Session Z33: Statistical and Nonlinear Physics

Synchronization of phase oscillators in large complex networks

It has been shown in recent years that many real world networks have a complex structure (e.g., scale-free networks). The effect of a complex interaction network on the dyamics of coupled dynamical systems is, therefore, of interest. An important aspect of the dynamics is the synchronization of coupled oscillators. I will present a generalization of the classical Kuramoto model of all-to-all coupled oscillators to the case of a general topology of the network of interactions. We find that for a large class of networks, there is still a transition from incoherence to coherent behavior at a critical coupling strength that depends on the largest eigenvalue of the adjacency matrix of the network. I will discuss the application of our theory to study the effect of heterogeneity in the degree distribution and degree-degree correlations in the network. Finally, I will comment on generalizations to more realistic dynamical systems.   

Weak Dynamic Links for Synchronizing Oscillator Arrays.

A novel synchronization mechanism observed in a model of coupled fiber laser arrays is explained [1]. The arrays can operate in a highly coherent way if some elements are driven more strongly than others. The synchronized state of such an inhomogeneous array, although sub-optimal relative to a uniformly pumped array, is far more robust with respect to parameter mismatch among the individual elements. Similar dynamical behavior might be useful for synchronizing more general coupled oscillator systems when amplitude dynamics is crucial.   

Competing Synchronization of Nonlinear Oscillators

Coupled nonlinear oscillators abound in nature and in man-made devices. Think for example of two neurons in the brain competing to get the attention of a third neuron, and eventually developing some sort of synchronization process. This is a common feature involving oscillators in general, and can be studied using numerical simulations and/or experimental setups. In this talk, results involving electronic circuits and plasma discharges will be presented showing interesting features related to the types of oscillators and to the types of couplings. In particular, for the case of two oscillators competing for synchronization with a third one, the target oscillator synchronizes alternately to one or the other of the competing oscillators. The time intervals of synchronous states vary in a random-like manner. Numerical and experimental results will be presented and the consistency between them will be discussed.   

Complexity, Parallel Computation and Statistical Physics

The intuition that a long history is required for the emergence of complexity in natural systems is formalized using the notion of {\em depth } (related to Bennett's {\em logical depth}). The depth of a system is defined in terms of the number of parallel computational steps needed to simulate it. Depth provides an objective, irreducible measure of history applicable to systems of the kind studied in statistical physics. The talk will review concepts of parallel computational complexity theory and then present results for the depth of several well-known model systems in non-equilibrium statistical physics. It is argued that physical complexity cannot occur in the absence of substantial depth and that depth is a useful proxy for physical complexity.   

Vortex-Phonon Interaction in the Kosterlitz-Thouless Theory

The ``canonical'' variables of the Kosterlitz-Thouless theory-- fields $\Phi_0({\bf r})$ and $\phi({\bf r})$, generally believed to stand for vortices and phonons (or their XY equivalents, like spin waves, etc.) turn out to be neither vortices and phonons, nor, strictly speaking, {\it canonical} variables. The latter fact explains paradoxes of (i) absence of interaction between $\Phi_0$ and $\phi$, and (ii) non-physical contribution of small vortex pairs to long-range phase correlations. We resolve the paradoxes by explicitly relating $\Phi_0$ and $\phi$ to canonical vortex-pair and phonon variables.   

Sticky Random Walks

Entangled strings is something we commonly observe. For example, wires for electrical appliances or communication network cords sometimes require us to disentangle them. We describe here a concept of sticky random walks to gain some insight into this phenomenon. The strings are represented by the trajectory of a random walker. This random walker leaves sticks or marks at certain time intervals. Therefore, a string is represented by this trajectory with these marks on it. By sending out multiple sticky random walkers, we obtained multiple sticky strings. Furthermore, a string is considered as entangled with another when these marks overlap at the same site in space, and not when they are simply crossed. Thus, the string is considered more sticky when there are more marks on it. We tested a situation having multiple sticky strings in a bounded two-dimensional square grid by sending out sticky random walks in this space. We found that in certain situations, the optimal balance between stickiness and number of strings gives most entangled situation.   

