Negative Heat Capacity in Self-Gravitating Systems

1 Negative Heat Capacity in Self-Gravitating Systems Jae Un Kim Department of Physics in Ajou University Course Title Professor Name October 27, 2025 2 Abstract Negative Heat Capacity in Self-Gravitating Systems The phenomenon of negative heat capacity—where a self-gravitating system becomes hotter as it loses energy—can be traced through a sequence of microscopic processes that connect thermal energy loss, density variation, and collisional dynamics. When external heat is released, the mean kinetic energy (K) decreases and the average particle velocity (v) diminishes. This reduction in velocity weakens the overall dynamic pressure and causes particles to remain within the same spatial region for a longer time. As a result, the local particle density (ρ) increases—not solely due to gravitational contraction but also because of kinematic stagnation, where slower particles accumulate simply by staying longer in one place. As the density rises, the gravitational potential energy (Ug) becomes more negative, indicating a deeper gravitational well and stronger confinement. Simultaneously, the number of particles per unit volume increases, which raises the collision probability (Pcoll) even if particle velocities are reduced. This means that collisions occur more frequently not only because the potential well deepens, but also because there are physically more particles within the same region. The enhanced collision frequency increases the rate of energy redistribution among particles, leading to a partial recovery of internal energy (U). Consequently, the average kinetic temperature (T) rises even though the total energy (E=K+Ug) continues to decrease. The process therefore yields a thermal inversion characteristic of self-gravitating systems, expressed as C=dE/dT <0. The overall sequence proceeds as follows: 1. External heat loss →Decrease in kinetic energy and particle velocity 2. Reduced velocity →Longer residence time →Local density increase 3. Density increase →Gravitational potential deepening and crowding of particles 4. Higher density →Collision probability increases independently of potential 5. More collisions →Enhanced energy redistribution →Internal energy recovery 3 6. Internal energy recovery →Temperature rise despite total energy loss ⇒C<0 This sequential framework clarifies the physical causality behind the temperature rise observed in self-gravitating ensembles. It highlights that both potential deepening and collisional amplification act together to produce the counterintuitive thermal behavior, providing a concrete microscopic basis for the emergence of negative heat capacity. Implications This counterintuitive behavior underlies the thermodynamic instability of self-gravitating systems. It explains why gravitationally bound systems tend to undergo core collapse and halo expansion, a process that eventually prevents them from reaching true thermal equilibrium. In summary, negative heat capacity arises because gravitational potential energy becomes increasingly negative during contraction, converting part of this potential energy into kinetic energy, which raises the system’s temperature even as total energy decreases. 4 Negative Heat Capacity in Self-Gravitating Systems Introductory text goes here. A wild example parenthetical reference appears (Contributor, 2023), followed by an in-text sample citation by Book Author (2021). Note that paragraphs are separated by a blank line in the editor. If you do not do this, it will continue as a single paragraph. This line is technically on a new line in the editor, but will print in the same paragraph. Method Participants Participant information goes here. Materials Words, along with a sample table (Table 1). Table 1 Sample words from this hypothetical experiment. First word Second word Yeet Yoink Hot Lit Measures Some words about the measures used. Design Some more words Procedure Description of the procedure. Results Descriptive Statistics Statistical words. A demonstrative Figure 1. 5 Figure 1 The mean vibes for each word type and counterbalanced condition order. Error bars represent one standard error of the mean. Inferential Statistics Analytical words. Discussion Discussion words. 6 References Book Author, F. P. (2021). Fancy Pants Whole Book. So & Fancy Publishing. Contributor, C. (2023). The Title of the Chapter. In F. Editor & S. Editor (Eds.), The Book Title (pp. 42–69). Other Publishing Place.