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

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(ua; va, vb) of the speed ua of the particle with mass Ma after the collision of two particles with mass Ma in speed va and mass Mb in speed vb. The function ψ (ua; va, vb) 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 β. The function ψ(ua; va, vb) is used in the equation below for the numerical iterations to get new distributions Pnew a (ua) from old distributions Pold a (va), and repeat with Pold a (va) = Pnew a (va), where na is the fraction of particles with mass Ma. Pnew 1 (u1) = n1 ∫ ∫ ψ (u1; v1, v1 )Pold 1 (v1) Pold 1 (v1 ′) ∞ 0 ∞ 0 dv1dv1 ′ + n2 ∫ ∫ ψ (u1; v1, v2)Pold 1 (v1)Pold 2 (v2) ∞ 0 ∞ 0 dv1dv2 Pnew 2 (u2) = n1 ∫ ∫ ψ (u2; v2, v1)Pold 2 (v2)Pold 1 (v1) ∞ 0 ∞ 0 dv2dv1 + n2 ∫ ∫ ψ (u2; v2, v2 ′ )Pold 2 (v2)Pold 2 (v2 ′ ) ∞ 0 ∞ 0 dv2dv2 ′ 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.

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 ( ; , ) of the speed of the particle with mass after the collision of two particles with mass in speed and mass in speed . The function ψ ( ; , ) 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 β.

Overview
James C. Maxwell (1860a,b) first provided the Maxwell speed distribution in 1860 on a statistical heuristic basis. Maxwell (1867) and Boltzmann (1872) carried out more investigations into the physical meaning of the distribution. Boltzmann (1877) derived the distribution again based on statistical thermodynamics. Nevertheless, none of their approaches were based on Newton's law of motion.
A way to prove the Maxwell-Boltzmann speed distribution is from a statistical view, beginning from the Boltzman distribution of the velocity in the x-direction as follows. , ℎ = √ 2 , is the Boltzmann constant, is the equilibrium temperature, and is the particle mass. Extending from the velocity to three independent velocities ( , , ) in three directions, and transferring it to spherical coordinates ( , , ) using = sin cos , = sin sin , and = cos gives Where ( ) is the Maxwell-Boltzmann speed distribution shown in Equation (4) (Brush, 1966, Landau et al., 1969, McQuarrie, 1976, Garrod, 1995, Maudlin, 2013.
In the Maxwell-Boltzmann speed distribution, the most probable speed, , is inversely proportional to the square root of the mass for fixed temperatures as follows Therefore, when two types of particles with mass 1 and 2 are mixed at the same temperature, the above equation gives the following mass-speed relationship where f(x, t)dx is the number of particles with speeds between x and x+dx, and similarly for f( ′ , t)d ′ , dn is the number of particles with speeds between ξ and ξ + dξ. In addition, the symbol ψ(ξ; x, x ′ ) represents the probability density function (PDF) of the resulting speed after a collision between two particles. The definition is excellent and meaningful, but the method used to calculate this PDF ψ has yet to be created. Following Boltzmann's work in 1872, we derived the PDF ψ based on Newton's law of motion in this paper. The PDF ψ for equal mass particles had been provided by Lin et al. (2019). To consider the collisions of unequal mass particles, we need to have four PDFs: ψ 11 ( 1 ; 1 , 1 ′ ), ψ 12 ( 1 ; 1 , 2 ), ψ 21 ( 2 ; 2 , 1 ), and ψ 22 ( 2 ; 2 , 2 ′ ). Where ψ ( ; , ) is the PDF of post-collision speed of a particle with a mass after the collision of two particles with mass in a pre-collision speed and mass in a pre-collision speed . For = , the PDF is identical to the collision of two equal mass particles. For ≠ , the PDF will be given in an integral form that will later be evaluated numerically for collision angles from 0 to .
After the PDF ψ ( ; , ) is derived, numerical iterations are used to get new distributions P ( ) from old distributions P ( ), and set P ( ) = P ( ) for the next iteration, where n is the fraction of particles with mass using the following equations.