Finite-Connectivity Spin-Glass Phase Diagrams and Low Density Parity Check Codes

We present phase diagrams of finite connectivity spin-glasses. We firstly compare the properties of the phase diagrams with the performance of low density parity check codes (LDPC) within the Replica Symmetric (RS) ansatz. We study the location of the dynamical and critical transition points within the one step Replica Symmetry Breaking theory (RSB), extending similar calculations that have been performed in the past for the Bethe spin-glass. The location of the dynamical transition line {\em does} change within the RSB theory, when comparing with the results obtained in the RS case. For LDPC decoding of messages transmitted over the binary erasure channel (BEC) we find, at zero temperature and rate $R=1/4$ an RSB transition point located at $p_c \simeq 0.7450 \pm 0.0050$, to be compared with the corresponding Shannon bound $1-R$. For the binary symmetric channel (BSC) we show that the low temperature reentrant dynamical transition boundary occurs at higher values of the channel noise when comparing with the RS case. Possible practical implications to improve the performance of the state-of-the-art error correcting codes are discussed.   

Novel Approach in Statistical Physics for Accurate Multiscale Materials Investigation

This paper discusses a novel scheme recently developed by the author [(i)Edgal, U. F., J. Chem. Phys. \textbf{94}, 8179, 1991; (ii)Edgal, U. F. and Huber, D. L., J. Phys. Chem. B, \textbf{108}, 13777, 2004; (iii)Edgal, U. F. and Huber, D. L., accepted for publication in the journal ``Physica A'', 2005] for \textbf{accurately} determining the free energy of arbitrary equilibrium classical and quantum (material) systems at arbitrary densities, temperatures, and interaction potentials. Nearest neighbor probability density functions are formulated. The scheme allows us avert the ``sign'' problem usually encountered in Fermion calculations. Extension to mixed systems, as well as a novel ensemble, the ``Nearest Neighbor'' ensemble, used to effect the computational component of the novel approach are briefly discussed. .   

Reversal of motion induced by mechanical coupling in Brownian motors

Many studies of Brownian ratchets have dealt with the asymmetric pumping of individual point-like particles. Here, we consider the transport of objects with internal structure in a flashing ratchet potential by investigating the overdamped behavior of a rod-like chain of evenly spaced point particles. In 1D, analytical arguments show that the current can reverse direction multiple times in response to changing the size of the chain or the temperature of the heat bath. However, if the rods are allowed to rotate freely in 3D, or if their length is much less than the spatial period of the ratchet potential, current reversal is no longer observed, and the qualitative behavior of single particle motion is recovered. All analytical predictions are confirmed by Brownian dynamics simulations. These results are relevant to the design of novel particle separation technology, and may provide the basis for simple, coarse-grained models of molecular motor transport that incorporate an object's size and internal degrees of freedom into the mechanism of transport.   

Optimal Resonance Forcing of Nonlinear Systems

We study the response of dynamical systems to additive forcing and find that, for a broad class of systems, the response is maximized by a pattern of forcing that mimics the time-reversed dynamics of the unforced system. Applying these results, we numerically construct families of optimal inputs and successfully perform spectroscopic system identification on our modeled systems.   

Quantum wave packets in hard-disk and hard-sphere billiards

Analysis of quantum dynamics in systems with classically chaotic analogs constitutes one of the main objectives for the field of {\it Quantum Chaos}. The quantum dynamics is known to be determined, to a large extent, by chaotic features of counterpart classical systems. We address time evolution of wave packets in open chaotic billiards, in which a quantum particle travels among a collection of fixed scatterers taken to be hard disks or hard spheres in two or three spatial dimensions respectively. By studying the autocorrelation function for the wave packets we provide a detailed analysis of the phenomenon of wave packet partial reconstruction in the course of the time evolution, and discuss a close connection between the reconstruction dynamics and such important properties of the counterpart classical systems as the Lyapunov exponents, the Kolmogorov-Sinai and the topological entropies.   

Universal Impedance, Admittance and Scattering Fluctuations in Quantum-chaotic Systems.

