Decade-long monitoring of blazars at optical and infrared (OIR) wavelengths with the Small and Moderate Aperture Research Telescope System (SMARTS) in Chile and in $gamma$-rays with the Fermi Large Area Telescope (LAT) has enabled the systematic study of their multi-wavelength long-term variability. In this work we investigate, from a theoretical perspective, the long-term variability properties of blazar emission by introducing an observationally motivated time-dependence to four main parameters of the one-zone leptonic model: injection luminosity of relativistic electrons, strength of magnetic field, Doppler factor, and external photon field luminosity. For the first time, we use both the probability density function and the power spectral density of the 10 year-long Fermi-LAT light curves to create variation patterns for the model parameters. Using as test beds two bright blazars from the SMARTS sample (PKS 2155-304 and 3C 273), we compute 10 year-long OIR, X-ray, and $gamma$-ray model light curves for different varying parameters. We compare the findings of our theoretical investigation with multi-wavelength observations using various measures of variability. While no single-varying parameter simulation can explain all multi-wavelength variability properties, changes in the electron luminosity and external radiation field in PKS 2155-304 and 3C 273, respectively, can account for most of them. Our results motivate future time-dependent studies with coupling between two or more physical parameters to describe the multi-wavelength long-term blazar variability.
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