## Abstract

In this expository article, we present a systematic formal derivation of the Kubo formula for the linear-response current due to a time-harmonic electric field applied to non-interacting, spinless charged particles in a finite volume in the quantum setting. We model dissipation in a transparent way by assuming a sequence of scattering events occurring at random-time intervals modeled by a Poisson distribution. By taking the large-volume limit, we derive special cases of the formula for free electrons, continuum and tight-binding periodic systems, and the nearest-neighbor tight-binding model of graphene. We present the analogous formalism with dissipation to derive the Drude conductivity of classical free particles.

Original language | English (US) |
---|---|

Pages (from-to) | 1765-1795 |

Number of pages | 31 |

Journal | Japan Journal of Industrial and Applied Mathematics |

Volume | 40 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2023 |

### Bibliographical note

Publisher Copyright:© 2023, The JJIAM Publishing Committee and Springer Nature Japan KK, part of Springer Nature.

## Keywords

- Dissipation
- Electrical conductivity
- Kubo formula
- Linear-response
- Trace formula
- von Neumann equation