Parity effect of the initial capital based on Parrondo’s games and the quantum interpretation

In our previous study [Zhu et al., Quantum game interpretation for a special case of Parrondo’s paradox, Physica A 390 (2011) 579], the capital-dependent Parrondo’s game where one game depends on the capital modulus $M=4$ was shown not to have a definite stationary probability distribution and that payoffs of the game depended on the parity of the initial capital. This paper presents a generalization of these results to even $M$ greater than 4. An intuitive explanation for producing this phenomenon is that the discrete-time Markov chain of the game is divided into two completely unrelated inner and outer rings. The process taking the inner ring or outer ring of the game is determined by the initial capital of parity and then a win or loss of the game is determined. Quantum game theory is used to further analyze the phenomenon. The results show that the explanation of the game corresponding to a stationary probability distribution is that the probability of the initial capital has reached parity.     

Simple form of the stationary distribution for 3D cellular automata in a special case

3D cellular automata can be analyzed by means of finite homogeneous Markov chains. If the automaton is allowed to change only one cell per iteration, and the transition probability depends linearly on the number of ones in the neighborhood, the Markov chain has two attractors at all zeroes and all ones. Otherwise-and this is the case we tackle-the chain is ergodic, thus allowing for the search of stationary distributions. This proves cumbersome in the general case, still, under detailed balance equation, the stationary distribution can be derived in closed form. The probability of a particular state is found to be exponential in the number of zero-one borders within the configuration.     

Quantum walk with jumps

We analyze a special class of 1-D quantum walks (QWs) realized using optical multi-ports. We assume non-perfect multi-ports showing errors in the connectivity, i.e. with a small probability the multi-ports can connect not to their nearest neighbor but to another multi-port at a fixed distance – we call this a jump. We study two cases of QW with jumps where multiple displacements can emerge at one timestep. The first case assumes time-correlated jumps (static disorder). In the second case, we choose the positions of jumps randomly in time (dynamic disorder). The probability distributions of position of the QW walker in both instances differ significantly: dynamic disorder leads to a Gaussian-like distribution, while for static disorder we find two distinct behaviors depending on the parity of jump size. In the case of even-sized jumps, the distribution exhibits a three-peak profile around the position of the initial excitation, whereas the probability distribution in the odd case follows a Laplace-like discrete distribution modulated by additional (exponential) peaks for long times. Finally, our numerical results indicate that by an appropriate mapping a universal functional behavior of the variance of the long-time probability distribution can be revealed with respect to the scaled average of jump size.     

Role of memory effect in the evolution of cooperation based on spatial prisoner's dilemma game

We investigate the impact of memory effect on the evolution of cooperation in the spatial prisoner's dilemma game, in which each player will record his own strategies during the previous M game rounds (here, M is also named as the memory length). At each time step, each individual will update his current strategy according to the Fermi-like probability which will be multiplied by a pre-factor, and this factor will be correlated with the fraction of previous strategy states identical with the individual strategy to be updated. The numerical simulation results demonstrate that the memory length will largely influence the cooperation level at the stationary state, and it is clearly shown that the intermediate value of M will optimally favor the emergence of cooperation and the dynamical evolution, and characteristic patterns also support these conclusions. In addition, we depict the full cooperation phase diagrams and find that the cooperation region will be broadened under the case of moderate M values. The current results also indicate that the limited memory may be enough for us to design the effective promotion mechanism and further understand the emergency of cooperation taking place upon many networked populations.     

Bipolar harmonics method in the semiclassical theory of sub-doppler cooling

The bipolar harmonics method is extended to the case of complex elliptic polarization vectors. The method is used to study, on the basis of the semiclassical theory, the multipole moments of the ground state of atoms under conditions of sub-Doppler cooling with a monochromatic light field possessing spatial gradients of the polarization. It is shown that for stationary atoms with an initial isotropic distribution over sublevels the multipole moments of rank κ decompose, in accordance with the parity κ of the rank, according to one of two minimal sets of bipolar harmonics with different symmetry under inversion. An expansion of the corrections, which are linear in the velocity, to the multipole moments with respect to the indicated minimal sets of bipolar harmonics is studied for a stationary state, and the expansion coefficients are analyzed. The orientation vector J of the atomic ensemble is studied on the basis of the proposed method for the dipole transition 1/2 → 1/2, and the light-induced forces for a specific 2D configuration of the light field, including radiation friction forces and Lorentz-type forces, are analyzed.     

