Standard Maxwell-Boltzmann distribution: Definition and Properties

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1 217-2 Standard Maxwell-Boltzmann distribution:, 53 Medellin, Colombia ORCID : doi: /RG Abstract In this report, a standard Maxwell-Boltzmann distribution (B) is defined by analogy to the concept of the standard Gaussian distribution. The most important statistical properties of B, as well as a simple method for generating random numbers from the standard Maxwell- Boltzmann distribution are presented. Given that the properties of B are already known, it is advantageous to describe any arbitrary Maxwell-Boltzmann distribution as a function of the standard Maxwell-Boltzmann distribution B. By using this approach, it is possible to demonstrate that the temperature of a material is a function only of the fluctuating component of the average molecular kinetic energy, and that it is independent of its macroscopic kinetic energy. Keywords Ideal gases; Maxwell-Boltzmann distribution; Molecular speed; Kinetic energy; Random numbers; Standard distribution; Temperature 1. Introduction Atoms and molecules are in permanent motion. Even for solid materials, they move and collide with their neighbors, although remaining almost at their same positions. The motion of molecules 1 in gases and liquids is perhaps easier to imagine. This motion can be characterized by their particular velocities. There is no reason to believe that all molecules in a material will move at exactly the same speed (which is the magnitude of the velocity vector). On the contrary, it seems more reasonable to think the opposite. In fact, experimental measurements demonstrate that molecules in a gas do move at different speeds. As an example, Figure 1 shows an experimental distribution of molecular speeds for potassium gas at 544 K.[1] Based 1 The concept of molecule used along this document includes the case of mono-atomic molecules. 22/2/217 Reports (1 / 16)

2 Frequency (a.u.) Standard Maxwell-Boltzmann distribution: on this premise, James Clerk Maxwell derived the molecular speed distribution of ideal gases in 186,[2] almost a century before it could be proved experimentally. Some years later, Ludwig Boltzmann was able to derive the same distribution using a more rigorous and formal approach.[3] The distribution they obtained is presently known as the Maxwell-Boltzmann distribution,[4] and it will be presented and briefly discussed in Section 2. The proposed standard Maxwell-Boltzmann distribution will be introduced in Section 3, along with their most important statistical properties. In Section 4, a mathematical procedure for obtaining random numbers based on the Standard Maxwell-Boltzmann distribution is described. These standard Maxwell-Boltzmann random numbers can be used to generate random numbers for any arbitrary Maxwell-Boltzmann distribution. Finally, in Section 5, the Standard Maxwell- Boltzmann distribution is used for analyzing the data presented in Fig. 1. 1,8,6,4, Molecular speed (m/s) Figure 1. Experimental molecular speed distribution for potassium gas at 544 K obtained using a vacuum oven coupled with a spiral velocity selector and a tungsten surface ionization detector.[1] The frequency is presented in arbitrary units. 2. The Maxwell-Boltzmann distribution According to Maxwell [2] and Boltzmann,[3] the (thermal) molecular speeds (v) of a pure, ideal gas can be described by the following probability distribution: f(v) = 2 v 2 e v2 2a 2 π a 3, v 22/2/217 Reports (2 / 16) (2.1)

3 Standard Maxwell-Boltzmann distribution: where a = k BT m is a characteristic speed, k B represents a conversion factor between energy and temperature (known as the Boltzmann constant ) which is equal to J/K, T is the temperature of the gas, and m is the mass of the gas molecules. Equation (2.1) represents a Gaussian distribution with respect to the kinetic energy of the molecules, which is proportional to the square molecular speed (v 2 ). For such speed distribution, the average speed can be calculated as: (2.2) E(v) = vf(v)dv = 1 a 2 3 π v3 e v2 = 1 a 3 2 π [ a2 v 2 e v2 = 8k BT πm 2a 2 2a 2 dv = 1 a 3 2 π v2 ve v2 + 2a 2 ve v2 2a 2 dv 2a 2 dv 2a 2 ] = 8 v2 a [ e π And the variance of the molecular speed following a Maxwell-Boltzmann distribution is: ] = 8 π a (2.3) Var(v) = E(v 2 ) (E(v)) 2 = 1 a 3 2 π = 1 a 3 2 π [ a2 v 3 e v2 = 3 a 2 π [ a2 ve v2 = 3a 2 8a2 π 2a 2 v4 e v2 2a 2 dv 2a 2 Consequently, the standard deviation is: 8a2 π + 3a 2 v 2 e v2 + a 2 e v2 2a 2 dv = a2 (3 8 π ) = (3 8 π ) k BT m 2a 2 dv ] 8a2 π ] 8a2 π = 3a 2 π [ 2a π 2 ] 8a2 π (2.4) 22/2/217 Reports (3 / 16)

