scholarly journals Mechanical Proof of the Maxwell-Boltzmann Speed Distribution With Numerical Iterations

2021 ◽  
Vol 10 (4) ◽  
pp. 21
Author(s):  
Hejie Lin ◽  
Tsung-Wu Lin
Keyword(s):  
Probability Density Function ◽  
Probability Distribution ◽  
Probability Density ◽  
Density Function ◽  
Speed Ratio ◽  
Square Root ◽  
Mean Square ◽  
Speed Distribution ◽  
Ideal Gases ◽  
Mechanical Proof

The Maxwell-Boltzmann speed distribution is the probability distribution that describes the speeds of the particles of ideal gases. The Maxwell-Boltzmann speed distribution is valid for both un-mixed particles (one type of particle) and mixed particles (two types of particles). For mixed particles, both types of particles follow the Maxwell-Boltzmann speed distribution. Also, the most probable speed is inversely proportional to the square root of the mass. This paper proves the Maxwell-Boltzmann speed distribution and the speed ratio of mixed particles using computer-generated data based on Newton’s law of motion. To achieve this, this paper derives the probability density function ψ^ab(u_a;v_a,v_b)  of the speed u_a of the particle with mass M_a after the collision of two particles with mass M_a in speed v_a and mass M_b in speed v_b. The function ψ^ab(u_a;v_a,v_b)  is obtained through a unique procedure that considers (1) the randomness of the relative direction before a collision by an angle α. (2) the randomness of the direction after the collision by another independent angle. The function ψ^ab(u_a;v_a,v_b) is used in the equation below for the numerical iterations to get new distributions P_new^a(u_a) from old distributions P_old^a(v_a), and repeat with P_old^a(v_a)=P_new^a(v_a), where n_a is the fraction of particles with mass M_a.   P_new^1(u_1)=n_1 ∫_0^∞ ∫_0^∞ ψ^11(u_1;v_1,v’_1) P_old^1(v_1) P_old^1(v’_1) dv_1 dv’_1                           +n_2 ∫_0^∞ ∫_0^∞ ψ^12(u_1;v_1,v_2) P_old^1(v_1) P_old^2(v_2) dv_1 dv_2 P_new^2(u_2)=n_1 ∫_0^∞ ∫_0^∞ ψ^21(u_2;v_2,v_1) P_old^2(v_2) P_old^1(v_1) dv_2 dv_1                           +n_2 ∫_0^∞ ∫_0^∞ ψ^22(u_2;v_2,v’_2) P_old^2(v_2) P_old^2(v’_2) dv_2 dv’_2 The final distributions converge to the Maxwell-Boltzmann speed distributions. Moreover, the square of the root-mean-square speed from the final distribution is inversely proportional to the particle masses as predicted by Avogadro’s law.

scholarly journals Mechanical Proof of the Maxwell Speed Distribution

2019 ◽  
Vol 8 (2) ◽  
pp. 90
Author(s):  
Tsung-Wu Lin ◽  
Hejie Lin
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Speed Distribution ◽  
Numerical Iteration ◽  
Analytical Integration ◽  
Mechanical Proof

This article derives the probability density function ψξ;x,x'  of the resulting speed ξ  from the collision of two particles with speeds x  and x' . This function had been left unsolved for about 150 years. Then uses two approaches to obtain the Maxwell speed distribution: (1) Numerical iteration: using the equation Pnewξ=0∞0∞ψξ;x,x' ∙Poldx∙Poldx' dxdx'   to get the new speed distribution from the old speed distribution. Also, after 9 iterations, the distribution converges to the Maxwell speed distribution. (2) Analytical integration: using the Maxwell speed distribution as the Poldx , and then getting Pnewξ  from the above integration. The result of Pnewξ  from analytical integration is proved to be exactly the Maxwell speed distribution.

scholarly journals Mechanical Proof of the Maxwell-Boltzmann Speed Distribution With Analytical Integration

2021 ◽  
Vol 10 (3) ◽  
pp. 135
Author(s):  
Hejie Lin ◽  
Tsung-Wu Lin
Keyword(s):  
Integral Form ◽  
Point Of View ◽  
Speed Ratio ◽  
Mass Ratio ◽  
Exact Integration ◽  
Speed Distribution ◽  
Ideal Gases ◽  
Mechanical Proof ◽  
Theoretical Limitation ◽  
New Distribution

The Maxwell-Boltzmann speed distribution is the probability distribution that describes the speeds of the particles of ideal gases. The Maxwell-Boltzmann speed distribution is valid for both un-mixed particles (one type of particle) and mixed particles (two types of particles). For mixed particles, both types of particles follow the Maxwell-Boltzmann speed distribution. Also, the most probable speed is inversely proportional to the square root of the mass. The Maxwell-Boltzmann speed distribution of mixed particles is based on kinetic theory; however, it has never been derived from a mechanical point of view. This paper proves the Maxwell-Boltzmann speed distribution and the speed ratio of mixed particles based on probability analysis and Newton’s law of motion. This paper requires the probability density function (PDF) ψ^ab(u_a; v_a, v_b) of the speed u_a  of the particle with mass M_a  after the collision of two particles with mass M_a  in speed v_a  and mass M_b  in speed v_b . The PDF ψ^ab(u_a; v_a, v_b)  in integral form has been obtained before. This paper further performs the exact integration from the integral form to obtain the PDF ψ^ab(u_a; v_a, v_b)  in an evaluated form, which is used in the following equation to get new distribution P_new^a(u_a)  from old distributions P_old^a(v_a) and P_old^b(v_b). When P_old^a(v_a) and P_old^b(v_b)  are the Maxwell-Boltzmann speed distributions, the integration P_new^a(u_a)  obtained analytically is exactly the Maxwell-Boltzmann speed distribution. P_new^a(u_a)=∫_0^∞ ∫_0^∞ ψ^ab(u_a;v_a,v_b) P_old^a(v_a) P_old^b(v_b) dv_a dv_b,    a,b = 1 or 2 The mechanical proof of the Maxwell-Boltzmann speed distribution presented in this paper reveals the unsolved mechanical mystery of the Maxwell-Boltzmann speed distribution since it was proposed by Maxwell in 1860. Also, since the validation is carried out in an analytical approach, it proves that there is no theoretical limitation of mass ratio to the Maxwell-Boltzmann speed distribution. This provides a foundation and methodology for analyzing the interaction between particles with an extreme mass ratio, such as gases and neutrinos.

