My main research topic both in graduate school and during my first two postdocs has been study of rigidity of elastic networks using methods of rigidity theory. Elastic networks can be thought of as networks of elastic springs connecting a set of sites. Mathematical rigidity theory deals with certain properties of these networks, such as the number of zero-frequency motions, sets of sites that do not deform in such motions (rigid clusters), stressed regions, etc., that depend only on the network geometry and often just on the topology, whereas the particular values of the spring constants and lengths are unimportant. This latter fact enabled the design of a very fast algorithm for analyzing network rigidity, the pebble game, developed by Don Jacobs, that essentially just does counting of elastic constraints on all length scales simultaneously. This algorithm was used, in particular, to study the phenomenon of rigidity percolation on diluted random elastic networks: as the constraint concentration increases, at some point a percolating rigid cluster emerges. My work in the area has been done in collaboration with M. F. Thorpe, N. Mousseau, J. C. Phillips, D. J. Jacobs, A. J. Rader and M.-A. Brière and has involved several subtopics:

**Rigidity percolation on Bethe lattices** Often, various statistical physics
models can be solved analytically on Bethe lattices, which are networks with
sites connected completely at random, regardless of the distance, and thus have
no finite rings in the thermodynamic limit. This is the case for rigidity
percolation as well. Following previous work in 2D (i.e., for networks with
sites having two degrees of freedom), we have solved the rigidity percolation
problem for 3D networks with both first- and second-neighbour constraints
(bond-bending networks) which are interesting as models of covalent glasses.
Unlike in both usual (connectivity) percolation on Bethe lattices and rigidity
percolation on regular 3D bond-bending lattices, the rigidity transition on
Bethe lattices is *first order*. We have obtained the phase diagram of the
problem as a function of both the bond concentration and the "chemical order"
parameter characterizing the propensity of sites with different number of
neighbours (modeling different valences of atoms in a glass network) to bind
together. See publication 3.

**Self-organization, self-organized criticality and the intermediate phase
in elastic networks**
The simplest percolation models involve studying randomly diluted networks
where one starts from a regular lattice and removes bonds completely at random.
But there is also interest in correlated percolation models. Many studies of
rigidity percolation are inspired by covalent glass networks, and these networks
are, of course, not totally random. Our self-organization models try to
capture one aspect of this non-randomness, the tendency to avoid excessive
mechanical stress. Being based on the pebble game constraint counting algorithm,
these models are necessarily simplistic, but we believe they capture some
important features of covalent glasses; indeed, there has been some experimental
confirmation of our main qualitative results. The idea is to start with an
empty lattice and then add bonds one by one as usual, but testing each bond for
whether it would cause stress in the network and rejecting those that do, until
introduction of stress becomes inevitable, after which bond insertion continues
completely at random. It turns out that in this model, there are two separate
percolation transitions, a rigidity transition and a stress transition, with
a rigid but stress-free intermediate phase between them. More recently,
recognizing that the original procedure is essentially an aggregation process,
as bonds are only inserted and never removed, and thus is biased towards certain
networks, we have introduced an additional equilibration step, consisting of
repeated removal and reinsertion of bonds after every bond addition, maintaining
the stress-free character of networks. If the
equilibration is long enough, the result is an unbiased distribution of
stress-free networks; this corresponds to equilibrium in the
*T* → 0 limit for
any statistical-mechanical model in which there is an energy cost associated
with stress. In this variant of the model, the intermediate phase is
still present, but it is unusual in that at any point within the intermediate
phase, the probability that there is rigidity percolation is non-trivial
(between 0 and 1). The distribution of finite cluster sizes is always
power-law within the whole intermediate phase; also, percolation in the
intermediate phase can often be created (destroyed) by a single bond addition
(removal). This suggests that the system is in the critical state on the verge
of percolation throughout the intermediate phase, an attribute of self-organized
criticality, but in an equilibrium system. We have also calculated the
bond-configurational entropy cost of self-organization using a Monte Carlo
procedure that somewhat resembles thermodynamic integration; this cost is just
a few percent of the total bond-configurational entropy, thus confirming that
self-organization effects are expected to be observed in real systems close to
the glass transition temperature. As mentioned, there is indeed some
experimental evidence of these effects. See publications
4, 7,
10, 15,
18. More recently, I have also extended
the model with equilibration above the upper boundary of the intermediate phase,
by equilibrating while keeping the number of stress-causing constraints to a
minimum, and also considered a finite-temperature analog of the model. These
results are still unpublished.

