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A Simple Molecular Machine in a Thermal Bath

We will assume that the energies in the independent ``pins'' of a molecular machine form a Boltzmann distribution (Assumption 3) so each ``pin'' acts like a simple harmonic oscillator in a thermal bath and

 

D = 2 d_space (17)

numbers are required to describe the machine velocities because each ``pin'' has a phase and an amplitude (i.e. two Fourier components x and y). At a given moment the energy of the j<sup>th</sup> such ``pin'' component is determined by its velocity and the ``pin's'' mass:

$$E_{j} = {\scriptstyle \frac{1}{2}}m_j v_j^2.$$ (18)

So that we will be able to easily compare ``pins'' with different masses we combine the velocity with the square root of the mass to define a new variable:

$$y_j = \sqrt{E_{j}} .$$ (19)

The assumption that energies of the ``pins'' have a Boltzmann distribution implies that

$$f({y_j}) = {\exp(- \beta {E}_{j})} / z,$$ (20)

where z is the ``partition function'', $z= \int_{-\infty}^{\infty} {\exp(-\beta {E}_{j})}dy_j$[Waldram, 1985]. Dividing by z assures us that the probabilities f(y_j) sum to 1. $\beta = {(k_{\mbox{\scriptsize B}}T)}^{-1}$where $k_{\mbox{\scriptsize B}}$ is Boltzmann's constant and T is the absolute temperature. Comparing (20) to (9), we find $\beta= \frac{1}{2 \sigma^{2}}$ so $\sigma^{2} = {\scriptstyle \frac{1}{2}}k_{\mbox{\scriptsize B}}T$.

Substituting for the energy by using (19) we find that

$$f({y_j}) = {\exp(-\beta {y_j}^{2})} / z.$$ (21)

The form of this equation shows that the set of y_j normalized velocity components have a Gaussian distribution.