The theory of statistical physics relies on ergodicity, whereby in large or interacting systems lacking integrability, trajectories eventually explore nearly all points in the phase space. It has been believed that ch...The theory of statistical physics relies on ergodicity, whereby in large or interacting systems lacking integrability, trajectories eventually explore nearly all points in the phase space. It has been believed that chaotic dynamics provide a possible pathway to ergodicity. Here, we examine the phase space density distributions and their recurrence in the harmonic oscillator, the linear and nonlinear Mathieu equations, the Lorenz attractor, and the Nosé–Hoover model. We show that in models with periodic or quasiperiodic dynamics, sharp peaks can be found in the phase space density distributions. However, for the chaotic dynamics, their distributions display totally different behaviors. We understand these differences using recurrence plots. Our results show that while chaotic dynamics provide an efficient way for the trajectory to explore a large portion of the phase space, which is necessary for ergodicity, the chaotic dynamics are not sufficient for this goal. For instance, despite the Nosé–Hoover model being chaotic, it is not sufficiently large for ergodicity. Therefore, our results may lead to an important conclusion, which is that ergodicity may be realized from large chaotic systems. These findings in these simple models can be explored in experiments in the future, which may provide some key insights into ergodic dynamics.展开更多
This paper investigates the ergodicity and weak convergence of transition probabilities for two-dimensional stochastic primitive equations driven by multiplicative noise.The existence of invariant measures is establis...This paper investigates the ergodicity and weak convergence of transition probabilities for two-dimensional stochastic primitive equations driven by multiplicative noise.The existence of invariant measures is established using the classical Krylov-Bogoliubov theory.The uniqueness of invariant measures and the weak convergence of transition probabilities are demonstrated through the application of the asymptotic coupling method and Foias-Prodi estimate.展开更多
In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of（pt）t≥0 to...In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of（pt）t≥0 to equilibrium uniform on any bounded subset in H.展开更多
In this paper, we show that a positive recurrent ?uid queue is automatically V-uniformly ergodic for some function V ≥ 1 but never uniformly ergodic. This reveals a similarity of ergodicity between a ?uid queue and a...In this paper, we show that a positive recurrent ?uid queue is automatically V-uniformly ergodic for some function V ≥ 1 but never uniformly ergodic. This reveals a similarity of ergodicity between a ?uid queue and a quasi-birth-and-death process. As a byproduct of V-uniform ergodicity, we derive computable bounds on the exponential moments of the busy period.展开更多
In this paper, we propose a new one-time one-key encryption algorithm based on the ergodicity of a skew tent chaotic map. We divide the chaotic trajectory into sub-intervals and map them to integers, and use this sche...In this paper, we propose a new one-time one-key encryption algorithm based on the ergodicity of a skew tent chaotic map. We divide the chaotic trajectory into sub-intervals and map them to integers, and use this scheme to encrypt plaintext and obtain ciphertext. In this algorithm, the plaintext information in the key is used, so different plaintexts or different total numbers of plaintext letters will encrypt different ciphertexts. Simulation results show that the performance and the security of the proposed encryption algorithm can encrypt plaintext effectively and resist various typical attacks.展开更多
We review the main aspects of the foundations of statistical mechanics. In particular we explain why many degrees of freedom are necessary, while chaos(in the sense of positive Lyapunov exponents) is only marginally r...We review the main aspects of the foundations of statistical mechanics. In particular we explain why many degrees of freedom are necessary, while chaos(in the sense of positive Lyapunov exponents) is only marginally relevant,for the emergence of statistical laws in macroscopic systems.展开更多
In this paper, we define robust weak ergodicity and study the relation between robust weak ergodicity and stable ergodicity for conservative partially hyperbolic systems. We prove that a C^r（r 〉 1） conservative par...In this paper, we define robust weak ergodicity and study the relation between robust weak ergodicity and stable ergodicity for conservative partially hyperbolic systems. We prove that a C^r（r 〉 1） conservative partially hyperbolic diffeomorphism is stably ergodic if it is robustly weakly ergodic and has positive （or negative） central exponents on a positive measure set. Furthermore, if the condition of robust weak ergodicity is replaced by weak ergodicity, then the diffeomophism is an almost stably ergodic system. Additionally, we show in dimension three, a C^r（r 〉 1） conservative partially hyperbolic diffeomorphism can be approximated by stably ergodic systems if it is robustly weakly ergodic and robustly has non-zero central exponents.展开更多
