Let us consider a gas that is heated by an external source, but is also able to cool. We assume that the heating and cooling rates depend only on the density and temperature of the gas, and denote them by $ H \left( \rho , T \right) $ and $ C \left( \rho, T\right) $. The net heat exchange is $ L \left( \rho, T \right) = H \left( \rho, T \right) - C \left( \rho, T\right) $. The gas will be in equilibrium if $ L=0 $. In principle, there may be multiple combinations of densities and temperature for which the gas will be in thermal equilibrium. However, not all equilibria are stable. When gas at an unstable equilibrium is perturbed, it will migrate to another equilibrium state. If the perturbation is not spatially uniform, then different parts of the gas will drift to different equilibrium points. This phenomenon may explain why some ISM sports material in two distinct phases.

The condition for equilibrium can be obtained from linear stability analysis of the hydrodynamic equations

$ \frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x} \left( \rho v \right) = 0 $

$ \frac{\partial v}{\partial t} + v \frac{\partial v}{\partial x} + \frac{1}{\rho} \frac{\partial p}{\partial x} = 0 $

$ \frac{1}{\gamma - 1} \left(\frac{\partial}{\partial t} + v \frac{\partial}{\partial x} \right )+\frac{\gamma}{\gamma-1} \left(\frac{\partial}{\partial t} +v \frac{\partial}{\partial x}\right )+\rho L = 0 $

$ p = \frac{k_b}{\mu} \rho T $


$ v = v_1 \epsilon $

$ T = T_0 + T_1 \epsilon $

$ \rho = \rho_0 + \rho_1 \epsilon $

$ \epsilon = \exp \left[ i \left( k x - \omega t \right) \right] $

and expanding the heat exchange function in Taylor series yields

$ L \left( \rho, T \right) = L \left( \rho_0, T_0 \right) +\left( \frac{\partial L}{\partial T} \right)_{\rho} \left( T - T_0 \right) + \left( \frac{\partial L}{\partial \rho}\right)_T \left( \rho - \rho_0 \right) $

(recalling that in equilibrium $ L \left( \rho_0, T_0 \right) = 0 $) yields a dispersion relation. This dispersion relation admits unstable modes if

$ \left( \frac{\partial L}{\partial T} \right)_{\rho} T_0 + \frac{1}{\gamma -1} \left( \frac{\partial L}{\partial \rho} \right)_T \rho_0 > 0 $