Consider a bath of Brownian particles at temperature $T$. If we sprinkle some larger particles in this (eg: pollen grains in water or dust motes in air), they'll diffuse with diffusion constant $D$ due to bombardments by the Brownian particles. For the same bombardments, any acceleration of these bigger particles due to an external force will die down to a terminal velocity $v_t=F/\gamma$, where $\gamma$ is a damping coefficient. The relation between their fluctuation and dissipation is given by a fluctuation-dissipation equation:
$\gamma D=k_BT$ (Einstein-Smoluchowski relation)
I now have a basic question regarding the behaviour of the individual terms on the left side. Suppose I were to slowly change just the temperature of the bath. That would change the product $\gamma D$. But how would $\gamma$ and $D$ separately change?
Drawing an analogy from the ideal gas state equation $PV = k_BT$, their individual behaviour might depend on the particular process in which I change $T$. So assume that my system (say a bath of water with pollen grains) remains at atmospheric pressure and at the same volume as I just notch up the temperature of the heat bath. How would $\gamma$ and $D$ change then?