At low temperatures the configurational phase space of a macroscopic complex system （e.g., a spin-glass） of N - 10^23 interacting particles may split into an exponential number Ωs - exp（const × N） of ergodic...At low temperatures the configurational phase space of a macroscopic complex system （e.g., a spin-glass） of N - 10^23 interacting particles may split into an exponential number Ωs - exp（const × N） of ergodic sub-spaces （thermodynamic states）. It is usually assumed that the equilibrium collective behavior of such a system is determined by its ground thermodynamic states of the minimal free-energy density, and that the equilibrium free energies follow the distribution of exponentied decay. But actually for some complex systems, the equilibrium free-energy values may follow a Gaussian distribution within an intermediate temperature range, and consequently their equilibrium properties are contributed by excited thermodynamic states. Based on this analysis, the re-weighting parameter y in the cavity approach of spin-glasses is easily understood. Depending on the free-energy distribution, the optimal y can either be equal to or be strictly less than the inverse temperature β.展开更多
Let X be a compact metric space and C（X） be the space of all continuous functions on X. In this article, the authors consider the Markov operator T ： C（X）N C（X）N defined by for any f = （f1,f2,… ,fN）, where ...Let X be a compact metric space and C（X） be the space of all continuous functions on X. In this article, the authors consider the Markov operator T ： C（X）N C（X）N defined by for any f = （f1,f2,… ,fN）, where （pij） is a N x N transition probability matrix and {wij } is an family of continuous transformations on X. The authors study the uniqueness, ergodicity and unidimensionality of T＊-invariant measures where T＊ is the adjoint operator of T.展开更多
The finite data estimates of the complex fourth-order moments of a signal consisting of random harmonics are analyzed. Conditions for the fourth-order stationarity and ergodicity are obtained. Explicit formulas for th...The finite data estimates of the complex fourth-order moments of a signal consisting of random harmonics are analyzed. Conditions for the fourth-order stationarity and ergodicity are obtained. Explicit formulas for the estimation error and its variance, as well as their limiting large sample values are derived. Finally, a special case relevant to cubic phase coupling is considered, and these results are stated for this case, the variance is shown to comprise an ergodic and a nonergodic part.展开更多
The IP P＋M/M/c queueing system has been extensively used in the modern communication system.The existence and uniqueness of stationary distribution of the queue length L（t）for IP P＋M/M/1 queue has been proved in[1...The IP P＋M/M/c queueing system has been extensively used in the modern communication system.The existence and uniqueness of stationary distribution of the queue length L（t）for IP P＋M/M/1 queue has been proved in[10].In this paper,we shall give the su？cient and necessary conditions of l-ergodicity,geometric ergodicity,and prove that they are neither uniformly polynomial ergodicity nor strong ergodicity.展开更多
Bandwidth,cutwidth,cyclic bandwidth,bandwidth sum and cyclic bandwidth sum are well-known indices about optimal labeling of graphs applied in VLSI design,network communications,and other areas involving the graph layo...Bandwidth,cutwidth,cyclic bandwidth,bandwidth sum and cyclic bandwidth sum are well-known indices about optimal labeling of graphs applied in VLSI design,network communications,and other areas involving the graph layout.To design the graphs with the given indices,we need to study the ergodicity.Let F be a set of graphs under consideration andφan integer-valued function defined on F,namely,φis an index,such as bandwidth and cutwidth.If there exists a graph G∈F such thatφ(G)=x for any integer x in the interval[a,b],where a and b are the minimum and maximum ofφon F,respectively,thenφis said to have ergodicity on F.Let Gnbe the set of simple connected graphs with order n and Tnthe set of trees with order n.In this paper,we investigate the ergodicity of bandwidth,cutwidth,cyclic bandwidth,the bandwidth sum and cyclic bandwidth sum on Tn and Gn.展开更多
In this paper, we investigate the flow of customers through queuing systems with randomly varying intensities. The analysis of the Kolmogorov-Chapman system of stationary equations for this model showed that it is not...In this paper, we investigate the flow of customers through queuing systems with randomly varying intensities. The analysis of the Kolmogorov-Chapman system of stationary equations for this model showed that it is not possible to construct a convenient symbolic solution. In this paper an attempt is made to circumvent this requirement by referring to the ergodicity theorems, which gives the conditions for the existence of the limit distribution in the service processes, but do not require knowledge of them.展开更多
In this paper, we give a survey on the PhD thesis of the first author. There theexistence and ergodicity on invariant measures of set-valued mappings are discused.
