Central Limit Theorems for the Energy Density in?the?Sherrington-Kirkpatrick Model

In this paper we consider central limit theorems for various macroscopic observables in the high temperature region of the Sherrington-Kirkpatrick spin glass model. With a particular focus on obtaining a quenched central limit theorem for the energy density of the system with non-zero external field, we show how to combine the mean field cavity method with Stein’s method in the quenched regime. The result for the energy density extends the corresponding result of Comets and Neveu in the case of zero external field.     

Driven response of time delay coupled limit cycle oscillators

We study the periodic forced response of a system of two limit cycle oscillators that interact with each other via a time delayed coupling. Detailed bifurcation diagrams in the parameter space of the forcing amplitude and forcing frequency are obtained for various interesting limits using numerical and analytical means. In particular, the effects of the coupling strength, the natural frequency spread of the two oscillators and the time delay parameter on the size and nature of the entrainment domain are delineated. For an appropriate choice of time delay, synchronization can occur with infinitesimal forcing amplitudes even at off-resonant driving. The system is also found to display a nonlinear response on certain critical contours in the space of the coupling strength and time delay. Numerical simulations with a large number of coupled driven oscillators display similar behavior. Time delay offers a novel tuning knob for controlling the system response over a wide range of frequencies and this may have important practical applications.     

Collective dynamics of delay-coupled limit cycle oscillators

Coupled limit cycle oscillators with instantaneous mutual coupling offer a useful but idealized mathematical paradigm for the study of collective behavior in a wide variety of biological, physical and chemical systems. In most real-life systems however the interaction is not instantaneous but is delayed due to finite propagation times of signals, reaction times of chemicals, individual neuron firing periods in neural networks etc. We present a brief overview of the effect of time-delayed coupling on the collective dynamics of such coupled systems. Simple model equations describing two oscillators with a discrete time-delayed coupling as well as those describing linear arrays of a large number of oscillators with time-delayed global or local couplings are studied. Analytic and numerical results pertaining to time delay induced changes in the onset and stability of amplitude death and phase-locked states are discussed. A number of recent experimental and theoretical studies reveal interesting new directions of research in this field and suggest exciting future areas of exploration and applications.     

The effect of nature and number of alkyl groups in tetraalkyl ammonium cations acting as templates during the synthesis of ZSM-5

Computer simulation studies have been applied to investigate the effect of alkyl groups in the structure directing property of the tetraalkyl ammonium cations (TAA) acting as templates during the synthesis of ZSM-5. The interaction between the model compounds has been studied to mimic the alkyl groups of the template molecule and the ZSM-5 framework as the nature and number of alkyl groups vary. Different cluster models were considered to study the extent of interaction with model compounds. The results of MNDO calculations show that the efficiency of TAA as template is dependent on the number and nature of alkyl groups present in them. The procedures used can successfully identify optimum templates for a given host and have applications in the synthesis of new microporous materials.     

Modeling and Simulation of Ionic Currents in Three-Dimensional Microfluidic Devices with Nanofluidic Interconnects

Electrokinetic fluid flow in nanocapillary array (NCA) membranes between vertically separated microfluidic channels offers an attractive alternative to using mechanical action to achieve fluidic communication between different regions of lab-on-a-chip devices. By adjusting the channel diameter, a, and the inverse Debye length, к, and applying the appropriate external potential, the nanochannel arrays, can be made to behave like digital fluidic switches, and the movement of molecules from one side of the array to the other side can be controlled. However, inherent differences in ionic mobility lead to non-equilibrium ion populations on the downstream side, which, in turn, shows up through transient changes in the microchannel conductance. Here we describe coupled calculations and experiments in which the electrical properties of a microfluidic–nanofluidic hybrid architecture are simulated by a combination of a compact model for the bulk electrical properties and iterative self-consistent solutions of the coupled Poisson, Nernst–Planck, and Navier–Stokes equations to recover the detailed ion motion in the nanopores. The transient electrical conductivity in the microchannel, after application of a forward bias pulse to the NCA membrane, is recovered in quantitative detail. The surface charge density of the nanopores and the capacitance of the membrane, which are critical determinants of electrokinetic flow through NCA, fall out of the analysis in a natural way, providing a clear mechanism to determine these critically important parameters.