Table of contents

Jarzynski's nonequilibrium work relation can be understood as the realization of the (hidden) time-generator reciprocal symmetry satisfied for the conditional probability function. To show this, we introduce the reciprocal process where the classical probability theory is expressed with real wave functions, and derive a mathematical relation using the symmetry. We further discuss that the descriptions by the standard Markov process from an initial equilibrium state are indistinguishable from those by the reciprocal process. Then the Jarzynski relation is obtained from the mathematical relation for the Markov processes described by the Fokker–Planck, Kramers and relativistic Kramers equations.

While the superconductor proximity effect is well understood in layered superconductor/normal-metal junctions, its understanding is quite limited in systems involving nanoparticles (NPs) and molecules. In recent studies, a unique inverse proximity effect phenomenon was found in which the critical temperatures of Nb films surprisingly increased upon the chemical attachment of gold NPs. Concomitantly, the tunneling density of states on and around the gold NPs was significantly modified, showing either zero-bias peaks or the development of proximity gaps in the NPs. These results seem to be related to the molecule-mediated coupling strength. Here, we study the strong molecular coupling regime of such an architecture, for which proximity gaps are induced in Au NPs. We show that significant pinning is induced in a periodic array of Au NPs coupled to a superconducting surface via organic molecules. The pinning potential in this case is stronger than the potential achieved through the direct proximity of Au or Ni islands to the superconducting surface. A matching field magnetoresistance signal can only be identified using the hybrid Au/organic-linker/Nb system. In this case, the matching vortex lattice density is higher than the saturation number. These results suggest that the NP-Nb electrical coupling through the molecules induces a resonance behavior, which modifies the local pairing amplitude.

Theoretic measures of information entropies like Shannon entropy and Fisher information are studied for multiple quantum well systems (MQWS). The effect of shape and number of wells in the MQWS is explored in detail. The shapes taken are: rectangular, parabolic and V-shape. Onicescu energy is an important tool to study the information content stored in the system, which is also found to depend on shape and number of wells of heterostructures. Statistical measure of complexity also shows noticeable dependence on these parameters.

A simultaneous variational principle is introduced that offers a novel avenue to the description of the equilibrium configurations, and at the same time of the elementary excitations, or undulations, of fluid lipid membranes, described by a geometric continuum free energy. The simultaneous free energy depends on the shape functions through the membrane stress tensor, and on an additional deformation spatial vector. Extremization of this free energy produces at once the Euler–Lagrange equations and the Jacobi equations, that describe elementary excitations, for the geometric free energy. As an added benefit, the energy of the elementary excitations, given by the second variation of the geometric free energy, is obtained without second variations. Although applied to the specific case of lipid membranes, this variational principle should be useful in any physical system where bending modes are dominant.

Network's synchronization destabilizes at large coupling strength when its nodes exchange information by scalar coupling (few coordinates) instead of using vector coupling (all coordinates). This issue is commonly tackled by modifying the network topology like a ring network to Small World or Scale Free networks which increases the coupling cost and decreases the robustness. In the present work, we show that without any structural alteration even a large ring network in the nearest neighborhood configuration using only a single coordinate in the coupling, could be made synchronizable. The primary condition is that the node dynamics should be given by a pair of oscillators (say, two oscillatory system TOS) rather than by a conventional way of single oscillator (say, single oscillatory system). It has been found that TOS not only stabilizes the chaotic synchronization but also the hyperchaotic synchronization manifold (a major challenge in the field of secure communication wherein multi parameter BK method is needed). The frameworks of drive-response system and master stability function have been used to study the TOS effect by varying TOS parameters with and without feedback (feedback means quorum sensing conditions). The TOS effect has been found numerically both in the chaotic (Rössler, Chua and Lorenz) and hyperchaotic (electrical circuit) systems. However, since threshold also increases as a side effect of TOS, the extent of β enhancement depends on the choice of oscillator model like larger for Rössler, intermediate for Chua and smaller for Lorenz.

The short pulse (SP) equation is an integrable equation. Multi-component generalizations of the SP equation are important for describing the polarization or anisotropic effects in optical fibers. An integrable semi-discretization of multi-component SP equation via Lax pair and Darboux transformation (DT) has been presented. We derive a Lax pair representation for the multi-component semi-discrete short pulse (sdSP) equation in the form of a block matrices by generalizing the 2 × 2 Lax pair matrices to the case of ${2}^{N}\times {2}^{N}$. A DT is studied for the multi-component sdSP equation and is used to compute soliton solutions of the system. Further, by expanding quasideterminants, we compute cuspon-soliton, smooth-soliton and loop-soliton solutions of the complex sdSP equation.

