Description

A variety of processes in physical chemistry or biology have irreversible dynamics of the following type: in a bounded region, particles of a finite size arrive at random and are adsorbed (stick to the surface) provided that they do not overlap with existing adsorbed particles. The process continues until there is no space remaining to accept particles, in which case we say that saturation or jamming has been achieved. While the dynamics are also of interest, mathematical work has focused on the probabilistic properties of the jamming configuration, such as the number of accepted particles and the statistics of the remaining vacant space. This process is called random sequential adsorption, random sequential packing, random parking, sequential inhibition, or irreversible monomer filling, among other names. The particles are called adatoms, cars, or monomers, depending on the context, and the region is a surface or substrate.

Some of the real-world processes that these models have been used to mimic include:

• Adsorption of proteins on surfaces, in which particles cannot stick on top of each other and so monolayers are formed, and in the regime where rates of desorption (unsticking) and surface diffusion are very slow compared to the rate of deposition (such as at very low temperature).
• Droplet adsorption at a surface in colloid science.
• More generally, processes of coagulation, condensation, or nucleation, where new events are inhibited by previous events. Examples include forest growth, where existing trees prevent new trees from growing nearby, or nest locations of gulls.
• Liquid structure in the "hard sphere liquid".

A scene of random ballistic deposition of cars from The Final Programme (1973).

The mathematical models that have been the subject of most analysis are lattice or continuum models in one-dimension, with an interval substrate and particles that are line segments of a fixed length and which arrive uniformly at random. In the continuum, Rényi's model has unit-length cars arriving at random on an interval, and parking if they do not overlap with cars already parked. The Flory–Page model is similar, but discrete (based on a one-dimensional lattice). One random quantity of interest is the fraction of occupied space (or the number of cars parked) once jamming is reached, and the behaviour of this as the length of the interval tends to infinity. For example, for his model Rényi proved that the fraction of occupied space converges to a constant around 0.747. Other problems of interest include the statistics of the gaps left between cars.

The project will involve investigating aspects of random parking, including analytical results in one dimension, analytical and/or simulation studies in higher dimensions, and potential applications.

Recommended prior knowledge

At least one of Markov Chains II or Probability II is necessary to understand the foundations of the project.

Stochastic Processes III is strongly recommended.

Students taking one or more of the 4H probability courses (Advanced Probability IV or Stochastic Analysis IV) may find them helpful, but they are not essential for the project.

For simulations, enthusiasm for R, python, or another suitable language will be necessary.

Resources

For some background on what may be involved, you should:

• consult resources on the web by searching for some of the key words in the above description;
• look at some of the recommended literature (or other literature you find) to see which look most interesting and/or helpful;
• think about how you might efficiently simulate random parking (a) in one dimension, and (b) in two dimensions.

Reading list. The references indicated with a * can be found through the library, either as books or e-journals. The others can be obtained from me at a later date! Some of the main mathematical results are in the following papers:

• A. Dvoretzky and H. Robbins, On the "parking" problem. Publ. Math. Inst. Hung. Acad. Sci. 9 (1964) 209–225.
• *Y. Itoh, On the minimum gaps generated by one-dimensional random packing. J. Appl. Probab. 17 (1980) 134–144.
• *A.G. Konheim and L. Flatto, Solution to problem 60-11. SIAM Review 4 (1962) 257–258.
• *E.S. Page, The distribution of vacancies on a line. J. Roy. Statist. Soc. Ser. B 21 (1959) 36–374.
• *M.D. Penrose, Random parking, sequential adsorption, and the jamming limit. Commun. Math. Phys. 218 (2001) 153–176.
• A. Rényi, On a one-dimensional problem concerning random space-filling. Publ. Math. Inst. Hung. Acad. Sci. 3 (1958) 109–127 (Hungarian). English translation in Selected Trans. Math. Stat. Prob. 4 (1963) 203–218.

• *J.D. Bernal, A geometrical approach to the structure of liquids. Nature 183 (1959) 141–147.
• *J.W. Evans, Random and cooperative sequential adsorption. Rev. Mod. Phys. 65 (1993) 1281–1330.
• J. Feder, Random sequential adsoption. J. Theor. Biol. 87 (1980) 237–254.
• *P.J. Flory, Intramolecular reaction between neighboring substituents of vinyl polymers. J. Amer. Chem. Soc. 61 (1939) 1518–1521.
• *P.L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics. Cambridge University Press, Cambridge, 2010.
• F. MacRitchie, Proteins at interfaces. Adv. Protein Chem. 32 (1978) 283–326.
• *M. Rabe, D. Verdes, and S. Seeger, Understanding protein adsorption phenomena at solid surfaces. Adv. Colloid Interface Sci. 162 (2011) 87–106.
• H. Solomon and H.J. Weiner, A review of the packing problem. Comm. Statist. Th. Meth. 15 (1986) 2571–2607.
• *S. Torquato, Perspective: Basic understanding of condensed phases of matter via packing models. J. Chem. Phys. 149 (2018) 020901.

Get in touch if you would be interested in doing some simulations and/or have any questions!