Velocity Diagram for Collision of Two Particles
A velocity diagram for a collision of two particles is used to derive the probability density function. Two concentric circles in the 2D plane can be constructed as a velocity diagram for a collision of two particles in 3D space. The concentric circles velocity diagram provides a geometric relationship between the pre-collision and post-collision speeds of a collision. The concentric circles velocity diagram was used by Maxwell in his study of the Maxwell-Boltzmann speed distribution and is explained and proved in this section.
Concentric circles velocity diagrams are based on two reference frames: a fixed reference frame (O) and a center-of-mass (CM) reference frame (C). Speeds can be transferred between the fixed frame (O) and the CM frame (C).

The Fixed Reference Frame
In the fixed reference frame, before a collision, two particles with mass 1 and 2 are moving at pre-collision velocities ⃗ 1 ( ⃗⃗⃗⃗⃗⃗ ) and ⃗ 2 ( ⃗⃗⃗⃗⃗⃗ ). After the collision, the post-collision velocities of the two particles change to ⃗⃗ 1 ( ⃗⃗⃗⃗⃗⃗ ) and ⃗⃗ 2 ( ⃗⃗⃗⃗⃗⃗⃗ ), as shown in Figure 2. It is important to note that the variables in the PDF ψ 12 ( 1 ; 1 , 2 ) are speeds, which are the magnitudes of the velocities. Note that the vector ⃗ without an arrow-hat indicates the length of the vector.

The Concentric Circles in Center-of-Mass Reference Frame
The center-of-mass (CM) reference frame uses the center of mass of two particles as its origin point (C). The velocity of the center-of-mass of two particles is related to the two masses 1 and 2 and their corresponding velocities ⃗ 1 and ⃗ 2 as where 1 and 2 are mass ratio and are defined as: 1 ≡ 1 1 + 2 Before the collision, the pre-collision velocities of particles 1 and 2 in the CM frame are Based on the conservation of momentum, the total momentum in the CM frame is zero. Since the CM frame is an inertial reference frame moving with the constant velocity ⃗ and the particle velocities are relative to the CM frame, The above equation guarantees two things: (1) Point C will be located in a straight line between point A and point B, and (2) the distance ratio from the center-of-mass point C to point A and point B is   Vol. 10, No. 4;2021 In the CM frame, two concentric circles can be constructed: Circle C-A: a circle center at point C with a radius of | ⃗⃗⃗⃗⃗⃗ | Circle C-B: a circle center at point C with a radius of | ⃗⃗⃗⃗⃗⃗ | Using these two concentric circles in the CM frame, constraints of the post-collision velocities can be formulated and visualized as shown in Figure 4. After the collision, the post-collision velocities of particle 1 and particle 2 change to ⃗⃗ 1 ( ⃗⃗⃗⃗⃗⃗ ) and ⃗⃗ 2 ( ⃗⃗⃗⃗⃗⃗ ) according to the following three rules: 1. Point P will be on Circle C-A. ∴ 1 = 1 2. Point Q will be on Circle C-B.

Derivation of the Probability Density Function
After any collision between two particles, the resulting speeds depend on two factors: (1) the random directions of the two pre-collision velocities ⃗ 1 and ⃗ 2 and (2) the random direction of the post-collision velocity ⃗⃗ 1 , which is related to the contact point of the two particles. These two random factors are used in the following equation for the derivation of the PDF ψ 12 .
The right-hand side of the above equation will be derived in the following sections. The final formula of the probability density function is shown in Equation (24).

