Ilia Rushkin - Academia.edu (original) (raw)

Papers by Ilia Rushkin

Journal of Physics: Condensed Matter, 2011

We consider a simple model of quantum disorder in two dimensions, characterized by a long-range s... more We consider a simple model of quantum disorder in two dimensions, characterized by a long-range site-to-site hopping. The system undergoes a metal-insulator transition -its eigenfunctions change from being extended to being localized. We demonstrate that at the point of the transition the eigenfunctions do not become fractal. Their density moments do not scale as a power of the system size. Instead, in one of the considered limits our result suggests a power of the logarithm of the system size. In this regard, the transition differs from a similar one in the one-dimensional version of the same system, as well as from the conventional Anderson transition in more than two dimensions. PACS numbers: 73.20.Fz, 72.15.Rn, 05.45.Df Critical wave functions have been a subject of intense theoretical research during the last two decades (see Ref.

Bulletin of the American …, 2006

Standard stochastic Loewner evolution (SLE) is driven by a continuous Brownian motion, which then... more Standard stochastic Loewner evolution (SLE) is driven by a continuous Brownian motion, which then produces a continuous fractal trace. If jumps are added to the driving function, the trace branches. We consider a generalized SLE driven by a superposition of a Brownian motion and a stable Lévy process. The situation is defined by the usual SLE parameter, κ, as well as α which defines the shape of the stable Lévy distribution. The resulting behavior is characterized by two descriptors: p, the probability that the trace self-intersects, andp, the probability that it will approach arbitrarily close to doing so. Using Dynkin's formula, these descriptors are shown to change qualitatively and singularly at critical values of κ and α. It is reasonable to call such changes "phase transitions". These transitions occur as κ passes through four (a well-known result) and as α passes through one (a new result). Numerical simulations are then used to explore the associated touching and near-touching events.

In the first chapter we consider critical curves - conformally invariant curves that appear at cr... more In the first chapter we consider critical curves - conformally invariant curves that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field theory (CFT). We also provide links between this description and the stochastic (Schramm-) Loewner evolution (SLE). The connection appears in the long-time limit of stochastic evolution of various SLE observables related to CFT primary fields. The multifractal spectrum of harmonic measure and other fractal characteristics of critical curves are obtained. In the second chapter a stochastic growth process of branching trees based on stochastic Loewner evolution is introduced by adding a stable Levy process to the forcing. Its behavior is analyzed analytically and numerically. The properties of the growth change qualitatively and singularly at critical values of parameters.

In this work we studied the random flow induced in a fluid by the motion of a dilute suspension o... more In this work we studied the random flow induced in a fluid by the motion of a dilute suspension of the swimming algae Volvox carteri. The fluid velocity in the suspension is a superposition of the flow fields set up by the individual organisms, which in turn have multipole contributions that decay as inverse powers of distance from the organism. Here we show that the conditions under which the central limit theorem guarantees a Gaussian probability distribution function of velocity fluctuations are satisfied when the leading force singularity is a Stokeslet. Deviations from Gaussianity are shown to arise from near-field effects. Comparison is made with the statistical properties of abiotic sedimenting suspensions. The experimental results are supplemented by extensive numerical studies.

Fractal geometry of critical curves appearing in 2D critical systems is characterized by their ha... more Fractal geometry of critical curves appearing in 2D critical systems is characterized by their harmonic measure. For systems described by conformal field theories with central charge c<=1 the scaling exponents of the harmonic measure have been computed by B. Duplantier [Phys. Rev. Lett. 84, 1363 (2000)] by relating the problem to boundary two-dimensional gravity. We present a simple argument connecting the harmonic measure of critical curves to operators obtained by fusion of primary fields, and compute characteristics of the fractal geometry by means of regular methods of conformal field theory. The method is not limited to theories with c<=1.

