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A Simple Harmonic Oscillator in a Thermal Bath

If a simple harmonic oscillator is immersed in a thermal bath, then impacts with neighboring atoms change the phase and energy in an irregular way. Equipartition of energy between the oscillator and the bath implies that each independent Fourier component of the velocity in (5) has a Boltzmann distribution [Waldram, 1985]:

$$f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-E_x/2\sigma^{2}}$$ (9)

and

$$f(y) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-E_y/2\sigma^{2}} ,$$ (10)

where

$$E_x = {\scriptstyle \frac{1}{2}}m x^{2} \;\;\;\;\;\mbox{and}\;\;\;\;\; E_y = {\scriptstyle \frac{1}{2}}m y^{2} .$$ (11)

The meaning of $\sigma$ will be discussed below. We use the Boltzmann distribution to introduce thermal noise into our Newtonian description of an oscillator. Substituting from (11) into (9) and (10) gives:

$$f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-m x^{2}/4\sigma^{2}}$$ (12)

and

$$f(y) = \frac{1}{\sigma \sqrt{2 \pi}} e^{-m y^{2}/4\sigma^{2}} ,$$ (13)

so the velocities x and yhave a normal or Gaussian distribution with a standard deviation proportional to $\sigma$. Since the oscillator is surrounded by a huge thermal bath and impacts from the bath are not predictable, the changes in motion of the oscillator are probabilistic. Maxwell's classical model for the velocity distribution of molecules in an ideal gas also uses a Gaussian velocity distribution [Wannier, 1966,Castellan, 1971,Waldram, 1985]. The normal distribution is graphed as the D=1 curve in Fig. 4.

  

Figure 4: High Dimensional Sphere Density.

The sphere probability density as a function of radius, f_D(r), is drawn for $D = 1, 2, 4, 8, \ldots, 1024$ dimensions (see Appendix 22). Except for the Gaussian curve (D = 1), which passes through the point (0,1), the curves are ``normalized'' so that their peaks pass through (1,1). At higher dimensions the curves approach the Gaussian distribution again and peak sharply. The dashed line is at e<sup>-1/2</sup>, which intercepts ``normalized'' Gaussian distributions at one standard deviation from the mean.

What is the probability f(x,y)that the oscillator will have the velocity components x and y? Since x and y are independent, we may write the probability density as

$$f(x,y) = f(x) f(y) = \frac{1}{\sigma^2 2 \pi} e^{-m (x^2 + y^... ...igma^{2}} = \frac{1}{\sigma^2 2 \pi} e^{-m r^2 /4\sigma^{2}} ,$$ (14)

where $r = \sqrt{x^2 + y^2}$ is the distance in velocity space from the origin to the point (x,y), as in Fig. 3. The probability of finding that the oscillator has velocities in a small region dx dy is f(x,y) dx dy. Since $dx dy = r dr d\phi$ [Thomas, 1968] we can convert to polar coordinates:

$$f(x,y) dx dy = \frac{1}{\sigma^2 2 \pi} r e^{-m r^2 /4\sigma^{2}} dr d\phi.$$ (15)

The total density at the radius r in an interval dr is therefore

$$f_{2}(r) dr = \int_0^{2 \pi} \frac{1}{\sigma^2 2 \pi} r e^{-... ... dr d\phi = \frac{1}{\sigma^2 } r e^{-m r^2 /4\sigma^{2}} dr .$$ (16)

The subscript ``2'' in ``f₂(r)'' indicates that two Gaussian distributions were used to obtain the density distribution. This ``Rayleigh'' distribution is graphed as the D=2 curve in Fig. 4 and shown as a smooth grey scale in Fig. 5. Notice that the distribution is radially symmetric and that the density in a thin ring around the origin approaches zero at the origin since r = 0 there.

  

Figure 5: The Rayleigh Distribution

The continuous grey-tone distribution represents the analytic probability density, f₂(r). Each small open circle ($\bigcirc$) represents the coordinates of two normally distributed values with mean 0 and standard deviation 1. Each normally distributed value was the sum of 100 pseudo-random numbers with a flat distribution.

We found in the previous section that when an oscillator is in a vacuum the total energy is constant so that the radius r is constant and the set of all possible states with energy r²is represented by a circle. In a heat bath the oscillator can exchange energy with the surrounding medium and the distribution is more spread out, according to the Rayleigh distribution. This ``open'' description of a simple harmonic oscillator allows for energy and phase changes.