Some current and less-current projects

Want to see this page translated into Ukranian? Click the Ukrainian translation by Domri team (which seems to be an interior design company; go figure!).

Here is a Slovenian translation of some of the material below , and here is some more translated material on the recombination reaction. These translations were created by Gasper Halipovich

A Kinetic View of Statistical Physics

This graduate level text, written in collaboration with Paul Krapivsky and Eli Ben-Naim was published by Cambridge University Press in December 2011.

Table of Contents and Preface. Current List of Errata.

A Guide to First-Passage Processes

This book was published by Cambridge University Press in 2001. The cover illustrates first-passage in a finite interval where death awaits at one boundary and a restaurant date awaits at the other. Table of contents and preface. Current list of errata. Short list of exercises.

Reviews of the book:
by Alan Bray in Journal of Statistical Physics, February 2002.
by Robert Dorfman in American Journal of Physics, November 2002.
by Paolo Laureti, in Econophysics Forum, November 2002.

Collisional Impact

            A single moving particle collides with one of the stationary particles in an infinite zero-temperature gas of elastic particles at rest. The result (left) is a collision cascade that propagates spherically outward. The number of moving particles (red) at time t and the number collisions up to time t grow as tξ and tη, with ξ=2d/(d+2), η=2(d+1)/(d+2), and d the spatial dimension. These growth laws are the same as those from a hydrodynamic theory for the shock wave emanating from an explosion. For a particle incident on a static gas that fills the half-space x>0 (right), the resulting backsplatter, or "break shot" on an infinite pool table, ultimately contains almost all the initial energy. For details, see Exciting Hard Spheres, T. Antal, P. L. Krapivsky, and S. Redner, Phys. Rev. E 78, 030301 (2008).

Coarsening in Ising ferromagnets

What happens when an Ising ferromagnet, with spins endowed with Glauber spin-flip dynamics, is suddenly cooled from high temperature to zero temperature? Surprisingly, this simple system gets "stuck" in one of the relatively large number of metastable states. In two dimensions, getting stuck occurs approximately 1/3 of the time, while in three dimensions, the ground state is never reached. Instead, the systems gets stuck in a complex "plumber's nightmare state" that is illustrated below:

A typical metastable of the three-dimensional Ising model on a finite cube with periodic boundary conditions in all directions. One phase of the spins are represented as blue unit-size cubic blocks. This spin cluster has a sponge-like topology so that there are no convex corners. The red blocks denote "blinker" spins which can flip indefinitely with no energy cost. For more information, see Freezing in Ising Ferromagnets, V. Spirin, P. L. Krapivsky, and S. Redner, Phys. Rev. E 65, 016119 (2001) Zero-Temperature Freezing in Three-Dimensional Kinetic Ising Model, J. Olejarz, P. L. Krapivsky, and S. Redner, Phys. Rev. E 83, 030104(R) (2011), and Zero-Temperature Relaxation of Three-Dimensional Ising Ferromagnets, J. Olejarz, P. L. Krapivsky, and S. Redner, Phys. Rev. E 83, 051104 (2011).


This interface is the result of depositing unit cubes inside a corner that represents the positive octant of 3-space. The resulting growing interface approaches a deterministic limiting shape in the long-time limit. The figure on the left shows the result of a simulation of the interface at t=140, and the right shows our theoretical prediction for the limiting shape that is based on on the generalizing the corresponding interface in two dimensions to three dimensions by accounting for the full symmetries of the system. For more information, see Growth Inside a Corner: The Limiting Interface Shape, J. Olejarz, P. L. Krapivsky, S. Redner, and K. Mallick, Phys. Rev. Lett. 108, 016102 (2012).