Journal of Statistical Mechanics: Theory and Experiment, 2008

Standard Schramm-Loewner evolution (SLE) is driven by a continuous Brownian motion which then pro... more Standard Schramm-Loewner evolution (SLE) is driven by a continuous Brownian motion which then produces a trace, a continuous fractal curve connecting the singular points of the motion. If jumps are added to the driving function, the trace branches. In a recent publication (Rushkin et al 2006 J. Stat. Mech. P01001 [cond-mat/0509187]) we introduced a generalized SLE driven by a superposition of a Brownian motion and a fractal set of jumps (technically a stable Lévy process). We then discussed the small scale properties of the resulting Lévy-SLE growth process. Here we discuss the same model, but focus on the global scaling behavior which ensues as time goes to infinity. This limiting behavior is independent of the Brownian forcing and depends upon only a single parameter, α, which defines the shape of the stable Lévy distribution. We learn about this behavior by studying a Fokker-Planck equation which gives the probability distribution for end points of the trace as a function of time. As in the short time case previously studied, we observe that the properties of this growth process change qualitatively and singularly at α = 1. We show both analytically and numerically that the growth continues indefinitely in the vertical direction for α > 1, goes as log t for α = 1, and saturates for α < 1. The probability density has two different scales corresponding to directions along and perpendicular to the boundary. In the former case, the characteristic scale is X(t) ∼ t 1/α . In the latter case the scale is Y (t) ∼ A + Bt 1−1/α for α = 1, and Y (t) ∼ ln t for α = 1. Scaling functions for the probability density are given for various limiting cases.

Journal of Physics A: Mathematical and Theoretical, 2007

We consider critical curves -conformally invariant curves that appear at critical points of two-d... more We consider critical curves -conformally invariant curves that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field theory (CFT). We also provide links between this description and the stochastic (Schramm-) Loewner evolution (SLE). The connection appears in the long-time limit of stochastic evolution of various SLE observables related to CFT primary fields. We show how the multifractal spectrum of harmonic measure and other fractal characteristics of critical curves can be obtained.

Journal of Physics A: Mathematical and Theoretical, 2008

Fractal geometry of random curves appearing in the scaling limit of critical two-dimensional stat... more Fractal geometry of random curves appearing in the scaling limit of critical two-dimensional statistical systems is characterized by their harmonic measure and winding angle. The former is the measure of the jaggedness of the curves while the latter quantifies their tendency to form logarithmic spirals. We show how these characteristics are related to local operators of conformal field theory and how they can be computed using conformal invariance of critical systems with central charge c 1.

Advanced Materials, 2012

The prediction and subsequent creation of artifi cially engineered metamaterials has opened the p... more The prediction and subsequent creation of artifi cially engineered metamaterials has opened the pathway to revolutionary effects in light-matter interactions. In such materials the properties of the dielectric and magnetic permittivities, ε and μ , are not governed by the response of the individual atoms in the presence of an electromagnetic fi eld, but are determined by the sub-wavelength structure of the material. In 1996 Pendry et al. predicted that a sparse cubic metal wire array with micrometerwide wires would have a signifi cantly reduced plasma frequency due to the reduced average electron density and the large selfinductance of the structure. A plethora of experimental work in this fi eld has demonstrated metamaterials from GHz up to yellow optical frequencies. Reaching higher optical frequencies has proven problematic, as it requires the manipulation of materials on the scale of just a few tens of nanometers over macroscopic areas. While fabrication techniques such as focused ion beam lithography, direct laser writing, and atomic layer deposition provide design fl exibility, they are limited with regard to accessible feature sizes as well as the scalability of samples.

Bulletin of the American Physical Society, Nov 23, 2010

In this work we studied the random flow induced in a fluid by the motion of a dilute suspension o... more In this work we studied the random flow induced in a fluid by the motion of a dilute suspension of the swimming algae {\ it Volvox carteri}. The fluid velocity in the suspension is a superposition of the flow fields set up by the individual organisms, which in turn have multipole contributions that decay as inverse powers of distance from the organism. Here we show that the conditions under which the central limit theorem guarantees a Gaussian probability distribution function of velocity fluctuations are satisfied when the leading force ...

Physical review letters, Oct 26, 2010

In dilute suspensions of swimming microorganisms the local fluid velocity is a random superpositi... more In dilute suspensions of swimming microorganisms the local fluid velocity is a random superposition of the flow fields set up by the individual organisms, which in turn have multipole contributions decaying as inverse powers of distance from the organism. Here we show that the conditions under which the central limit theorem guarantees a Gaussian probability distribution function of velocities are satisfied when the leading force singularity is a Stokeslet, but are not when it is any higher multipole. These results are confirmed by numerical studies and by experiments on suspensions of the alga Volvox carteri, which show that deviations from Gaussianity arise from near-field effects.

