Why is the adiabatic coefficient 3 for longitudinal plasma waves and ( D + 2)/ D for sound waves in neutral gases?

The above question may be answered as follows. A typical longitudinal plasma wave for which the collision frequency, nu ν ν , is low relative to the angular wave frequency, omega ω ω , only interacts with the degree of freedom, upper D D D , related to motion parallel to the wave vector. This gives an adiabatic coefficient corresponding to upper D equals 1 D = 1 $D = 1$ , i.e. 3. At high nu divided by omega ν / ω ν /ω , relevant to neutral gases, collisions cause even longitudinal sound waves to interact with all active upper D D D , yielding an adiabatic coefficient of left parenthesis upper D plus 2 right parenthesis divided by upper D ( D + 2 ) / D $(D+2)/D$ . We present a minimal example illustrating the above transition based on linear analysis of a non-relativistic, isotropic, homogeneous, Maxwellian one-component system with a Bhatnagar–Gross–Krook collision operator. Macroscopic forces, mainly included in plasma physics, are essential for the transition at low nu divided by omega ν / ω ν /ω . Additionally, the tensor nature of pressure and collision operators satisfying mass, momentum and energy conservation must be invoked to obtain the correct response. Our analysis yields a polytropic index at arbitrary