4 Relative frequency Standard Maxwell-Boltzmann distribution: σ(v) = Var(v) = (3 8 π ) k BT m (2.5) A particular characteristic of this type of distribution is that the coefficient of variation (CV) is a constant: CV(v) = σ(v) 8 (3 E(v) = π ) k BT m 8 = 3π k B T 8 1 π m Figure 2 shows the predicted speed distributions of an ideal gas following the Maxwell- Boltzmann distribution at 3 different temperatures. The points correspond to the experimental data presented in Figure 1. Even though the fit is not perfect, the similitude is quite remarkable; this difference is can be explained by the effect of vacuum in the measuring apparatus, which causes a flow of gas towards the vacuum pump. This effect will be discussed in further detail in Section 5. (2.6) 3E-3 2E-3 2E-3 T=3 K T=544 K T=1 K T=544 K (Exp) 1E-3 5E-4 E Molecular speed (m/s) Figure 2. Molecular speed distributions for potassium gas obtained from the Maxwell- Boltzmann distribution at 3 K, 544 K and 1 K (solid lines), compared to the experimental speeds measured at 544 K (data points).[1] 22/2/217 Reports (4 / 16)

5 Standard Maxwell-Boltzmann distribution: 3. Standard Maxwell-Boltzmann distribution: Let us now define a reduced molecular speed (B) as the ratio between the molecular speed and the average molecular speed of the distribution: B = v(t) E(v(T)) = v(t) 8k BT πm The variable B is a Maxwell-Boltzmann random variable, described by a Maxwell-Boltzmann distribution where E(v(T)) = 1 (in the corresponding units of speed being used), or equivalently, T = πm, or a = π. Then, its frequency distribution will be (from Eq. 2.1): 8k B 8 f(b) = 32 π 2 b2 e 4b 2 π, b The most important characteristic of this dimensionless standard Maxwell-Boltzmann variable (B) is that its average value is: E(B) = bf(b)db = 32 π 2 b3 e 4b 2 π db = 4 π b2 ( 8 π ) be 4b2 π db = 4 π [b2 e 4b2 π + π 4 ( 8 π ) be 4b2 π db] = e 4b2 π = 1 Similarly, it can also be found that the variance, standard deviation and coefficient of variation of B are, respectively: (3.1) (3.2) (3.3) Var(B) = 3π 8 1 (3.4) σ(b) = 3π 8 1 (3.5) CV(B) = 3π 8 1 (3.6) The reduced molecular speed has been denoted as standard Maxwell-Boltzmann distribution by analogy with the standard Gaussian distribution.[5] 22/2/217 Reports (5 / 16)

6 Standard Maxwell-Boltzmann distribution: From Eq. (3.1), it is possible to relate any arbitrary Maxwell-Boltzmann distribution of molecular speeds to the standard Maxwell-Boltzmann variable B by: In this way, v(t) = 8k BT πm B (3.7) E[v(T)] = E ( 8k BT πm B) = 8k BT E(B) = πm 8k BT πm (3.8) and Var[v(T)] = Var ( 8k BT πm B) = 8k BT πm which are consistent with Eq. (2.3) and (2.4). Var(B) = 8k BT πm (3π 1) = k BT (3 8 ) (3.9) 8 m π The advantage of defining the dimensionless standard Maxwell-Boltzmann variable B, is that it is now possible to determine all additional properties for B, and they can then be used to describe any Maxwell-Boltzmann distribution by using Eq. (3.7). For example, the expected values for different powers of B are: E(B 2 ) = b 2 f(b)db = 32 π 2 b3 be 4b 2 π db = 4 π b3 ( 8 π ) be 4b2 π db = 4 π [b3 e 4b2 π 3 b be 4b2 π db] = 3 2 [ b ( 8 π ) be 4b2 π db] This result is obtained given that = 3 2 [be 4b2 π e 4b2 π db] = 3π 8 e 4b2 π db = π. 4 (3.1) 22/2/217 Reports (6 / 16)

