Fundamentally, the gyrokinetic model assumes the plasma is strongly magnetized ( ρ ρ i ≪ ≪ L plasma ), the perpendicular spatial scales are comparable to the gyroradius ( k ⊥ ⊥ ρ ρ i ∼ ∼ 1 ), and the behavior of interest has low frequencies ( ω ω ≪ ≪ Ω Ω i ≪ ≪ Ω Ω e ). We must also expand the distribution function, f s = f s 0 + f s 1 + ⋯ ⋯ , and assume the perturbation is small compared to the background ( f s 1 ≪ ≪ f s 0 ). The starting point is the Fokker–Planck equation and Maxwell's equations. The first step is to change spatial variables from the particle position r to the guiding center position R . Then, we change velocity coordinates from ( v x , v y , v z ) to the velocity parallel v ∥ ∥ ≡ ≡ v ⋅ ⋅ b ^ ^ , the magnetic moment μ μ ≡ ≡ m s v ⊥ ⊥ 2 2 B , and the gyrophase angle φ φ . Here parallel and perpendicular are relative to b ≡ ≡ B / B , the direction of the magnetic field, and m s is the mass of the particle. Now, we can average over the gyrophase angle at constant guiding center position, denoted by ⟨ … … ⟩ φ φ , yielding the gyrokinetic equation.