Journal of Physics: Condensed Matter, 2011

We consider a simple model of quantum disorder in two dimensions, characterized by a long-range s... more We consider a simple model of quantum disorder in two dimensions, characterized by a long-range site-to-site hopping. The system undergoes a metal-insulator transition -its eigenfunctions change from being extended to being localized. We demonstrate that at the point of the transition the eigenfunctions do not become fractal. Their density moments do not scale as a power of the system size. Instead, in one of the considered limits our result suggests a power of the logarithm of the system size. In this regard, the transition differs from a similar one in the one-dimensional version of the same system, as well as from the conventional Anderson transition in more than two dimensions. PACS numbers: 73.20.Fz, 72.15.Rn, 05.45.Df Critical wave functions have been a subject of intense theoretical research during the last two decades (see Ref.

Bulletin of the American …, 2006

Standard stochastic Loewner evolution (SLE) is driven by a continuous Brownian motion, which then... more Standard stochastic Loewner evolution (SLE) is driven by a continuous Brownian motion, which then produces a continuous fractal trace. If jumps are added to the driving function, the trace branches. We consider a generalized SLE driven by a superposition of a Brownian motion and a stable Lévy process. The situation is defined by the usual SLE parameter, κ, as well as α which defines the shape of the stable Lévy distribution. The resulting behavior is characterized by two descriptors: p, the probability that the trace self-intersects, andp, the probability that it will approach arbitrarily close to doing so. Using Dynkin's formula, these descriptors are shown to change qualitatively and singularly at critical values of κ and α. It is reasonable to call such changes "phase transitions". These transitions occur as κ passes through four (a well-known result) and as α passes through one (a new result). Numerical simulations are then used to explore the associated touching and near-touching events.

In the first chapter we consider critical curves - conformally invariant curves that appear at cr... more In the first chapter we consider critical curves - conformally invariant curves that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field theory (CFT). We also provide links between this description and the stochastic (Schramm-) Loewner evolution (SLE). The connection appears in the long-time limit of stochastic evolution of various SLE observables related to CFT primary fields. The multifractal spectrum of harmonic measure and other fractal characteristics of critical curves are obtained. In the second chapter a stochastic growth process of branching trees based on stochastic Loewner evolution is introduced by adding a stable Levy process to the forcing. Its behavior is analyzed analytically and numerically. The properties of the growth change qualitatively and singularly at critical values of parameters.

In this work we studied the random flow induced in a fluid by the motion of a dilute suspension o... more In this work we studied the random flow induced in a fluid by the motion of a dilute suspension of the swimming algae Volvox carteri. The fluid velocity in the suspension is a superposition of the flow fields set up by the individual organisms, which in turn have multipole contributions that decay as inverse powers of distance from the organism. Here we show that the conditions under which the central limit theorem guarantees a Gaussian probability distribution function of velocity fluctuations are satisfied when the leading force singularity is a Stokeslet. Deviations from Gaussianity are shown to arise from near-field effects. Comparison is made with the statistical properties of abiotic sedimenting suspensions. The experimental results are supplemented by extensive numerical studies.

Fractal geometry of critical curves appearing in 2D critical systems is characterized by their ha... more Fractal geometry of critical curves appearing in 2D critical systems is characterized by their harmonic measure. For systems described by conformal field theories with central charge c<=1 the scaling exponents of the harmonic measure have been computed by B. Duplantier [Phys. Rev. Lett. 84, 1363 (2000)] by relating the problem to boundary two-dimensional gravity. We present a simple argument connecting the harmonic measure of critical curves to operators obtained by fusion of primary fields, and compute characteristics of the fractal geometry by means of regular methods of conformal field theory. The method is not limited to theories with c<=1.

