• Volume 73, Issue 3

September 2009,   pages  215-1

• Preface

• Some recent developments in non-equilibrium statistical physics

We first recall the laws of classical thermodynamics and the fundamental principles of statistical mechanics and emphasize the fact that the fluctuations of a system in macroscopic equilibrium, such as Brownian motion, can be explained by statistical mechanics and not by thermodynamics. In the vicinity of equilibrium, the susceptibility of a system to an infinitesimal external perturbation is related to the amplitude of the fluctuations at equilibrium (Einstein’s relation) and exhibits a symmetry discovered by Onsager. We shall then focus on the mathematical description of systems out of equilibrium using Markovian dynamics. This will allow us to present some remarkable relations derived during the last decade and valid arbitrarily far from equilibrium: the Gallavotti–Cohen fluctuation theorem and Jarzynski’s non-equilibrium work identities. These recent results will be illustrated by applying them to simple systems such as the Brownian ratchet model for molecular motors and the asymmetric exclusion process which is a basic example of a driven lattice gas.

• Chaotic systems in complex phase space

This paper examines numerically the complex classical trajectories of the kicked rotor and the double pendulum. Both of these systems exhibit a transition to chaos, and this feature is studied in complex phase space. Additionally, it is shown that the short-time and long-time behaviours of these two $\mathcal{PT}$ -symmetric dynamical models in complex phase space exhibit strong qualitative similarities.

• The structure of states and maps in quantum theory

The structure of statistical state spaces in the classical and quantum theories are compared in an interesting and novel manner. Quantum state spaces and maps on them have rich convex structures arising from the superposition principle and consequent entanglement. Communication channels (physical processes) in the quantum scheme of things are in one-to-one correspondence with completely positive maps. Positive maps which are not completely positive do not correspond to physical processes. Nevertheless they prove to be invaluable mathematical tools in establishing or witnessing entanglement of mixed states. We consider some of the recent developments in our understanding of the convex structure of states and maps in quantum theory, particularly in the context of quantum information theory.

• Entanglement in non-Hermitian quantum theory

Entanglement is one of the key features of quantum world that has no classical counterpart. This arises due to the linear superposition principle and the tensor product structure of the Hilbert space when we deal with multiparticle systems. In this paper, we will introduce the notion of entanglement for quantum systems that are governed by non-Hermitian yet $\mathcal{PT}$ -symmetric Hamiltonians. We will show that maximally entangled states in usual quantum theory behave like non-maximally entangled states in $\mathcal{PT}$ -symmetric quantum theory. Furthermore, we will show how to create entanglement between two $\mathcal{PT}$ qubits using non-Hermitian Hamiltonians and discuss the entangling capability of such interaction Hamiltonians that are non-Hermitian in nature.

• Minimal classical communication and measurement complexity for quantum information splitting of a two-qubit state

We investigate the usefulness of the highly entangled five-partite cluster and Brown states for the quantum information splitting (QIS) of a special kind of two-qubit state using remote state preparation. In our schemes, the information that is to be shared is known to the sender. We show that, QIS can be accomplished with just two classical bits, as opposed to four classical bits, when the information that is to be shared is unknown to the sender. The present algorithm, demonstrated through the cluster and Brown states is deterministic as compared to the previous works in which it was probabilistic.

• Transition from Poisson to circular unitary ensemble

Transitions to universality classes of random matrix ensembles have been useful in the study of weakly-broken symmetries in quantum chaotic systems. Transitions involving Poisson as the initial ensemble have been particularly interesting. The exact two-point correlation function was derived by one of the present authors for the Poisson to circular unitary ensemble (CUE) transition with uniform initial density. This is given in terms of a rescaled symmetry breaking parameter Λ. The same result was obtained for Poisson to Gaussian unitary ensemble (GUE) transition by Kunz and Shapiro, using the contour-integral method of Brezin and Hikami. We show that their method is applicable to Poisson to CUE transition with arbitrary initial density. Their method is also applicable to the more general $\ell$CUE to CUE transition where CUE refers to the superposition of $\ell$ independent CUE spectra in arbitrary ratio.

• Random matrix ensembles with random interactions: Results for $EGUE(2)-SU (4)$

We introduce in this paper embedded Gaussian unitary ensemble of random matrices, for m fermions in 𝛺 number of single particle orbits, generated by random two-body interactions that are $SU(4)$ scalar, called EGUE$(2)-SU (4)$. Here the $SU (4)$ algebra corresponds to Wigner’s supermultiplet $SU (4)$ symmetry in nuclei. Formulation based on Wigner–Racah algebra of the embedding algebra $U (4\Omega) \supset U (\Omega) \bigotimes SU (4)$ allows for analytical treatment of this ensemble and using this analytical formulas are derived for the covariances in energy centroids and spectral variances. It is found that these covariances increase in magnitude as we go from EGUE(2) to EGUE(2)-s to EGUE(2)-$SU (4)$ implying that symmetries may be responsible for chaos in finite interacting quantum systems.

• A random matrix approach to RNA folding with interaction

In the matrix model of RNA [G Vernizzi, H Orland and A Zee, Phys. Rev. Lett. 94, 168103 (2005)] we introduce external interactions on n bases in the action of the partition function where $n \leq L$ and 𝐿 is the length of the polymer chain. The RNA structures found in the model can be separated into two regimes: (i) $0 \leq \alpha \leq 1$, $n &lt; L$ and $0 \leq \alpha &lt; 1$, $n = L$ where unpaired and paired base structures exist and (ii) $\alpha = 1$, $n = L$ with only completely paired base structures. Figures for the genus distribution functions show differences at small lengths. We consider the situation when the strength of external perturbation is different on different bases in the polymer chain.