We calculate ground state configurations of a mixed Ising model on a square lattice where spins ${S}_{i}^{A}=\pm 2,\pm 1,0$ in one A sublattice are in alternating sites with spins ${\sigma }_{j}^{B}=\pm 1,0$, located on the other B sublattice.The Hamiltonian of the system includes nearest-neighbors interactions between the S_i^A and ${\sigma }_{j}^{B}$ spins, next-nearest-neighbors interactions between the S_i^A spins, and between the ${\sigma }_{j}^{B}$ spins, single-ion anisotropy, and an external magnetic field.

In this paper we show what seems to be the (very simple) key for quantization of classical systems. Given a manifold M, each pair given by a riemannian metric (nondegenerate, of arbitrary signature) and a linear connection, canonically determine a quantization rule or 'Correspondence Principle', which assigns to each classical magnitude (function in TM, subject to certain conditions) a differential operator in ${{ \mathcal C }}^{\infty }(M)$. The issue about the order in which the p' and q' are to be taken in quantization loses all meaning, when the general rule has been fixed. Once specified the Correspondence Principle, each 'classical state' of the system, understood as a vector field on M, determines a wave equation for each magnitude. The Schrödinger equation is a particular example of these wave equations.

We study the system of spin-orbit (SO) coupled Bose–Einstein condensates (BEC) with Rabi coupling in quasi-two dimensions characterized by unequal Rashba and Dresselhaus couplings. The ground state properties and the phase diagram of the system are studied within the mean-field approximation at T = 0. The energy-momentum dispersion relation corresponding to the single-particle ground state exhibits an infinitely degenerate Rashba ring, whose degeneracy is destroyed by the Rabi coupling, leading to a single-point lowest energy. The effect of Rabi coupling and interaction in the system show up three different phases, namely stripe, plane wave and zero momentum phase. At a particular parametric condition, the system admits a critical point separating all three different phases, known as the tri-critical point. For further insight, the momentum distribution, the energy and the longitudinal/transverse spin polarization corresponding to the quantum phases are comprehensively discussed.

E M Purcell showed that a body has to perform non-reciprocal motion in order to propel itself in a highly viscous environment. The swimmer with one degree of freedom is bound to do reciprocal motion, whereby the center of mass of the swimmer will not be able to propel itself due to the Scallop theorem. In the present study, we are proposing a new artificial swimmer called the one-hinge swimmer. Here we will show that flexibility plays a crucial role in the breakdown of Scallop theorem in the case of one-hinge swimmer or two-dimensional scallop at low Reynolds number. To model a one-hinge artificial swimmer, we use bead spring model for two arms joined by a hinge with bending potential for the arms in order to make them semi-flexible. The fluid is simulated using a particle based mesoscopic simulation method called the multi-particle collision dynamics with Anderson thermostat. Here, we show that when our swimmer has rigid arms, the center of mass of the swimmer is not able to propel itself as expected from the Scallop theorem. When we introduce flexibility in the arms, the time reversal symmetry breaks in the case of the one-hinged swimmer without the presence of a head contrary to the one-armed super paramagnetic swimmer which required a passive head in order to swim. The reduced velocity of the swimmer is studied using a range of parameters like flexibility, beating frequency and the amplitude of the beat, where we obtain similar scaling as that of the one-armed swimmer. We also calculate the dimensionless Sperm number for the swimmer and we get the maximum velocity when the Sperm number is around ∼1.8

Using x-ray magnetic circular dichroism and ab-initio calculations, we explore the La_1−xSr_xMnO₃/LaAlO₃/SrTiO₃ (001) heterostructure as a mean to induce transfer of spin polarized carriers from ferromagnetic La_1−xSr_xMnO₃ layer into the 2DEG (two-dimensional electron gas) at the LaAlO₃/SrTiO₃ interface. By out-of-plane transport measurements, the tunneling across the LaAlO₃ barrier is also analyzed. Our results suggest small or vanishing spin-polarization for the 2DEG: magnetic dichroism does not reveal a neat signal on Ti atoms, while calculations predict, for the pristine stoichiometric interface, a small spin-resolved mobile charge of 2.5 × 10¹³ cm⁻² corresponding to a magnetic moment of 0.038 μ_B per Ti atom, tightly confined within the single SrTiO₃ layer adjacent to LaAlO₃. Such a small magnetization is hard to be detected experimentally and perhaps not robust enough to survive to structural disorder, native doping, or La_1−xSr_xMnO₃ dead-layer effects. Our analysis suggests that, while some spin-diffusion cannot be completely ruled out, the use of ferromagnetic La_1−xSr_xMnO₃ epilayers grown on-top of LaAlO₃/SrTiO₃ is not effective enough to induce robust spin-transport properties in the 2DEG. The examined heterostructure is nevertheless an excellent test-case to understand some fundamental aspects of the spin-polarized charge transfer in 2D wells.

