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Accounting for the effects of stellar magnetic phenomena is indispensable to fully exploit radial velocities (RVs). Correlated time variations are often mitigated by Gaussian processes (GP). They rely on fitting kernel functions that are motivated on mathematical grounds, and whose physical interpretation is often elusive. We aim to establish a connection between stellar activity affecting RVs and their correlations with physical parameters, and compare this connection with kernels used in the literature. We use simple activity models to investigate the relationship between the physical processes generating the signals and the covariances typically found in data, and to demonstrate the qualitative behaviour of this relationship. We use the StarSim code to calculate RVs of an M dwarf with different realistic evolving spot configurations. Their auto-correlation functions (ACF) show a very specific behaviour and are related to the kernel. GP regression is performed using a quasi-periodic (QP) and harmonic oscillator kernels. Comparison of the resulting kernels with the exact ACFs allows us to cross-match the kernel hyper-parameters with the introduced physical values, study the overall capabilities of the kernels, and improve their definition. We find that the QP kernel provides a more straightforward interpretation of the physics. It is able to consistently recover both the introduced rotation period P and the spot lifetime. Our study indicates that the performance can be enhanced by fixing the form factor w and adding a physically motivated cosine term with period P/2, where the contribution to the ACF for the different spot configurations differs significantly. The new quasi-periodic with cosine kernel leads to significantly better model likelihoods, can potentially distinguish between different spot configurations, and can thereby improve the sensitivity of RV exoplanet searches.