The Randomness of the Directions in 3D Before a Collision
For two given pre-collision speeds 1 and 2 , the post-collision speeds depend on the directions of the two pre-collision velocities ⃗ 1 and ⃗ 2 . The two pre-collision velocities both have random directions. The directions of these two pre-collision velocities determine the radius of circle C-A and circle C-B. The randomness of the two velocities can be reduced to one random angle which considers only the relative direction between ⃗ 1 and ⃗ 2 and is defined as the angle between the two pre-collision velocities ⃗ 1 and ⃗ 2 (in the fixed frame) as shown. For fixed magnitudes of 1 and 2 , if ⃗ 2 = ̅̅̅̅ is also fixed in the direction, but the direction of ⃗ 1 = ̅̅̅̅ is changed, then point A will be located on a spherical surface O-A, as shown in Figure 6. And the probability of point A on the surface is uniformly distributed since ⃗ 1 has equal opportunity in any direction. Therefore the velocity vector of mass 1 before the collision has a random direction in 3D space. The sphere surface O-A is centered at point O and its radius is 1 . The band area between and + d on the sphere surface O-A is 2 ( 1 sin )( 1 dα), the total area of the sphere surface is 4 1 2 , and the ratio is

The Randomness of the Directions in 3D after a Collision
Besides the pre-collision directions defined by α (representing the relative moving direction before the collision), the location of the contact point also determines the post-collision velocity. The random location of the contact point is related to the location of Mass 1 center before the collision. It can be observed that Point P is always located on the sphere surface C-P and that the probability is uniform on the surface. The reasons are based on the following factors: (1) The band-area in Figure 7(a) is 8 1 2 sin cos .
(3) The ratio of the two areas is 4 1 2 / 2 , it is not dependent on .
(4) When the Center of Mass 1 hits inside the ring of the disk, Point P will be located inside the band on the sphere surface.
(5) It has an equal opportunity to hit any point inside the disk.
Similar to the pre-collision directions defined by α (representing the relative moving direction before the collision), the location of Point P will be defined by , as the angle between ⃗ and ⃗⃗ 1 , as shown in Figure 8, such that the same angle will result in the same magnitude of the different velocity ⃗⃗ 1 . The band area between and + d on the sphere surface C-P is 2 ( 1 sin )( 1 dβ), the total area of the sphere surface is 4 1 2 , and the ratio is 1 2 sin . Therefore the probability density of Point P located on the sphere surface C-P at angle is

Computational Proof by Numerical Iterations
Both the Maxwell-Boltzmann speed distribution and the speed ratio of two types of particles can be validated using computer-generated data based on the PDF derived in the previous section. The detailed methodology and results are shown below.

Setup of the Numerical Iterations
Assume that particle type 1 has n 1 particles with mass 1 and particle type 2 has 2 particles with mass 2 . For this example: it is given that (n 1 , n 2 ) = (0.7,0. ) , ( 1 , 2 ) = (0. ,0. ) . Note that the total number of particles and the particles' exact mass are irrelevant to the results. Also, the size of the monotonic particle is irrelevant to the results. The colliding speeds are discretized into 81 equally spaced speeds beginning from 1 = 0.5 in increments of 1.0 and ending at 81 = 80.5. The probabilities are assumed to be zero for speeds over 80.5.

Conclusion and Outlook
It is essential to get the probability density function (PDF) ψ ( ; , ) since from it, the Maxwell-Boltzmann speed distribution and the speed ratio can be obtained. From the derivation of the PDF ψ ( ; , ), we reveal the basic mechanism behind the macroscopic phenomenon. The mechanics of the collision of particles is a bridge between microscopic behavior and macroscopic phenomenon. Another example is the macroscopic pressure and the microscopic velocity of the particles. We can see clearly that the universe does play dice with the random angles based on mechanics at the microscopic level for individual particles. By the way, on the other hand, group behaviors such as pressure waves or mass flows are deterministic at the macroscopic level.
The results, 1 1, 2 = 2 2, 2 , obtained from Figure 10, not only prove the Maxwell-Boltzmann speed distribution, but also Avogadro's law based on the mechanical behavior of the collisions of the particles.
This article gives mechanical proof of the Maxwell-Boltzmann speed distribution and the speed ratio for monatomic gases only. The same procedures can be applied to polyatomic gases. The PDF ψ ( ; , ) must be extended to including the rotation (or spin) of molecules unless the rotation is small and its effect can be neglected. Moreover, the procedures may also be applied to charged particles. It is possible that the chemical characteristic of molecules could be revealed from the speed and spin distributions induced by the mutual action of two different kinds of molecules. Then, the unique procedure provided may have a significant impact on this kind of research.