x.2
u
c
1-
cos(x.1),x.1UniformDistribution[{-π,π}],x.2UniformDistribution-1,2
x.2
u
c
Therefore, the standard deviation of 12 is calculated.
In[]:=
σ
u
Out[]=
2
c
2
u
3
The standard deviation of 34 is the same:
In[]:=
Simplify[StandardDeviation[34],Assumptions0<u<c]
Out[]=
2
c
2
u
3
Then, the ratio between and the velocity components on the x-axis of the particles in can be obtained.
σ
u
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0
In[]:=
Simplify[/StandardDeviation[],Assumptions0<u<c]
σ
u
Out[]=
2
c
2
u
c
The same factor can also be obtained by evaluating the ratio of the standard deviation of 34 to the standard deviation of the velocity components on the z-axis in .
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0
In[]:=
Simplify[StandardDeviation[34]/StandardDeviation[UniformDistribution[{-c,c}]],Assumptions0<u<c]
Out[]=
2
c
2
u
c
This result implies that when a subset of the particles in the reference frame composed of particles moving at the same speed (such as c) and in (spatial) random directions forms a reference frame moving at speed u, the speed of 1-particles or their forming k-particles in will be relatively decreased, with a degree of deceleration corresponding to the value determined by the scale factor -.
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u
2
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c
Based on the above conclusions, the following result will be easily obtained:
When observing all of the moving particles in from , all the randomly moving particles in can be considered to have an additional velocity component u along the z-axis. Then, according to cosine theorem, the probability density of the particles in observed in can be expressed as (where k is the number of the particles, u is the speed of and v is the norm of momentum of these k particles observed from ):
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u
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In[]:=
Clear["Global`*"];=TransformedDistribution+-2kuvCos[ArcCos[η]],vMaxwellDistribution-,ηUniformDistribution[{-1,1}];FullSimplify[PDF[,x],Assumptionsc>0∧0<u<c]
2
(ku)
2
v
k
2
c
2
u
3
Out[]=
|
The meaningful part (first branch) is selected to be verified. Note that the sampling with the replacement method in the particle swarm with a mean speed of u can simulate all of the cases of the particle swarm with a mean speed of u. (The following code takes averagely 108 + 77 minutes)