Vibrational spectrum of Granular packings with random matrices.

The vibrational spectrum of granular packings can be used as a signature of the jamming transition, with the density of states at zero frequency becoming nonzero at the transition. It has been proposed previously that the vibrational spectrum of granular packings can be approximately obtained from random matrix theory. Here, we show that the autocorrelation function of the density of states shows good agreement between dynamical numerical simulations of frictionless bead packs near the jamming point and the analytic predictions of the Laguerre orthogonal ensemble of random matrices; there is clear disagreement with the Gaussian orthogonal ensemble, establishing that the Laguerre ensemble correctly reproduces the universal statistical properties of jammed granular matter and excluding the Gaussian orthogonal ensemble. We also present a random lattice model which is a physically motivated variant of the random matrix ensemble. Numerical calculations reveal that this model reproduces the known features of the vibrational density of states of frictionless granular matter, while also retaining the correlation structure seen in the Laguerre random matrix theory.

Thermal transport across membranes and the Kapitza length from photothermal microscopy.

An analytical model is presented for light scattering associated with heat transport near a cell membrane that divides a complex system into two topologically distinct half-spaces. Our analysis is motivated by experiments on vibrational photothermal microscopy which have not only demonstrated remarkably high contrast and resolution, but also are capable of providing label-free local information of heat transport in complex morphologies. In the first Born approximation, the derived Green's function leads to the reconstruction of a full 3D image with photothermal contrast obtained using both amplitude and phase detection of periodic excitations. We show that important fundamental parameters including the Kapitza length and Kapitza resistance can be derived from experiments. Our goal is to spur additional experimental studies with high-frequency modulation and heterodyne detection in order to make contact with recent theoretical molecular dynamics calculations of thermal transport properties in membrane systems.

Conjugate Acid-Base Interaction Driven Phase Transition at a 2D Air-Water Interface.

A lattice model is described to explain a recent striking Sum Frequency Generation (SFG) observation of a cooperative surface adsorption effect for an organic acid system at an air-water interface. The reported anomalous pH-dependent enhancement in p-methylbenzoic acid (pmBA) arises from an interaction between the acid (HA) and its conjugate base anion (A-), which competes with strong Coulombic repulsion between the conjugate bases (A--A -). Using a statistical mechanical approach, this lattice gas model reveals an analogy to well-studied magnetic systems in which the attraction between the two different molecular species leads to a phase transition to a two-dimensional checkerboard phase consisting of a network of anion-acid complexes formed at the low-dielectric air-water interface. Cooperative acid-anion interactions that control partitioning at solution and aerosol interfaces are of interest to fields ranging from oceanic and atmospheric chemistry, pharmacology, and chemical engineering.

Glassy dynamics in disordered oscillator chains.

The escape of energy injected into one site in a disordered chain of nonlinear oscillators is examined numerically. When the disorder has a "fractal" pattern, the decay of the residual energy at the injection site can be fit to a stretched exponential with an exponent that varies continuously with the control parameter. At low temperature, we see evidence that energy can be trapped for an infinite time at the original site, i.e., classical many body localization.

Driven inelastic Maxwell gas in one dimension.

A lattice version of the driven inelastic Maxwell gas is studied in one dimension with periodic boundary conditions. Each site i of the lattice is assigned with a scalar "velocity," v_{i}. Nearest neighbors on the lattice interact, with a rate τ_{c}^{-1}, according to an inelastic collision rule. External driving, occurring with a rate τ_{w}^{-1}, sustains a steady state in the system. A set of closed coupled equations for the evolution of the variance and the two-point correlation is found. Steady-state values of the variance, as well as spatial correlation functions, are calculated. It is shown exactly that the correlation function decays exponentially with distance, and the correlation length for a large system is determined. Furthermore, the spatiotemporal correlation C(x,t)=〈v_{i}(0)v_{i+x}(t)〉 can also be obtained. We find that there is an interior region -x^{*}x^{*}, the correlation function remains the same as the initial form. C(x,t) exhibits second-order discontinuity at the transition points x=±x^{*}, and these transition points move away from the x=0 with a constant speed.

