Mathematical aspects of the Kubo formula for electrical conductivity with dissipation

Alexander B. Watson, Dionisios Margetis, Mitchell Luskin

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)1765-1795
Number of pages31
JournalJapan Journal of Industrial and Applied Mathematics
Issue number3
StatePublished - Sep 2023

Bibliographical note

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


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


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