Symmetry Approach in the Problem of Gas Expansion into Vacuum


A brief review of the results on the expansion of quantum and classical gases into vacuum based on the use of symmetries is presented. For quantum gases in the Gross–Pitaevskii (GP) approximation, additional symmetries arise for gases with a chemical potential μ that depends on the density n powerfully with exponent ν = 2/D, where D is the space dimension. For gas condensates of Bose atoms at temperatures T → 0, this symmetry arises for two-dimensional systems. For D = 3 and, accordingly, ν = 2/3, this situation is realized for an interacting Fermi gas at low temperatures in the so-called unitary limit (see, for example, L. P. Pitaevskii, Phys. Usp. 51, 603 (2008)). The same symmetry for classical gases in three-dimensional geometry arises for gases with the adiabatic exponent γ = 5/3. Both of these facts were discovered in 1970 independently by Talanov [V. I. Talanov, JETP Lett. 11, 199 (1970).] for a two-dimensional nonlinear Schrödinger (NLS equation, which coincides with the Gross–Pitaevskii equation), describing stationary self-focusing of light in media with Kerr nonlinearity, and for classical gases, by  Anisimov and Lysikov [S. I. Anisimov and Yu. I. Lysikov, J. Appl. Math. Mech. 34, 882 (1970)]. In the quasiclassical limit, these GP equations coincide with the equations of the hydrodynamics of an ideal gas with the adiabatic exponent γ = 1 + 2/D. Self-similar solutions in this approximation describe the angular deformations of the gas cloud against the background of an expanding gas by means of Ermakov-type equations. Such changes in the shape of an expanding cloud are observed in numerous experiments both during the expansion of gas after exposure to powerful laser radiation, for example, on metal, and during the expansion of quantum gases into vacuum.

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The authors are grateful to A.V. Turlapov and A.M. Kamchatov for many useful discussions.


The work of one of the authors (E.A.K.) was carried out with the support of the Russian Science Foundation (Grant no. 19-72-30028). Another author (M.Yu.K.) thanks for the support of the Program of Basic Research of the National Research University “Higher School Economies” and expresses gratitude to the Russian Foundation for Basic Research (grant no. 20-02-00015). The authors made equal contributions to this work.

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Correspondence to E. A. Kuznetsov or M. Yu. Kagan.

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Contribution for the JETP special issue in honor of I.E. Dzyaloshinskii’s 90th birthday

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Kuznetsov, E.A., Kagan, M.Y. Symmetry Approach in the Problem of Gas Expansion into Vacuum. J. Exp. Theor. Phys. 132, 704–713 (2021).

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