Linear response formula for open systems.

An exact expression for the finite frequency response of open classical systems coupled to reservoirs is obtained. The result is valid for any conserved current. No assumption is made about the reservoirs apart from thermodynamic equilibrium. At nonzero frequencies, the expression involves correlation functions of boundary currents and cannot be put in the standard Green-Kubo form involving currents inside the system.

Linear-response formula for finite-frequency thermal conductance of open systems.

An exact linear-response expression is obtained for the heat current in a classical Hamiltonian system coupled to heat baths with time-dependent temperatures. The expression is equally valid at zero and finite frequencies. We present numerical results on the frequency dependence of the response function for three different one-dimensional models of coupled oscillators connected to Langevin baths with oscillating temperatures. For momentum conserving systems, a low-frequency peak is seen that is higher than the zero-frequency response for large systems. For momentum nonconserving systems, there is no low-frequency peak. The momentum nonconserving system is expected to satisfy Fourier's law; however, at the single bond level, we do not see any clear agreement with the predictions of the diffusion equation even at low frequencies. We also derive an exact analytical expression for the response of a chain of harmonic oscillators to a (not necessarily small) temperature difference; the agreement with the linear-response simulation results for the same system is excellent.

Large-scale curvature of networks.

Understanding key structural properties of large-scale networks is crucial for analyzing and optimizing their performance and improving their reliability and security. Here, through an analysis of a collection of data networks across the globe as measured and documented by previous researchers, we show that communications networks at the Internet protocol (IP) layer possess global negative curvature. We show that negative curvature is independent of previously studied network properties, and that it has a major impact on core congestion: the load at the core of a finite negatively curved network with N nodes scales as N(2), as compared to N(1.5) for a generic finite flat network.

Scaling of load in communications networks.

We show that the load at each node in a preferential attachment network scales as a power of the degree of the node. For a network whose degree distribution is p(k)∼k{-γ} , we show that the load is l(k)∼k{η} with η=γ-1 , implying that the probability distribution for the load is p(l)∼1/l{2} independent of γ . The results are obtained through scaling arguments supported by finite size scaling studies. They contradict earlier claims, but are in agreement with the exact solution for the special case of tree graphs. Results are also presented for real communications networks at the IP layer, using the latest available data. Our analysis of the data shows relatively poor power-law degree distributions as compared to the scaling of the load versus degree. This emphasizes the importance of the load in network analysis.

Continuum and lattice heat currents for oscillator chains.

We show that two commonly used definitions for the heat current give different results-through the Kubo formula-for the heat conductivity of oscillator chains. The difference exists for finite chains, and is expected to be important more generally for small structures. For a chain of N particles that are tethered at the ends, the ratio of the heat conductivities calculated with the two currents differs from unity by O(1/N). For a chain held at constant pressure, the difference from unity decays more slowly, and is consistent with O(1/Neta) with 1>eta>0.5.

Exact identity for nonlinear wave propagation.

An exact identity for nonlinear sound wave propagation in one-dimensional media is shown to apply under a wide variety of conditions. The evidence suggests that this is true whenever there is a single input and output channel on each side of the medium, which can generally be achieved using monochromatic filters. It is shown mathematically that the identity cannot be derived from this condition and the known symmetries of the system. The question of what hidden symmetries in the system lead to the identity, and whether it applies even more generally to light waves, remains unresolved.

Equilibration and universal heat conduction in fermi-pasta-ulam chains.

It is shown numerically that for Fermi-Pasta-Ulam (FPU) chains with alternating masses and heat baths at slightly different temperatures at the ends, the local temperature (LT) on small scales behaves paradoxically in steady state. This expands the long established problem of equilibration of FPU chains. A well-behaved LT appears to be achieved for equal mass chains; the thermal conductivity is shown to diverge with chain length N as N(1/3), relevant for the much debated question of the universality of one-dimensional heat conduction. The reason why earlier simulations have obtained systematically higher exponents is explained.

