Figure 3: Main graph: Thermal noise error of the phase as a function of the (relative) oscillation amplitude a$_{os}$/a$_{th}$ . The black, solid, thin line corresponds to the relation obtained from the relation known in the literature, which diverges for small oscillation amplitude. The black, dotted line corresponds to the relation Δ ϕ$_{th}$ = $^{√}$〈 (tan − $^{1}$( x/y )) $^{2}$〉 (11), which is not correct at the singular point { x, y } = { 0 , 0 } . The red, thick, solid line shows the correct relation calculated from the probability distribution 26 discussed in the appendix B. Finally, the red, thin, dotted line corresponds to the approximation Δ ϕ$_{th}$ = $^{√}$a 2 $_{th}$/ (2 a 2 os + 3 a 2 $_{th}$/π $^{2}$) (relation 13), which has the correct low and large amplitude limits. Inset: Probability distributions p$_{ϕ}$ ( ϕ ) for different (relative) oscillation amplitude a$_{os}$/a$_{th}$ . The probability distributions have been calculated for the range of oscillation amplitudes a$_{os}$/a$_{th}$ = 0 − 2. The probability distribution p$_{ϕ}$ ( ϕ ) for a$_{os}$/a$_{th}$ = 0 is flat while that for a$_{os}$/a$_{th}$ = 2 is essentially Gaussian and has the highest peak at ϕ = 0. 21