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An RF phase modulation whose frequency is set close to a resonant condition with the synchrotron tune can be used to form resonance islands in the longitudinal phase space. A slow variation of the modulation amplitude and frequency can be performed to trap particles inside the island structure and to realise adiabatic transport, according to adiabatic theory for Hamiltonian systems. However, when dealing with electron beams, energy damping and quantum excitation effects cannot be neglected, and the adiabatic trapping into resonance can be described as an isothermal transformation of an ensemble of particles that evolves according to an adiabatically modulated Hamiltonian. In this study, we establish the theoretical framework for describing longitudinal motion under slow modulation of the RF phase using stochastic dynamical system theory. We also present numerical simulations with parameters of realistic lepton rings to characterise beam splitting between the core region and a $1:1$ resonant island. Special attention has been given to the thermal properties of the two beamlets, which define their equilibrium emittances and their lifetimes.
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