Oded Agam

Normal random matrix ensemble as a growth problem

In general or normal random matrix ensembles, the support of eigenvalues of large size matrices is a planar domain (or several domains) with a sharp boundary. This domain evolves under a change of parameters of the potential and of the size of matrices. The boundary of the support of eigenvalues is a real section of a complex curve. Algebro-geometrical properties of this curve encode physical properties of random matrix ensembles. This curve can be treated as a limit of a spectral curve which is canonically defined for models of finite matrices. We interpret the evolution of the eigenvalue distribution as a growth problem, and describe the growth in terms of evolution of the spectral curve. We discuss algebro-geometrical properties of the spectral curve and describe the wave functions (normalized characteristic polynomials) in terms of differentials on the curve. General formulae and emergence of the spectral curve are illustrated by three meaningful examples.

Lineage correlations of single cell division time as a probe of cell-cycle dynamics

Stochastic processes in cells are associated with fluctuations in mRNA, protein production and degradation, noisy partition of cellular components at division, and other cell processes. Variability within a clonal population of cells originates from such stochastic processes, which may be amplified or reduced by deterministic factors. Cell-to-cell variability, such as that seen in the heterogeneous response of bacteria to antibiotics, or of cancer cells to treatment, is understood as the inevitable consequence of stochasticity. Variability in cell-cycle duration was observed long ago; however, its sources are still unknown. A central question is whether the variance of the observed distribution originates from stochastic processes, or whether it arises mostly from a deterministic process that only appears to be random. A surprising feature of cell-cycle-duration inheritance is that it seems to be lost within one generation but to be still present in the next generation, generating poor correlation between mother and daughter cells but high correlation between cousin cells. This observation suggests the existence of underlying deterministic factors that determine the main part of cell-to-cell variability. We developed an experimental system that precisely measures the cell-cycle duration of thousands of mammalian cells along several generations and a mathematical framework that allows discrimination between stochastic and deterministic processes in lineages of cells. We show that the inter- and intra-generation correlations reveal complex inheritance of the cell-cycle duration. Finally, we build a deterministic nonlinear toy model for cell-cycle inheritance that reproduces the main features of our data. Our approach constitutes a general method to identify deterministic variability in lineages of cells or organisms, which may help to predict and, eventually, reduce cell-to-cell heterogeneity in various systems, such as cancer cells under treatment.

Viscous Fingering and the Shape of an Electronic Droplet in the Quantum Hall Regime

We show that the semiclassical dynamics of an electronic droplet, confined in a plane in a quantizing inhomogeneous magnetic field in the regime where the electrostatic interaction is negligible, is similar to viscous (Saffman-Taylor) fingering on the interface between two fluids with different viscosities confined in a Hele-Shaw cell. Both phenomena are described by the same equations with scales differing by a factor of up to 10(-9). We also report the quasiclassical wave function of the droplet in an inhomogeneous magnetic field.

Quantum chaos, irreversible classical dynamics, and random matrix theory.

