Why is the adiabatic coefficient 3 for longitudinal plasma waves and ( <i>D</i> + 2)/ <i>D</i> 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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>ν</mml:mi> </mml:math> $\nu$ , is low relative to the angular wave frequency, omega <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>ω</mml:mi> </mml:math> $\omega$ , only interacts with the degree of freedom, upper D <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>D</mml:mi> </mml:math> $D$ , related to motion parallel to the wave vector. This gives an adiabatic coefficient corresponding to upper D equals 1 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:math> $D = 1$ , i.e. 3. At high nu divided by omega <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>ν</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>ω</mml:mi> </mml:math> $\nu /\omega$ , relevant to neutral gases, collisions cause even longitudinal sound waves to interact with all active upper D <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>D</mml:mi> </mml:math> $D$ , yielding an adiabatic coefficient of left parenthesis upper D plus 2 right parenthesis divided by upper D <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>D</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>D</mml:mi> </mml:math> $(D+2)/D$ . 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 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:mi>ν</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>ω</mml:mi> </mml:math> $\nu /\omega$ . 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