Reports 2016-1 Modelling the effect of fluctuation in nonlinear systems using variance algebra - Application to light scattering of ideal gases, Medellin 050030, Colombia ORCID : 0000-0002-7634-7161 doi: 10.13140/RG.2.2.36501.52969 Abstract The main purpose of this paper is to illustrate the use of the properties and algebra of variance as a useful tool for modelling the effect of normal random fluctuations in nonlinear systems. Any arbitrary differentiable nonlinear function of normal random variables can be expressed as a power series expansion of standard normal random variables. Given that the properties and algebra of the expected value, variance and covariance of are known, it is possible to obtain a simplified mathematical model for the expected value and variance of the dependent variable as a function of the normal random fluctuation of the independent variables. The proposed method is used for describing the effect of fluctuations in temperature and pressure on the density of an ideal gas, and on the intensity of light scattered by the ideal gas. The results of the model obtained are compared to Monte Carlo simulations of the system. Keywords Fluctuation; Ideal Gas; Light Scattering; Monte Carlo Simulation; Variance Algebra 1. Introduction Random fluctuations occur everywhere in Nature [1]. The law of large numbers indicates that by increasing the frequency of occurrences of events, the number of measurements or the size of a system, the effect of fluctuations is reduced [2]. However, fluctuations remain as an important part of our World. Conventional modelling techniques are based on the average behavior of the systems. This type of modelling is also known as deterministic modelling. But fluctuation plays a key role in certain phenomena, such as for example, the scattering of light, turbulence, the onset of phase changes, polymer growth, enzymatic reactions, gene expression, nuclear and exothermic reactors, the weather, and many others [3-13]. Most importantly, these systems where fluctuation is relevant are complex highly nonlinear systems, 05/12/2016 Reports 2016-1 (1 / 19)
so the effect of the fluctuation in most cases cannot be easily determined by conventional deterministic modelling methods. The purpose of this paper is to present a basic mathematical tool proposed for modelling the effect of random fluctuation in nonlinear systems. It is therefore required to understand how random variables behave, and this is not only regarding the expected value but mainly the variance. The mathematical basics of variance algebra are presented in Section 2. An example of application to the scattering of light in ideal gases is described and analyzed in Section 3. 2. Variance Algebra Definition 1. The variance of a random variable is defined as the second moment with respect to the mean value, and it is expressed as: [14] [ ] (2.1) where the symbol E() represents the expected value operator. Definition 2. For a discrete random variable, the expected value is defined as: (2.2) where is the probability of occurrence for the value, and. Definition 3. For a continuous random variable, the expected value is defined as: (2.3) where is the probability density function of, and represents all possible realizations of. Equivalently, the probability density function has the following property: Definition 4. Expected value of a function of of a single continuous random variable : (2.4). The following definition is valid for a function [ ] (2.5) In the present work, the effect of continuous random variables on the fluctuation of nonlinear systems is analyzed, although some of the results obtained may also be valid for discrete random variables. 05/12/2016 Reports 2016-1 (2 / 19)
For understanding variance algebra, it is important first to understand the basic algebra of the expected value operator. These results are obtained by applying Definition 4. Addition of a constant value to a random variable. (2.6) can be either positive or negative, so the following expression is equivalent: (2.7) Multiplication of a constant value to a random variable. (2.8) Assuming, the following remains true: (2.9) Sum of random variables : (2.10) Or, in the more general case: (2.11) On the other hand, we have variance algebra. Using Definitions 1 and 3, as well as the basic algebra of the expected value operator, we obtain that: [ ] [ ] [ ] (2.12) And similarly to the expected value, the variance of a general function of a random variable can be determined. Definition 5. The variance of a function of a single random variable is: [ ] ([ ] ) [ ( )] (2.13) Thus, the basic algebra of the variance operator is: Addition of a constant value to a random variable : ( ) [ ] [( ) ] Similarly, ( ) (2.14) 05/12/2016 Reports 2016-1 (3 / 19)