The evolution of acoustic waves can be evaluated in two ways: either as a temporal, or a spatial propagation. Propagating in space provides the considerable advantage of being able to handle dispersion and propagation across interfaces with remarkable efficiency; but propagating in time is more physical and gives correctly behaved reflections and scattering without effort. Which should be chosen in a given situation, and what compromises might have to be made? Here the natural behaviors of each choice of propagation are compared and contrasted for an ordinary second order wave equation, the time-dependent diffusion wave equation, an elastic rod wave equation, and the Stokes'/ van Wijngaarden's equations, each case illuminating a characteristic feature of the technique. Either choice of propagation axis enables a partitioning the wave equation that gives rise to a directional factorization based on a natural 'reference' dispersion relation. The resulting exact coupled bidirectional equations then reduce to a single unidirectional first-order wave equation using a simple 'slow evolution' assumption that minimizes effect of subsequent approximations, while allowing a direct term-to-term comparison between exact and approximate theories.

A simple and accurate numerical technique for finding eigenvalues, node structure, and expectation values of ${ \mathcal P }{ \mathcal T }$-symmetric potentials is devised. The approach involves expanding the solution to the Schrödinger equation in series involving powers of both the coordinate and the energy. The technique is designed to allow one to impose boundary conditions in ${ \mathcal P }{ \mathcal T }$-symmetric pairs of Stokes sectors. The method is illustrated by using many examples of ${ \mathcal P }{ \mathcal T }$-symmetric potentials in both the unbroken- and broken-${ \mathcal P }{ \mathcal T }$-symmetric regions.

The characterization of mobile charge carriers of semiconductor materials has spurred the development of numerous two dimensional carrier profiling tools. Here, we investigate the mobile charge carriers of several samples by multi-harmonic electrostatic force microscopy (MH-EFM) and scanning microwave impedance microscopy (sMIM). We present the basic principles and experiment setups of these two methods. And then several typical samples, i.e. a standard n-type doped Si sample, mechanical exfoliation and chemical vapor deposition grown molybdenum disulfide (MoS₂) layers are systemically investigated by sMIM and MH-EFM. The difference and (dis)advantages of these two modes are discussed. Both modes can provide carrier concentration profiles and have sub-surface sensitivity. They also have advantages in sample preparation in which contact electrodes are not required and insulating or electrically isolated samples can readily be studied. The basic mode, physics quantities extracted, dielectric response form and parasitic charges in scanning environment result in difference in experiment results for these two kinds of methods. The techniques described in this study will effectively promote research on basic science and semiconductor applications.

Bursting neurons are characterized by an alternation of active and inactive state, during the active state they exhibit train of spikes and quiescent behavior in the inactive state. When bursting neurons are coupled they can synchronize depending on the coupling strength. If the coupling strength is increased above a critical value, a transition from an asynchronous state to a partial synchronized state can be observed. This transition has the same type observed in coupled Kuramoto oscillators. When the natural frequency distribution of the oscillators is properly defined, the bursting neurons and the Kuramoto oscillators describe the same phase transition for synchronization.

In the calculation of hot-plasma atomic structure, the continuum wavefunctions are characterized by phase shifts, which therefore determine the scattering cross-sections. In this short paper, we propose a recurrence relation for the phase shifts in the case of a particular type of parametric potentials widely used in atomic-structure codes. These potentials have to be linear combinations of static screened Coulomb potentials (Yukawa-type potentials) multiplied by polynomial functions.