Coarsening of the kinetic Ising model on a 1024x1024 square lattice with periodic boundary conditions at times 200, 1000, 5000, and 25000 following a quench from infinite temperature to zero temperature. A spanning domain, which eventually coarsens into a vertical stripe, is highlighted white. In Freezing into Stripe States in Two-Dimensional Ferromagnets and Crossing Probabilities in Critical Percolation, K. Barros, P. L. Krapvisky, and S. Redner, Phys. Rev. E 80, 040401 (2009), we predict the probability to reach the stripe state to be 0.3388 by making a connection to exactly-known crossing probabilities in percolation.

Long-lived "diagonal stripe" configuration on the square lattice with periodic boundary conditions. This configuration corresponds to a stripe that winds once around the torus equatorially and once toroidally, and arises in roughly 5% of all configurations. This diagonal stripe eventually and always reaches the ground state in a time that scales as L3.5. For details see Fate of Zero-Temperature Ising Ferromagnets, V. Spirin, P. L. Krapivsky, and S. Redner, Phys. Rev. E 63, 036118 (2001); and Freezing in Ising Ferromagnets, V. Spirin, P. L. Krapivsky, and S. Redner, Phys. Rev. E; 65, 016119 (2001).

A typical metastable state of the homogeneous ferromagnetic Ising model with Glauber dynamics on the 3-coordinated Cayley tree. At zero temperature these metastable states are actually stable. Shown are the first 4 levels of the tree. Spins in squares have their state uniquely determined by the states of the two "daughter" spins (black for + spins, red for - spins). Spins in circles are determined by the spin state of their parent (blue for + spins, magenta for - spins). For details, see Freezing in Ising Ferromagnets, V. Spirin, P. L. Krapivsky, and S. Redner, Phys. Rev. E; 65, 016119 (2001).

Dynamics of Social Balance

How can one characterize social networks with both friendly and unfriendly relations? A crucial concept is that of social balance, where each relationship triad contains an even number of friendly links. Such a triad fulfils the adage:

How do social networks evolve when both friendly and unfriendly relations exist? A familiar example is attempting to remain friendly with a married couple that gets divorced. It can be difficult to remain friendly with both former spouses if they dislike each other and the simplest solution may be to remain friends with only one of the former spouses. That is, change the sense of a link to eliminate imbalanced triads --- those that contain 1 or 3 unfriendly links.

A more compelling example is the evolution of relations among the protagonists of World War I. These relationship changes gradually led to a reorganization of alliances between European nations into a socially balanced state. While social balance is a natural outcome, it is not necessarily a good one!

For details, see Dynamics of Social Balance on Networks, T. Antal, P. L. Krapivsky, and S. Redner, Phys. Rev. E 72, 036121 (2005), and Social Balance on Networks: The Dynamics of Friendship and Enmity, T. Antal, P. L. Krapivsky, and S. Redner, Phyisca D 224, 130 (2006); (physics/0605183).

Spatial pattern formation in A+B--->0

Cooling of inelastic gases

Space-time snapshot of the particles.

Anomalous decay in asynchronous ballistic annihilation

Space-time diagram of A+A--->0 in one dimension when each particle equiprobably has a velocity +V or -V, as well as a superimposed diffusion with diffusion coefficient D.

Space-time snapshot of the particles.

"Golden" random walk

This is a symmetric random walk in one dimension in which the length of the nth step equal to g-n, where g=(1+sqrt(5))/2 is the golden ratio. The probability distribution of this process is shown below.

Probability distribution at resolution 64 bins. Probability distribution at resolution 256 bins. Probability distribution at resolution 1024 bins.

Ballistic Deposition at Glancing Angles

Snapshots of the evolution of a growing surface when particles are ballistically incident at angles of +7.125 or -7.125 degrees (arctan(1/8)) from the horizontal and stick irreversibly to the deposit upon first contact.

Snapshot after 2 million particles. Snapshot after 10 million particles. Snapshot after 50 million particles.

Distribution of Citations

Distribution of City Populations

Baseball Data

Sidney Redner <>
Last modified: Thu Nov 6 12:08:14 EST 2003