7 Standard Maxwell-Boltzmann distribution: E(B 3 ) = b 3 f(b)db = 32 π 2 b4 be 4b 2 π db = 4 π b4 ( 8 π ) be 4b2 π db = 4 π [b4 e 4b2 π 4 b 2 be 4b2 π db] = 2 [ b 2 ( 8 π ) be 4b2 π db] = 2 [b 2 e 4b2 π 2 be 4b2 π db] = π 2 [ ( 8 π ) be 4b2 π db] = π 2 [e 4b2 π ] = π 2 (3.11) E(B 4 ) = b 4 f(b)db = 32 π 2 b5 be 4b 2 π db = 4 π b5 ( 8 π ) be 4b2 π db = 4 π [b5 e 4b2 π 5 b 3 be 4b2 π db] = 5 2 [ b3 ( 8 π ) be 4b2 π db] = 5 2 [b3 e 4b2 π 3 b be 4b2 π db] = 15π 16 [ b ( 8 π ) 4b 2 be π db] = 15π 4b 2 16 [be π Generalizing these results, it can be found that: e 4b2 π db] = 15π2 64 (3.12) E(B n ) = { 1 2 (π 8 ) (n 1)/2 (n 1)/2 (2i + 2) i= n/2 ( π 8 )n/2 (2i + 1) i= n odd n even (3.13) On the other hand, the variances of the powers of B are: Var(B 2 ) = E(B 4 ) (E(B 2 )) 2 = 15π2 9π2 = 3π Var(B 3 ) = E(B 6 ) (E(B 3 )) 2 = 15π3 π (3.14) (3.15) Generalizing these results, it can be found that: 22/2/217 Reports (7 / 16)

8 Var(B n ) = { n ( π 8 )n (2i + 1) 1 4 (π 8 )(n 1) (2i + 2) 2 i= n Standard Maxwell-Boltzmann distribution: (n 1)/2 i= n odd ( π 8 )n [ (2i + 1) (2i + 1) 2 ] n even i= n/2 i= (3.16) Using these results, it is now possible to analyze any nonlinear function (g) of an arbitrary Maxwell-Boltzmann distribution (v), using a Taylor series expansion approach as follows:[6] n g(v) = g(n) () ( 8k BT n! πm ) B n n= Then, the expected value of the nonlinear function will be: n E[g(v)] = g(n) () ( 8k BT n! πm ) E(B n ) n= which can be readily determined by using Eq. (3.13). A similar approach can be used to find the variance of the nonlinear function: where Var[g(v)] = ( g(n) () ) n! n=1 2 ( 8k BT πm ) 2n Var(B n ) n+m + g(n) ()g (m) () ( 8k BT n! m! πm ) Cov(B n, B m ) n=1 m n (3.17) (3.18) (3.19) 22/2/217 Reports (8 / 16)

9 f - Probability denisty function Cov(B n, B m ) = { (n+m)/2 ( π 8 )(n+m)/2 [ (2i + 1) π 2 i= (n+m)/2 (n 1)/2 Standard Maxwell-Boltzmann distribution: (2i + 2) (2i + 2) ] n odd, m odd i= (m 1)/2 i= n odd, m even n even, m odd ( π (n+m)/2 8 ) [ (2i + 1) (2i + 1) (2i + 1) ] n even, m even i= n/2 i= m/2 i= (3.2) Figure 3 shows the probability density function for the standard Maxwell-Boltzmann distribution B, which corresponds to the reduced molecular speed distribution. 1,9,8,7,6,5,4,3,2,1,5 1 1,5 2 2,5 3 B - Standard Maxwell-Boltzmann random variable Figure 3. Probability density function f for the standard Maxwell-Boltzmann random variable B (reduced molecular speed distribution) It is also possible to determine the mode of the standard Maxwell-Boltzmann random variable B, by setting the derivative of f(b) with respect to b equal to zero, as follows: df(b) db = d db (32 π 2 b2 e 4b 2 π ) = 32 4b 2 π 2 (2be π 8 π b3 e 4b2 π ) = (3.21) The solution of Eq. (3.21) yields the following non-trivial result: 22/2/217 Reports (9 / 16)