We present a compact current sensor based on a superconducting microwave lumped-element resonator with a nanowire kinetic inductor, operating at 4.2 K. The sensor is suitable for multiplexed readout in GHz range for large-format arrays of cryogenic detectors. The device consists of a lumped-element resonant circuit, fabricated from a single 4 nm-thick superconducting layer of niobium nitride. Thus, the fabrication and operation is significantly simplified in comparison to state-of-the-art current readout approaches. Because the resonant circuit is inductively coupled to the feed line the current to be measured can directly be injected without having the need of an impedance matching circuit, reducing the system complexity. With the proof-of-concept device we measured a current noise floor δI_min of 10 pA/Hz^1/2 at 10 kHz. Furthermore, we demonstrate the ability of our sensor to amplify a pulsed response of a superconducting nanowire single-photon detector using a GHz-range carrier for effective frequency-division multiplexing.

The recent article 'Entropic Updating of Probability and Density Matrices' [1] derives and demonstrates the inferential origins of both the standard and quantum relative entropies in unison. Operationally, the standard and quantum relative entropies are shown to be designed for the purpose of inferentially updating probability distributions and density matrices, respectively, when faced with incomplete information. We call the inferential updating procedure for density matrices the 'quantum maximum entropy method'. Standard inference techniques in probability theory can be criticized for lacking concrete physical consequences in physics; but here, because we are updating quantum mechanical density matrices, the quantum maximum entropy method has direct physical and experimental consequences. The present article gives a new derivation of the Quantum Bayes Rule, and some generalizations, using the quantum maximum entropy method while discuss some of the limitations the quantum maximum entropy method puts on the measurement process in Quantum Mechanics.

In n-doped diluted magnetic semiconductors the (s-d)-exchange-induced shift of the donor electron levels in an external B-field is expected to be subject to local fluctuations due to the statistical distribution of the magnetic ions. We investigated this phenomenon by resonant electron spin-flip Raman spectroscopy (ESFRS) on (Zn,Mn)Se:Cl samples with Mn contents up to 0.07 and n-dopant concentrations up to 1.3 × 10¹⁸ cm⁻³. The profile of the ESFRS efficiency versus photon energy shows a pronounced resonance, comprising two contributions due to the donor-bound exciton (D⁰,X) and the free-exciton (X⁰) as the intermediate states for the ESFRS process. The exchange-induced widening of the electron energy distribution is clearly reflected in the spectral width of the (D⁰,X) resonance contribution. A distinct B-field-induced broadening occurs, typically by a factor of two between B ∼1 T and B = 7 T. By employing the statistics of the local Mn concentration fluctuations, the radius of the donor-bound exciton (D⁰,X) is evaluated from this broadening, yielding (D⁰,X) radii from 1.75 nm to 5.90 nm and showing a decrease with increasing Mn content as well as dopant concentration.

The emergence and survival of cooperation is one of the hardest problems still open in science. Several factors such as the existence of punishment, repeated interactions, topological effects and the formation of prestige may all contribute to explain the counter-intuitive prevalence of cooperation in natural and social systems. The characteristics of the interaction networks have been also signaled as an element favoring the persistence of cooperators. Here we consider the invasion dynamics of cooperative behaviors in complex topologies. The invasion of a heterogeneous network fully occupied by defectors is performed starting from nodes with a given number of connections (degree) k₀. The system is then evolved within a Prisoner's Dilemma game and the outcome is analyzed as a function of k₀ and the degree k of the nodes adopting cooperation. Carried out using both numerical and analytical approaches, our results show that the invasion proceeds following preferentially a hierarchical order in the nodes from those with higher degree to those with lower degree. However, the invasion of cooperation will succeed only when the initial cooperators are numerous enough to form a cluster from which cooperation can spread. This implies that the initial condition has to be a suitable equilibrium between high degree and high numerosity. These findings have potential applications to the problem of promoting pro-social behaviors in complex networks.

Building upon the recently developed formalism of Kinetic Field Theory (KFT) for cosmic structure formation by Bartelmann et al, we investigate a kinematic relationship between diffusion and gravitational interactions in cosmic structure formation. In the first part of this work we explain how the process of structure formation in KFT can be separated into three processes, particle diffusion, the accumulation of structure due to initial momentum correlations and interactions relative to the inertial motion of particles. We study these processes by examining the time derivative of the non-linear density power spectrum in the Born approximation. We observe that diffusion and accumulation are delicately balanced because of the Gaussian form of the initial conditions, and that the net diffusion, resulting from adding these two counteracting contributions, approaches the contributions from the interactions in amplitude over time. This hints at a kinematic relation between diffusion and interactions in KFT. Indeed, in the second part, we show that the response of the system to arbitrary gradient forces is directly related to the evolution of particle diffusion in the form of kinematic fluctuation-dissipation relations (FDRs). This result is independent of the interaction potential. We show that this relationship roots in a time-reversal symmetry of the underlying generating functional. Furthermore, our studies demonstrate how FDRs originating from purely kinematic arguments can be used in theories far from equilibrium.