(2.15) Multiplication of a constant value to a random variable : ( ) ( ) [ ( ) ] Correspondingly, (2.16) (2.17) It can be summarized that for a general function of the form value and variance are:, the expected (2.18) (2.19) Equations (2.18) and (2.19) are valid for any arbitrary probability distribution. Sum of random variables : ( ) [ ] where [ ] [( ) ( ) ] ( ) ( ) (2.20) Definition 6. is the covariance of and, defined as: [( )( )] (2.21) Notice that: ( ) (2.22) In general, it can be found that (2.23) In natural processes, the normal distribution is the most common distribution, as a direct result of the Central Limit Theorem [15], which establishes that the sum of different independent random variables tends toward a distribution with the following probability density function: 05/12/2016 Reports 2016-1 (4 / 19)
(2.24) where is the mean or the expected value of the distribution, and is the standard deviation of the distribution, equivalent to the square root of the variance. For any arbitrary random variable, the mean and standard deviation are: (2.25) (2.26) The distribution described by Eq. (2.24), known as Gaussian distribution or bell curve, is a normal distribution of mean, and standard deviation. The standard version of the normal distribution (denoted as ) is characterized by a mean value of zero and a standard deviation of one. And from Eq. (2.25) and (2.26): (2.27) (2.28) Definition 7. Any normal random variable with mean and standard deviation, can be related to a standard normal random variable, by: From Eq. (2.18)-(2.19) and (2.27)-(2.29), it can be shown that: (2.29) (2.30) (2.31) This is a very interesting result, as the behavior of any normal random variable can be described in terms of the standard normal distribution. In order to obtain the variance of any arbitrary function of a normal random variable, it is useful to describe the function as a series expansion around the mean as follows: (2.32) where is the n-th derivative of with respect to. Replacing Eq. (2.29) in Eq. (2.32): (2.33) 05/12/2016 Reports 2016-1 (5 / 19)
Proposition 1. Any arbitrary differentiable nonlinear function of a normal random variable with mean and standard deviation can be expressed as the infinite series expansion (2.33) Thus, the expected value of is: [ ] [ ] (2.34) and the variance of is: [ ] [ ] (2.35) By using Eq. (2.23), Eq. (2.35) can be transformed into: [ ] ( ) where (2.36) [ ] (2.37) and (2.38) Proposition 2. The variance of any arbitrary differentiable nonlinear function of a normal random variable with mean and standard deviation, can be expressed as a function (2.36) of the expected values of the powers of a standard normal random variable. Let us now understand the behavior of expected value of is:. From Definition 4 and 7, and Eq. (2.24), the (2.39) If is an odd number, then by symmetry: (2.40) 05/12/2016 Reports 2016-1 (6 / 19)
If is an even number, then by symmetry: Modelling the effect of fluctuation in nonlinear systems using (2.41) The integral in Eq. (2.41) can be solved by multiple partial substitutions considering the following useful equations [16]: (2.42) (2.43) For example, for : (2.44) For : (2.45) Generalizing for any value of (2.46) In summary, { (2.47) Similarly, from Eq. (2.37) and (2.47) the variance of will be given by: { (2.48) and from Eqs. (2.38), (2.47) and (2.48), the covariance between and will be given by: 05/12/2016 Reports 2016-1 (7 / 19)
{ (2.49) By using Definition 7, it is possible to transform any arbitrary differentiable nonlinear function of a normal random variable, into a function of the standard normal variable : (2.50) and therefore: [ ] [ ] (2.51) [ ] [ ] ( ) (2.52) We have now all the elements required to determining the expected value and the variance of any given arbitrary differentiable nonlinear function of a normal random variable. 3. Application to the light scattering of an ideal gas Let us first consider an ideal gas at constant temperature and pressure. Under these conditions, the density of the gas can be determined by [17]: where is the molecular weight of the gas, and is the Universal gas constant. Assuming that pressure is a normal random variable whereas the temperature is constant, from Definition 7 it is possible to describe the fluctuating pressure as: (3.1) (3.2) 05/12/2016 Reports 2016-1 (8 / 19)