**Exactly mean-field conductivity in a correlated resistor network
model** As an offshoot of our work on self-organization in rigidity percolation,
we have considered its direct analog in usual, connectivity percolation,
the loopless percolation model. This model has been studied extensively
before, as it is the *q* → 0 limit of the *q*-state Potts model; the novelty of
our work is that by analogy with our rigidity model, we also consider what
happens when bonds continue to be inserted at random after the network can
no longer remain loopless (i.e., above the spanning tree limit). We have been
able to first show numerically and then prove that when links in such a network
are replaced by resistances, the conductivity changes exactly linearly as a
function of bond concentration, just like in mean-field theory. The proof is
based on the Kirchhoff theorem giving an expression for the conductance of a
resistor network in terms of the number of trees that can be built on it. See
publication 12.

**Algorithms for rigidity analysis of general 3D networks**
While the theory underlying the pebble game algorithm is valid for 3D
bond-bending networks, it is invalid in general for 3D elastic networks that are
not bond-bending. But the question of practical interest is how frequent the
errors would be in a pebble-game-type algorithm based on this theory. We have
constructed such an algorithm, as well as a slower but exact algorithm based on
network relaxation that we use to test the new pebble game. It turns out that
at least in some cases of interest, errors in the new pebble game are extremely
rare and thus the algorithm can be regarded as operationally exact in these
cases. However, we have also developed a procedure that generates systematically
the rare counterexamples and thus makes it possible to estimate their
frequency. See publication
19.

**Rigidity percolation transition in diluted 3D central-force networks**
One case when the pebble game for 3D networks is essentially exact is
randomly diluted central-force regular lattices. Applying the pebble game to
this case, we have found that the percolation transition is first order on
the BCC and FCC bond-diluted networks and the BCC site-diluted networks, but
second order on the FCC site-diluted network, contrary to what one might expect
based on universality arguments. The results for first order transitions are
especially spectacular: there are only very small rigid clusters
(≲ 10 sites)
below the transition, and then, upon a single bond addition, a huge percolating
cluster taking up more than 90% of the network emerges. While the evidence we
have is rather strong, the non-universality claim is certainly extraordinary
enough that our results should be obtained by other means in the future. Exotic
alternative explanations are possible. See
publication 19.

**Rigidity percolation in chemically ordered networks**
This is yet another
case when the rigidity transition is first order. Take a bond-bending 3D network all sites
of which have 3 neighbours (are 3-coordinated), but which otherwise is arbitrary.
Then start breaking up bonds into two adding a site in the middle of the bond,
thus introducing 2-coordinated sites. Never put two 2-coordinated sites next to
each other, until that becomes necessary, after which never put three
2-coordinated sites next to each other, etc. In this way, maximum "chemical
order" is achieved. We have proved that in this case, the whole network
initially consists of a single rigid cluster, up until there are just enough
2-coordinated sites to always have a 2-coordinated site next to a 3-coordinated
site and vice versa (at this point the fraction of 2-coordinated sites is 60%).
This single cluster situation persists until 6 more sites are added, but when
one more site is added, the huge rigid cluster breaks up into as many tiny
clusters as there are sites in the network. This is thus the sharpest rigidity
transition imaginable. See publication 11.