Khan and Penrose obtained an exact solution for colliding plane impulsive gravitational waves with the remarkable feature that the spacetime before the collision is flat and after the collision is not only curved, but develops a future curvature singularity. The spacetime has been extensively investigated analytically by various authors using the Newman-Penrose equations. We feel that a separate insight is provided by using the method of Weber and Wheeler to calculate the momentum imparted to test particles by the colliding waves. In this paper we probe the curvature by considering the momentum imparted to test particles by the colliding gravitational waves so as to try to visualise the way the curvature increases in the chosen coordinates as the singularity is approached.

We have developed a procedure to determined analytical expressions for the electromagnetic field and impedance of a circular induction coil having an arbitrary orientation in a conductive tube. Initially we express the field of a circular current filament in free space in terms of the global coordinate system referenced to tube axis. This is done by representing the filament field as a single layer potential and forming an integral over the layer using the source coordinates of a static Green's kernel. A coordinate transform to local source coordinates referenced to the filament axis allows one to determine the coil source function from that of the filament via integration. The effects of induced current in the tube wall on the coil impedance can then be determined for an internal coil. By extension of the present results one can predict the arbitrary orientated coil impedance for an external coil and the impedance variations of a tilted coil due to flaws in tubes.

Static behavior of fine particle cloud in fine particle (dusty) plasmas under gravity is analyzed numerically on the basis of the drift-diffusion equations. The system is assumed to have one-dimensional structure and the effect of the ion drag force is taken into account. The results confirm the analytically obtained prediction that we have an enhanced charge neutrality in fine particle clouds under gravity as well as under microgravity. They are also compared with a simple model based on the enhanced charge neutrality in clouds to show its applicability.

We study the relativistic version of the d-dimensional isotropic quantum harmonic oscillator based on the spinless Salpeter equation. This has no exact analytical solutions. We use perturbation theory to obtain compact formulas for the first and second-order relativistic corrections; they are expressed in terms of two quantum numbers and the spatial dimension d. The formula for the first-order correction is obtained using two different methods and we illustrate how this correction splits the original energy into a number of distinct levels each with their own degeneracy. Previous authors obtained results in one and three dimensions and our general formulas reduce to them when d = 1 and d = 3 respectively. Our two-dimensional results are novel and we provide an example that illustrates why two dimensions is of physical interest. We also obtain results for the two-dimensional case using a completely independent method that employs ladder operators in polar coordinates. In total, three methods are used in this work and the results all agree.

The force per unit area on the surface of a colloidal particle is a fundamental dynamical quantity in the mechanics and statistical mechanics of colloidal suspensions. Here we compute it in the limit of slow viscous flow for a suspension of N spherical active colloids in which activity is represented by surface slip. Our result is best expressed as a set of linear relations, the 'generalized Stokes laws', between the coefficients of a tensorial spherical harmonic expansion of the force per unit area and the surface slip. The generalized friction tensors in these laws are many-body functions of the colloidal configuration and can be obtained to any desired accuracy by solving a system of linear equations. Quantities derived from the force per unit area—forces, torques and stresslets on the colloids and flow, pressure and entropy production in the fluid—have succinct expressions in terms of the generalized Stokes laws. Most notably, the active forces and torques have a dissipative, long-ranged, many-body character that can cause phase separation, crystallization, synchronization and a variety of other effects observed in active suspensions. We use the results above to derive the Langevin and Smoluchowski equations for Brownian active suspensions, to compute active contributions to the suspension stress and fluid pressure, and to relate the synchrony in a lattice of harmonically trapped active colloids to entropy production. Our results provide the basis for a microscopic theory of active Brownian suspensions that consistently accounts for momentum conservation in the bulk fluid and at fluid-solid boundaries.

An algorithm for finding the bound-state eigenvalues and eigenfunctions of a Hermitian Hamiltonian operator using Green's method, developed by Waxman [1], has been extended to include non-Hermitian Hamiltonian operators.

