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title, date, draft, tags
| title | date | draft | tags | |||
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| Research Paper: The Thermodynamic Bias Toward Manifolds in Causal Sets: Mean-Field Prerequisites for Lorentz Invariance (Letter) | 2026-06-01T08:00:00Z | false |
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Abstract: The extraction of the Minkowski metric from discrete causal graphs is complicated by the Kleitman-Rothschild (KR) order collapse. We introduce a thermodynamic partition function governed by the discrete Benincasa-Dowker action augmented with a non-local volume penalty. By evaluating the partition function using a mean-field approximation, we explicitly calculate the critical topological temperature \beta_c and demonstrate a thermodynamic phase transition that strictly suppresses highly entropic non-manifold KR-orders. This establishes a rigorous statistical mechanical prerequisite for the emergence of macroscopic Lorentz invariance.
The Partition Function and Mean-Field Phase Transition
Let \Omega_N be the space of causal sets of N elements. The canonical partition function is:
Z = \sum_{\mathcal{C} \in \Omega_N} e^{-S_{BD}(\mathcal{C}) - \beta V(\mathcal{C})}
where S_{BD} is the Benincasa-Dowker action. The volume penalty V(\mathcal{C}) = \sum_{x \prec y} | \{ z \in \mathcal{C} \mid x \prec z \prec y \} |.
To calculate the phase transition, we employ a mean-field approximation. Let p be the probability of a relation x \prec y. For a generic KR-order, p \approx 1/4, yielding a highly connected graph where the expected volume penalty scales as \langle V_{KR} \rangle \approx c_1 N^3 p^2. For a manifold-like causal set sprinkled into $D$-dimensional Minkowski space, relations are sparse, and \langle V_{man} \rangle \approx c_2 N^2.
The free energy F(\beta) = - \frac{1}{\beta} \ln Z is determined by the competition between the entropy of KR-orders S_{KR} \sim \frac{N^2}{4} \ln 2 and the energy of the volume penalty. Evaluating the saddle point of the mean-field partition function:
Z \approx \int dp \, e^{N^2 \left( \frac{\ln 2}{4} - \beta c_1 N p^2 \right)}
we find a critical inverse temperature \beta_c \propto \frac{\ln 2}{c_1 N}. For \beta \gt \beta_c, the extensive \mathcal{O}(N^3) energetic penalty dominates the \mathcal{O}(N^2) entropy, driving a first-order topological phase transition. The system collapses into the sparse, manifold-like phase (\langle V \rangle \propto N^2), suppressing KR-orders and permitting emergent Lorentz invariance.
References
- [Surya2019] S. Surya, Living Rev. Relativ. 22, 5 (2019).
- [Kleitman1975] D. Kleitman, B. Rothschild, Trans. Am. Math. Soc. 205, 205 (1975).