The Heat Capacity of Random Tournaments
A study in nonequilibrium statistical mechanics
S.P.C. Zegers (TU Delft - Applied Sciences)
M.S. Bauer – Mentor (TU Delft - Applied Sciences)
J. Komjáthy – Mentor (TU Delft - Electrical Engineering, Mathematics and Computer Science)
A.R. Akhmerov – Graduation committee member (TU Delft - Applied Sciences)
A.B.T. Barbaro – Graduation committee member (TU Delft - Electrical Engineering, Mathematics and Computer Science)
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Abstract
This thesis characterises the behaviour of the nonequilibrium heat capacity C_N(β) on random tournament graphs T_N without an energy landscape for large N. In this context, the heat capacity C_N(β) is completely determined by the geometry of the underlying network and is derived from the quasipotential. The quasipotential is obtained by solving a discrete Poisson equation involving the random generator L_N of a non-reversible Markov chain over the tournament.
Numerical results suggest a self-averaging property, leading to our main conjecture:
N · C_N(β) = 4C₂(β) (1 + N⁻¹/² · ξ_C)
where C₂(β) is the heat capacity of a fundamental two-level system and the random variable ξ_C → N(0, τ_C) in distribution.
To prove the conjecture, we provide three main contributions:
1. We derive a coarse-grained generator Ḡ_N in score-space that represents the system in an idealised situation where the score bins B_s (which count the number of vertices with a certain score) attain their expected value. We solve its corresponding Poisson problem and prove that it captures the ensemble-averaged 4C₂(β) limit.
2. We shift focus to the statistics of the score bins, which are globally dependent random variables. Using a change of measure, we establish a concentration result and a local Central Limit Theorem (CLT) for the sizes of the score bins N_s.
3. We introduce an auxiliary generator G_N that bridges the vertex and score spaces, linking Ḡ_N and L_N. We demonstrate that G_N satisfies the conjectured Gaussian scaling by deriving the pseudoinverse Ḡ_N† and solving the Poisson problem for G_N perturbatively.
We conclude by providing a viable route to proving that C_N(β) is close to the heat capacity of G_N.