Age Reporting of Very Old Samples

Radiocarbon ◽  
1980 ◽  
Vol 22 (4) ◽  
pp. 1021-1027 ◽  
Author(s):  
Adam Walanus ◽  
Mieczysław F Pazdur
Keyword(s):  
Probability Density Function ◽  
Probability Distribution ◽  
Probability Density ◽  
Density Function ◽  
Statistical Interpretation ◽  
Very Old ◽  
Radiocarbon Age ◽  
Fiducial Probability ◽  
Limiting Value

Problems of the statistical interpretation of radiocarbon age measurements of old samples are discussed, based on the notion of fiducial probability distribution. A probability density function of age has been given. A detailed discussion of different facets of the probability distribution of age has led us to the confirmation of the use of 2σ as the best limiting value between the regions of finite and infinite dates. It has been proposed to make use of the principle of constant probability P = 0.68 in the regions of both finite and infinite ages instead of the criterion N + kσ.

scholarly journals Appropriate Probability Density Function of Convex Bodies.

2021 ◽  
pp. 130-139
Author(s):  
Khalid A Ateia ◽  
Tarig A Abdelhaleem
Keyword(s):  
Probability Density Function ◽  
Probability Distribution ◽  
Convex Body ◽  
Probability Density ◽  
Density Function ◽  
Random Vector ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Convex Bodies ◽  
Concentration Of Measure

We investigate under the notion of Large Deviation Principle & Concentration of Measure as a technique,the ability of estimating the probability density function of any random vector in the space Rn. We found that an appropriate probability distribution for any convex body in the space is sub – Gaussian.

Weibull Probability Distribution of Wind Speed for Gaza Strip for 10 Years

2019 ◽  
Vol 892 ◽  
pp. 284-291
Author(s):  
Ahmed S.A. Badawi ◽  
Nurul Fadzlin Hasbullah ◽  
Siti Hajar Yusoff ◽  
Sheroz Khan ◽  
Aisha Hashim ◽  
...  
Keyword(s):  
Wind Speed ◽  
Probability Density Function ◽  
Probability Distribution ◽  
Wind Energy ◽  
Probability Density ◽  
Density Function ◽  
Gaza Strip ◽  
Actual Data ◽  
Average Wind Speed ◽  
Average Wind

The need of clean and renewable energy, as well as the power shortage in Gaza strip with few wind energy studies conducted in Palestine, provide the importance of this paper. Probability density function is commonly used to represent wind speed frequency distributions for the evaluation of wind energy potential in a specific area. This study shows the analysis of the climatology of the wind profile over the State of Palestine; the selections of the suitable probability density function decrease the wind power estimation error percentage. A selection of probability density function is used to model average daily wind speed data recorded at for 10 years in Gaza strip. Weibull probability distribution function has been estimated for Gaza based on average wind speed for 10 years. This assessment is done by analyzing wind data using Weibull probability function to find out the characteristics of wind energy conversion. The wind speed data measured from January 1996 to December 2005 in Gaza is used as a sample of actual data to this study. The main aim is to use the Weibull representative wind data for Gaza strip to show how statistical model for Gaza Strip over ten years. Weibull parameters determine by author depend on the pervious study using seven numerical methods, Weibull shape factor parameter is 1.7848, scale factor parameter is 4.3642 ms-1, average wind speed for Gaza strip based on 10 years actual data is 2.95 ms-1 per a day so the behavior of wind velocity based on probability density function show that we can produce energy in Gaza strip.

The cost of a carrier-borne epidemic

1974 ◽  
Vol 11 (4) ◽  
pp. 642-651 ◽  
Author(s):  
D. Jerwood
Keyword(s):  
Probability Density Function ◽  
Probability Distribution ◽  
Probability Density ◽  
Density Function ◽  
Expected Number ◽  
Stochastic Epidemic ◽  
Definition Of ◽  
The Cost ◽  
General Stochastic

In this paper, the cost of the carrier-borne epidemic is considered. The definition of duration, as used by Weiss (1965) and subsequent authors, is generalised and the probability distribution for the number of located carriers is obtained. One component of cost, namely the area generated by the trajectory of carriers, is examined and its probability density function derived. The expected area generated is then shown to be proportional to the expected number of carriers located during the epidemic, a result which has an analogue in the general stochastic epidemic.

scholarly journals Mathematical approach to human character

2020 ◽  
Author(s):  
Takuya Yabu
Keyword(s):  
Probability Density Function ◽  
Probability Distribution ◽  
Probability Density ◽  
Density Function ◽  
Negative Emotions ◽  
The Other ◽  
Mathematical Approach ◽  
Fixed Probability ◽  
To Receive

I thought about whether to receive positive or negative emotions from an event from the perspective of human character. Regarding the human character, I define it as a process of selecting one's emotion x so that the received emotion x becomes x=0 with respect to the event X and the reaction of the other party when one's thoughts and reactions occur as the accompanying reactions. Mathematically modeled it, the probability density function of how much to select an emotion has a fixed probability distribution. I also described how to deal with one's character as an application of this model.

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