In many condensed matter and molecular systems, motions of constituent atoms
span a wide range of time scales. For instance, in protein molecules, the time
scale of the folding process exceeds the fastest time scale of vibrational
motions of atoms by at least 6 orders of magnitude and often more.
Straightforward molecular dynamics (MD) simulations run into trouble, as the time
step should correspond to the fastest time scale, but the total simulation time
should be as long as the longest time. One possible approach is based on the
fact that the configuration space of these systems can be separated into basins
of attraction to potential energy minima, and the dynamics consists of
high-frequency vibrations within these basins interrupted by much more rare
jumps between them. One could then coarse-grain the dynamics by
concentrating on the interbasin jumps (or activated events). This
would still reproduce the correct long-time dynamics, provided that the
relative frequencies of possible jumps are reproduced correctly. There have been
many methods developed mostly by the computational chemistry community that
allow one to find the pathways and then calculate the rates of the transitions.
Most of these methods assume that the initial and the final states are known at
least roughly. In many situations, this is not the case: for a given initial
state, many different jumps are possible and the final states are not known
*a priori*. Barkema and Mousseau have developed a method,
the activation-relaxation technique (ART),
for generating different possible jumps. In this method, the
system starts at a minimum and gets displaced in a random direction until
reaching a certain threshold (the "basin boundary") defined in terms of the
lowest eigenvalue of the Hessian; once the threshold is reached, the system
moves in the direction of the eigenvector corresponding to this eigenvalue
relaxing in the perpendicular direction; this will often bring the system in the
vicinity of a saddle point, thus completing the *activation* part, and
then the *relaxation* brings the system to another minimum. Coupled
with methods for calculating the rates, ART should allow, at
least in principle, finding the rates of all jumps, executing them with appropriate
relative frequencies and thus reproducing the dynamics. However, this approach
requires the generation of a reasonably complete list of possible transitions
for each visited minimum, which is computationally expensive. The problem arises from the fact that the probabilities of
generating certain events in the ART method and other similar
methods are not known in advance. My work has involved some attempts towards
solving this problem and has been done in collaboration with
N. Mousseau,
G. T. Barkema
and H. Vocks.

**Combining intrabasin MD with ART: POP-ART**
The properly obeying probability ART (POP-ART) method modifies ART in such a way
that the transitions between basins occur with relative probabilities obeying
the detailed balance condition appropriate for the given temperature. This
guarantees the correct distribution over different basins. The simulation is
carried out at a constant energy (i.e., in the microcanonical ensemble),
although coupling with a thermostat is possible. First of all, the random push
within the initial basin is replaced by an MD run — this guarantees that
points on the basin boundary (defined as in the original ART) are sampled with
the correct probabilities. Once the boundary crossing occurs, the activation is
carried out. This activation step is similar to that in ART, but it is continued
until another basin boundary is reached and modified so as to ensure
reversibility. If the activation step is viewed as a coordinate transformation,
the detailed balance condition requires that the Jacobian of this transformation
be equal to 1, or else an additional acceptance/rejection step is required. The
Jacobian can be calculated on the fly during the activation. We have calculated
the Jacobian for a model potential showing that when it is averaged over
possible trajectories, it corresponds to the free energy change between the
basins, having both energetic and entropic contributions. We have tested the
method comparing it to MD for crystalline Si with an interstitial using the
Stillinger-Weber potential showing that different interstitial configurations
are reproduced with correct probabilities; simulations for a vacancy diffusion
showed that the method can be much more efficient than MD at low enough
temperatures. See publication 14.

**Using memory of transitions in event-based simulations**
Even when the dynamics is coarse-grained by eliminating vibrations and
concentrating on activated events, the sampling of the configuration space may
still be inefficient. This is because in complex systems, there are often
groups of basins, often called metabasins, such that transitions between basins
within the metabasin are relatively frequent, but transitions to outside the
metabasin are much more rare. The system will then visit the same states over
and over again before moving elsewhere. For optimization problems, when the only
purpose is finding the global energy minimum, there is a well-known approach,
tabu search, in which a memory of recently visited states is introduced and
returning to these states is prohibited for some time afterwards. In general,
this is not a suitable approach when not just the lowest-energy state needs to
be found, but reproducing the correct distribution over the states at a given
temperature is required. However, using a toy model, we have found that if the
memory of states is replaced by the memory of transitions, the obtained
distribution is nearly exact. See publication
16.

Over the years, I have also been involved in research on other topics. While in some cases this has not resulted in anything publishable, this has been a useful experience, as it has given me a chance to familiarize myself with very different areas of physics.

**Non-critical phase matchings in nonlinear optics**
This was
one of my undergraduate research projects, under supervision of N. E. Kornienko
from the Kiev Shevchenko University. Nonlinear processes in optics,
such as the second harmonic generation or the sum and difference frequency
generation, are only efficient when the waves involved phase-match so that
constructive interference is achieved everywhere along the path. This would be
difficult to achieve in an isotropic medium with dispersion, but is possible in
an anisotropic crystal, when sending the light in a particular direction and
using waves of different polarizations. In some cases, the matchings are
non-critical, meaning that the matching condition
is approximately satisfied over a wide range of frequencies and/or angles, which
is desirable for some applications. My work involved calculating conditions of
such matchings for a particular material. See publication 1.

**Cluster expansion methods for electronic structure of substitutional alloys**
Electronic structure calculations for periodic solids utilize the periodicity
by using Bloch's theorem and considering a single unit cell. For amorphous
solids, this is, of course, impossible. In the intermediate case of a
substitutional alloy having a periodic lattice but with a disorder in the
occupancy of the lattice sites, there is still no perfect periodicity, but one
can consider a periodic solid with parameters found in a self-consistent way as
the zeroth approximation. The standard approach, the coherent potential
approximation (CPA), uses the Green's function formalism and finds the
self-consistent potential by requiring the vanishing averaged single-site scattering
matrix. One can then improve the result by developing cluster expansions —
one particular way is described in publication 2. As my undergraduate project,
under supervision of S. P. Repetsky
(the page is in Ukrainian) from the Kiev Shevchenko University, I have
applied the CPA to a toy model with a
continuous, rather than discrete, set of possible on-site parameters, and also
have made an attempt to develop a self-consistent cluster expansion for this
case.

**Linear conversion of waves at a plasma boundary**
In a plasma
in a magnetic field, several modes of electromagnetic waves are possible, with
dispersion relations depending on both the angle between the direction of
propagation and the magnetic field and the polarization. Just like in a
birefringent crystal an ordinary wave can convert partially into an
extraordinary wave upon reflection from the surface, conversion between
different modes is possible at the plasma boundary. As yet another undergraduate
project, under supervision of K. P. Shamrai from the
Kiev Institute for Nuclear Research, I have considered analytically the
conversion of helicon waves at the plasma boundary in the limits of both a very
sharp and a very fuzzy boundary.

**Stochastic models of population dynamics** During my stay at
Arizona State University,
I had a chance to interact with
T. J. Newman (now at Dundee), whose
research is in the area of applications of statistical physics to biological
systems and especially the effect of fluctuations in population dynamics,
biochemical reactions, etc. I was involved, in particular, in the study of
properties of the stochastic logistic model of population dynamics.
Unfortunately, it turned out that most results obtained by me were obtained
before, so this research has not resulted in any publications.

**Kinetic Monte Carlo simulations of pattern formation due to a standing wave**
While this was not a research project that I can call my own, I participated in
it extensively by aiding a graduate student in developing a kinetic Monte Carlo
code for simulating pattern formation on a surface irradiated by a standing
electromagnetic wave due to both inhomogeneous heating of the surface and
steering of atoms being deposited on the surface by the electromagnetic field.