where represents the mean pressure and denotes the standard deviation of the fluctuations in pressure. Thus, the density of the gas considering normal fluctuations in pressure will be: The gas density is now a function of the standard normal random variable (3.3) and therefore: (3.4) Since Eq. (3.4) is already a linear function of there is no need to make the series expansion, and the basic algebra of expected values and variance can be used. The expected value of the gas density would be: [ ] (3.5) and its variance: [ ] (3.6) Gas density fluctuations are important because they cause fluctuations in the relative permittivity of the gas, which are responsible for the scattering of light by the atmosphere (Rayleigh scattering) [18]. Let us now assume that temperature fluctuates around its mean value following a normal random distribution while the pressure is constant. Again from Definition 7, the temperature can be expressed as: represents the mean temperature, whereas denotes the standard deviation of the fluctuations in temperature. Thus the density of the gas considering fluctuations in temperature will be: (3.7) (3.8) The basic algebra of the expected value and the variance is not straightforward now, and the series expansion is required (Proposition 1): (3.9) 05/12/2016 Reports 2016-1 (9 / 19)
The expected value of the gas density will be: Modelling the effect of fluctuation in nonlinear systems using [ ] (3.10) [ ] (3.11) For small temperature fluctuations ( after the second term as: ), Eq. (3.11) can be approximated by truncating [ ] (3.12) It is observed that the mean gas density is not affected by fluctuations in pressure, but increases as the fluctuations in temperature increase. This is the result of the nonlinearity of the effect of temperature. Negative fluctuations in temperature increase gas density more than positive fluctuations reduce it, and the net effect is an increase of the mean gas density as a function of the relative magnitude of the fluctuations. Furthermore, this effect is more significant at lower temperatures. The variance of the gas density under a fluctuating temperature will be: [ ] [ ] (3.13) For small temperature fluctuations ( term: ), truncating the summations after the second ( ) (3.14) A validation of these results can be done by Monte Carlo (MC) simulation [19]. Normal random numbers are generated for describing the fluctuation in temperature, whereas density is calculated according to Eq. (3.8). The following values were considered in the simulation:,,,. Different values of were assumed. Simulated mean density and standard deviation are obtained from 2000 realizations. The comparison between the simulated values and the values predicted from Eq. (3.12) and (3.14) are presented in Figures 1 and 2. In both cases, the predicted behavior of the gas density is consistent with the simulation results, especially for small relative temperature 05/12/2016 Reports 2016-1 (10 / 19)
Mean gas density, (kg/m 3 ) Modelling the effect of fluctuation in nonlinear systems using fluctuations (<12%). The relative fluctuation is also known as the coefficient of variation (CV) [14]. Figure 1 also shows the 95% confidence limits for the mean gas density, calculated by, where is the number of realizations. Deviations observed in Figure 2 for larger fluctuations (CV > 12%) are mainly due to the truncation of the expansion, performed in Eq. (3.14). For larger fluctuations, additional terms of the series should be considered. 1.210 1.205 1.200 1.195 1.190 1.185 1.180 1.175 1.170 1.165 MC Simulation Calculated (3.12) 95% Confidence limits 1.160 0.00 0.05 0.10 0.15 Relative temperature fluctuation, T / T Figure 1. Comparison of mean gas density results, considering a normal fluctuation in temperature, between Monte Carlo simulation (2000 realizations) and Equation (3.12). 95% confidence intervals for Eq. (3.12) are included. An additional degree of complexity is found when both pressure and temperature are considered to fluctuate simultaneously. In this case, from Definition 7 and Eq. (3.1): (3.15) It is important noticing that each variable can be described as a function of a standard normal random variable, but they are different and independent. For this reason two standard normal random variables are introduced: and. 05/12/2016 Reports 2016-1 (11 / 19)
Gas density fluctuation, (kg/m 3 ) Modelling the effect of fluctuation in nonlinear systems using 0.25 0.20 MC Simulation Calculated (3.14) 0.15 0.10 0.05 0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 Relative temperature fluctuation, T / T Figure 2. Comparison of the fluctuation in gas density, considering a normal fluctuation in temperature, between Monte Carlo simulation (2000 realizations) and Equation (3.14). It is also important to consider the following properties of two independent standard normal random variables: { (3.16) [ ] [ ] { (3.17) (3.18) Now, transforming Eq. (3.15) into a series expansion (Proposition 1): (3.19) 05/12/2016 Reports 2016-1 (12 / 19)
The expected value of the gas density will be: Modelling the effect of fluctuation in nonlinear systems using [ ] Neglecting the terms for which, (3.20) [ ] (3.21) That is, the mean gas density depends on the temperature fluctuation but not on the pressure fluctuation. And the variance for this case is (Proposition 2): [ ] [ [ ] [ ] ] Considering up to the second term in the summations: (3.22) [ ( )] (3.23) Figure 3 and 4 present the results of MC simulation for the mean and standard deviation of the gas density, considering the simultaneous fluctuation of temperature and pressure compared to the values obtained from Eq. (3.21) and (3.23). The same conditions as in the previous simulation were used, with a value of, and changing the relative fluctuation in pressure. Again, the proposed mathematical approximation is in good agreement with the MC simulation. 05/12/2016 Reports 2016-1 (13 / 19)
Mean gas density, (kg/m 3 ) Modelling the effect of fluctuation in nonlinear systems using 1.200 1.195 1.190 1.185 1.180 1.175 1.170 1.165 MC Simulation Calculated (3.21) 95% Confidence limits 1.160 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 Relative pressure fluctuation, P / P Figure 3. Comparison of mean gas density results, considering normal fluctuations in temperature and pressure, between Monte Carlo simulation (2000 realizations) and Equation (3.21). 95% confidence limits for Eq. (3.21) are included. Relative temperature fluctuation = 10%. Notice that Eq. (3.21) is equivalent to Equations (3.5) and (3.12), when and, respectively. Similarly, Eq. (3.23) is equivalent to Equations (3.6) and (3.14) when and, respectively. It can also be concluded that considering up to the second term in the summations is a fairly good approximation for small relative fluctuations (CV < 12%). As it was previously mentioned, light scattering takes place as a result of fluctuations in the relative permittivity of the atmosphere caused by density fluctuations. The intensity of the light scattered at a distance and an angle, is proportional to the variance of the relative permittivity:[18] ( ) (3.24) where is the intensity of the incident light with a wavelength on a region of volume. 05/12/2016 Reports 2016-1 (14 / 19)
Gas density fluctuation, (kg/m 3 ) Modelling the effect of fluctuation in nonlinear systems using 0.24 0.22 0.20 0.18 0.16 0.14 0.12 MC Simulation Calculated (3.23) 0.10 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 Relative pressure fluctuation, P / P Figure 4. Comparison of the fluctuation in gas density, considering normal fluctuations in temperature and pressure, between Monte Carlo simulation (2000 realizations) and Equation (3.23). Relative temperature fluctuation = 10%. In addition, the relative permittivity can be related to the density by the Clausius-Mossotti equation: [18] (3.25) where is a constant and is characteristic for each gas. Rearranging Eq. (3.25): (3.26) Considering that the gas density is a normal random variable with mean deviation, then and standard (3.27) Expanding in series (Proposition 1), it can be found that: 05/12/2016 Reports 2016-1 (15 / 19)
Relative permittivity minus 1, -1 Modelling the effect of fluctuation in nonlinear systems using (3.28) Truncating the series from the second term forward: (3.29) The approximated expected value will then be: (3.30) And its variance (Proposition 2): (3.31) Or equivalently, in terms of : (3.32) 1.18E-03 MC simulation Calculated (3.30) 1.17E-03 1.16E-03 1.15E-03 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Relative density fluctuation, Figure 5. Comparison of the values for the mean relative permittivity minus one, considering normal fluctuations in density, between Monte Carlo simulation (2000 realizations) and Equation (3.30). 95% confidence limits for Eq. (3.30) are included. 05/12/2016 Reports 2016-1 (16 / 19)