10 Standard Maxwell-Boltzmann distribution: Mode(B) = π.886 (3.22) 4 and the value of its probability density function is: f(mode(b)) = 8 π e (3.23) Table 1 summarizes key descriptive statistics indicators for variable B. Table 1. Descriptive statistics summary of the standard Maxwell-Boltzmann distribution (B) Mean 1. Median.9639 Mode.8862 First quartile.69 Third quartile Interquartile range.581 Variance.1781 Standard deviation.422 Coefficient of variation Standard Maxwell-Boltzmann distribution: Random numbers generation Random numbers belonging to the standard Maxwell-Boltzmann distribution can be obtained from its cumulative probability function, assuming that the cumulative function behaves as a uniform random number between and 1. The cumulative probability function of B is given by: b F(b) = f(b)db b = 32 π 2 b2 e 4b 2 π db = 4 π b ( 8 π ) be 4b2 π db = 4 b b π [be 4b2 π e 4b2 π db] = 4 π b 22/2/217 Reports (1 / 16) b b [ e 4b2 π db be 4b2 π ] Given that the integral e 4b2 π db does not have an exact analytical solution, it has to be solved numerically. In order to reduce the computational load of integrating the distribution during the generation of random numbers, the cumulative probability distribution of B can be approximated by the following empirical function: (4.1) F(b) ( e 3.265b ) (4.2)

11 F - Cumulative probability function Standard Maxwell-Boltzmann distribution: Figure 4 shows the numerical solution of F(b) as well as the empirical approximation presented in Eq. (4.2). 1,9,8,7,6,5,4,3,2,1 Numerical solution Empirical approximation (Eq. 4.2),5 1 1,5 2 2,5 3 B - Standard Maxwell-Boltzmann random variable Figure 4. Comparison between the cumulative probability function F for the standard Maxwell- Boltzmann random variable B (solid line) and the empirical approximation given in Eq. (4.2) (dotted line). Figure 5. Histogram of 1. realizations of the standard Maxwell-Boltzmann random variable B using Eq. (4.3), compared to the corresponding probability density function B (solid line) 22/2/217 Reports (11 / 16)

12 From Eq. (4.2) it is now possible to generate random realizations of B as: b(ξ) =.363ln ( (ξ +.5) ) where ξ is a uniform random number between and 1. Standard Maxwell-Boltzmann distribution: Figure 5 shows a histogram of 1. random realizations of B generated from Eq. (4.3). (4.3) 5. Application Example: Potassium gas in vacuum oven As an example of application for the standard Maxwell-Boltzmann variable B, let us consider the data presented in Figure 1 for the distribution of molecular speeds of potassium gas in a vacuum oven at 544 K.[1] Previously, it was mentioned that the data resembled the shape of the Maxwell-Boltzmann distribution but the fit was not perfect. In this section, it will be shown that if the macroscopic flow of gas towards the vacuum pump is considered, the fit of the data significantly improves. But first, let us determine the average kinetic energy K of the potassium molecules reaching the detector: K = m 2 f iv i 2 where m is the molecular mass (m = kg for potassium), and f i is the relative frequency of molecular speed v i. For the data presented in Figure 1, the average kinetic energy of potassium molecules is found to be J. If the temperature of the gas is calculated as:[7] i T = 2 K 3k B then, the temperature of the gas reaching the detector would be 812 K, which represents an increase in temperature of 268 K, almost 5% higher than the temperature of the oven (544 K). Let us now consider that the molecular velocity of the gas (v) is composed by a macroscopic velocity and a fluctuating velocity following a Maxwell-Boltzmann distribution. In that case, the molecular velocity can be expressed as: (5.1) (5.2) 22/2/217 Reports (12 / 16)

13 Standard Maxwell-Boltzmann distribution: v = v + 8k BT πm Bu where v is the macroscopic velocity of the gas, and u is a random unit direction vector. Given that the detector only receives gas molecules in a single direction (x), then it is possible to conclude that the molecular speed reaching the detector (v x ) is given by: (5.3) v x = v x + 8k BT πm B Considering that each result (v x,i ) measured by the detector corresponds to a single realization (b i ) of B, then: (5.4) and rearranging Eq. (5.5): v x,i = v x + 8k BT πm b i (5.5) b i = v x,i v x 8k BT πm The value of the probability density function for each particular realization is: f(b i ) = 4m πk B T (v x,i v x ) 2 e m 2k B T (v x,i v x ) 2, b i There are two unknown variables in this expression: v x and T. A generalized reduced gradient (GRG) nonlinear optimization algorithm,[8] is used for finding the unknown values by minimizing the square error between the experimental frequencies (adjusted so that the maximum value of the experimental frequency is 8 π e 1, corresponding to the probability (5.6) (5.7) 22/2/217 Reports (13 / 16)

