Statistical physics of two-temperature rotational energy distributions in stationary plasmas

Two-temperature rotational energy distributions from rarefied diatomic molecules are very often observed in laboratory plasmas. There has been much debate over the years about the physical meaning of this kind of rotational energy distributions and the associated statistical physics. We show here that under certain reasonable assumptions and constraints the condition of Shannon-Jaynes entropy maximization may produce a two-temperature distribution. This may happen, for instance, when a system is simultaneously coupled to different thermal baths. In plasmas this is possible because rarefied molecular species may be immersed in a medium where electrons and the dominant atomic species are quasidecoupled, each of them acting as distinct thermal baths. Considering that molecular species may interact both with electrons and heavy neutral species, we may ask what should be the resulting molecular energy distribution. We answer this question in this paper and give some examples on how this can be used to interpret experimental molecular distribution from partially ionized plasmas.

Figure 1

System $\mathfrak{S}$ is coupled to thermal baths ${\mathfrak{T}}_1$ and ${\mathfrak{T}}_2$. The total mean energy of system $\mathfrak{S}$ is $U=U_1+U_2$, where $U_1$ and $U_2$ are the partial contributions from thermal baths ${\mathfrak{T}}_1$ and ${\mathfrak{T}}_2$ to the total internal energy $U$.

Figure 2

Plot of the distribution given by Eq. (23), where $T=300,T_e=10000\mathrm{K},{\xi }_0=100.0,E_0=1$, and ${\sigma }_0=0.2\mathrm{eV}$.

Figure 3

Boltzmann plot built from a state-to-state algorithm applied to a high resolution spectra of the 0-0 band of the OH violet system. The fitting based on the two bath theory is represented by the full blue line curve whereas the fitting based on the two-exponential sum is represented by the dashed red curve. (a) Plot on the linear scale. (b) Plot on the logarithmic scale.

Figure 4

Spectra fitting of a high resolution spectra 0-0 band of the OH violet system. Plot (a) fitting based on the two bath theory. Plot (b) fitting based on the two-exponential sum model.

Figure 5

Representation of the energy exchange between system $\mathfrak{S}$ and baths ${\mathfrak{T}}_1$ and ${\mathfrak{T}}_2$. We assume that for a given small time interval, heat amounts $\delta Q_1^{(+)}$ and $\delta Q_2^{(+)}$ are introduced into the system by thermal baths ${\mathfrak{T}}_1$ and ${\mathfrak{T}}_2$, and heat amounts $\delta Q_1^{(-)}$ and $\delta Q_2^{(-)}$ are removed from the system by thermal baths ${\mathfrak{T}}_1$ and ${\mathfrak{T}}_2$.