This award supports theoretical research and education to apply non-equilibrium statistical physics to fundamental natural processes, including interfaces in coarsening magnetic systems, stochastically-driven search algorithms, diffusion of Brownian particles in complex media, and evolutionary biology.
The geometry of coarsening interfaces will be studied in the simplest geometrical setting of a single infinite right angle corner that separates two broken-symmetry ordered phases. In two dimensions, the smoothing of this corner interface can be exactly solved by constructing an isomorphism between the corner interface and an exclusion process. One of the goals of this work is to extend this approach to the challenging case of the three-dimensional corner by developing a mapping between the interface evolution and a 3-species exclusion process on the hexagonal lattice and then solving the latter model.
The PI will construct optimized strategies for a searcher that seeks a target by detecting tracer particles that are steadily emitted by the target. The searcher adjusts its trajectory when tracers are detected so as to move closer to the source. A range of strategies will be studied in order to find the optimal search process as a function of fundamental parameters, such as the nature of the tracer motion, the tracer lifetime and its concentration. Another research project is to apply first-passage propagation as an efficient way to simulate diffusion in porous media and, as a byproduct, provide a novel way to reconstruct microscopic Brownian particle trajectories in a porous medium from first-passage information at widely separated time intervals. Such information would provide a non-invasive way to determine the micro-structure of porous media in a non-invasive manner.
A complementary research effort will be focused on determining the number of species of a given mass in related animal groups. From basic features of fossil data, a minimalist diffusion-reaction model can be distilled that accounts for the observed broad distribution of the number of species of a given mass. The time dependence of this distribution will now be studied, both to describe the species abundance distribution in different epochs, and to understand how punctuated evolution manifests itself in the development of species abundance distributions.
The research will contribute substantially to graduate education by providing the basis for the Ph.D. thesis research of several graduate students. Some of the research projects will be performed in collaboration with colleagues at the Santa Fe Institute (SFI) and at the newly-relocated Schlumberger research lab in Cambridge, Massachusetts. The project on species abundance has brought a more quantitative dimension to current SFI efforts to understand macro-scale phenomena within systems biology. The projects on first-passage propagation and trajectory reconstruction could provide Schlumberger scientists with new theoretical tools to help predict the physical properties of oil-bearing rocks.
NONTECHNICAL SUMMARY This award supports theoretical research and education to advance statistical physics to encompass physical and material systems that are far from equilibrium. Many important processes ranging from materials growth to the physical processes that support life are nonquilibrium systems that lack the intrinsic balance of (thermodynamic) forces that characterize equilibrium systems, a pillar in the conceptual foundations of statistical physics. Statistical physics has a broad reach and the research encompasses several problems, among them is to better understand of the process of coarsening in which interfaces that separate different phases in a material shrink and eventually disappear. This process is ubiquitous in materials science. The PI will also study diffusion through porous media with an aim to develop insights that will enable one to deduce the nature of the porous material without damaging the material. Finally, the PI will also apply the methods of nonequilibrium statistical mechanics to problems at the interface of the physical sciences with biology.
Inspired in large part by fundamental problems in materials science, this research contributes to the further development of the powerful framework of statistical physics enabling insights into a wide range of problems that extend beyond materials science.
The research will contribute substantially to graduate education by providing the basis for the Ph.D. thesis research of several graduate students. Some of the research projects will be performed in collaboration with colleagues at the Santa Fe Institute (SFI) and at the newly-relocated Schlumberger research lab in Cambridge, Massachusetts. The project on species abundance has brought a more quantitative dimension to current SFI efforts to understand macro-scale phenomena within systems biology. The projects on first-passage propagation and trajectory reconstruction could provide Schlumberger scientists with new theoretical tools to help predict the physical properties of oil-bearing rocks.
", 08/01/09--07/31/12) investigated kinetic spin systems, cooperative exclusion processes, biologically-inspired physical phenomena, and social dynamics. A major outcome of this project was the elucidation of new phenomenology in kinetic spin systems. According to conventionally-accepted understanding, a ferromagnet that is suddenly cooled (quenched) from above the critical temperature, where the magnetization is zero, to very low temperatures, undergoes coarsening. Coarsening is the process whereby the initial zero-magnetization state, which consists randomly-oriented atomic spin, evolves into a mosaic of growing single-phase domains after the quench (see the accompanying figure). The process is often investigated within the kinetic Ising model, in which each atom in a ferromagnet contains an elemental spin that can either point up or down. Nearest-neighbor interactions between spins favor their alignment so that ferromagnetism can arise. In two dimensions, we have shown that the coarsening patterns are equivalent to continuum percolation. That is, the black and white domains---which correspond to areas where all the spins either all point up or all point down---can be viewed as landmass and ocean in a random landscape. In this landscape, the elevation is a random function of position with average elevation equal to zero. The level of the oceans in this landscape corresponds to zero height, so that exactly half the area corresponds to land and half to seas. This equivalence allows us to quantify the topology of the coarsening patterns. Specifically, we determined the probabilities for a cluster to wind m times horizontally and n times vertically; the upper panel set in the first figure shows a coarsening pattern that ultimately winds around the square one horizontally and once vertically, while the lower panel set shows a coarsening pattern that ultimately winds around the square twice horizontally and once vertically. In three dimensions, the coarsening pattern is remarkably complex and the ground state is never reached in large systems. Instead, an initially disordered system that is quenched to zero temperature always freezes into a mixed-phase state of two strongly-interpenetrating domains, one that consists of spins that all point up and the other of spins that all point down. This complex topology also arises in micellar systems (mixtures of oil, water, and surfactant) and is known as a gyroid phase or, more picturesquely, as the "plumber's nightmare". A typical example of such a system is depicted in the accompanying figure. Here each cube represents a spin that point up, with the spin at the cube center, while the empty space represents the spins that point down. The resulting domain topology has many holes (see the second figure) and and is a discrete analog of the gyroid phase in micellar systems. An intriguing feature of the long-time state is that it contains "blinker" spins (highlighted in the figure). These lie on the domain interfaces and can flip freely. This behavior arises because a blinker spin is alway surrounded by equal numbers of up and down spins. Such spin-flip events therefore can occurs with no cost in energy. Once such a blinker spin flips, neighboring interface spins are now in a zero-energy environment and can flip freely, leading to additional zero-energy spin flips, etc. Thus the spin system wanders within a subset of states that all have the same energy and leads to ultra-slow evolution. These two features of a complex domain topology and anomalously slow relaxation show that the accepted picture of power-law coarsening does not apply to the kinetic Ising model in three dimensions. The broader impacts of our research effort include the training of three graduate students, one of whom completed his Ph.D. in 2012 and, more significantly, the completion of a graduate text in non-equilibrium statistical physics, "A Kinetic View of Statistical Physics" (Cambridge University Press, 2010), co-authored by Paul Krapivsky and Eli Ben-Naim. This graduate text emphasizes analytical problem-solving methods in a self-contained manner for a broad and paradigmatic set of non-equilibrium phenomena. Topics covered include diffusion, collision phenomena, symmetric and asymmetric exclusion, aggregation and fragmentation kinetics, adsorption phenomena, the dynamics of spin systems, theories of coarsening, kinetics of disorder and hysteresis, reaction kinetics, and the properties of complex networks.