We experimentally investigate fluctuations in the eigenvalues of the impedance, admittance and scattering matrices of wave chaotic systems using a microwave analog of a quantum chaotic infinite square well potential. We consider a 2-D, time-reversal symmetric chaotic microwave resonator driven by two non-ideally coupled ports. The system-specific coupling effects are removed using the measured radiation impedance matrix ($\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over {Z}} _{Rad} )$ [1] of the two ports. A normalized impedance matrix ($\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over {z}} )$ is thus obtained, and the Probability Density Function (PDF) of its eigenvalues is predicted to be universal depending only on the cavity loss. We observe remarkable agreement between the statistical properties of $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over {z}} $ and $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over {y}} =\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over {z}} ^{-1}$ for all degrees of loss, which is in accordance with [1, 2] and Random Matrix Theory (RMT). We compare the joint PDF of the eigenphases of the normalized scattering matrix ($\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over {s}} )$ with that obtained from RMT for varying degrees of loss. We study the joint PDF of the eigenvalues of $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over {s}} \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over {s}} ^{\dag }$ and find good agreement with [3]. [1] X. Zheng, \textit{et al.,} -- Electromagnetics (in press); condmat/0408317; S. Hemmady, \textit{et al}., Phys. Rev. Lett. \textbf{94}, 014102 (2005).[2] Y. V. Fyodorov, \textit{et al.},-- condmat/0507016.[3] P. W. Brouwer and C. W. J Beenakker -- PRB \textbf{55}, 4695 (1997). Work supported by DOD MURI AFOSR Grant F496200110374, DURIP Grants FA95500410295 and FA95500510240.   

Controlling transitions in a Duffing oscillator by sweeping the driving frequency.

We consider a high-$Q$ Duffing oscillator in a weakly non-linear regime with the driving frequency $\sigma$ varying in time between $\sigma_i$ and $\sigma_f$ at a characteristic rate $r$. We found that the frequency sweep can cause controlled transitions between two stable states of the system. Moreover, these transitions are accomplished via a transient that lingers for a long time around the third, unstable fixed point of saddle type. We propose a simple explanation for this phenomenon and find the transient life-time to scale as $-(\ln {|r-r_c|})/\lambda_r$ where $r_c$ is the critical rate necessary to induce a transition and $\lambda_r$ is the repulsive eigenvalue of the saddle. The same type of phenomena is expected to hold for a large class of driven nonlinear oscillators which are describable by a two-basin model.   

Structure and Complexity in Rule Ensemble Cellular Automata

Individual elementary cellular automata (ECA) rules are attractive models for a range of non-equilibrium physical systems, but rule ensembles remain poorly understood. This paper presents the first known analysis of the equally weighted ensemble of all ECA rules. Ensemble dynamics reveal persistent, localized, non-interacting structures strongly correlated by velocity and reminiscent of solitons, instead of equilibration. Dispersion from a single initial site generates peaks traveling at low-denominator fractional velocities, some of which are not discernable in individual rules, implying collective excitation. Principal component analysis of the rule space shows the ECA are dense (with $\sim $111 eigenrules out of 128 ECA rules, up to symmetry), but can be transformed to a simple basis set that is quasi-linear in initial conditions. These results suggest that the ECA, often considered to be the simplest nontrivial set of ``short program'' models for self-assembly, might be approximated well by computationally simpler models. This work also shows, surprisingly, that structure can develop without favoring a single evolution rule.   

Universality away from Critical Points: Collapse of Observables in a Thermostatistical Model

The $p$-state clock model in two dimensions is a discrete model exhibiting, for $p>4$, a quasi-liquid phase in a region $T_14$ and all $T>T_{\rm eu}$, all thermal averages become identical to those of the continuous, planar rotor model ($p=\infty$). This {\em collapse of thermodynamic observables} amounts to an emergent symmetry, not present in the Hamiltonian. For $p \ge 8$, the collapse starts in the quasi-liquid phase and makes the transition at $T_2$ indistinguishable from the Berezinskii-Kosterlitz-Thouless (BKT) transition of the planar rotor. For $p \le 6$, we find $T_{\rm eu} > T_2$, and the transition at $T_2$ is no longer BKT. The results include a detailed analysis of the critical properties at $T_1$ and $T_2 $. Broader implications are discussed.