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An investigation on Stirling approximation and entropy
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Abstract
The Stirling approximation is a mathematical formula developed by James Stirling in the early 18th century, which estimates the value of factorials for large numbers. The formula has had a significant impact on many fields of study, including being used to estimate the entropy of a system. Entropy is a fundamental concept in thermodynamics that describes the degree of disorder or randomness in a system which refers to the number of ways in which the system’s particles can be arranged. The flow of entropy is a concept that describes the direction and rate of change of entropy in a system. Mechanical and thermal equilibrium is when no net flow of energy between the systems so that the systems do not undergo any further changes in their state. Joule expansion involves the expansion of a gas in a container with a movable piston. In this process, the gas expands into a larger volume, and as a result, either temperature or pressure changes. The Stirling approximation can be used to estimate the change in entropy of the gas during this process and can be used to calculate the amount of work done by the gas. This essay summarizes the importance of the Stirling approximation and finds that it is a fundamental calculation method for explaining and solving thermodynamics-related problems which can be used in various subtopics and fields. It is not only a calculation method in mathematics but also a key factor in further developing physics research.
Keywords
Stirling approximation, number of states, entropy flow, mechanical and thermal equilibrium, Joule expansion.
[1]. Weisstein, Eric W. "Stirling's Approximation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StirlingsApproximation.html
[2]. Derek Teaney. (n.d.). https://derek-teaney.github.io/S23_Phy306/lectures/math_bignumbers_ probability.pdf, page 3-4.
[3]. Le Cam, L. (1986), "The central limit theorem around 1935", Statistical Science, 1 (1): 78–96, doi:10.1214/ss/1177013818, JSTOR 2245503, MR 0833276; see p. 81, "The result, obtained using a formula originally proved by de Moivre but now called Stirling's formula, occurs in his 'Doctrine of Chances' of 1733."
[4]. Wehrl, Alfred (1 April 1978). "General properties of entropy". Reviews of Modern Physics. 50 (2): 221–260. Bibcode:1978RvMP...50..221W. doi:10.1103/RevModPhys.50.221.
[5]. Derek Teaney. (n.d.). https://derek-teaney.github.io/S23_Phy306/lectures/EntropyAndIrrever-sibility.pdf, page 2.
[6]. Thermodynamic equilibrium. Thermodynamic_equilibrium. (n.d.). https://www.chemeurope. com/en/encyclopedia/Thermodynamic_equilibrium.html
[7]. Schekochihin, A. (n.d.). Basic Thermodynamics - University of Oxford Department of Physics. https://www-thphys.physics.ox.ac.uk/people/AlexanderSchekochihin/A1/2011/handout3.pdf, page 1.
Cite this article
Zhu,J. (2023). An investigation on Stirling approximation and entropy. Theoretical and Natural Science,11,232-238.
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Volume title: Proceedings of the 2023 International Conference on Mathematical Physics and Computational Simulation
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