$$a = \left(\frac{3NR^*}{4E^*}\right)^{1/3},\quad \delta = \frac{a^2}{R^*} = \left(\frac{9N^2}{16R^*E^{*2}}\right)^{1/3},\quad p(x,y) = \frac{3P}{2\pi a^2} \sqrt{1-\frac{x^2+y^2}{a^2}}.$$

![\[fig:setup\]Contact between two elastic spheres with rough surfaces and the equivalent problem of indentation of an effective elastic and rough half-space by a rigid smooth sphere of effective radius.](setup){width="100.00000%"}

Roughness
---------

To model the surface roughness we use a white-noise filtering technique [@hu1992ijmtm; @yastrebov2017jmps]. This technique enables us to control accurately the spectrum of the surface and also to preserve the stochastic aspect of real roughness. Namely, we control (1) the Hurst exponent $H$, which determines the power-law decay (at least for short wavelengths $\lambda$) of spectral energy $\Phi \sim \lambda^{2(1+H)}$, and also (2) the lower and (3) the upper cut-off wavelengths $\lambda_s$ and $\lambda_l$, respectively, where lower indices $s$ and $l$ stand for “short” and “long”. At the same time the spectrum can be made bi-fractal: a power-law increase with the wavelength in the interval $\lambda\in[\lambda_s,\lambda_l]$ can be followed by a plateau for longer wavelengths $\lambda\in(\lambda_l,L]$, where $L$ is the sample length.

Based on this definition, we can readily conclude that for loads implying the value of the macroscopic contact radius ${a_{\mbox{\tiny macro}}}$ which is comparable with the shortest roughness wavelength ${a_{\mbox{\tiny macro}}}\sim\lambda_s$, the contact will 