BSU Bulletin. Mathematics, Informatics
Bibliographic description:
, , , , FREDHOLM INTEGRAL EQUATION OF SECOND KIND IN STATISTICAL PHYSICS OF FLUIDS // BSU Bulletin. Mathematics, Informatics. - 2020. №3. . - С. 32-41.
Title:
FREDHOLM INTEGRAL EQUATION OF SECOND KIND IN STATISTICAL PHYSICS OF FLUIDS
Financing:
Работа выполнена при частичной финансовой поддержке РФФИ, грант №18-02-00523а. Авторы выражают благодарность Д. С. Сандитову и Б. Б. Дамдинову за обсуждение и полезные комментарии к статье. Авторы признательны А. А. Гаврилюку за интерес, проявленный к работе.
Codes:
Annotation:
The article analyzes the applicability of algorithms for solving various approximations for nonlinear equations of statistical physics of fluids to the solution of the linear integral Fredholm equation of second kind, which has been proposed earlier for description of
surface phenomena in liquids. We have considered a molecular system of solid spheres bordering a solid surface. In the Percus–Yevick approximation for the kernel and right-hand side,
the solution is sought in the class of piecewise continuous functions. We formulate the method
of analytical calculation for each interval in the domain of function. For other approximations,
the core of the equation and the right-hand side are calculated numerically. Fredholm equation
should be also solved numerically. To solve it, we propose the Labik–Malijevsky algorithm,
which is a standard of precision in modern physics of fluids. It is proposed to use this algorithm to calculate the two-particle function of meta-stable states distribution in the theory of
the first kind chaotic phase transition of supercooled liquid – ideal glass, which will allow
describing surface phenomena in amorphous films.
surface phenomena in liquids. We have considered a molecular system of solid spheres bordering a solid surface. In the Percus–Yevick approximation for the kernel and right-hand side,
the solution is sought in the class of piecewise continuous functions. We formulate the method
of analytical calculation for each interval in the domain of function. For other approximations,
the core of the equation and the right-hand side are calculated numerically. Fredholm equation
should be also solved numerically. To solve it, we propose the Labik–Malijevsky algorithm,
which is a standard of precision in modern physics of fluids. It is proposed to use this algorithm to calculate the two-particle function of meta-stable states distribution in the theory of
the first kind chaotic phase transition of supercooled liquid – ideal glass, which will allow
describing surface phenomena in amorphous films.
Keywords:
supercooled liquid; ideal glass; partial distribution functions; replicas; chaotic first-kind phase transition; Fredholm equation of second kind.
List of references:
1. Martinov G. A. Fundamental Theory of Liquids; Method of Distribution Functions. Bristol, 1992, 470 p.
2. Vompe A. G., Martynov G. A. The Self-Consistent Statistical Theory of Condensation. The Journal of Chemical Physics. 1997. Vol. 106, no. 14. Pp. 6095–6101. DOI: https://doi.org/10.1063/1.473272.
3. Parisi G., Zamponi F. The Ideal Glass Transition of Hard Spheres. The Journal of Chemical Physics. 2005. Vol. 123, no. 14. Pp. 144–501. DOI: https://doi.org/10.1063/1.2041507.
4. Rogers F. J., Young D. A. New, Thermodynamically Consistent, Integral Equation for Simple Fluids. Physical Review A. 1984. Vol. 30, no. 2. P. 999. DOI: https://doi.org/10.1103/PhysRevA.30.999.
5. Tikhonov D. A., Kiselyov O. E., Martynov G. A., Sarkisov G. N. Singlet Integral Equation in the Statistical Theory of Surface Phenomena in Liquids. J. of Mol. Liquids. 1999. Vol. 82. Pp. 3–17. DOI: https://doi.org/10.1016/S0167-7322(99)00037-9.
6. Agrafonov Yu., Petrushin I. Two-Particle Distribution Function of a Non-Ideal Molecular System near a Hard Surface. Physics Procedia. 2015. Vol. 71. Pp. 364–368. DOI: https://doi.org/10.1016/j.phpro.2015.08.353.
7. Agrafonov Yu. V., Petrushin I. S. Random First Order Transition from a Supercooled Liquid to an Ideal Glass. Condensed Matter and Interphases. 2020. Vol. 22. No. 2. Pp. 291–302. DOI: https://doi.org/10.17308/kcmf.2020.22/2959.
8. Agrafonov Y. V., Petrushin I. S. Using Molecular Distribution Functions to Calculate the Structural Properties of Amorphous Solids. Bull. Russ. Acad. Sci. Phys. 2020. Vol. 8. Pp. 783–787. DOI: https://doi.org/10.3103/S1062873820070035.
9. Wertheim M. S. Exact solution of the Percus–Yevick Integral Equation for Hard Spheres. Phys. Rev. Letters. 1963. Vol. 10, no. 8. Pp. 321–323. DOI:
https://doi.org/10.1103/PhysRevLett.10.321.
