Abstract

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.

中文翻译：

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气体膨胀成真空问题的对称性方法

摘要

简要回顾了基于对称性的使用量子和经典气体膨胀到真空中的结果。对于 Gross-Pitaevskii (GP) 近似中的量子气体，具有化学势 μ 的气体会产生额外的对称性，该化学势 μ强烈依赖于密度n，指数 ν = 2/ D，其中D是空间维度。对于温度T  → 0下玻色原子的气体凝聚物，二维系统会出现这种对称性。对于D = 3，相应地， ν = 2/3，这种情况是在低温下在所谓的单一极限下相互作用的费米气体实现的（例如，参见 L. P. Pitaevskii, Phys. Usp. 51，603（2008））。对于具有绝热指数 γ = 5/3 的气体，三维几何中的经典气体具有相同的对称性。这两个事实都是在 1970 年由 Talanov 独立发现的 [VI Talanov, JETP Lett. 11 , 199 (1970).] 用于二维非线性薛定谔（NLS 方程，与 Gross-Pitaevskii 方程一致），描述了具有克尔非线性的介质中光的平稳自聚焦，以及对于经典气体，由 Anisimov 和Lysikov [SI Anisimov 和 Yu. I. Lysikov, J. Appl。数学。机械。34 , 882 (1970)]。在准经典极限下，这些 GP 方程与绝热指数 γ = 1 + 2/ D的理想气体的流体动力学方程重合. 这种近似中的自相似解通过 Ermakov 型方程描述了气体云在膨胀气体背景下的角变形。在许多实验中都观察到膨胀云形状的这种变化，在暴露于强激光辐射后的气体膨胀期间，例如在金属上，以及在量子气体膨胀到真空期间。