Journal of Statistical Mechanics: Theory and Experiment, 2008

Standard Schramm-Loewner evolution (SLE) is driven by a continuous Brownian motion which then pro... more Standard Schramm-Loewner evolution (SLE) is driven by a continuous Brownian motion which then produces a trace, a continuous fractal curve connecting the singular points of the motion. If jumps are added to the driving function, the trace branches. In a recent publication (Rushkin et al 2006 J. Stat. Mech. P01001 [cond-mat/0509187]) we introduced a generalized SLE driven by a superposition of a Brownian motion and a fractal set of jumps (technically a stable Lévy process). We then discussed the small scale properties of the resulting Lévy-SLE growth process. Here we discuss the same model, but focus on the global scaling behavior which ensues as time goes to infinity. This limiting behavior is independent of the Brownian forcing and depends upon only a single parameter, α, which defines the shape of the stable Lévy distribution. We learn about this behavior by studying a Fokker-Planck equation which gives the probability distribution for end points of the trace as a function of time. As in the short time case previously studied, we observe that the properties of this growth process change qualitatively and singularly at α = 1. We show both analytically and numerically that the growth continues indefinitely in the vertical direction for α > 1, goes as log t for α = 1, and saturates for α < 1. The probability density has two different scales corresponding to directions along and perpendicular to the boundary. In the former case, the characteristic scale is X(t) ∼ t 1/α . In the latter case the scale is Y (t) ∼ A + Bt 1−1/α for α = 1, and Y (t) ∼ ln t for α = 1. Scaling functions for the probability density are given for various limiting cases.

Journal of Physics A: Mathematical and Theoretical, 2007

We consider critical curves -conformally invariant curves that appear at critical points of two-d... more We consider critical curves -conformally invariant curves that appear at critical points of two-dimensional statistical mechanical systems. We show how to describe these curves in terms of the Coulomb gas formalism of conformal field theory (CFT). We also provide links between this description and the stochastic (Schramm-) Loewner evolution (SLE). The connection appears in the long-time limit of stochastic evolution of various SLE observables related to CFT primary fields. We show how the multifractal spectrum of harmonic measure and other fractal characteristics of critical curves can be obtained.

Journal of Physics A: Mathematical and Theoretical, 2008

Fractal geometry of random curves appearing in the scaling limit of critical two-dimensional stat... more Fractal geometry of random curves appearing in the scaling limit of critical two-dimensional statistical systems is characterized by their harmonic measure and winding angle. The former is the measure of the jaggedness of the curves while the latter quantifies their tendency to form logarithmic spirals. We show how these characteristics are related to local operators of conformal field theory and how they can be computed using conformal invariance of critical systems with central charge c 1.

Advanced Materials, 2012

The prediction and subsequent creation of artifi cially engineered metamaterials has opened the p... more The prediction and subsequent creation of artifi cially engineered metamaterials has opened the pathway to revolutionary effects in light-matter interactions. In such materials the properties of the dielectric and magnetic permittivities, ε and μ , are not governed by the response of the individual atoms in the presence of an electromagnetic fi eld, but are determined by the sub-wavelength structure of the material. In 1996 Pendry et al. predicted that a sparse cubic metal wire array with micrometerwide wires would have a signifi cantly reduced plasma frequency due to the reduced average electron density and the large selfinductance of the structure. A plethora of experimental work in this fi eld has demonstrated metamaterials from GHz up to yellow optical frequencies. Reaching higher optical frequencies has proven problematic, as it requires the manipulation of materials on the scale of just a few tens of nanometers over macroscopic areas. While fabrication techniques such as focused ion beam lithography, direct laser writing, and atomic layer deposition provide design fl exibility, they are limited with regard to accessible feature sizes as well as the scalability of samples.

Bulletin of the American Physical Society, Nov 23, 2010

In this work we studied the random flow induced in a fluid by the motion of a dilute suspension o... more In this work we studied the random flow induced in a fluid by the motion of a dilute suspension of the swimming algae {\ it Volvox carteri}. The fluid velocity in the suspension is a superposition of the flow fields set up by the individual organisms, which in turn have multipole contributions that decay as inverse powers of distance from the organism. Here we show that the conditions under which the central limit theorem guarantees a Gaussian probability distribution function of velocity fluctuations are satisfied when the leading force ...

Physical review letters, Oct 26, 2010

In dilute suspensions of swimming microorganisms the local fluid velocity is a random superpositi... more In dilute suspensions of swimming microorganisms the local fluid velocity is a random superposition of the flow fields set up by the individual organisms, which in turn have multipole contributions decaying as inverse powers of distance from the organism. Here we show that the conditions under which the central limit theorem guarantees a Gaussian probability distribution function of velocities are satisfied when the leading force singularity is a Stokeslet, but are not when it is any higher multipole. These results are confirmed by numerical studies and by experiments on suspensions of the alga Volvox carteri, which show that deviations from Gaussianity arise from near-field effects.