Adiabatic Cooling

An adiabatic process is one in which no heat enters or leaves the system.

The thermdynamic first law takes the form

\[ dU=-PdV ,\]

since $dQ=0$.

Note that $C_v=(\partial U/ \partial T)_V$ , but the internal energy of an ideal gas depends only on the temperature and is independent of the volume (because there are no intermolecular forces). Thus for an ideal gas, we have

\[ dU=C_v dT = PdV \]

\[ PV= RT=（C_p-C_v)T  \]

\[ \gamma = C_p/C_v \]

Then we have

\[ TV^{\gamma-1}=const  \]

\[ d\ln(T)/dt = -(\gamma-1) \nabla \cdot u\]

where we have use 

\[ \frac{1}{V} \frac{dV}{dt} = \nabla \cdot  u \]

where $u$ is the flow speed.

Since $K_B T \propto E$, here $E$ is thermal energy. We have 

\[ d\ln(E)/dt = -(\gamma-1) \nabla \cdot u \]

For non-relativistic particle, $\gamma=5/3$, $E=p^2/m$.

For relativistic particle, $\gamma=4/3$, $E=pc$,

we have 

\[ \frac{dp}{dt} = -\frac{p}{3} \nabla \cdot u \]

The bulk of particles lose energy by adiabatic cooling due to the work that protons exert on the expanding material. Particles (pressure ?) do (positive) work on the expanding medium cause their bulk energy reduce.


Reference:

https://github.com/bolverk/theoretical_physics_digest

https://theoretical-physics-digest.fandom.com/wiki/The_passage_of_energetic_charged_particles_through_interplanetary_space