In this paper,a stochastic framework with a general nonmonotonic response function is formulated to investigate the competition dynamics between two species in a chemostat environment.The model incorporates both white...In this paper,a stochastic framework with a general nonmonotonic response function is formulated to investigate the competition dynamics between two species in a chemostat environment.The model incorporates both white noise and telegraph noise,the latter being described by Markov process.The existence of a unique global positive solution for the stochastic chemostat model is established.Subsequently,by using the ergodic theory of Markov process and utilizing techniques of stochastic analysis,the critical value differentiating between persistence in mean and extinction for the microorganism species is explored.Moreover,the existence of a unique stationary distribution is proved by using stochastic Lyapunov analysis.Finally,numerical simulations are introduced to support the obtained results.展开更多
针对智能反射面(reconfigurable intelligent surface,RIS)不具备复杂的感知和信号处理能力,使得信道状态信息(channel state information,CSI)估计存在误差的情况,构建了不完美Nakagami-m衰落信道下的RIS辅助端到端的无线通信系统模型...针对智能反射面(reconfigurable intelligent surface,RIS)不具备复杂的感知和信号处理能力,使得信道状态信息(channel state information,CSI)估计存在误差的情况,构建了不完美Nakagami-m衰落信道下的RIS辅助端到端的无线通信系统模型,分析了此模型下系统的中断概率(outage probability,OP)和各态历经容量(ergodic capacity,EC).首先基于矩匹配法推导了端到端信干噪比(signal to interference plus noise ratio,SINR)的累积分布函数(cumulative distribution function,CDF),然后基于此CDF导出了OP与EC的解析闭合表达式,最后推导了高SINR条件下OP与EC的近似表达式.仿真实验结果表明:解析结果与蒙特卡罗仿真数据高度吻合,且在高SINR情况下与近似曲线逼近;完美CSI的OP值会随着SINR增大而持续降低,而不完美CSI会出现地板现象,且相关系数ρ越小,OP值越大;通过增加RIS的反射元件数量可有效减少信道估计误差带来的性能损失.展开更多
对D2D(Device to Device)通信系统的隐蔽通信问题进行研究,提出了一种基于智能反射面(Intelligent Reflecting Surfaces,IRS)的D2D通信系统,该系统采用双IRS模式。首先分析了各信号的信干噪比(SINR)和各自的期望值,并据此推导出系统遍...对D2D(Device to Device)通信系统的隐蔽通信问题进行研究,提出了一种基于智能反射面(Intelligent Reflecting Surfaces,IRS)的D2D通信系统,该系统采用双IRS模式。首先分析了各信号的信干噪比(SINR)和各自的期望值,并据此推导出系统遍历总容量的解形式,接着针对隐蔽通信建立了二元假设问题,推导出误检测率的解形式,并据此计算出平均最小误检测率。除此之外,还分析了无IRS系统的遍历总容量和平均最小误检测率,与双IRS系统进行对比。仿真结果表明,双IRS系统的遍历总容量和平均最小误检测率优于无IRS系统,双IRS系统能够比传统的无IRS系统获得更高的容量和更低的误检测率。展开更多
基金supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No.XDB0500000)the Innovation Program for Quantum Science and Technology (Grant Nos. 2021ZD0301200 and2021ZD0301500)the Alliance of International Science Organizations (ANSO)。
文摘The theory of statistical physics relies on ergodicity, whereby in large or interacting systems lacking integrability, trajectories eventually explore nearly all points in the phase space. It has been believed that chaotic dynamics provide a possible pathway to ergodicity. Here, we examine the phase space density distributions and their recurrence in the harmonic oscillator, the linear and nonlinear Mathieu equations, the Lorenz attractor, and the Nosé–Hoover model. We show that in models with periodic or quasiperiodic dynamics, sharp peaks can be found in the phase space density distributions. However, for the chaotic dynamics, their distributions display totally different behaviors. We understand these