SZ
S.P.C. Zegers
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The Heat Capacity of Random Tournaments
A study in nonequilibrium statistical mechanics
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. ...
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. ...
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.
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.
Bachelor thesis
(2022)
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S.P.C. Zegers, A.B.T. Barbaro, C.R. Kleijn, H.E.A. van den Akker, F.H.J. Redig
In several experiments, enzymes have shown an in increase in diffusivity in the presence of their substrate. The enhancement in diffusivity ranged from as low as 28% for urease to 80% in the case of alkaline phosphatase. There are two main competing theories. One asserts that catalytically driven boosts propel the enzyme forward in ‘leaps’, while the other argues that phoretic activity due to attractive or repulsive surface interactions on the enzyme are responsible for the enhanced diffusivity. At the moment of writing, no consensus has been reached on the mode of enhancement. A novel agent-based lattice model for the diffusion of enzymes was derived in this Bachelor’s Thesis in accordance to the phoretic theory of enhanced diffusion. The model that was developed is a stochastic model which describes the interactions at the particle level. The model showed that phoresis produces enhanced diffusion in the order that was expected for the enzyme urease. Furthermore, the model showed pattern formation in the case of repulsive interactions between enzyme and substrate. The instability of this phase transition was investigated using linear stability analysis on the continuum limit of the lattice model. This yielded a restriction on the strength of the interactions for which pattern formation could occur.
...
In several experiments, enzymes have shown an in increase in diffusivity in the presence of their substrate. The enhancement in diffusivity ranged from as low as 28% for urease to 80% in the case of alkaline phosphatase. There are two main competing theories. One asserts that catalytically driven boosts propel the enzyme forward in ‘leaps’, while the other argues that phoretic activity due to attractive or repulsive surface interactions on the enzyme are responsible for the enhanced diffusivity. At the moment of writing, no consensus has been reached on the mode of enhancement. A novel agent-based lattice model for the diffusion of enzymes was derived in this Bachelor’s Thesis in accordance to the phoretic theory of enhanced diffusion. The model that was developed is a stochastic model which describes the interactions at the particle level. The model showed that phoresis produces enhanced diffusion in the order that was expected for the enzyme urease. Furthermore, the model showed pattern formation in the case of repulsive interactions between enzyme and substrate. The instability of this phase transition was investigated using linear stability analysis on the continuum limit of the lattice model. This yielded a restriction on the strength of the interactions for which pattern formation could occur.