to the cantilever in order to prevent the system to be near the origin of the { x ( t ) , y ( t ) } phase space. Then, the mean phase is 〈 ϕ ( t ) 〉$_{τ}$ = − π/ 2[27] and the fluctuation of the phase is Δ ϕ$_{th}$ = $^{√}$〈 ϕ $^{2}$( t ) 〉$_{τ}$ − 〈 ϕ ( t ) 〉 2 τ = $^{√}$〈 (tan − $^{1}$( x ( t ) /y ( t ))) $^{2}$〉$_{τ}$ . Using the relation 〈 f $^{2}$( z$_{0}$ + z ) 〉 = f $^{2}$( z$_{0}$ ) + | f $^{′}$( z$_{0}$ ) | 2 〈 z $^{2}$〉 we have, with f ( z ) = tan − $^{1}$( z ), z ( t ) = x ( t ) /y ( t ) and z$_{0}$ = 〈 x ( t ) /y ( t ) 〉 = 0, And finally, with 〈 y $^{2}$( t ) 〉$_{τ}$ = ( a 2 os + a 2 $_{th}$/ 2) and relation 9 we find for the phase noise measured around a center frequency ν$_{c}$ with a bandwidth bw = 1 /τ : For large oscillation amplitude, the phase noise is therefore Δ a$_{th}$/ ( √ 2 a$_{os}$ ), which can be interpreted as a variation of the phase due to thermal fluctuation of magnitude a$_{th}$/ √ 2 of the x -component when the y -component has an oscillation amplitude a$_{os}$ . For very small oscillation amplitude, relation 12 would give Δ ϕ$_{th}$ = 1. However, as discussed above, at the origin of the { x ( t ) , y ( t ) } phase space the phase is mathematically not well defined, relation (11) cannot be used, and relation 12 is not accurate. On physical arguments one would expect a uniform distribution of phase, that is, a (normalized) probability distribution p ( ϕ ) = 1 / (2 π ), which has a mean deviation Δ ϕ = π/ √ 3. As shown in appendix B this is correct in the limit of vanishing oscillation (see inset of figure 3). The correct relation for the phase noise has no simple functional relation with the oscillation amplitude, therefore we propose as approximation for the correct phase fluctuation which has the correct large and low oscillation behavior. Figure 3 shows the known large oscillation behavior for phase noise, the correct relation calculated in the appendix as well as the approximations according to relations 11 and relation 13 with the correct small and large oscillation limits. With relation 10 we finally obtain for the total frequency noise: Δ ϕ$_{th}$ = √ √ 1 1 + 〈 x ( t ) /y ( t ) 〉 2 τ 〈 $^{(}$x ( t ) y ( t )$^{)}$$^{2}$$^{〉}$ τ = $^{√}$〈 x $^{2}$( t ) 〉$_{τ}$ 〈 y $^{2}$( t ) 〉$_{τ}$ (11) Δ ϕ$_{th}$ = √ Δ a 2 $_{th}$( ν$_{c}$, bw ) / 2 a 2 os + Δ a 2 $_{th}$( ν$_{c}$, bw ) / 2 (12) Δ ϕ$_{th}$ = √ Δ a 2 $_{th}$( ν$_{c}$, bw ) 2 a 2 os + 3Δ a 2 $_{th}$( ν$_{c}$, bw ) /π 2 (13) Δ ν$_{th}$ = ν$_{0}$ 2 Q$^{√}$ Δ a 2 $_{th}$( ν$_{c}$, bw ) 2 a 2 os + 3Δ a 2 $_{th}$( ν$_{c}$, bw ) /π 2 (14) 10