Journal of Physical Studies 26(3), Article 3003 [12 pages] (2022) DOI: https://doi.org/10.30970/jps.26.3003 MANDELSTAM PROBLEM V. P. Lesnikov  Odesa Polytechnic National University, 1, Shevchenko Ave., Odesa, UA–65044, Ukraine, e-mail: lesnikov@op.edu.ua Received 14 March 2022; in final form 08 July 2022; accepted 11 July 2022; published online 17 September 2022

The theory of thermal fluctuations in open hydrodynamic steady states (OHSS) is presented exclusively within the framework of hydrodynamics. The history of studies of fluctuations in a continuous medium with a stationary flux is described. It is shown that the application the fluctuation-dissipation theorem (FDT) to the OHSS with the requirement of fulfilling the Onsager's reciprocal relations (fluctuating hydrodynamics), is erroneous. The reason is that the flux, changing the dynamics and initial values of the fluctuations, violates the detailed balance existing in equilibrium. This is demonstrated by the example of the Mandelstam problem on fluctuations in a medium with a heat flux. For this problem, the structure dynamic factor is calculated for an isotropic solid and a liquid. The loss of time symmetry by the correlation functions of fluctuations and the asymmetry of their spectral representations in this problem is due to the spatial temperature variation, which determines the flux. In order to show the generality of this result for all OHSS with spatial heterogeneity, the Kelvin problem on thermal fluctuations of the interface displacements between two liquids is also considered. The upper moving liquid has velocity potential changes as the temperature in the Mandelstam problem. Reciprocal relations for both the Mandelstam and the Kelvin problems are pointed out.

Key words: Mandelstam, open hydrodynamic steady state (OHSS), fluctuations, flux, reciprocal relations, fluctuating hydrodynamics.

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1. L. Onsager, Phys. Rev. 15, 405, 2265 (1931);
   Crossref
2. L. I. Mandelstam, Dokl. Akad. Nauk SSSR 11, 219 (1934); L. I. Mandelstam, Complete Works (Publishing House of the Acad. Sci. USSR, Moscow, 1948–1950).
3. M. A. Leontovich, Zh. Eksp. Teor. Fiz. 9, 1314 (1939); M. A. Leontovich, Selected Works (Nauka, Moscow, 1985).
4. V. V. Vladimirskii, Dokl. Akad. Nauk SSSR 38, 229 (1943).
5. I. L. Fabelinskii, Molecular Scattering of Light (Vysshaja shkola, Moscow, 1965).
6. M. A. Leontovich, Dokl. Akad. Nauk SSSR 1, 97 (1935); M. A. Leontovich Selected Works (Nauka, Moscow, 1985).
7. G. S. Landsberg, A. A. Shubin, Zh. Eksp. Teor. Fiz. 9, 1309 (1939); G. S. Landsberg, Selected Works (Publishing House of the Acad. Sci. USSR, Moscow, 1958).
8. D. Beysens, Y. Garrabos, G. Zalczer, Phys. Rev. Lett. 45, 403 (1980);
   Crossref
9. I. L. Fabelinskii, Phys.-Uspekhi 164, 897, (1994);
   Crossref
10. G. E. Uhlenbeck, Usp. Fiz. Nauk 103, 275 (1971);
    Crossref
11. R. F. Fox, G. E. Uhlenbeck, Phys. Fluids 13, 1893 (1970);
    Crossref
12. R. F. Fox, G. E. Uhlenbeck, Phys. Fluids 13, 2881 (1970);
    Crossref
13. M. Bixon, R. Zwanzig, Phys. Rev. 187, 267 (1969);
    Crossref
14. F. L. Hinton, Phys. Fluids 13, 857 (1970);
    Crossref
15. J. Keizer, Statistical Thermodynamics of Nonequilibrium Processes (Springer, New York, 1987);
    Crossref
16. R. Zwanzig, Nonequilibrium Statistical Mechanics (Oxford University Press, 2001).
17. L. D. Landau, E. M. Lifshitz, Sov. Phys. JETP 5, 512 (1957).
18. S. M. Rytov, Theory of Electrical Fluctuations and Heat Radiation (Publishing House of the Acad. Sci. USSR, Moscow, 1953).
19. V. M. Zaitsev, M. I. Shliomis, Sov. Phys. JETP 32, 866 (1971).
20. J. M. Ortiz de Zarate, J. V. Sengers, Hydrodynamic fluctuations in fluids and fluid mixtures (Amsterdam, Elseveir, 2006);
    Crossref
21. V. P. Lesnikov, I. Z. Fisher, Sov. Phys. JETP 40, 667 (1975).
22. Experimental Thermodynamics Vol. X: Non-equilibrium Thermodynamics with Applications, edited by D. Bedeaux, S. Kjelstrup, J. Sengers (Royal Society of Chemistry, Cambridge, 2016);
    Crossref
23. N. N. Bogoliubov, Problems of Dynamical Theory in Statistical Physics (Gostekhizdat, Moscow, 1946); Selected Works on Statistical Physics (Moscow University Press, Moscow, 1979).
24. L. D. Landau, E. M. Lifshitz, Course of Theoretical Physics. Volume 5: Statistical Physics (Addison-Wesley, Reading, MA, 1973).
25. V. P. Lesnikov, J. Phys. Stud. 1, 208 (1997);
    Crossref
26. V. P. Lesnikov, Ukr. J. Phys. 64, 126 (2019);
    Crossref
27. V. P. Lesnikov, Ukr. J. Phys. 49, 279 (2004).
28. A.-M. S. Tremblay, M. Arai, E. D. Siggia, Phys. Rev. A 23, 1451 (1981);
    Crossref
29. H. Kiefte, V. J. Clouter, R. Penney, Phys. Rev. B 30, 4017 (1984);
    Crossref
30. D. S. Chung, K. Y. Lee, E. Mazur, Phys. Lett. A 145, 348 (1990);
    Crossref
31. L. D. Landau, E. M. Lifshitz, Fluid Mechanics (Pergamon, London, 1959).
32. R. D. Mountain, Rev. Mod. Phys. 38, 205 (1966);
    Crossref
33. D. McIntyre, J. V. Sengers, in Physics of Simple Liquids, edited by H. N. V. Tamperley, J. S. Rowlinson, G. S. Rushbrooke (North-Holland Publishing Company, Amsterdam, 1968).