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If you want to restrict yourself to physically realistic systems (that are not inherently stochastic), we can restrict ourselves to ordinary differential equations, say $$\dot{x} = F(x); \qquad x ∈ ℝ^n$$ with $F$ being smooth and not ridiculously complex (e.g., not a polynomial with hundred terms). The dynamics of this system is governed by invariant ...

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If you are cooling your object that you wish to hear, then the exact sound will depend on the exact temperature (as given by yuki96's answer at 17nK). However, any temperature above the nanoKelvin temperature scale will sound the same, but the volume will increase with temperature (according to the Stefan-Boltzmann law). The sound of a warm blackbody (such ...

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This problem can be solved with noise-shaping. Since the shape of the spectrum is known, it can be used as a base for the power spectral density: $$P(f,T)=\frac{ 2 h f^3}{c^2} \frac{1}{e^\frac{h f}{k_\mathrm{B}T} - 1}$$ where $k_\mathrm{B}$ is the Boltzmann constant, $h$ is the Planck constant, and $c$ is the speed of light. This outputs the relative ...

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