Table of contents

Using the connection with the Frobenius manifold (FM) structure, we study the matrix model description of minimal Liouville gravity (MLG) based on the Douglas String equation. Our goal is to find an exact discrete formulation of the $(q,p)$ MLG model that intrinsically contains information about the conformal selection rules. We discuss how to modify the FM structure appropriately for this purposes. We propose a modification of the construction for Lee–Yang series involving the ${{A}_{p-1}}$ algebra instead of the previously used A₁ algebra. With the new prescription, we calculate correlators on the sphere up to four points and find full agreement with the continuous approach without using resonance transformations.

We discuss general features of the operator product expansion and use it to infer multi-point manifestations of the energy cascade in turbulence. We calculate explicitly the possible form of the three-point velocity correlation function when one distance is smaller than two others. We elucidate manifestation of direct and inverse energy cascades in the three-point velocity correlation function.

We present a descriptive review of physical problems dealing with extreme values in several fields of physics. We consider different physical situations involving random variables that are correlated or not, and study the statistics of extremal variables, which is relevant for situations where height fluctuations, catastrophic events such as material failure, or power outrage occur. We describe the general theory and relate the cumulative limit distributions that can be accessible in experiments to microscopic models. In many cases however, the random variables are correlated, in interface problems for example, and the characteristics of the interaction are revealed in the asymptotic behavior of the limit distribution.

A simple stochastic model of a self regulating gene that displays bistable switching is analyzed. While on, a gene transcribes mRNA at a constant rate. Transcription factors can bind to the DNA and affect the gene's transcription rate. Before an mRNA is degraded, it synthesizes protein, which in turn regulates gene activity by influencing the activity of transcription factors. Protein is slowly removed from the system through degradation. Depending on how the protein regulates gene activity, the protein concentration can exhibit noise induced bistable switching. An asymptotic approximation of the mean switching rate is derived that includes the pre exponential factor, which improves upon a previously reported logarithmically accurate approximation. With the improved accuracy, a uniformly accurate approximation of the stationary probability density, describing the gene, mRNA copy number, and protein concentration is also obtained.

We consider the dynamics and thermodynamics of a pair of magnetic dipoles interacting via their magnetic fields. We consider only the 'spin' degrees of freedom; the dipoles are fixed in space. With this restriction it is possible to provide the general solution of the equations of motion in analytical form. Thermodynamic quantities, such as the specific heat and the zero field susceptibility are calculated analytically or by combining low temperature asymptotic series and a complete high temperature expansion. The thermal expectation value of the autocorrelation function is determined for the low temperature regime and short times including terms linear in T. Furthermore, we have performed Monte Carlo simulations for the system under consideration and compared our analytical results with these.

We introduce a new superintegrable Kepler–Coulomb system with non-central terms in N-dimensional Euclidean space. We show this system is multiseparable and allows separation of variables in hyperspherical and hyperparabolic coordinates. We present the wave function in terms of special functions. We give an algebraic derivation of spectrum of the superintegrable system. We show how the $so(N+1)$ symmetry algebra of the N-dimensional Kepler–Coulomb system is deformed to a quadratic algebra with only three generators and structure constants involving a Casimir operator of $so(N-1)$ Lie algebra. We construct the quadratic algebra and the Casimir operator. We show this algebra can be realized in terms of deformed oscillator and obtain the structure function which yields the energy spectrum.

We rigorously define the self-adjoint one-dimensional Salpeter Hamiltonian perturbed by an attractive $\delta -interaction,$ of strength $\beta ,$ centred at the origin, by explicitly providing its resolvent. Our approach is based on a 'coupling constant renormalization', a technique used first heuristically in quantum field theory and implemented in the rigorous mathematical construction of the self-adjoint operator representing the negative Laplacian perturbed by the $\delta -interaction$ in two and three dimensions. We show that the spectrum of the self-adjoint operator consists of the absolutely continuous spectrum of the free Salpeter Hamiltonian and an eigenvalue given by a smooth function of the parameter $\pi /\beta .$ The method is extended to the model with two twin attractive deltas symmetrically situated with respect to the origin in order to show that the discrete spectrum of the related self-adjoint Hamiltonian consists of two eigenvalues, namely the ground state energy and that of the excited antisymmetric state. We investigate in detail the dependence of these two eigenvalues on the two parameters of the model, that is to say both the aforementioned strength $\beta$ and the separation distance. With regard to the latter, a remarkable phenomenon is observed: differently from the well-behaved Schrödinger case, the 1D-Salpeter Hamiltonian with two identical Dirac distributions symmetrically situated with respect to the origin does not converge, as the separation distance shrinks to zero, to the one with a single $\delta -interaction$ centred at the origin having twice the strength. However, the expected behaviour in the limit (in the norm resolvent sense) can be achieved by making the coupling of the twin deltas suitably dependent on the separation distance itself.

