The vibrational energy flow transition in organic molecules: Theory meets experiment

Most large dynamical systems are thought to have ergodic dynamics, whereas small systems may not have free interchange of energy between degrees of freedom. This assumption is made in many areas of chemistry and physics, ranging from nuclei to reacting molecules and on to quantum dots. We examine the transition to facile vibrational energy flow in a large set of organic molecules as molecular size is increased. Both analytical and computational results based on local random matrix models describe the transition to unrestricted vibrational energy flow in these molecules. In particular, the models connect the number of states participating in intramolecular energy flow to simple molecular properties such as the molecular size and the distribution of vibrational frequencies. The transition itself is governed by a local anharmonic coupling strength and a local state density. The theoretical results for the transition characteristics compare well with those implied by experimental measurements using IR fluorescence spectroscopy of dilution factors reported by Stewart and McDonald [Stewart, G. M. & McDonald, J. D. (1983) J. Chem. Phys. 78, 3907–3915].

Asymptotic properties of stationary sequences.

Bibliography: p. 146.

Birth-death processes with disaster and instantaneous resurrection

A new structure with the special property that instantaneous resurrection and mass disaster are imposed on an ordinary birth-death process is considered. Under the condition that the underlying birth-death process is exit or bilateral, we are able to give easily checked existence criteria for such Markov processes. A very simple uniqueness criterion is also established. All honest processes are explicitly constructed. Ergodicity properties for these processes are investigated. Surprisingly, it can be proved that all the honest processes are not only recurrent but also ergodic without imposing any extra conditions. Equilibrium distributions are then established. Symmetry and reversibility of such processes are also investigated. Several examples are provided to illustrate our results.

Existence and uniqueness of Q-processes with a given finite μ-invariant measure

Let Q be a stable and conservative Q-matrix over a countable state space S consisting of an irreducible class C and a single absorbing state 0 that is accessible from C. Suppose that Q admits a finite mu-subinvariant measure in on C. We derive necessary and sufficient conditions for there to exist a Q-process for which m is mu-invariant on C, as well as a necessary condition for the uniqueness of such a process.

Subgeometric rates of convergence for a class of continuous-time Markov process

Let (Phi(t))(t is an element of R+) be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure pi. We investigate the rates of convergence of the transition function P-t(x, (.)) to pi; specifically, we find conditions under which r(t) vertical bar vertical bar P-t (x, (.)) - pi vertical bar vertical bar -> 0 as t -> infinity, for suitable subgeometric rate functions r(t), where vertical bar vertical bar - vertical bar vertical bar denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

On the existence of uni-instantaneous Q-processes with a given finite mu-invariant measure

Let S be a countable set and let Q = (q(ij), i, j is an element of S) be a conservative q-matrix over S with a single instantaneous state b. Suppose that we are given a real number mu >= 0 and a strictly positive probability measure m = (m(j), j is an element of S) such that Sigma(i is an element of S) m(i)q(ij) = -mu m(j), j 0 b. We prove that there exists a Q-process P(t) = (p(ij) (t), i, j E S) for which m is a mu-invariant measure, that is Sigma(i is an element of s) m(i)p(ij)(t) = e(-mu t)m(j), j is an element of S. We illustrate our results with reference to the Kolmogorov 'K 1' chain and a birth-death process with catastrophes and instantaneous resurrection.

Calculation of the microcanonical temperature for the classical Bose field

The ergodic hypothesis asserts that a classical mechanical system will in time visit every available configuration in phase space. Thus, for an ergodic system, an ensemble average of a thermodynamic quantity can equally well be calculated by a time average over a sufficiently long period of dynamical evolution. In this paper, we describe in detail how to calculate the temperature and chemical potential from the dynamics of a microcanonical classical field, using the particular example of the classical modes of a Bose-condensed gas. The accurate determination of these thermodynamics quantities is essential in measuring the shift of the critical temperature of a Bose gas due to nonperturbative many-body effects.

Symbolic Dynamics in the Free-Fall Equal-Mass Three-Body Problem

The paper has been presented at the 12th International Conference on Applications of Computer Algebra, Varna, Bulgaria, June, 2006

Maximal Number of Successors in a Nginar(1) Process

2000 Mathematics Subject Classification: 60J80, 60J20, 60J10, 60G10, 60G70, 60F99.

Achievable rates and outage probability of cognitive radio with dynamic frequency hopping under imperfect spectrum sensing

