Velocity Averaging for the Wigner Kinetic Equation in the Semiclassical Regime
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866910872662704128 |
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| author | Golse, François Möller, Jakob |
| author_facet | Golse, François Möller, Jakob |
| contents | This paper discusses the possibility of applying the velocity averaging theorems in [F. Golse, P.-L. Lions, B. Perthame, R. Sentis: J. Funct. Anal. 76(1):110--125, 1988] to the Wigner equation governing the quantum evolution of the Wigner transform of quantum density operators. Our first main results address the case of the Wigner function of a special class of density operators associated to mixed states, whose Hilbert-Schmidt norm is of order $\hbar^{d/2}$, where $d$ is the space dimension and $\hbar$ the reduced Planck constant. In space dimension $d=1$, we prove that the density function belongs to the Sobolev space $H^s(\mathbb R)$ for some $s>0$. In the case of pure states, we first obtain a characterization of the Wigner transform of rank-one quantum density operators, and apply this characterization (1) to analyze a rather general setting in which velocity averaging cannot apply to the Wigner functions of a family of rank-one density operators whose evolution is governed by the von Neumann equation, and (2) to obtain a quick derivation of Madelung's system of quantum hydrodynamic equations. This derivation provides a physical explanation of one key assumption used in the proof of the negative result (1) described above. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_09924 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Velocity Averaging for the Wigner Kinetic Equation in the Semiclassical Regime Golse, François Möller, Jakob Analysis of PDEs Mathematical Physics 81S30, 81Q20, 35B65 This paper discusses the possibility of applying the velocity averaging theorems in [F. Golse, P.-L. Lions, B. Perthame, R. Sentis: J. Funct. Anal. 76(1):110--125, 1988] to the Wigner equation governing the quantum evolution of the Wigner transform of quantum density operators. Our first main results address the case of the Wigner function of a special class of density operators associated to mixed states, whose Hilbert-Schmidt norm is of order $\hbar^{d/2}$, where $d$ is the space dimension and $\hbar$ the reduced Planck constant. In space dimension $d=1$, we prove that the density function belongs to the Sobolev space $H^s(\mathbb R)$ for some $s>0$. In the case of pure states, we first obtain a characterization of the Wigner transform of rank-one quantum density operators, and apply this characterization (1) to analyze a rather general setting in which velocity averaging cannot apply to the Wigner functions of a family of rank-one density operators whose evolution is governed by the von Neumann equation, and (2) to obtain a quick derivation of Madelung's system of quantum hydrodynamic equations. This derivation provides a physical explanation of one key assumption used in the proof of the negative result (1) described above. |
| title | Velocity Averaging for the Wigner Kinetic Equation in the Semiclassical Regime |
| topic | Analysis of PDEs Mathematical Physics 81S30, 81Q20, 35B65 |
| url | https://arxiv.org/abs/2503.09924 |