= 1 πa 2 th ∫ ∫ dx dy e − ( x/a$_{th}$ ) $^{2}$e − (( y − a$_{os}$ ) /a$_{th}$ ) 2 δ ( ϕ − tan − $^{1}$( x/y )) = 1 πa 2 th ∫ ∫ dϑrdr e − ( r sin( ϑ ) /a$_{th}$ ) $^{2}$e − (( r cos( ϑ ) − a$_{os}$ ) /a$_{th}$ ) 2 δ ( ϕ − tan − $^{1}$( r sin ( ϑ ) /r cos ( ϑ ))) = 1 πa 2 th ∫ ∫ dϑrdr e − ( r $^{2}$− 2 r cos( ϑ ) a$_{os}$ + a 2 $_{os}$) /a 2 $_{th}$) δ ( ϕ − ϑ ) = 1 πa 2 th e − a 2 os sin $^{2}$( ϕ ) /a$_{th}$ ∞ ∫ 0 rdr e − ( r − cos( ϕ ) a$_{os}$ ) $^{2}$/a2 $_{th}$) = 1 πa 2 th e − a 2 os sin $^{2}$( ϕ ) /a 2 th   ∞ ∫ 0 dr ( r − cos ( ϕ ) a$_{os}$ ) e − ( r − cos( ϕ ) a$_{os}$ ) $^{2}$/a2 $_{th}$) + cos ( ϕ ) a$_{os}$ ∞ ∫ 0 dr e − ( r − cos( ϕ ) a$_{os}$ ) $^{2}$/a2 $_{th}$)   = 1 πa 2 th e − a 2 os sin $^{2}$( ϕ ) /a 2 th $^{(}$a 2 th 2 e − cos $^{2}$( ϕ ) a 2 $_{os}$/a 2 $_{th}$) + cos ( ϕ ) a$_{os}$ √ π 2 a$_{th}$ ( 1 + Erf $^{[}$a$_{os}$ a$_{th}$ cos ( ϕ )$^{]))}$ = 1 2 π ( e − a 2 $_{os}$/a 2 th + √ π a$_{os}$ a$_{th}$ e − a 2 os sin $^{2}$( ϕ ) /a 2 th cos ( ϕ )$^{(}$1 + Erf $^{[}$a$_{os}$ a$_{th}$ cos ( ϕ )$^{]))}$ (26) p$_{ϕ}$ ( ϕ ) ≃ 1 √ πϕ$_{th}$ e − ϕ $^{2}$/ϕ2 th p$_{ϕ}$ ( ϕ ) ≃ 1 2 π ( 1 + 2 √ π ϕ$_{th}$ cos ( ϕ )$^{)}$ where Erf[ x ] = 2 / √ π $^{∫}$dx e − x 2 is the normalized Error Function (Erf[ ∞ ] = 1). This prob- ability distribution is plotted in the inset of figure 3 for the range of oscillation amplitudes a$_{os}$/a$_{th}$ = 0 − 2. For large oscillation amplitude a$_{os}$ >> a$_{th}$ the first term can be neglected, in addition we can assume Erf[ ... ] ≃ 1, and only very small angles contribute to the probability amplitude (sin ( ϕ ) ≃ ϕ ; cos ( ϕ ) ≃ 1) then, with ϕ$_{th}$ ≡ a$_{th}$/a$_{os}$ , we find which is a normalized Gaussian probability distribution of the angle ϕ with variance Δ ϕ = ϕ$_{th}$/ √ 2 = a$_{th}$/ ( √ 2 a$_{os}$ ), in agreement with the high excitation limit of eq. 12. We note that the ratio ϕ$_{th}$ ≡ a$_{th}$/a$_{os}$ can be interpreted as the fluctuation of the phase due to thermal variation a$_{th}$/ √ 2 of the x-component when the y-component of the oscillation is fixed at a$_{os}$ (for a$_{os}$ >> a$_{th}$ thermal fluctuation of the y-component essentially gives no contribution to phase noise). For small oscillation amplitudes the second term in 26 is small and we find, to first order in 1 /ϕ$_{th}$ , The angle probability distribution therefore becomes non-Gaussian and ultimately uniform (see inset of figure 3), as is expected for vanishing oscillation amplitude. The mean value 17