# What is an expression/physical law that relates high frequency thermal fluctuations to gas pressure?

When a gas is compressed the 'ideal gas law' can predict what the increase in gas temperature will be. But that's just a mean temperature, right?

At a quantum level the frequency of molecular collisions (Brownian motion) increases to give a rise in temperature and so consider smaller and smaller time scales (higher and higher frequency) the energy and local temperature will fluctuate with larger amplitude about the mean internal energy or temperature.

Is there a mathematical expression (physical law) that relates the size of high frequency thermal fluctuations (or local temperature) to the applied pressure?

If we consider temperature to be due to translational motion of the molecules and we assume the system has reached equilibrium, then the velocity distribution of the molecules is given by the Maxwell distribution:

$$f(v) = \sqrt{\left(\frac{m}{2\pi k T}\right)^3} 4 \pi v^2 \exp\left(\frac{m v^2}{2 k T}\right)$$

which will give you the velocity distribution of the molecules based on the thermodynamic temperature. So as you apply pressure, you can use the ideal gas law to determine the new equilibrium thermodynamic temperature which you can then use to generate the velocity distribution.

To compute some metrics, let's first look at the distribution of speed rather than velocity:

$$\chi(v) = \sqrt{\left(\frac{m}{2\pi k T}\right)^3} 4 \pi v^2 \exp\left(-\frac{m v^2}{2 k T}\right)$$

Now, we can find the most probable speed:

$$v_{mp} = \left(\frac{2 k T}{m}\right)^{1/2}$$

and we can find the RMS speed

$$\left(\overline{v^2}\right)^{1/2} = \left(\frac{3 k T}{m}\right)^{1/2} \approx 1.22 v_{mp}$$

• Can I not just substitute $$\frac{P}{\rho R}$$ as $T$ into this equation for an expression in $P$ ? – docscience Feb 27 '15 at 20:43
• ..and furthermore integrate the expression with respect to $v$ to get maybe some single metric (would that be something like power?) rather than a distribution? – docscience Feb 27 '15 at 20:48
• @docscience Yes, you could substitute in an expression for $T$, that should be fine. However, that expression is a probability distribution function so if you integrated with respect to $v$ you would get the cumulative distribution function which is almost definitely not what you want. I don't know what you're trying to simulate exactly or how this ties in to your governing equations, but you may just want to assign initial velocities to your particles by drawing from the distribution and giving it an random direction vector. – tpg2114 Feb 27 '15 at 21:11
• I'm looking for a scalar metric that provides a rough measure of the size of the thermal 'noise' as a function of the gas pressure. Since the expression above is a PDF, then maybe the FWHM of this function might be a suitable choice? Thanks. – docscience Feb 27 '15 at 21:21
• @docscience I added expressions for the most probable and RMS speeds, that may be what you are looking for. – tpg2114 Feb 27 '15 at 21:28