A Monte Carlo simulation of coagulation

A Monte Carlo simulation technique is described for the study of the coagulation of suspended particles. The method is computationally efficient since the particle trajectories are not used to determine coagulations. Instead, pairs of particles are assigned probabilities to coagulate and the evolution is computed as a stochastic Markov game. We also describe a simple analytic method to obtain the stationary distribution of sizes for the various mechanisms of relative particle motion. It is demonstrated that the simulation yields the correct stationary size distribution independent of initial condition.     

The paradox of group behaviors based on Parrondo’s games

We assume a multi-agent model based on Parrondo’s games. The model consists of game A between individuals and game B. In game A, two behavioral patterns are defined: competition and inaction. A controlled alternation strategy of behavioral pattern that gives a single player the highest return is proposed when game A+B is played randomly. Interesting phenomena can be found in collective games where a large number of individuals choose the behavioral pattern by voting. When game B is the capital-dependent version, the outcome can be better for the players to vote randomly than to vote according to their own capital. An explanation of such counter-intuitive phenomena is given by noting that selfish voting prevents the competition behavior of game A that is essential for the total capital to grow. However, if game B is the history-dependent version, this counter-intuitive phenomenon will not happen. The reason is selfish voting results in the competition behavior of game A, and finally it produces the winning results.     

Das Fluktuationsspektrum von Elektronen in einem stationären Nichtgleichgewichtszustand

The Boltzmann equation is used to calculate the time correlation function and the fluctuation spectrum for electrons obeying classical statistics. The stationary joint distribution for one electron to be initially ink ₀=k(0) and finally ink=k(t) is given by the product of the conditional probability and the stationary distribution. These quantities can be found from the Boltzmann equation if there exists, for any initial distribution, a unique solution which satisfies the Markov equation and tends to a stationary solution for large times under stationary conditions. It is proved that these conditions hold for linear collision operators and in the relaxation approximation. General operator expressions for the fluctuation spectrum and the differential conductivity in a stationary electric field are given, which can be evaluated within the usual approximation schemes known for the stationary, nonequilibrium solutions of the Boltzmann equation. In equilibrium they reproduce the classical fluctuation dissipation theorem. In a nonequilibrium state they define a noise temperature depending on the field. In the relaxation approximation and for polynomial band structure the exact solution can be found. For parabolic and biparabolic spherical bands the result is discussed explicitly.     

Time-space white noise eliminates global solutions in reaction-diffusion equations

We prove that perturbing the reaction-diffusion equation u_t=u_xx+(u₊)^p (p>1), with time-space white noise produces that solutions explodes with probability one for every initial datum, opposite to the deterministic model where a positive stationary solution exists.     

Complete Characterization of Convergence to Equilibrium for an Inelastic Kac Model

Pulvirenti and Toscani introduced an equation which extends the Kac caricature of a Maxwellian gas to inelastic particles. We show that the probability distribution, solution of the relative Cauchy problem, converges weakly to a probability distribution if and only if the symmetrized initial distribution belongs to the standard domain of attraction of a symmetric stable law, whose index ?? is determined by the so-called degree of inelasticity, p>0, of the particles: $\alpha=\frac{2}{1+p}$ . This result is then used: (1) To state that the class of all stationary solutions coincides with that of all symmetric stable laws with index ??. (2) To determine the solution of a well-known stochastic functional equation in the absence of extra-conditions usually adopted.     

The transient solution of the laser Fokker-Planck equation

The transient solution of the laser Fokker-Planck equation is investigated in the threshold region. Especially for the initial condition, that no photons are present att=0, the transient of the laser distribution function, of the mean intensity and of the mean squared deviation (variance) into the stationary values are shown. Finally we give a physical explanation of the effects involved and we discuss some features of the transient photon counting probability.     

Derivation of the Boltzmann equation for a Fermi gas

Point scatterers are placed on the real line such that the distances between scatterers are independent identically distributed random variables (stationary renewal process). For a fixed configuration of scatterers a particle performs the following random walk: The particle starts at the pointx with velocityυ, ¦υ¦=1. In between scatterers the particle moves freely. At a scatterer the particle is either transmitted or reflected, both with probability 1/2. For given initial conditions of the particle the velocity autocorrelation function is a random variable on the scatterer configurations. If this variable is averaged over the distribution of scatterers, it decays not faster thant ^?3/2.     

Integral quadratic functionals defined in a normal Markov two-dimensional field and their statistics

We consider the problem of distribution of an integral quadratic functional defined in a stationary two-dimensional random normal Markov field. The generating function of the random-value probability distribution of the functional is found and an approximate expression for the probability distribution is obtained. The effect of the parameters of the two-dimensional field on the statistical properties of the distribution of the considered functional is analyzed. A generalization of the solution to the case of a multidimensional stationary normal Markov field is proposed. Kharkov State Polytechnical University, Kharkov, Ukraine Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 43, No. 6, pp. 562–572, June, 2000.     