The Bohigas-Giannoni-Schmit conjecture stating that the statistical spectral properties of systems which are chaotic in their classical limit coincide with random matrix theory (RMT) is proved. A new semiclassical field theory for individual chaotic systems is constructed in the framework of a nonlinear s model. The low lying modes are shown to be associated with the Perron-Frobenius (PF ) spectrum of the underlying irreversible classical dynamics. It is shown that the existence of a gap in the PF spectrum results in RMT behavior. Moreover, our formalism offers a way of calculating system specific corrections beyond RMT. [S0031-9007(96)00191-3] PACS numbers: 05.45.+b Random matrix theory (RMT) [1] emerged from the need to characterize complex quantum systems in which knowledge of the Hamiltonian is minimal, e.g., complex nuclei. The basic hypothesis is that the Hamiltonian may be treated as one drawn from an ensemble of random matrices with appropriate symmetries. It has been proposed by invoking the complexity of systems which have many degrees of freedom with unknown interaction coupling among them. The study of the statistical quantum properties of systems with a small number of degrees of freedom and their relation to RMT has developed along two lines. The first was by considering an ensemble of random systems such as disordered metallic grains [2]. Randomness in this case is introduced on the level of the Hamiltonian itself, often as a consequence of an unknown impurity configuration. In the second approach, RMT was used in order to understand the level statistics of nonstochastic systems which are chaotic in their classical limit such as the Sinai or the stadium billiards [3]. Here “randomness” is generated by the underlying deterministic classical dynamics itself. Nevertheless, it has been conjectured [3] that “spectrum fluctuations of quantal time-reversalinvariant systems whose classical analogs are strongly chaotic have the Gaussian orthogonal ensembles pattern.” Despite being supported by extensive numerical studies, the origin of the success of RMT as well as its domain of validity are still not completely resolved. In this Letter we show that, in the semiclassical limit, this conjecture is indeed valid for systems with exponential decay of classical correlation functions in time. Moreover, the formalism which we introduce offers a way of calculating system specific corrections beyond RMT. So far the main attempts to establish the relationship between nonstochastic chaotic systems and RMT have been based on periodic orbit theory [4]. Gutzwiller’s trace formula expresses the density of states (DOS) as sum over the classical periodic orbits of the system. However, the number of relevant periodic orbits is exponentially large and clearly contains information that is redundant from the quantum mechanical point of view. This detailed information conceals the way of drawing a connection between the quantum behavior of chaotic systems and RMT. Indeed, the success of the periodic orbit theory approach in reproducing RMT results [5,6] appears to be limited. Here we develop a new semiclassical approach in which the basic ingredients are global modes of the time evolution of the underlying classical system rather than periodic orbits. It is possible to construct a field theory in which the effective action is associated with the classical flow in phase space. We argue that the statistical quantum properties of the system are intimately related to the irreversible classical dynamics or, more precisely, to the Perron-Frobenius (PF) modes in which a disturbance in the classical probability density of a chaotic system relaxes into the ergodic distribution. These modes decay at different rates. This enables a description of the system at levels of increasing complexity by incorporating higher and higher modes. The “zero mode” manifests the conservation of classical probability and corresponds to a uniform distribution over the energy shell. Taking into account only this mode one obtains RMT. Deviations from the universal RMT behavior emerge from the consideration of the higher PF modes. Our approach is analogous to that of disordered systems where the diffusion modes account for the classical relaxation. However, in the field theoretic description of disordered systems [7] averaging is performed over an ensemble. By contrast, in order to characterize individual systems, only energy averaging will be employed here. To establish the Bohigas-Giannoni-Schmit (BGS) conjecture we first show that quantum statistical correlators are described by a functional nonlinear s model. Its low lying modes are identified with the PF eigenmodes of the underlying classical dynamics. We then argue that, provided classical correlation functions decay exponentially in time, there is an energy domain where the zero mode

Chaos, Interactions, and Nonequilibrium Effects in the Tunneling Resonance Spectra of Ultrasmall Metallic Particles

We explain the observation of clusters in the tunneling resonance spectra of small metallic particles of few nanometer size. Each cluster of resonances is identified with one excited single--electron state of the metal particle, shifted as a result of the different nonequilibrium occupancy configurations of the other single--electron states. Assuming the underlying classical dynamics of the electrons to be chaotic, we determine the typical shift to be

Shot noise in chaotic systems: "classical" to quantum crossover.

\Delta/g

Adaptation of autocatalytic fluctuations to diffusive noise.

where

Semiclassical field theory approach to quantum chaos

g

“Quantum phase transitions” in classical nonequilibrium processes☆

is the dimensionless conductance of the grain.

Semiclassical evolution of the spectral curve in the normal random matrix ensemble as Whitham hierarchy

This paper is devoted to study of the classical-to-quantum crossover of the shot noise in chaotic systems. This crossover is determined by the ratio of the particle dwell time in the system, tau(d), to the characteristic time for diffraction t(E) approximately lambda(-1)|lnh, where lambda is the Lyapunov exponent. The shot noise vanishes when t(E)>>tau(d), while it reaches a universal value in the opposite limit. Thus, the Lyapunov exponent of chaotic mesoscopic systems may be found by shot noise measurements.