10. Labik S., Malijevsky A., Vonka P. A Rapidly Convergent Method of Solving the OZ Equation. Molecular Physics. 1985. Vol. 56, no. 3. Pp. 709–715. DOI: https://doi.org/10.1080/00268978500102651.
11. Malijevsky A., Labik S. The Bridge Function for Hard Spheres. Molecular Physics. 1987. Vol. 60, no. 3. Pp. 663–669. DOI: https://doi.org/10.1080/00268978700100441.
12. Franz S., Mezard M., Parisi G., Peliti L. The Response of Glassy Systems to Random Perturbations: A Bridge between Equilibrium and Off-Equilibrium. Journal of Statistical Physics. 1999. Vol. 97, no. 3–4. Pp. 459–488. DOI: https://doi.org/10.1023/A:1004602906332.
13. Mezard M., Parisi G. Thermodynamics of Glasses: a First Principles Computation. J. of Phys.: Condens. Matter. 1999. Vol. 11. Pp. A157–A165. DOI:
https://doi.org/10.1088/0953-8984/11/10A/011.
14.Robles M., López de Haro M., Santos A., Bravo Yuste S. Is There a Glass Transition for Dense Hard-Sphere Systems? The Journal of Chemical Physics. 1998. Vol. 108, no. 3. Pp. 1290–1291. DOI: https://doi.org/10.1063/1.475499.
2. Vompe A. G., Martynov G. A. The Self-Consistent Statistical Theory of Condensation. The Journal of Chemical Physics. 1997. Vol. 106, no. 14. Pp. 6095–6101. DOI: https://doi.org/10.1063/1.473272.
3. Parisi G., Zamponi F. The Ideal Glass Transition of Hard Spheres. The Journal of Chemical Physics. 2005. Vol. 123, no. 14. Pp. 144–501. DOI: https://doi.org/10.1063/1.2041507.
4. Rogers F. J., Young D. A. New, Thermodynamically Consistent, Integral Equation for Simple Fluids. Physical Review A. 1984. Vol. 30, no. 2. P. 999. DOI: https://doi.org/10.1103/PhysRevA.30.999.
5. Tikhonov D. A., Kiselyov O. E., Martynov G. A., Sarkisov G. N. Singlet Integral Equation in the Statistical Theory of Surface Phenomena in Liquids. J. of Mol. Liquids. 1999. Vol. 82. Pp. 3–17. DOI: https://doi.org/10.1016/S0167-7322(99)00037-9.
6. Agrafonov Yu., Petrushin I. Two-Particle Distribution Function of a Non-Ideal Molecular System near a Hard Surface. Physics Procedia. 2015. Vol. 71. Pp. 364–368. DOI: https://doi.org/10.1016/j.phpro.2015.08.353.
7. Agrafonov Yu. V., Petrushin I. S. Random First Order Transition from a Supercooled Liquid to an Ideal Glass. Condensed Matter and Interphases. 2020. Vol. 22. No. 2. Pp. 291–302. DOI: https://doi.org/10.17308/kcmf.2020.22/2959.
8. Agrafonov Y. V., Petrushin I. S. Using Molecular Distribution Functions to Calculate the Structural Properties of Amorphous Solids. Bull. Russ. Acad. Sci. Phys. 2020. Vol. 8. Pp. 783–787. DOI: https://doi.org/10.3103/S1062873820070035.
9. Wertheim M. S. Exact solution of the Percus–Yevick Integral Equation for Hard Spheres. Phys. Rev. Letters. 1963. Vol. 10, no. 8. Pp. 321–323. DOI:
https://doi.org/10.1103/PhysRevLett.10.321.
10. Labik S., Malijevsky A., Vonka P. A Rapidly Convergent Method of Solving the OZ Equation. Molecular Physics. 1985. Vol. 56, no. 3. Pp. 709–715. DOI: https://doi.org/10.1080/00268978500102651.
11. Malijevsky A., Labik S. The Bridge Function for Hard Spheres. Molecular Physics. 1987. Vol. 60, no. 3. Pp. 663–669. DOI: https://doi.org/10.1080/00268978700100441.
12. Franz S., Mezard M., Parisi G., Peliti L. The Response of Glassy Systems to Random Perturbations: A Bridge between Equilibrium and Off-Equilibrium. Journal of Statistical Physics. 1999. Vol. 97, no. 3–4. Pp. 459–488. DOI: https://doi.org/10.1023/A:1004602906332.
13. Mezard M., Parisi G. Thermodynamics of Glasses: a First Principles Computation. J. of Phys.: Condens. Matter. 1999. Vol. 11. Pp. A157–A165. DOI:
https://doi.org/10.1088/0953-8984/11/10A/011.
14.Robles M., López de Haro M., Santos A., Bravo Yuste S. Is There a Glass Transition for Dense Hard-Sphere Systems? The Journal of Chemical Physics. 1998. Vol. 108, no. 3. Pp. 1290–1291. DOI: https://doi.org/10.1063/1.475499.
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