• Resonance eigenfunctions in chaotic scattering systems

We study the semiclassical structure of resonance eigenstates of open chaotic systems. We obtain semiclassical estimates for the weight of these states on different regions in phase space. These results imply that the long-lived right (left) eigenstates of the non-unitary propagator are concentrated in the semiclassical limit $\hbar \rightarrow 0$ on the backward (forward) trapped set of the classical dynamics. On this support the eigenstates display a self-similar behaviour which depends on the limiting decay rate.

• Probabilistic interpretation of resonant states

We provide probabilistic interpretation of resonant states. We do this by showing that the integral of the modulus square of resonance wave functions (i.e., the conventional norm) over a properly expanding spatial domain is independent of time, and therefore leads to probability conservation. This is in contrast with the conventional employment of a bi-orthogonal basis that precludes probabilistic interpretation, since wave functions of resonant states diverge exponentially in space. On the other hand, resonant states decay exponentially in time, because momentum leaks out of the central scattering area. This momentum leakage is also the reason for the spatial exponential divergence of resonant state. It is by combining the opposite temporal and spatial behaviours of resonant states that we arrive at our probabilistic interpretation of these states. The physical need to normalize resonant wave functions over an expanding spatial domain arises because particles leak out of the region which contains the potential range and escape to infinity, and one has to include them in the total count of particles.

• Statistics of resonances in one-dimensional continuous systems

We study the average density of resonances (DOR) of a disordered one-dimensional continuous open system. The disordered system is semi-infinite, with white-noise random potential, and it is coupled to the external world by a semi-infinite continuous perfect lead. Our main result is an integral representation for the DOR which involves the probability density function of the logarithmic derivative of the wave function at the contact point.

• Effective mass theory of a two-dimensional quantum dot in the presence of magnetic field

The effective mass of electrons in low-dimensional semiconductors is position-dependent. The standard kinetic energy operator of quantum mechanics for this position-dependent mass is non-Hermitian and needs to be modified. This is achieved by imposing the BenDaniel–Duke (BDD) boundary condition. We have investigated the role of this boundary condition for semiconductor quantum dots (QDs) in one, two and three dimensions. In these systems the effective mass m i inside the dot of size R is different from the mass m o outside. Hence a crucial factor in determining the electronic spectrum is the mass discontinuity factor $\beta = m_{i} /m_{o}$ . We have proposed a novel quantum scale, 𝜎, which is a dimensionless parameter proportional to $\beta{2}R^{2}V_{0}$ , where $V_{0}$ represents the barrier height. We show both by numerical calculations and asymptotic analysis that the ground state energy and the surface charge density, $(\rho(R))$, can be large and dependent on 𝜎. We also show that the dependence of the ground state energy on the size of the dot is infraquadratic. We also study the system in the presence of magnetic field 𝐵. The BDD condition introduces a magnetic length-dependent term $(\sqrt{\hbar /eB})$ into 𝜎 and hence the ground state energy. We demonstrate that the significance of BDD condition is pronounced at large 𝑅 and large magnetic fields. In many cases the results using the BDD condition is significantly different from the non-Hermitian treatment of the problem.

• Re-appraisal of the $P$, $T$ -odd interaction constant $W_{d}$ in YbF: Relativistic configuration interaction approach

Restricted active space (RAS) configuration interaction (CI) approach is employed to compute the $P$, $T$ -odd interaction constant $W_{d}$ for the ground ($^{2} \sum_{1/2}$ ) state of YbF molecule. The present estimate of $W_{d} = −1.164 \times 10^{25}$ Hz/e-cm is expected to provide a reliable limit on the electron's electric dipole moment (EDM), $d_{e}$.

• On the concept of spectral singularities

In this paper, we discuss the concept of spectral singularities for non-Hermitian Hamiltonians. We exihibit spectral singularities of some well-known concrete Hamiltonians with complex-valued coefficients.

• Octonion wave equation and tachyon electrodynamics

The octonion wave equation is discussed to formulate the localization spaces for subluminal and superluminal particles. Accordingly, tachyon electrodynamics is established to obtain a consistent and manifestly covariant equation for superluminal electromagnetic fields. It is shown that the true localization space for bradyons (subluminal particles) is $R^{4}$ - (three space and one time dimensions) space while that for the description of tachyons is $T^{4}$ - (three time and one space dimensions) space.

• Quantum information paradox: Real or fictitious?

One of the outstanding puzzles of theoretical physics is whether quantum information indeed gets lost in the case of black hole (BH) evaporation or accretion. Let us recall that quantum mechanics (QM) demands an upper limit on the acceleration of a test particle. On the other hand, it is pointed out here that, if a Schwarzschild BH exists, the acceleration of the test particle would blow up at the event horizon in violation of QM. Thus the concept of an exact BH is in contradiction with QM and quantum gravity (QG). It is also reminded that the mass of a BH actually appears as an integration constant of Einstein equations. And it has been shown that the value of this integration constant is actually zero! Thus even classically, there cannot be finite mass BHs though zero mass BH is allowed. It has been further shown that during continued gravitational collapse, radiation emanating from the contracting object gets trapped within it by the runaway gravitational field. As a consequence, the contracting body attains a quasi-static state where outward trapped radiation pressure gets balanced by inward gravitational pull and the ideal classical BH state is never formed in a finite proper time. In other words, continued gravitational collapse results in an `eternally collapsing object' which is a ball of hot plasma and which is asymptotically approaching the true BH state with $M = 0$ after radiating away its entire mass energy. And if we include QM, this contraction must halt at a radius suggested by the highest QM acceleration. In any case no event horizon (EH) is ever formed and in reality, there is no quantum information paradox.

• List of participants

• Organizing Committee

• # Pramana – Journal of Physics

Current Issue
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December 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019