Stabilities of one-dimensional stationary states of Bose-Einstein condensates

We explore the dynamical stabilities of a quasi-one-dimensional (1D) Bose-Einstein condensate (BEC) consisting of fixed N atoms with time-independent external potential. For the stationary states with zero flow density the general solution of the perturbed time evolution equation is constructed, and the stability criterions concerning the initial conditions and system parameters are established. Taking the lattice potential case as an example, the stability and instability regions on the parameter space are found. The results suggest a method for selecting experimental parameters and adjusting initial conditions to suppress the instabilities.     

A fluctuation problem in particle transport theory II

A balance equation is formulated for the probability that a particle injected into an infinite, amorphous medium will have suffered N collisions and have given rise to n new particles in a given energy range at time t. The method of regeneration points has been employed and this leads, in the case of two particle production, to a non-linear, integro-differential equation for the probability generating function. This equation is solved for the case of foreign particles slowing down, in which case it becomes linear and results are obtained which include the effects of electronic stopping and absorption, thus generalizing the work in part I. In the cascade problem, a single particle gives rise to two new particles in every collision and it is shown, for a simple hard-sphere model with 1/v scattering and absorption, how the non-linear equation may be solved. The probability for the number of particles and the number of collisions suffered to absorption is obtained in the case of zero absorption, the probability law is shown to obey a Furry distribution. The limitations of the method described in part I for dealing with cascades are highlighted.     

氢原子波函数的经典极限分析

由电子绕原子核的经典轨道运动出发,对电子的椭圆轨道初始坐标参数求平均,得到空间密度分布。这分布与经典极限条件下氢原子中定态几率密度相同。类似的比较也推广到相空间进行。由这一比较得出结论:定态氢原子波函数不描述单个原子而描述一个系综。 关键词：     

Time-Delay Induced Reentrance Phenomenon in?a?Triple-Well Potential System Driven by?Cross-Correlated Noises

The effects of the time delay on the stationary properties of a triple-well potential system driven by the cross-correlated multiplicative and additive noises are investigated. The stationary probability distribution function (SPDF) is obtained by means of a numerical simulation method. The results indicate that: (i) The time delay can induce the reentrance phenomenon with increasing the delay time; (ii) The cross-correlation between the multiplicative and additive noises induces the symmetry breaking of the SPDF.     

On the effects of a general form of the initial runout on the oscillation frequencies and critical speeds of a spinning disk

This paper is concerned with the effect of the lack of flatness on the dynamics of spinning disks. Of particular interest is the study of the effect of an unsymmetrical initial runout on the oscillation frequencies and the critical speeds of the spinning disks. It is assumed that a spinning disk is initially warped in an initial asymmetric shape. Using Von Karman's plate theory, the equations of motion are derived in a rotating frame. Taking advantage of the orthogonal properties of the eigenfunctions of a stationary disk, the lateral displacement and the stress function are expressed as their summation. Since these eigenfunctions produce an orthonormal space, any shape and level of initial runout can be written as their summation also. Using Galerkin's method, the equations of motion are discretized and a set of coupled linear equations are found taking into account the effect of an initial arbitrary runout. The numerical results are found to have a good agreement with those obtained using ANSYS. It is found that the natural frequencies of the spinning disk calculated in the space fixed frame increases due to an initial runout. When the initial runout is assumed to be asymmetric, it is found that, due to asymmetric stress distributed in the disk, a frequency split between the backward and forward traveling waves of a given preferential mode of a stationary disk occurs. It is also seen that in some cases the initial runout in the form of the (n,m) mode has the least effect on the natural frequencies of the backward traveling waves of the same mode. As such it is observed that in the case the critical speed the (n,m) mode is less sensitive to an initial runout in the form of the (n,m) mode. To verify the accuracy of the numerical predictions, experiments were conducted on two disks, one which was flat and the other that had an initial nonflatness in the form of the (0,2) mode. The numerical and experimental results indicate a close correspondence.     

Efficient error estimation in quantum key distribution

In a quantum key distribution(QKD)system,the error rate needs to be estimated for determining the joint probability distribution between legitimate parties,and for improving the performance of key reconciliation.We propose an efficient error estimation scheme for QKD,which is called parity comparison method(PCM).In the proposed method,the parity of a group of sifted keys is practically analysed to estimate the quantum bit error rate instead of using the traditional key sampling.From the simulation results,the proposed method evidently improves the accuracy and decreases revealed information in most realistic application situations.