  We propose a new scenario to simultaneously explain a large lepton asymmetry
and a small baryon asymmetry. We consider a rolling scalar field and its
derivative coupling to the lepton number current. The presence of an effective
nonzero time derivative of the scalar field leads to $CPT$ violation so that
the lepton asymmetry can be generated even in thermal equilibrium as pointed
out by Cohen and Kaplan. In this model, the lepton asymmetry varies with time.
In particular, we consider the case where it grows with time. The final lepton
asymmetry is determined by the decoupling of the lepton number violating
interaction and can be as large as order unity. On the other hand, if the
decoupling takes place after the electroweak phase transition, a small baryon
asymmetry is obtained from the small lepton asymmetry at that time through
sphaleron effects. We construct a model in which a rolling scalar field is
identified with a quintessence field.