In this study, the authors propose simple methods to evaluate the achievable rates and outage probability of a cognitive radio (CR) link that takes into account the imperfectness of spectrum sensing. In the considered system, the CR transmitter and receiver correlatively sense and dynamically exploit the spectrum pool via dynamic frequency hopping. Under imperfect spectrum sensing, false-alarm and miss-detection occur which cause impulsive interference emerged from collisions due to the simultaneous spectrum access of primary and cognitive users. That makes it very challenging to evaluate the achievable rates. By first examining the static link where the channel is assumed to be constant over time, they show that the achievable rate using a Gaussian input can be calculated accurately through a simple series representation. In the second part of this study, they extend the calculation of the achievable rate to wireless fading environments. To take into account the effect of fading, they introduce a piece-wise linear curve fitting-based method to approximate the instantaneous achievable rate curve as a combination of linear segments. It is then demonstrated that the ergodic achievable rate in fast fading and the outage probability in slow fading can be calculated to achieve any given accuracy level.

Optimal Power Allocation for Multiuser Secure Communication in Cooperative Relaying Networks

We consider a cooperative relaying network in which a source communicates with a group of users in the presence of one eavesdropper. We assume that there are no source-user links and the group of users receive only retransmitted signal from the relay. Whereas, the eavesdropper receives both the original and retransmitted signals. Under these assumptions, we exploit the user selection technique to enhance the secure performance. We first find the optimal power allocation strategy when the source has the full channel state information (CSI) of all links. We then evaluate the security level through: i) ergodic secrecy rate and ii) secrecy outage probability when having only the statistical knowledge of CSIs.

Beamforming and interference cancellation in D2D communication underlaying cellular networks

This paper presents an analytical performance investigation of both beamforming (BF) and interference cancellation (IC) strategies for a device-to-device (D2D) communication system underlaying a cellular network with an M-antenna base station (BS). We first derive new closed-form expressions for the ergodic achievable rate for BF and IC precoding strategies with quantized channel state information (CSI), as well as, perfect CSI. Then, novel lower and upper bounds are derived which apply for an arbitrary number of antennas and are shown to be sufficiently tight to the Monte-Carlo results. Based on these results, we examine in detail three important special cases including: high signal-to-noise ratio (SNR), weak interference between cellular link and D2D link, and BS equipped with a large number of antennas. We also derive asymptotic expressions for the ergodic achievable rate for these scenarios. Based on these results, we obtain valuable insights into the impact of the system parameters, such as the number of antennas, SNR and the interference for each link. In particular, we show that an irreducible saturation point exists in the high SNR regime, while the ergodic rate under IC strategy is verified to be always better than that under BF strategy. We also reveal that the ergodic achievable rate under perfect CSI scales as log2M, whilst it reaches a ceiling with quantized CSI.

Beamforming and interference cancellation schemes for D2D communications

We investigate device-to-device (D2D) communication underlaying cellular networks with M-antenna base stations. We consider both beamforming (BF) and interference cancellation (IC) strategies under quantized channel state information (CSI), as well as, perfect CSI. We derive tight closed-form approximations of the ergodic achievable rate which hold for arbitrary transmit power, location of users and number of antennas. Based on these approximations, we derive insightful asymptotic expressions for three special cases namely high signal-to-noise (SNR), weak interference, and large M. In particular, we show that in the high SNR regime a ceiling effect exists which depends on the received signal-to-interference ratio and the number of antennas. Moreover, the achievable rate scales logarithmically with M. The ergodic achievable rate is shown to scale logarithmically with SNR and the antenna number in the weak interference case. When the BS is equipped with large number of antennas, we find that the ergodic achievable rate under quantized CSI reaches a saturated value, whilst it scales as log2M under perfect CSI.

Massive MIMO with I/Q imbalance: Performance analysis and compensation

In this paper, we consider the uplink of a single-cell massive multiple-input multiple-output (MIMO) system with inphase and quadrature-phase imbalance (IQI). This scenario is of particular importance in massive MIMO systems, where the deployment of lower-cost, lower-quality components is desirable to make massive MIMO a viable technology. Particularly, we investigate the effect of IQI on the performance of massive MIMO employing maximum-ratio combining (MRC) receivers. In order to study how IQI affects channel estimation, we derive a new channel estimator for the IQI-impaired model and show that IQI can substantially downgrade the performance of MRC receivers. Moreover, a low-complexity IQI compensation scheme, suitable for massive MIMO, is proposed which is based on the IQI coefficients' estimation and it is independent of the channel gain. The performance of the proposed compensation scheme is analytically evaluated by deriving a tractable approximation of the ergodic achievable rate and providing the asymptotic power scaling laws assuming transmission over Rayleigh fading channels with log-normal large-scale fading. Finally, we show that massive MIMO effectively suppresses the residual IQI effects, as long as, the compensation scheme is applied.