We analyze the uniform conductivity of a one-dimensional degenerate fermion system placed in a random disorder potential so smooth that backward scattering can be neglected. We use the nonlinear Luttinger liquid model to consider effects of both interaction and the curvature of fermionic dispersion. The finite-frequency conductivity, calculated in the lowest order of disorder potential, consists of two parts. The first one is the elastic contribution, largely independent of temperature and interaction. The second one is the inelastic contribution, strongly dependent on temperature and frequency and appearing upon the simultaneous presence of curvature, disorder, and interaction. We argue that apart from such finite-frequency conductivity, there should always remain the $\delta$-function peak of conductivity at zero frequency, whose weight is weakly dependent on the disorder.

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• Electrical conductivity
• Disordered systems
• Strongly correlated systems
• Diagrammatic methods
• Luttinger liquid model