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I have been pondering a theoretical question for a while and though I think I have a good idea of what is happening, I would like to see what others think.

One way to define the adiabatic index, $\gamma$ is by the number of degrees of freedom, $f$ we get the following expression:

$$ \gamma = \frac{f+2}{f} $$

And we know that the lower bound on $\gamma$ is 1.

I am interested in the case of a gas that does not conserve energy due to exchanges with the surroundings and what regimes we can consider. So I proceed by asymptotic analysis of the cases when $\gamma \rightarrow 1$ and $\gamma \rightarrow \infty$.

1. $\gamma \rightarrow 1$

In this case, $f \rightarrow \infty$ so the gas is very energetic. Would it be correct to say that here the cooling is very efficient since the large number of degrees of freedom leads to more energy transferred to the surroundings? From my analysis it makes sense that energy conservation is worse the larger $f$.

2. $\gamma \rightarrow \infty$

This should then be the opposite of the other case. The number of degrees of freedom is now very small so I believe that very little energy should be transferred to the surroundings from the gas. The cooling in this regime is inefficient.

Am I correct to picture this as two regimes with a smooth transition to better energy conservation as $f$ decreases or am I missing a concept?

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    $\begingroup$ What exactly do you mean by "dissipation"? When you say "a gas that does not conserve energy" do you mean "a gas that is exchanging energy with its surroundings? $\endgroup$ Jul 13, 2018 at 12:22
  • $\begingroup$ Yes, you understood correctly. I updated the question to be more clear. $\endgroup$
    – fhorrobin
    Jul 13, 2018 at 12:26

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I don't think the picture you have is correct. I am not aware of any particularly direct connection between $f$, the number of quadratic degrees of freedom per particle, and the rate at which heat is transferred out of system. The reason for this is that for heat dissipation what matters is not so much the number of degrees of freedom but how strongly those degrees of freedom interact with the outside world, to which the answer may be "essentially not at all". For example in an Einstein-solid $f = 6$. In this model of a solid each atom has 3 kinetic degrees of freedom and 3 position degrees of freedom (which, unlike in a gas, contribute here due to the forces between neighbouring atoms). However this can be directly related to the rate of heat loss from the solid as most of these degrees of freedom are buried deep within the solid and have no direct link with the outside world.

If $f$ was directly linked to the rate of heat loss then, if we increased the size of our system while keeping the density constant, the rate of heat loss would go like $\sim Nf \sim Vf$, but in reality it is normally proportional to surface area rather than volume.

What is true is that $f$ is related to the heat capacity via the equipartition theorem, so increasing $f$ will increase the amount of heat that the system needs to loose in order to reduce the temperature a given amount.

In terms of what the 2 limits you thought about mean, in the limit that $\gamma \rightarrow 1$ then an adiabatic change in the system will look like an isothermal change. We can interpret this as saying that as $f\rightarrow \infty$ the system starts to act as its own reservoir. The heat capacity of the system is proportional to $f$ and so is also diverging, so each individual degree of freedom sees itself as coupled to a thermal reservoir at the temperature of the system.

The opposite limit is a degenerate case. It does not make sense to talk about a system with no degrees of freedom and putting $\gamma$ into the adiabatic equation gives unphysical answers.


A quick footnote about the applicability of this analysis. $f$ only really have a physical meaning in the context of systems that obey the euipartition theorem, but it is important to note that many systems do not. In particular quantum systems generally do not obey the requirements of the theorem and so it is not really meaningful to define $f$ in these cases, except for in the high temperature limit.

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