14 Probability density function Standard Maxwell-Boltzmann distribution: density function of the mode of B given by Eq. 3.23) and the probability density function predicted by Eq. (5.7). The results obtained after the optimization are: v x = m/s (5.8) T = K (5.9) By this route, the temperature obtained from the experimental molecular speeds is only 4.58% higher than the temperature of the oven, which is within a reasonable experimental error. Figure 6 shows the experimental distribution of molecular speeds, compared to the results predicted by Eq. (5.5) using the values obtained in Eq. (5.8) and (5.9). 1,,9,8,7,6,5,4,3,2,1, Predicted by Eq. (5.7) Experimental [1] Molecular speed (m/s) Figure 6. Predicted (solid line) vs. experimental (data points) distributions of molecular speeds for potassium gas in a vacuum oven at 544 K. The flaw behind the initial calculations (Eq. 5.2) is the assumption that the temperature is proportional to the total average kinetic energy of the molecules, whereas only the contribution of fluctuating velocities to the kinetic energy is related to the temperature of the system. Let us consider again the general expression for the molecular velocity presented in Eq. (5.3). From Eq. (5.3), the total kinetic energy of a molecule becomes 2 : 2 Neglecting internal motion such as rotation or any type of molecular deformation. 22/2/217 Reports (14 / 16)

15 Standard Maxwell-Boltzmann distribution: K = mv v 2 = m v mk BT π B v u + 4k BT π B2 The expected value of the total molecular kinetic energy will then be: E(K) = K = m v k BT 2 (5.1) (5.11) given that E(u) = and E(B 2 ) = 3π. The first term corresponds to the average macroscopic 8 kinetic energy of the gas K m, whereas the second term corresponds to the average fluctuating kinetic energy which will be denoted as the average thermal kinetic energy K t. It is then clear that the temperature is not related to the overall average kinetic energy of the molecules, but only to their fluctuating component, the thermal component. 6. Conclusion Based on the idea of the standard Gaussian distribution, a standard Maxwell-Boltzmann distribution (B) is defined with the purpose of generalizing the properties of the Maxwell- Boltzmann distribution, and facilitating the analysis of functions involving Maxwell-Boltzmann random variables, in particular, the speed of molecules. The most important property of such dimensionless standard Maxwell-Boltzmann distribution is that its expected value is 1. Random numbers following a standard Maxwell-Boltzmann distribution can be easily generated simply by using an empirical algebraic expression as a function of a uniform random number. By considering the experimental speeds of potassium gas in a vacuum oven, it was possible to exemplify the use of the standard Maxwell-Boltzmann distribution for describing the results. It was also possible to show that the temperature of a system is a function only of the fluctuating component of the average molecular kinetic energy, and that the macroscopic kinetic energy has no effect on temperature. Acknowledgments The author gratefully acknowledges the helpful discussions with Prof. Jaime Aguirre (Universidad Nacional de Colombia). 22/2/217 Reports (15 / 16)

16 Standard Maxwell-Boltzmann distribution: This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] Miller, R. C., & Kusch, P. (1955). Velocity distributions in potassium and thallium atomic beams. Physical Review, 99(4), [2] Maxwell, J. C. (186). Illustrations of the dynamical theory of gases. - Part I. On the motions and collisions of perfectly elastic spheres. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 19(124), [3] Boltzmann, L. (1872). Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. K. Acad. Wiss.(Wein) Sitzb., II Abt, 66. [4] Simon, D. A., & McQuarrie, J. D. (1997). Physical Chemistry: A Molecular Approach. University Science Books. [5] Devore, J. L. (215). Probability and Statistics for Engineering and the Sciences. Cengage Learning. [6] H. Hernandez (216). Modelling the effect of fluctuation in nonlinear systems using variance algebra - Application to light scattering of ideal gases, Reports, 216-1, doi: /RG [7] Halliday, D., Walker, J., & Resnick, R. (21). Fundamentals of Physics, Chapter 21. John Wiley & Sons. [8] Lasdon, L. S., Waren, A. D., Jain, A., & Ratner, M. (1978). Design and testing of a generalized reduced gradient code for nonlinear programming. ACM Transactions on Mathematical Software (TOMS), 4(1), /2/217 Reports (16 / 16)