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Thermal Noise in Y space

An oscillator in equilibrium with a thermal bath has an average energy of $k_{\mbox{\scriptsize B}}T$ joules [Waldram, 1985]. Since a molecular machine has d_space ``pins'', each of which is assumed to be equivalent to an oscillator, the total energy is

$$N_y = k_{\mbox{\scriptsize B}}T \times d_{space} \;\;\;\;\;\;\mbox{(joules)} .$$ (30)

This expression also gives the average energy of a molecule with d_spacedegrees of freedom [Castellan, 1971]. From equations (30) and (17) we see that

$$N_y = {\scriptstyle \frac{1}{2}}k_{\mbox{\scriptsize B}}T \times D \;\;\;\;\;\;\mbox{(joules)}$$ (31)

so each of the y_j variables has an average energy of ${\scriptstyle \frac{1}{2}}k_{\mbox{\scriptsize B}}T$. Combining equation (30) with equation (25), we find

$$r_y = \sqrt{k_{\mbox{\scriptsize B}}T d_{space}} .$$ (32)

Thus, thermal noise displaces the configuration of the machine away from the sphere center by an amount related to the absolute temperature. For this reason we may regard the gumballs of Fig. 1 as representing ``thermal noise spheres''.