Journal of Nuclear Science, Engineering and Technology (JONSAT)
In cooperation with the Iranian Nuclear Society

Solving Random Equations Related to Electric Current and Radiation of Radioactive Materials with Effect of White Noise Perturbation

Document Type : Scientific Note

Authors

A. Khalili Golmankhaneh 1
R. Abaszade 1
A. Pishkoo 2

1 Department of Physics, Urmia Branch, Islamic Azad University, P.O.Box: 969, Urmia, Iran.

2 Physics and Accelerators Research School, Nuclear Science and Technology Research Institute, AEOI, P.O.Box: 14155-1339, Tehran-Iran

https://doi.org/10.24200/nst.2021.1206
Abstract
physical systems are mostly modeled by certain differential equations. However, in most cases random effects are omitted and therefore, the solutions are not in agreement with the experimental results. In the present work, we have investigated the effect of white noise perturbation on two physical models. At first, we have reviewed the random variables and processes and then, we have solved the Langevin equation, which is a general form of a random equation with a random white noise perturbation. We have also proposed a stochastic model for both the electric current and radiation of radioactive materials. This is done by considering the white noise perturbation sentence in the corresponding ordinary differential equations. By solving these random equations, we have obtained the mean value function, the variance function, and the random process function, as well. Finally, the results have been simulated and the corresponding diagrams presented. In order to simulate the desired random movement, the Monte Carlo method in Microsoft Excel environment was used.

Highlights

1.   A. Papoulis, S. U. Pillai, Probability, Random Variables and Stochastic Processes,‌ 4th ed.  McGraw-Hill Europe, 2002.

 

2.     N. W. Ashcroft and N. D. Mermin, Advanced Solid State Physics, 1st edition, Cengage Learning, 1976.

 

3.    P. W. Jones and P. Smith, Stochastic Processes: An Introduction, 3rd ed. Chapman and Hall/CRC, 2020.

 

4.    A. Parsian, Basics of Probability and Statistics for Science and Engineering Students, 3rd ed. Isfahan University of Technology, 1397.

 

5.    G.‌A. Parham, Random Processes,‌ Shahid Chamran University of Ahvaz, 1389.

 

6.     M. Mohseni, Statistical Mechanics, Payame Noor University Press, Tehran, 1384.

 

7.    S.M. Ross, Stochastic Processes, 2nd ed. Wiley, 1996.

 

8.  W.E. Meyerhof, Elements of Nuclear Physics, McGraw-Hill, 1967.

 

9.     M. Arkani, et al. Design and construction of a two-channel data acquisition system for random processes based on FPGA, Journal of Nuclear Science and Technology, 36(2),  29 (1394)

 

10.   R. Rezaian, R. Farnoush, Numerical comparison of the solution of the stochastic differential equation with Gaussian and Poisson white noise, Journal of Operational Research and Its Applications, 7‌(1),  93 (2010) (In Presian)

 

11.  R. Rezaian, R. Farnoush, and G. Yari, Effects of white and mixed noise permutation on numerical solution of stochastic differential equation related to bio-mathematical model, Journal of Operational Research and Its Applications, 6(23), 19, (1388) (In Persian)

 

12. D. S. Lemons, An Introduction to stochastic processes in physics, JHU Press, (2002).
 
13.   S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys., 15(1) (1943).

 

14. D.T. Gillespie, The mathematics of Brownian motion and Johson noise, ‌Am. J. Phys., 64 (3) 225 (1996).

 

15.  M. Kac, Random walk and theory of Brownian motion, Am.  Math.  Mon., 54(7) 369, (1947).

 

16. I. Karatzas, S Shreve, Brownian motion and stochastic calculus, 2nd ed. Springer-Verlag New York, (2012).

 

17.   M. Kijma, Stochastic processes with applications in finance, Chapman and Hall/CRC, (2002).

 

18.   C. H. Eab, S. C. Lim, Ornstein–Uhlenbeck process with fluctuating damping. Physica A, 492, 790 (2018).

 

19.   A. S. Balankin, et al., Noteworthy fractal features and transport properties of Cantor tartans, Phys. Lett. A, 382(23), 1534, (2018).

