Permutation invariant matrix quantum thermodynamics and negative specific heat capacities in large N systems
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2024
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| author | O'Connor, Denjoe Ramgoolam, Sanjaye |
| author_facet | O'Connor, Denjoe Ramgoolam, Sanjaye |
| contents | We study the thermodynamic properties of the simplest gauged permutation invariant matrix quantum mechanical system of oscillators, for general matrix size $N$. In the canonical ensemble, the model has a transition at a temperature $T$ given by $x = e^{ -1/ T } \sim x_c=e^{-1/T_c}=\frac{\log N}{N}$, characterised by a sharp peak in the specific heat capacity (SHC), which separates a high temperature from a low temperature region. The peak grows and the low-temperature region shrinks to zero with increasing $N$. In the micro-canonical ensemble, for finite $N$, there is a low energy phase with negative SHC and a high energy phase with positive SHC. The low-energy phase is dominated by a super-exponential growth of degeneracies as a function of energy which is directly related to the rapid growth in the number of directed graphs, with any number of vertices, as a function of the number of edges. The two ensembles have matching behaviour above the transition temperature. We further provide evidence that these thermodynamic properties hold in systems with $U(N)$ symmetry such as the zero charge sector of the 2-matrix model and in certain tensor models. We discuss the implications of these observations for the negative specific heat capacities in gravity using the AdS/CFT correspondence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_13150 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Permutation invariant matrix quantum thermodynamics and negative specific heat capacities in large N systems O'Connor, Denjoe Ramgoolam, Sanjaye High Energy Physics - Theory Mathematical Physics Combinatorics We study the thermodynamic properties of the simplest gauged permutation invariant matrix quantum mechanical system of oscillators, for general matrix size $N$. In the canonical ensemble, the model has a transition at a temperature $T$ given by $x = e^{ -1/ T } \sim x_c=e^{-1/T_c}=\frac{\log N}{N}$, characterised by a sharp peak in the specific heat capacity (SHC), which separates a high temperature from a low temperature region. The peak grows and the low-temperature region shrinks to zero with increasing $N$. In the micro-canonical ensemble, for finite $N$, there is a low energy phase with negative SHC and a high energy phase with positive SHC. The low-energy phase is dominated by a super-exponential growth of degeneracies as a function of energy which is directly related to the rapid growth in the number of directed graphs, with any number of vertices, as a function of the number of edges. The two ensembles have matching behaviour above the transition temperature. We further provide evidence that these thermodynamic properties hold in systems with $U(N)$ symmetry such as the zero charge sector of the 2-matrix model and in certain tensor models. We discuss the implications of these observations for the negative specific heat capacities in gravity using the AdS/CFT correspondence. |
| title | Permutation invariant matrix quantum thermodynamics and negative specific heat capacities in large N systems |
| topic | High Energy Physics - Theory Mathematical Physics Combinatorics |
| url | https://arxiv.org/abs/2405.13150 |