Statistical Thermodynamics of Clustered Populations

We present a thermodynamic theory for a generic population of $M$ individuals distributed into $N$ groups (clusters). We construct the ensemble of all distributions with fixed $M$ and $N$, introduce a selection functional that embodies the physics that governs the population, and obtain the distribution that emerges in the scaling limit as the most probable among all distributions consistent with the given physics. We develop the thermodynamics of the ensemble and establish a rigorous mapping to thermodynamics. We treat the emergence of a so-called "giant component" as a formal phase transition and show that the criteria for its emergence are entirely analogous to the equilibrium conditions in molecular systems. We demonstrate the theory by an analytic model and confirm the predictions by Monte Carlo simulation.Comment: Minor edits to tex

Quantum Probability and the Born Ensemble

We formulate a discrete two-state stochastic model with elementary rules that give rise to Born statistics and reproduce the probabilities of the Schrodinger equation under an associated Hamiltonian matrix, which we identify. We define the probability to observe a state, classical or quantum, in proportion to the number of events at that state--number of ways the walker may materialize at a point of observation at time t, starting from known initial state at t=0. The quantum stochastic process differs from its classical counterpart in that the quantum walker is a pair of qubits, each transmitted independently through all possible paths to a point of observation, and whose recombination may produce a positive or negative event. We represent the state of the walker via a square matrix of recombination events, interpret the indeterminacy of the qubit state as rotations of this matrix, and show that the Born rule counts the number elements on this matrix that remain invariant over a full rotation.Comment: 13 pages including appendi

Monomer-addition growth with a slow initiation step: A growth model for silica particles from alkoxides

A simplified monomer-addition model with a first-order activation step is developed to describe the dynamics of growth of silica particles from alkoxides. In the limit of slow hydrolysis, we obtain expressions for the evolution of the particle mass and particle polydispersity, as well as an expression for the particle size as a function of the hydrolysis rate constant, the polymerization rate constant, and the initial concentration of the orthosilicate. We find that the formation of the particles is adequately modeled by a reaction limited growth.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27723/1/0000113.pd

Dynamics of growth of silica particles from ammonia-catalyzed hydrolysis of tetra-ethyl-orthosilicate

The NH3-catalyzed formation of colloidal silica particles from tetra-ethyl-orthosilicate (TEOS) in methanol and ethanol is studied by means of light scattering and Raman spectroscopy. We find that the growth is characterized by an incubation period after which no significant nucleation takes place. The particles have uniform, non-fractal structure and show low polydispersity. In the presence of excess water, the rate-limiting step is the hydrolysis, which is a first-order process in the orthosilicate concentration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/27243/1/0000250.pd