differences using recurrence plots. Our results show that while chaotic dynamics provide an efficient way for the trajectory to explore a large portion of the phase space, which is necessary for ergodicity, the chaotic dynamics are not sufficient for this goal. For instance, despite the Nosé–Hoover model being chaotic, it is not sufficiently large for ergodicity. Therefore, our results may lead to an important conclusion, which is that ergodicity may be realized from large chaotic systems. These findings in these simple models can be explored in experiments in the future, which may provide some key insights into ergodic dynamics.
基金supported in part by the NSFC(12171084,12326367)the Jiangsu Provincial Scientific Research Center of Applied Mathematics(BK20233002)the fundamental Research Funds for the Central Universities(RF1028623037)。
文摘This paper investigates the ergodicity and weak convergence of transition probabilities for two-dimensional stochastic primitive equations driven by multiplicative noise.The existence of invariant measures is established using the classical Krylov-Bogoliubov theory.The uniqueness of invariant measures and the weak convergence of transition probabilities are demonstrated through the application of the asymptotic coupling method and Foias-Prodi estimate.
基金supported by the National Science Foundation of China(1067121290820302)the National Science Foundation of Hunan Province
文摘In this article,we first prove the existence and uniqueness of the solution to the stochastic generalized porous medium equation perturbed by Lévy process,and then show the exponential convergence of（pt）t≥0 to equilibrium uniform on any bounded subset in H.
基金Supported by the National Natural Science Foundation of China(11571372,11771452)the Innovation Program of Central South University(10900-50601010)
文摘In this paper, we show that a positive recurrent ?uid queue is automatically V-uniformly ergodic for some function V ≥ 1 but never uniformly ergodic. This reveals a similarity of ergodicity between a ?uid queue and a quasi-birth-and-death process. As a byproduct of V-uniform ergodicity, we derive computable bounds on the exponential moments of the busy period.
基金supported by the National Natural Science Foundation of China (Grant Nos.61173183,60973152,and 60573172)the Superior University Doctor Subject Special Scientific Research Foundation of China (Grant No.20070141014)the Natural Science Foundation of Liaoning Province,China (Grant No.20082165)
文摘In this paper, we propose a new one-time one-key encryption algorithm based on the ergodicity of a skew tent chaotic map. We divide the chaotic trajectory into sub-intervals and map them to integers, and use this scheme to encrypt plaintext and obtain ciphertext. In this algorithm, the plaintext information in the key is used, so different plaintexts or different total numbers of plaintext letters will encrypt different ciphertexts. Simulation results show that the performance and the security of the proposed encryption algorithm can encrypt plaintext effectively and resist various typical attacks.