We present the full classification of wave patterns evolving from an initial step-like discontinuity for arbitrary choice of boundary conditions at the discontinuity location in the DNLS equation theory. In this non-convex dispersive hydrodynamics problem, solutions of the Whitham modulation equations are mapped to parameters of a modulated wave by two-valued functions what makes situation much richer than that for a convex case of the NLS equation type. In particular, new types of simple-wave-like structures, such as trigonometric shocks and combined shocks, appear as building elements of the whole wave pattern observable in the long-time evolution of pulses after the wave-breaking point. The developed theory can find applications to propagation of light pulses in fibers and to the theory of Alfvén dispersive shock waves.

In this paper we report on generalized Lorenz models. Five- and six-dimensional Lorenz models are investigated, which are obtained by considering respectively two and three additional Fourier modes in addition to the modes included in the derivation of the classical three-dimensional Lorenz model. Parameter planes, bifurcation diagrams, and attractors in the phase-space are used, in order to investigate the influence of the additional Fourier modes on solutions, when compared with the solutions for the classical Lorenz model. It is shown that for parameters σ and b kept fixed, a larger parameter r results for the onset of chaos in five-and six-dimensional Lorenz models. Also it is shown that the shape of bifurcation diagrams, periodic, and chaotic attractors is preserved in both generalized Lorenz models. Additionally, it is shown that hyperchaos is observed only in the six-dimensional Lorenz model, at least in the parameter ranges here investigated.

In this work we present a semi-classical approach to solve the inverse spectrum problem for one-dimensional wave equations for a specific class of potentials that admits quasi-stationary states. We show how inverse methods for potential wells and potential barriers can be generalized to reconstruct significant parts for the combined potentials. For the reconstruction one assumes the knowledge of the complex valued spectrum and uses the exponential smallness of its imaginary part. Analytic spectra are studied and a recent application of the method in the literature for gravitational wave physics is discussed. The method allows for a simple reconstruction of quasi-stationary state potentials from a given spectrum. Thus it might be interesting for different branches of physics and related fields.

Considering linear and nonlinear optical effects like group velocity dispersion, higher-order dispersion, Kerr nonlinearity, self-steepening, stimulated Raman scattering we obtain a higher-order nonlinear Schrödinger equation describing the propagation of ultra-short pulse in optical fiber. We construct exact bright and dark solitary wave solutions of the generalized obtained equation, obeying to some constraint relations between coefficient's equation via the Bogning-Djeumen Tchaho-Kofané method (BDKm). The generalized higher-order nonlinear Schrödinger equation is obtained by affecting coefficients n_i(i = 0, 1, .., 5) to different terms of non modified equation. New solutions are obtained, and the term or higher-order dispersion can be considered as the new selector of solitary wave-type propagating in the higher-order nonlinear optical fiber.

The magnetic phases induced by the interplay between disorder acting only on particles with a given spin projection ('spin-dependent disorder') and a local repulsive interaction is explored. To this end the magnetic ground state phase diagram of the Hubbard model at half-filling is computed within dynamical mean-field theory combined with the geometric average over disorder, which is able to describe Anderson localization. Five distinct phases are identified: a ferromagnetically polarized metal, two types of insulators, and two types of spin-selective localized phases. The latter four phases possess different long-range order of the spins. The predicted phase diagram may be tested experimentally using cold fermions in optical lattices subject to spin-dependent random potentials.

We study two coupled optomechanical systems interact mutually through an optical fiber and a phonon tunneling, which are controlled by two switches K₁ and K₂. Compared to the quantum synchronization of the optomechanical systems without modification, we find that a proper periodic modification by different switches setups can achieve a better quantum synchronization. In addition, we also analyze the robustness of the periodically modified system against the bath's mean temperature or the oscillators' frequency difference, respectively.

There are growing interests on dynamics of phase-singularities (PSs) in complex systems such as ventricular fibrillation, defect in fluids and liquid crystals, living creatures, quantum vortex and so on. A master equation approach on the number of PS for studying birth-death dynamics of PSs is invented first by Gil, Lega and Meunier. Although their approach is applied to various complex systems including non-linear birth-death rates, time-dependent solution of related master equation is obtained only rarely. Even a master equation with full linear birth-death rates, time-dependent solution is not also given due to the analytical complexity and the existence of singularity in the probability generating function. In this paper, an approximate time-dependent solution of the master equation and the associated waiting time distribution are obtained explicitly with the aid of the method of the Poisson transform. Numerical evaluation of the obtained approximate solution teaches us that there exists the universal scaling law in the waiting time distribution.