The experimental realization of successive non-demolition measurements on single microscopic systems brings up the question of ergodicity in quantum mechanics (QM). We investigate whether time averages over one realization of a single system are related to QM averages over an ensemble of similarly prepared systems. We adopt a generalization of the von Neumann model of measurement, coupling the system to N 'probes', with a strength that is at our disposal, and detecting the latter. The model parallels the procedure followed in experiments on quantum electrodynamic cavities. The modification of the probability of the observable eigenvalues due to the coupling to the probes can be computed analytically and the results compare qualitatively well with those obtained numerically by the experimental groups. We find that the problem is not ergodic, except in the case of an eigenstate of the observable being studied.

Quantum correlation lies at the very heart of almost all of the non-classical phenomena exhibited by quantum systems composed of two or more subsystems. In recent times it has been pointed out that there is a kind of quantum correlation, namely discord, which is more general than entanglement. Some authors have investigated the phenomenon that for certain initial states the quantum correlations as well as the classical correlations exhibit sudden change under simple Markovian noise. We show that this dynamical behavior of the correlations of both types can be explained using the idea of complementary correlations. We also show that though a certain class of mixed entangled states can resist the monotonic decay of quantum correlations, this is not true for all mixed states. Moreover, pure entangled states of two qubits will never exhibit such sudden change.

We investigate the finite temperature expectation values of the charge and current densities for a massive fermionic field with nonzero chemical potential, μ, in the geometry of a straight cosmic string with a magnetic flux running along its axis. These densities are decomposed into the vacuum expectation values and contributions coming from the particles and antiparticles. The charge density is an even periodic function of the magnetic flux with a period equal to the quantum flux and an odd function of the chemical potential. The only nonzero component of the current density corresponds to the azimuthal current. The latter is an odd periodic function of the magnetic flux and an even function of the chemical potential. At high temperatures, the parts of the charge density and azimuthal current induced by the planar angle deficit and magnetic flux are exponentially small. The asymptotic behavior at low temperatures crucially depends on whether the value $|\mu |$ is larger or smaller than the mass of the field quanta, m. For $|\mu |\lt m$ the charge density and the contributions to the azimuthal current from the particles and antiparticles are exponentially suppressed at low temperatures. In the case $|\mu |\gt m$, the charge and current densities receive two contributions from the vacuum expectation values and from the particles or antiparticles (depending on the sign of the chemical potential). At large distances from the string the latter exhibits a damping oscillatory behavior with the amplitude inversely proportional to the square of the distance.

Tensionless string theory on ${\rm Ad}{{{\rm S}}_{3}}\times {{{\rm S}}^{3}}\times {{\mathbb{T}}^{4}}$, as captured by a free symmetric product orbifold, has a large set of conserved currents which can be usefully organized in terms of representations of a $\mathcal{N}=(4,4)$ supersymmetric higher spin algebra. In this paper we focus on the single particle currents which generate the asymptotic stringy symmetry algebra on ${\rm Ad}{{{\rm S}}_{3}}$, and whose wedge modes describe the unbroken gauge symmetries of string theory in this background. We show that this global subalgebra contains two distinct higher spin algebras that generate the full algebra as a 'higher spin square'. The resulting unbroken stringy symmetry algebra is exponentially larger than the two individual higher spin algebras.

Two non-Hermitian fermion models are proposed and analyzed by using Foldy–Wouthuysen transformations. One model has Lorentz symmetry breaking and the other has a non-Hermitian mass term. It is shown that each model has real energies in a given region of parameter space, where they have a locally conserved current.

Different variants of hybrid kinetic-fluid models are considered for describing the interaction of a bulk fluid plasma obeying magnetohydrodynamics (MHD) and an energetic component obeying a kinetic theory. Upon using the Vlasov kinetic theory for energetic particles, two planar Vlasov-MHD models are compared in terms of their stability properties. This is made possible by the Hamiltonian structures underlying the considered hybrid systems, whose infinite number of invariants makes the energy-Casimir method effective for determining stability. Equilibrium equations for the models are obtained from a variational principle and in particular a generalized hybrid Grad–Shafranov equation follows for one of the considered models. The stability conditions are then derived and discussed with particular emphasis on kinetic particle effects on classical MHD stability.