Energy dissipation and fluctuation response for particles in fluids.

An equality was recently proved relating energy dissipation to the difference of the response and velocity correlation functions for a class of Langevin equations. We generalize this for the physically important case of particles in a fluid, where bath fluctuations are nonlocal in time due to hydrodynamic modes. We also show that the inclusion of a mass term does not alter the result and provide a simple physical interpretation of the original equality.

Universality of one-dimensional heat conductivity.

We show analytically that the heat conductivity of oscillator chains diverges with system size N as N(1/3), which is the same as for one-dimensional fluids. For long cylinders, we use the hydrodynamic equations for a crystal in one dimension. This is appropriate for stiff systems such as nanotubes, where the eventual crossover to a fluid only sets in at unrealistically large . Despite the extra equation compared to a fluid, the scaling of the heat conductivity is unchanged. For strictly one-dimensional chains, we show that the dynamic equations are those of a fluid at all length scales even if the static order extends to very large . The discrepancy between our results and numerical simulations on Fermi-Pasta-Ulam chains is discussed.

Hysteresis multicycles in nanomagnet arrays.

We predict two physical effects in arrays of single-domain nanomagnets by performing simulations using a realistic model Hamiltonian and physical parameters. First, we find hysteretic multicycles for such nanomagnets. The simulation uses continuous spin dynamics through the Landau-Lifshitz-Gilbert (LLG) equation. In some regions of parameter space, the probability of finding a multicycle is as high as approximately 0.6 . We find that systems with larger and more anisotropic nanomagnets tend to display more multicycles. Our results also demonstrate the importance of disorder and frustration for multicycle behavior. Second, we show that there is a fundamental difference between the more realistic vector LLG equation and scalar models of hysteresis, such as Ising models. In the latter case spin and external field inversion symmetry is obeyed, but in the former it is destroyed by the dynamics, with important experimental implications.

Return to return point memory.

We describe a new class of systems exhibiting return point memory (RPM), different from those discussed before in the context of ferromagnets. We show numerically that one-dimensional random Ising antiferromagnets have exact RPM when evolving from a large field, but not when started at finite field, unlike the ferromagnetic case. This implies that the standard approach to understanding ferromagnetic RPM will fail for this case. We also demonstrate RPM with a set of variables that keeps track of spin flips at each site. Conventional RPM for the spins is a projection of this result, suggesting that spin flip variables might be a more fundamental representation of the dynamics. We also present a mapping that embeds the antiferromagnetic chain in a two-dimensional ferromagnet, and prove RPM for spin-exchange dynamics in the interior of the chain with this mapping.

Correlations and scaling in one-dimensional heat conduction.

We examine numerically the full spatiotemporal correlation functions for all hydrodynamic quantities for the random collision model introduced recently. The autocorrelation function of the heat current, through the Kubo formula, gives a thermal conductivity exponent of 1/3 in agreement with the analytical prediction and previous numerical work. Remarkably, this result depends crucially on the choice of boundary conditions: for periodic boundary conditions (as opposed to open boundary conditions with heat baths) the exponent is approximately 1/2. All primitive hydrodynamic quantities scale with the dynamic critical exponent predicted analytically.

Subharmonics and aperiodicity in hysteresis loops.

We show that it is possible to have hysteretic behavior for magnets that does not form simple closed loops in steady state, but cycles multiple times before returning to its initial state. We show this by studying the low temperature dynamics of the 3D Edwards-Anderson spin glass. The specific multiple varies from system to system and is often quite large and increases with system size. The last result suggests that the magnetization could be aperiodic in the large system limit for some realizations of randomness. It should be possible to observe this phenomenon experimentally.

One-dimensional heat conductivity exponent from a random collision model.

We obtain numerically the thermal conductivity of a quasi-one-dimensional classical chain of hard sphere particles as a function of the length of the chain, introducing a fresh model for this problem. The conductivity scales as a power law of the length over two decades, with an exponent very close to the analytical prediction of 1/3.