The Grand Canonical Ensemble: Theory and Applications

Main Article Content

Yulianto Agung Rezeki
Hanung Vernanda Putri
Erlina Fatkhur Rohmah
Yuriz Ananda Santika
Dimas Pandu Mahardika

##article.abstract##

The grand canonical ensemble (GCE) is an assembly structure with walls that the system or energy can penetrate. The ensemble is used in an open system that allows power and mass to exist with the environment. Only constant temperature and volume can limit the energy and number of systems in a GCE. As a result, the GCE differs in the number of systems and the amount of energy in each assembly. However, the total number of assemblies and the total amount of energy are constant. The purpose of this article is to explain the concept of the GCE and explain the implementation of the concept of the GCE in various fields. The application of the GCE concept in the physics study topics i.e. the equilibrium between the system and the particle energy reservoir, the grand partition function, the average number of occupations, semi-classical particles in integral formulations, the Bose-Einstein grand partition function, the Fermi-Einstein grand partition function, alternative formulation, and calculate the average number of particles. In addition to the application of the concept in the field of physics, there is also the implementation of the concept of a GCE in the field of chemistry, including determining the general equation of an ideal gas, a GCE approach to understanding the topic of electrochemical-thermodynamic studies, measuring fluctuations in the number of systems in the assembly and its relation to ion collisions, and the grand canonical Monte Carlo simulation. Each implementation has its use or purpose.

Article Details

How to Cite
Rezeki, Y. A., Putri, H. V., Rohmah, E. F., Santika, Y. A., & Mahardika, D. P. (2021). The Grand Canonical Ensemble: Theory and Applications. Advanced Sustainable Engineering, 1(1), 24-36. Retrieved from https://ukischolarsnetwork.org/index.php/ase/article/view/50
##section.section##
Articles

References

M. Abdullah, Mekanika Statistik. Bandung: Institut Teknologi Bandung Press, 2017.

V. Vovchenko, O. Savchuk, R. V. Poberezhnyuk, M. I. Gorenstein, and V. Koch, “Connecting fluctuation measurements in heavy-ion collisions with the grand-canonical susceptibilities,” Phys. Lett. Sect. B Nucl. Elem. Part. High-Energy Phys., vol. 811, p. 135868, 2020.

K. Jimenez and J. Reslen, “Thermodynamic signatures of an underlying quantum phase transition: A grand canonical approach,” Phys. Lett. Sect. A Gen. At. Solid State Phys., vol. 380, no. 34, pp. 2603–2607, 2016.

F. Calvo and D. J. Wales, “Harmonic superposition method for grand-canonical ensembles,” Chem. Phys. Lett., vol. 623, pp. 17–21, 2015.

M. M. Melander, “Grand canonical ensemble approach to electrochemical thermodynamics, kinetics, and model Hamiltonians,” Curr. Opin. Electrochem., vol. 29, p. 100749, 2021.

A. Groß, “Grand-canonical approaches to understand structures and processes at electrochemical interfaces from an atomistic perspective,” Curr. Opin. Electrochem., vol. 27, p. 100684, 2021.

H. Q. Pang, S. Li, and Z. Y. Li, “Grand canonical Monte Carlo simulations of hydrogen adsorption in carbon aerogels,” Int. J. Hydrogen Energy, vol. 46, no. 70, pp. 34807–34821, 2021.

S. Lamperski and R. Górniak, “Inverse grand-canonical Monte Carlo investigation of the activity coefficient of polar fluids,” Fluid Phase Equilib., vol. 295, no. 2, pp. 255–263, 2010.

T. Al-Mulla, R. J. M. Pellenq, and F. J. Ulm, “Griffith’s postulate: Grand Canonical Monte Carlo approach for fracture mechanics of solids,” Eng. Fract. Mech., vol. 199, no. February, pp. 544–554, 2018.

T. Zimmermann, F. Šebesta, and J. V. Burda, “A new grand-canonical potential for the thermodynamic description of the reactions in solutions with constant pH,” J. Mol. Liq., vol. 335, 2021.

