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?