The Semiclassical Limit of the 2D Dirac--Hartree Equation with Periodic Potentials
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918259149766656 |
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| author | Lee, Jinyeop Qi, Kunlun |
| author_facet | Lee, Jinyeop Qi, Kunlun |
| contents | We study the semiclassical limit of the two-dimensional Dirac--Hartree equation in the presence of a periodic external potential. The spinor dynamics are formulated using the matrix-valued Wigner transform together with spectral projectors onto the positive and negative energy bands. Under suitable assumptions on the initial data and the potentials, we rigorously derive Vlasov-type transport equations describing the evolution of the band-resolved phase-space densities in both the massive and massless regimes. In the massless case, the limiting dynamics propagate ballistically with constant speed, while in the massive case the velocity is relativistic. Our analysis justifies the emergence of relativistic Vlasov equations from Dirac--Hartree dynamics in the semiclassical regime. As a corollary, we recover the relativistic Vlasov--Poisson equation from the Dirac equation with a regularized Coulomb interaction when the regularization vanishes together with the semiclassical parameter. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_19362 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Semiclassical Limit of the 2D Dirac--Hartree Equation with Periodic Potentials Lee, Jinyeop Qi, Kunlun Mathematical Physics 35Q40, 81Q05, 82C10 We study the semiclassical limit of the two-dimensional Dirac--Hartree equation in the presence of a periodic external potential. The spinor dynamics are formulated using the matrix-valued Wigner transform together with spectral projectors onto the positive and negative energy bands. Under suitable assumptions on the initial data and the potentials, we rigorously derive Vlasov-type transport equations describing the evolution of the band-resolved phase-space densities in both the massive and massless regimes. In the massless case, the limiting dynamics propagate ballistically with constant speed, while in the massive case the velocity is relativistic. Our analysis justifies the emergence of relativistic Vlasov equations from Dirac--Hartree dynamics in the semiclassical regime. As a corollary, we recover the relativistic Vlasov--Poisson equation from the Dirac equation with a regularized Coulomb interaction when the regularization vanishes together with the semiclassical parameter. |
| title | The Semiclassical Limit of the 2D Dirac--Hartree Equation with Periodic Potentials |
| topic | Mathematical Physics 35Q40, 81Q05, 82C10 |
| url | https://arxiv.org/abs/2512.19362 |