That is, the expected value of relative permittivity is practically independent of the fluctuations in density, but the variance of relative permittivity is proportional to the variance of gas density. Therefore, the intensity of light scattering is also proportional to the variance of gas density. Figures 5 and 6 show the expected value and variance of the relative permittivity calculated with Eq. (3.30) and (3.31), respectively, compared to the values obtained by MC simulation for 2000 realizations. Different values for the standard deviation of gas density were considered. The characteristic constant of the gas was assumed to be m 3 /kg. The mean density of the gas was assumed to be 1.176 kg/m 3. Again, the equations derived are in good agreement with the simulation results. Relative permittivity fluctuation, 1.6E-04 1.4E-04 1.2E-04 1.0E-04 8.0E-05 6.0E-05 4.0E-05 2.0E-05 0.0E+00 MC simulation Calculated (3.31) 0 0.02 0.04 0.06 0.08 0.1 0.12 Relative density fluctuation, Figure 6. Comparison of the fluctuation in relative permittivity, considering normal fluctuations in density, between Monte Carlo simulation (2000 realizations) and Equation (3.31). Now, combining Equations (3.21), (3.23) and (3.32), the fluctuation in the relative permittivity can be expressed as a function of small fluctuations in temperature and pressure, assuming an ideal gas: [ ] [ ( )] [ ] (3.32) 05/12/2016 Reports 2016-1 (17 / 19)
Thus, from (3.24) and (3.32), the intensity of light scattered by an ideal gas can be expressed as: ( ) [ ] [ ( )] [ ] (3.33) So, the intensity of the light scattered by an ideal gas will be influenced in a complex way, by the fluctuations in both temperature and pressure, which on the other hand, depend on the scale considered (i.e. volume ). This is an effect that can be used, for example, for relating thermal diffusivity with scattering intensity [20]. In conclusion, variance algebra and power series expansions can be used to model the effect of random variables in nonlinear systems. This was exemplarily shown for studying the effect of temperature and pressure perturbations on the density and on the scattering of light for an ideal gas. Monte Carlo simulations confirmed the validity of the proposed approach. Acknowledgments The author gratefully acknowledges the helpful discussions with Prof. Dr. Silvia Ochoa (Universidad de Antioquia) and Prof. Jaime Aguirre (Universidad Nacional de Colombia). This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References [1] F. Beichelt, Applied Probability and Stochastic Processes, 2 nd ed., CRC Press, Boca Raton, 2016. [2] I. Hacking, Nineteenth century cracks in the concept of determinism, J. History of Ideas 44 (1983) 455-475. [3] M. Kerker, The scattering of light and other electromagnetic radiation, Academic press, New York, 1969. [4] H. Tennekes, J. L. Lumley, A first course in turbulence, MIT press, Boston, 1972. [5] L. E. Reichl, I. Prigogine, A modern course in statistical physics, University of Texas press, Austin, 1980. 05/12/2016 Reports 2016-1 (18 / 19)
[6] A. Michalak, T. Ziegler, Stochastic simulations of polymer growth and isomerization in the polymerization of propylene catalyzed by Pd-based diimine catalysts, J. Am. Chem. Soc. 124 (2002) 7519-7528. [7] H. F. Hernandez, K. Tauer, Stochastic simulation of imperfect mixing in free radical polymerization, Macromol. Symp. 271 (2008) 64-74). [8] H. Qian, E. L. Elson, Single-molecule enzymology: stochastic Michaelis Menten kinetics, Biophys. Chem. 101 (2002) 565-576. [9] D. T. Gillespie, Stochastic simulation of chemical kinetics, Annu. Rev. Phys. Chem. 58 (2007) 35-55. [10] M. B. Elowitz, A. J. Levine, E. D. Siggia, P. S. Swain, Stochastic gene expression in a single cell, Science 297 (2002) 1183-1186. [11] M. M. R. Williams, Random processes in nuclear reactors, Pergamon Press, Oxford, 2013. [12] H. U. Onken, E. Wicke, Statistical Fluctuations of Temperature and Conversion at the Catalytic CO Oxidation in an Adiabatic Packed Bed Reactor, Berichte der Bunsengesellschaft für Physik. Chem. 90 (1986) 976 981. [13] K. Hasselmann, Stochastic climate models part I. Theory, Tellus 28 (1976) 473-485. [14] J. L. Devore, Probability and Statistics for Engineering and the Sciences, 9 th ed., Cengage Learning, Boston, 2016. [15] R. Durrett, Probability: theory and examples, Cambridge University Press, Cambridge, 2010. [16] I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, Academic press, San Diego, 1980. [17] J. C. Kotz, P. M. Treichel, J. Townsend, Chemistry and chemical reactivity, 8 th ed., Cengage Learning, Belmont, 2012. [18] D. A. McQuarrie, Statistical Mechanics, Harper Collins Publishers, New York, 1976. [19] R. Y. Rubinstein, D. P. Kroese, Simulation and the Monte Carlo method, 2 nd ed., John Wiley & Sons, Hoboken, 2008. [20] B. Kruppa, J. Straub, Dynamic light scattering: An efficient method to determine thermal diffusivity of transparent fluids, Experim. Thermal Fluid Sci. 6 (1993) 28-38. 05/12/2016 Reports 2016-1 (19 / 19)