 

20.  A. K Golmankhaneh, A. S. Balankin, Sub-and super-diffusion on Cantor sets: Beyond the paradox, Phys. Lett. A, 382‌(14), 960 (2018).

 

21.  A.  Elisa, D. Nualart, F. Viens, Stochastic heat equation with white-noise drift, Annales de l'Institut Henri Poincare (B) Probability and Statistics, 36(2)‌, 181‌ (2000).

Keywords

Stochastic process
Langevin equation
Stochastic perturbation
White noise
1.   A. Papoulis, S. U. Pillai, Probability, Random Variables and Stochastic Processes,‌ 4th ed.  McGraw-Hill Europe, 2002.
 
2.     N. W. Ashcroft and N. D. Mermin, Advanced Solid State Physics, 1st edition, Cengage Learning, 1976.
 
3.    P. W. Jones and P. Smith, Stochastic Processes: An Introduction, 3rd ed. Chapman and Hall/CRC, 2020.
 
4.    A. Parsian, Basics of Probability and Statistics for Science and Engineering Students, 3rd ed. Isfahan University of Technology, 1397.
 
5.    G.‌A. Parham, Random Processes,‌ Shahid Chamran University of Ahvaz, 1389.
 
6.     M. Mohseni, Statistical Mechanics, Payame Noor University Press, Tehran, 1384.
 
7.    S.M. Ross, Stochastic Processes, 2nd ed. Wiley, 1996.
 
8.  W.E. Meyerhof, Elements of Nuclear Physics, McGraw-Hill, 1967.
 
9.     M. Arkani, et al. Design and construction of a two-channel data acquisition system for random processes based on FPGA, Journal of Nuclear Science and Technology, 36(2),  29 (1394)
 
10.   R. Rezaian, R. Farnoush, Numerical comparison of the solution of the stochastic differential equation with Gaussian and Poisson white noise, Journal of Operational Research and Its Applications, 7‌(1),  93 (2010) (In Presian)
 
11.  R. Rezaian, R. Farnoush, and G. Yari, Effects of white and mixed noise permutation on numerical solution of stochastic differential equation related to bio-mathematical model, Journal of Operational Research and Its Applications, 6(23), 19, (1388) (In Persian)
 
12. D. S. Lemons, An Introduction to stochastic processes in physics, JHU Press, (2002).
 
13.   S. Chandrasekhar, Stochastic problems in physics and astronomy, Rev. Mod. Phys., 15(1) (1943).
 
14. D.T. Gillespie, The mathematics of Brownian motion and Johson noise, ‌Am. J. Phys., 64 (3) 225 (1996).
 
15.  M. Kac, Random walk and theory of Brownian motion, Am.  Math.  Mon., 54(7) 369, (1947).
 
16. I. Karatzas, S Shreve, Brownian motion and stochastic calculus, 2nd ed. Springer-Verlag New York, (2012).
 
17.   M. Kijma, Stochastic processes with applications in finance, Chapman and Hall/CRC, (2002).
 
18.   C. H. Eab, S. C. Lim, Ornstein–Uhlenbeck process with fluctuating damping. Physica A, 492, 790 (2018).
 
19.   A. S. Balankin, et al., Noteworthy fractal features and transport properties of Cantor tartans, Phys. Lett. A, 382(23), 1534, (2018).
 
20.  A. K Golmankhaneh, A. S. Balankin, Sub-and super-diffusion on Cantor sets: Beyond the paradox, Phys. Lett. A, 382‌(14), 960 (2018).
 
21.  A.  Elisa, D. Nualart, F. Viens, Stochastic heat equation with white-noise drift, Annales de l'Institut Henri Poincare (B) Probability and Statistics, 36(2)‌, 181‌ (2000).

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