  In this paper we model the high-energy tail of observed solar energetic
proton energy distributions with a kappa distribution function. We employ a
technique for deriving the thermodynamic parameters of solar energetic proton
populations measured by the Parker Solar Probe (PSP) Integrated Science
Investigation of the Sun (IS$\odot$IS) EPI-Hi high energy telescope (HET), over
energies from 10 - 60 MeV. With this technique we explore, for the first time,
the characteristic thermodynamic properties of the solar energetic protons
associated with an interplanetary coronal mass ejection (ICME) and its driven
shock. We find that (1) the spectral index, or equivalently, the thermodynamic
parameter kappa of solar energetic protons ($\kappa_{\rm EP}$) gradually
increases starting from the pre-ICME region (upstream of the CME-driven shock),
reaching a maximum in the CME ejecta ($\kappa_{\rm EP} \approx 3.5$), followed
by a gradual decrease throughout the trailing portion of the CME; (2) solar
energetic proton temperature and density ($T_{\rm EP}$ and $n_{\rm EP}$) appear
anti-correlated, a behavior consistent to sub-isothermal polytropic processes;
and (3) values of $T_{\rm EP}$ and $\kappa_{\rm EP}$ appear are positively
correlated, indicating an increasing entropy with time. Therefore, these proton
populations are characterized by a complex and evolving thermodynamic behavior,
consisting of multiple sub-isothermal polytropic processes, and a large-scale
trend of increasing temperature, kappa, and entropy. This study and its
companion study by Livadiotis et al. (2024) open a new set of procedures for
investigating the thermodynamic behavior of energetic particles and their
shared thermal properties.
