Classical Density Functional Theory: The Local Density Approximation
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866909296334209024 |
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| author | Jex, Michal Lewin, Mathieu Madsen, Peter |
| author_facet | Jex, Michal Lewin, Mathieu Madsen, Peter |
| contents | We prove that the lowest free energy of a classical interacting system at temperature $T$ with a prescribed density profile $ρ(x)$ can be approximated by the local free energy $\int f_T(ρ(x))dx$, provided that $ρ$ varies slowly over sufficiently large length scales. A quantitative error on the difference is provided in terms of the gradient of the density. Here $f_T$ is the free energy per unit volume of an infinite homogeneous gas of the corresponding uniform density. The proof uses quantitative Ruelle bounds (estimates on the local number of particles in a large system), which are derived in an appendix. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_18028 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Classical Density Functional Theory: The Local Density Approximation Jex, Michal Lewin, Mathieu Madsen, Peter Mathematical Physics Statistical Mechanics We prove that the lowest free energy of a classical interacting system at temperature $T$ with a prescribed density profile $ρ(x)$ can be approximated by the local free energy $\int f_T(ρ(x))dx$, provided that $ρ$ varies slowly over sufficiently large length scales. A quantitative error on the difference is provided in terms of the gradient of the density. Here $f_T$ is the free energy per unit volume of an infinite homogeneous gas of the corresponding uniform density. The proof uses quantitative Ruelle bounds (estimates on the local number of particles in a large system), which are derived in an appendix. |
| title | Classical Density Functional Theory: The Local Density Approximation |
| topic | Mathematical Physics Statistical Mechanics |
| url | https://arxiv.org/abs/2310.18028 |