Non-Boltzmann/Gibbs Distribution for Non-Hermitian Steady States at Finite Temperature

The Boltzmann/Gibbs distribution is a fundamental concept in statistical physics that governs the distribution of different equilibrium states at a particular temperature. For non-Hermitian (NH) systems at finite temperature, the equilibrium state becomes a steady state and the Boltzmann/Gibbs distribution is deformed. In this paper we showed a universal feature for NH steady states at finite temperature -- the non-Boltzmann/Gibbs distribution. To make it clear, we took a two-level NH systems as an example and developed the quantum Liouvillian statistical theory to characterize it. The density matrix for the two-level NH system at finite temperature is effectively described by that for a two-level Hermitian system with certain Liouvillian Hamiltonian. In particular, according to the non-Boltzmann/Gibbs distribution, non-thermalization effect for steady states at high temperature was explored that is quite different from thermalization effect for usual equilibrium states in Hermitian systems. This discovery will open a door to novel physics for NH systems at finite temperature.
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