F. W. Sears and G. L. Salinger, Thermodynamic, Kinetic Theory, and Statistical Thermodynamic 3rd Edition. Canada: Addison-Wesley Publishing Company, 1975.

M. Khalfaoui, S. Knani, M. A. Hachicha, and A. Ben Lamine, “New theoretical expressions for the ?ve adsorption type isotherms classi?ed by BET based on statistical physics treatment,” J. Colloid Interface Sci., vol. 263, no. 2, pp. 350–356, 2003.

S. Knani, M. Mathlouthi, and A. Ben Lamine, “Modeling of the psychophysical response curves using the grand canonical ensemble in statistical physics,” Food Biophys., vol. 2, no. 4, pp. 183–192, 2007.

B. Diu, C. Guthmann, D. Lederer, and B. Roulet, Physique Statistique. Paris: Hermann, 1989.

[15] A. Ben Lamine and Y. Bouazra, “Application of statistical thermodynamics to the olfaction mechanism,” Chem. Senses, vol. 22, no. 1, pp. 67–75, 1997.

S. Knani, M. Khalfaoui, M. A. Hachicha, A. Ben Lamine, and M. Mathlouthi, “Modelling of water vapour adsorption on foods products by a statistical physics treatment using the grand canonical ensemble,” Food Chem., vol. 132, no. 4, pp. 1686–1692, 2012.

L. Couture and R. Zitoun, Physique Statistique. Ellipses, 1992.

S. E. Schmickler W, Interfacial electrochemistry. 2nd ed. Berlin: Springer, 2010.

A. Groß and S. Sakong, “Modelling the electric double layer at electrode/electrolyte interfaces,” Curr. Opin. Electrochem., vol. 14, pp. 1–6, 2019.

M. M. Melander, “Grand Canonical Rate Theory for Electrochemical and Electrocatalytic Systems I: General Formulation and Proton-coupled Electron Transfer Reactions,” J. Electrochem. Soc., vol. 167, no. 11, 2020.

M. M. Melander, M. J. Kuisma, T. E. K. Christensen, and K. Honkala, “Grand-canonical approach to density functional theory of electrocatalytic systems: Thermodynamics of solid-liquid interfaces at constant ion and electrode potentials,” J. Chem. Phys., vol. 150, no. 4, 2019.

S. Hamali, “Simulasi Monte Carlo,” 2017. https://bbs.binus.ac.id/management/2017/12/simulasi-monte-carlo/.

[23] Z. Y. Zhang, X. H. Liu, and H. Li, “The grand canonical Monte Carlo simulation of hydrogen adsorption in single-walled carbon nanotubes,” Int. J. Hydrogen Energy, vol. 42, no. 7, pp. 4252–4258, 2017.

G. E. Norman and V. S. Filinov, “Investigations of phase transitions by a Monte-Carlo method,” High Temp., vol. 7, no. 2, 1969.

M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids. Oxford: Clarendon Press, 1994.

S. Lamperski, “The individual and mean activity coefficients of an electrolyte from the inverse GCMC simulation,” Mol. Simul., vol. 33, no. 15, pp. 1193–1198, 2007.[27] S. Lamperski and M. P?uciennik, “The activity coefficient of high density systems with hard-sphere interactions: the application of the IGCMC method.,” Mol. Simul., vol. 36, no. 2, pp. 111–117, 2010.

S. Lamperski, C. W. Outhwaite, and L. B. Bhuiyan, “The electric double-layer differential capacitance at and near zero surface charge for a restricted primitive model electrolyte,” J. Phys. Chem. B, vol. 113, no. 26, pp. 8925–8929, 2009.

S. Lamperski and J. K?os, “Grand canonical Monte Carlo investigations of electrical double layer in molten salts,” J. Chem. Phys., vol. 129, no. 16, 2008.

A. Malasics, D. Gillespie, and D. Boda, “Simulating prescribed particle densities in the grand canonical ensemble using iterative algorithms,” J. Chem. Phys., vol. 128, no. 12, 2008.