### Abstract

In the semi-classical limit the relativistic quantum mechanics of a stationary beam of counter-streaming (negatively charged) electrons and one species of positively charged ions is described by a nonlinear system of finite-temperature Thomas-Fermi equations. In the high temperature/low density limit these Thomas-Fermi equations reduce to the (semi-)conformal system of Bennett equations discussed earlier by Lebowitz and the author. With the help of a sharp isoperimetric inequality it is shown that any hypothetical particle density function which is not radially symmetric about and decreasing away from the beam's axis would violate the virial theorem. Hence, all beams have the symmetry of the circular cylinder.

Original language | English (US) |
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Pages (from-to) | 607-626 |

Number of pages | 20 |

Journal | Communications In Mathematical Physics |

Volume | 226 |

Issue number | 3 |

DOIs | |

State | Published - Apr 1 2002 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

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*Communications In Mathematical Physics*, vol. 226, no. 3, pp. 607-626. https://doi.org/10.1007/s002200200625

**Symmetry results for finite-temperature, relativistic Thomas-Fermi equations.** / Kiessling, Michael.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Symmetry results for finite-temperature, relativistic Thomas-Fermi equations

AU - Kiessling, Michael

PY - 2002/4/1

Y1 - 2002/4/1

N2 - In the semi-classical limit the relativistic quantum mechanics of a stationary beam of counter-streaming (negatively charged) electrons and one species of positively charged ions is described by a nonlinear system of finite-temperature Thomas-Fermi equations. In the high temperature/low density limit these Thomas-Fermi equations reduce to the (semi-)conformal system of Bennett equations discussed earlier by Lebowitz and the author. With the help of a sharp isoperimetric inequality it is shown that any hypothetical particle density function which is not radially symmetric about and decreasing away from the beam's axis would violate the virial theorem. Hence, all beams have the symmetry of the circular cylinder.

AB - In the semi-classical limit the relativistic quantum mechanics of a stationary beam of counter-streaming (negatively charged) electrons and one species of positively charged ions is described by a nonlinear system of finite-temperature Thomas-Fermi equations. In the high temperature/low density limit these Thomas-Fermi equations reduce to the (semi-)conformal system of Bennett equations discussed earlier by Lebowitz and the author. With the help of a sharp isoperimetric inequality it is shown that any hypothetical particle density function which is not radially symmetric about and decreasing away from the beam's axis would violate the virial theorem. Hence, all beams have the symmetry of the circular cylinder.

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U2 - 10.1007/s002200200625

DO - 10.1007/s002200200625

M3 - Article

VL - 226

SP - 607

EP - 626

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -