# Publications

At low temperatures the dynamical degrees of freedom in amorphous solids are tunneling two-level systems (TLSs). Concentrating on these degrees of freedom, and taking into account disorder and TLS-TLS interactions, we obtain a “TLS glass,” described by the random-field Ising model with random 1/r^3 interactions. In this paper we perform a self-consistent mean-field calculation, previously used to study the electron-glass (EG) model [A. Amir *et al.*, Phys. Rev. B **77**, 165207 (2008)]. Similarly to the electron glass, we find a 1/λ distribution of relaxation rates λ, leading to logarithmic slow relaxation. However, with increased interactions the EG model shows slower dynamics whereas the TLS-glass model shows faster dynamics. This suggests that given system-specific properties, glass dynamics can be slowed down or sped up by the interactions.

All organisms control the size of their cells. We focus here on the question of size regulation in bacteria, and suggest that the quantitative laws governing cell size and its dependence on growth rate may arise as byproducts of a regulatory mechanism which evolved to support multiple DNA replication forks. In particular, we show that the increase of bacterial cell size during Lenski’s long-term evolution experiments is a natural outcome of this proposal. This suggests that, in the context of evolution, cell size may be a 'spandrel'

We observe nonmonotonic aging and memory effects, two hallmarks of glassy dynamics, in two disordered mechanical systems: crumpled thin sheets and elastic foams. Under fixed compression, both systems exhibit monotonic nonexponential relaxation. However, when after a certain waiting time the compression is partially reduced, both systems exhibit a nonmonotonic response: the normal force first increases over many minutes or even hours until reaching a peak value, and only then is relaxation resumed. The peak time scales linearly with the waiting time, indicating that these systems retain long-lasting memory of previous conditions. Our results and the measured scaling relations are in good agreement with a theoretical model recently used to describe observations of monotonic aging in several glassy systems, suggesting that the nonmonotonic behavior may be generic and that athermal systems can show genuine glassy behavior.

Bacteria tightly regulate and coordinate the various events in their cell cycles to duplicate themselves accurately and to control their cell sizes. Growth of Escherichia coli, in particular, follows a relation known as Schaechter’s growth law. This law says that the average cell volume scales exponentially with growth rate, with a scaling exponent equal to the time from initiation of a round of DNA replication to the cell division at which the corresponding sister chromosomes segregate. Here, we sought to test the robustness of the growth law to systematic perturbations in cell dimensions achieved by varying the expression levels of mreB and ftsZ. We found that decreasing the mreB level resulted in increased cell width, with little change in cell length, whereas decreasing the ftsZ level resulted in increased cell length. Furthermore, the time from replication termination to cell division increased with the perturbed dimension in both cases. Moreover, the growth law remained valid over a range of growth conditions and dimension perturbations. The growth law can be quantitatively interpreted as a consequence of a tight coupling of cell division to replication initiation. Thus, its robustness to perturbations in cell dimensions strongly supports models in which the timing of replication initiation governs that of cell division, and cell volume is the key phenomenological variable governing the timing of replication initiation. These conclusions are discussed in the context of our recently proposed “adder-per-origin” model, in which cells add a constant volume per origin between initiations and divide a constant time after initiation.

The proliferation of a growing microbial colony is well characterized by the population growth rate. However, at the single-cell level, isogenic cells often exhibit different cell-cycle durations. For evolutionary dynamics, it is thus important to establish the connection between the population growth rate and the heterogeneous single-cell generation time. Existing theories often make the assumption that the generation times of mother and daughter cells are independent. However, it has been shown that to maintain a bounded cell size distribution, cells that grow exponentially at the single-cell level need to adopt cell size regulation, leading to a negative correlation of mother-daughter generation time. In this work, we construct a general framework to describe the population growth in the presence of size regulation. We derive a formula for the population growth rate, which only depends on the variability of single-cell growth rate, independent of other sources of noises. Our work shows that a population can enhance the population growth by reducing the growth rate variability, which may be the reason for the mild growth rate variability observed in many microorganisms.

Memory is one of the unique qualities of a glassy system. The relaxation of a glass to equilibrium contains information on the sample’s excitation history, an effect often refer to as “aging.” We demonstrate that under the right conditions a glass can also possess a different type of memory. We study the conductance relaxation of electron glasses that are fabricated at low temperatures. Remarkably, the dynamics are found to depend not only on the ambient measurement temperature but also on the maximum temperature to which the system was exposed. Hence the system “remembers” its highest temperature. This effect may be qualitatively understood in terms of energy barriers and local minima in configuration space and therefore may be a general property of the glass state.

One-dimensional photonic crystals with slowly varying, i.e. "chirped", lattice period are responsible for broadband light reflectance in many diverse biological contexts, ranging from the shiny coatings of various beetles to the eyes of certain butterflies. We present a quantum scattering analogy for light reflection from these adiabatically chirped photonic crystals (ACPCs) and apply a WKB-type approximation to obtain a closed-form expression for the reflectance. From this expression we infer several design principles, including a differential equation for the chirp pattern required to elicit a given reflectance spectrum and the minimal number of bilayers required to exceed a desired reflectance threshold. Comparison of the number of bilayers found in ACPCs throughout nature and our predicted minimal required number also gives a quantitative measure of the optimality of chirped biological reflectors. Together these results elucidate the design principles of chirped reflectors in nature and their possible application to future optical technologies.

In science, as in life, `surprises' can be adequately appreciated only in the presence of a null model, what we expect a priori. In physics, theories sometimes express the values of dimensionless physical constants as combinations of mathematical constants like pi or e. The inverse problem also arises, whereby the measured value of a physical constant admits a `surprisingly' simple approximation in terms of well-known mathematical constants. Can we estimate the probability for this to be a mere coincidence, rather than an inkling of some theory? We answer the question in the most naive form.

We consider a class of biologically-motivated stochastic processes in which a unicellular organism divides its resources (volume or damaged proteins, in particular) symmetrically or asymmetrically between its progeny. Assuming the final amount of the resource is controlled by a growth policy and subject to additive and multiplicative noise, we derive the "master equation" describing how the resource distribution evolves over subsequent generations and use it to study the properties of stable resource distributions. We find conditions under which a unique stable resource distribution exists and calculate its moments for the class of affine linear growth policies. Moreover, we apply an asymptotic analysis to elucidate the conditions under which the stable distribution (when it exists) has a power-law tail. Finally, we use the results of this asymptotic analysis along with the moment equations to draw a stability phase diagram for the system that reveals the counterintuitive result that asymmetry serves to increase stability while at the same time widening the stable distribution. We also briefly discuss how cells can divide damaged proteins asymmetrically between their progeny as a form of damage control. In the appendix, motivated by the asymmetric division of cell volume in Saccharomyces cerevisiae, we extend our results to the case wherein mother and daughter cells follow different growth policies.

We explore the spectra and localization properties of the N-site banded one-dimensional non-Hermitian random matrices that arise naturally in sparse neural networks. Approximately equal numbers of random excitatory and inhibitory connections lead to spatially localized eigenfunctions and an intricate eigenvalue spectrum in the complex plane that controls the spontaneous activity and induced response. A finite fraction of the eigenvalues condense onto the real or imaginary axes. For large N, the spectrum has remarkable symmetries not only with respect to reflections across the real and imaginary axes but also with respect to 90∘ rotations, with an unusual anisotropic divergence in the localization length near the origin. When chains with periodic boundary conditions become directed, with a systematic directional bias superimposed on the randomness, a hole centered on the origin opens up in the density-of-states in the complex plane. All states are extended on the rim of this hole, while the localized eigenvalues outside the hole are unchanged. The bias-dependent shape of this hole tracks the bias-independent contours of constant localization length. We treat the large-Nlimit by a combination of direct numerical diagonalization and using transfer matrices, an approach that allows us to exploit an electrostatic analogy connecting the “charges” embodied in the eigenvalue distribution with the contours of constant localization length. We show that similar results are obtained for more realistic neural networks that obey “Dale's law” (each site is purely excitatory or inhibitory) and conclude with perturbation theory results that describe the limit of large directional bias, when all states are extended. Related problems arise in random ecological networks and in chains of artificial cells with randomly coupled gene expression patterns.

Random walks, and in particular, their first passage times, are ubiquitous in nature. Using direct enumeration of paths, we find the first-return-time distribution of a one-dimensional random walker, which is a heavy-tailed distribution with infinite mean. Using the same method, we find the last-return-time distribution, which follows the arcsine law. Both results have a broad range of applications in physics and other disciplines. The derivation presented here is readily accessible to physics undergraduates and provides an elementary introduction into random walks and their intriguing properties.

The electromagnetically induced transparency (EIT) spectrum of atoms diffusing in and out of a narrow beam is measured and shown to manifest the two-dimensional δ-function anomaly in a classical setting. In the limit of small-area beams, the EIT line shape is independent of power, and equal to the renormalized local density of states of a free particle Hamiltonian. The measured spectra for different powers and beam sizes collapses to a single universal curve with a characteristic logarithmic Van Hove singularity close to resonance.

Various bacteria such as the canonical gram negative *Escherichia coli* or the well-studied gram positive *Bacillus subtilis* divide symmetrically after they approximately double their volume. Their size at division is not constant, but is typically distributed over a narrow range. Here, we propose an analytically tractable model for cell size control, and calculate the cell size and interdivision time distributions, as well as the correlations between these variables. We suggest ways of extracting the model parameters from experimental data, and show that existing data for *E. coli*supports partial size control, and a particular explanation: a cell attempts to add a constant volume from the time of initiation of DNA replication to the next initiation event. This hypothesis accounts for the experimentally observed correlations between mother and daughter cells as well as the exponential dependence of size on growth rate.

Characterizing the frequency-dependent response of amorphous systems and glasses can provide important insights into their physics. Here, we study the response of an electron glass, where Coulomb interactions are important and have previously been shown to significantly modify the conductance and lead to memory effects and aging. We propose a model which allows us to take the interactions into account in a self-consistent way, and explore the frequency-dependent conduction at all frequencies. At low frequencies conduction occurs on the percolation backbone, and the model captures the variable-range-hopping behavior. At high frequencies conduction is dominated by localized clusters. Despite the difference in physical mechanisms at low and high frequency, we are able to approximately scale all numerical data onto a single curve, using two parameters: the DC conduction and the DC dielectric constant. The behavior follows the universal scaling that is experimentally observed for a large class of amorphous solids.

Interference phenomena are the source of some of the spectacular colors of animals and plants in nature. In some of these systems, the physical structure consists of an ordered array of layers with alternating high and low refractive indices. This periodicity leads to an optical band structure that is analogous to the electronic band structure encountered in semiconductor physics; namely, specific bands of wavelengths (the stop bands) are perfectly reflected. Here, we present a minimal model for optical band structure in a periodic multilayer and solve it using recursion relations. We present experimental data for various beetles, whose optical structure resembles the proposed model. The stop bands emerge in the limit of an infinite number of layers by finding the fixed point of the recursive relations. In order for these to converge, an infinitesimal amount of absorption needs to be present, reminiscent of the regularization procedures commonly used in physics calculations. Thus, using only the phenomenon of interference and the idea of recursion, we are able to elucidate the concepts of band structure and regularization in the context of experimentally observed phenomena, such as the high reflectance and the iridescent color appearance of structurally colored beetles.

We study, theoretically and numerically, a minimal model for phonons in a disordered system. For sufficient disorder, the vibrational modes of this classical system can become Anderson localized, yet this problem has received significantly less attention than its electronic counterpart. We find rich behavior in the localization properties of the phonons as a function of the density, frequency, and spatial dimension. We use a percolation analysis to argue for a Debye spectrum at low frequencies for dimensions higher than one, and for a localization-delocalization transition (at a critical frequency) above two dimensions. We show that in contrast to the behavior in electronic systems, the transition exists for arbitrarily large disorder, albeit with an exponentially small critical frequency. The structure of the modes reflects a divergent percolation length that arises from the disorder in the springs without being explicitly present in the definition of our model. Within the percolation approach, we calculate the speed of sound of the delocalized modes (phonons), which we corroborate with numerics. We find the critical frequency of the localization transition at a given density and find good agreement of these predictions with numerical results using a recursive Green-function method that was adapted for this problem. The connection of our results to recent experiments on amorphous solids is discussed.