文摘We review the main aspects of the foundations of statistical mechanics. In particular we explain why many degrees of freedom are necessary, while chaos(in the sense of positive Lyapunov exponents) is only marginally relevant,for the emergence of statistical laws in macroscopic systems.
基金supported by National Natural Science Foundation of China(11001284)Natural Science Foundation Project of CQ CSTC(cstcjjA00003)Fundamental Research Funds for the Central Universities(CQDXWL2012008)
文摘In this paper, we define robust weak ergodicity and study the relation between robust weak ergodicity and stable ergodicity for conservative partially hyperbolic systems. We prove that a C^r（r 〉 1） conservative partially hyperbolic diffeomorphism is stably ergodic if it is robustly weakly ergodic and has positive （or negative） central exponents on a positive measure set. Furthermore, if the condition of robust weak ergodicity is replaced by weak ergodicity, then the diffeomophism is an almost stably ergodic system. Additionally, we show in dimension three, a C^r（r 〉 1） conservative partially hyperbolic diffeomorphism can be approximated by stably ergodic systems if it is robustly weakly ergodic and robustly has non-zero central exponents.
基金supported by National Natural Science Foundation of China under Grant No.10774150
文摘At low temperatures the configurational phase space of a macroscopic complex system （e.g., a spin-glass） of N - 10^23 interacting particles may split into an exponential number Ωs - exp（const × N） of ergodic sub-spaces （thermodynamic states）. It is usually assumed that the equilibrium collective behavior of such a system is determined by its ground thermodynamic states of the minimal free-energy density, and that the equilibrium free energies follow the distribution of exponentied decay. But actually for some complex systems, the equilibrium free-energy values may follow a Gaussian distribution within an intermediate temperature range, and consequently their equilibrium properties are contributed by excited thermodynamic states. Based on this analysis, the re-weighting parameter y in the cavity approach of spin-glasses is easily understood. Depending on the free-energy distribution, the optimal y can either be equal to or be strictly less than the inverse temperature β.
文摘Let X be a compact metric space and C（X） be the space of all continuous functions on X. In this article, the authors consider the Markov operator T ： C（X）N C（X）N defined by for any f = （f1,f2,… ,fN）, where （pij） is a N x N transition probability matrix and {wij } is an family of continuous transformations on X. The authors study the uniqueness, ergodicity and unidimensionality of T＊-invariant measures where T＊ is the adjoint operator of T.
文摘The finite data estimates of the complex fourth-order moments of a signal consisting of random harmonics are analyzed. Conditions for the fourth-order stationarity and ergodicity are obtained. Explicit formulas for the estimation error and its variance, as well as their limiting large sample values are derived. Finally, a special case relevant to cubic phase coupling is considered, and these results are stated for this case, the variance is shown to comprise an ergodic and a nonergodic part.
基金Supported by the Chinese Universities Scientific Fund(BUPT2009RC0707,BUPT2011RC0703)
文摘The IP P＋M/M/c queueing system has been extensively used in the modern communication system.The existence and uniqueness of stationary distribution of the queue length L（t）for IP P＋M/M/1 queue has been proved in[10].In this paper,we shall give the su？cient and necessary conditions of l-ergodicity,geometric ergodicity,and prove that they are neither uniformly polynomial ergodicity nor strong ergodicity.
基金Supported by Science and Technology Program of Guangzhou(Grant No.202002030183)Natural Science Foundation of Guangdong(Grant No.2021A1515012045)+1 种基金National Natural Science Foundation of China(Grant No.12161073)Natural Science Foundation of Qinghai(Grant No.2020-ZJ-924)。
文摘Bandwidth,cutwidth,cyclic bandwidth,bandwidth sum and cyclic bandwidth sum are well-known indices about optimal labeling of graphs applied in VLSI design,network communications,and other areas involving the graph layout.To design the graphs with the given indices,we need to study the ergodicity.Let F be a set of graphs under consideration andφan integer-valued function defined on F,namely,φis an index,such as bandwidth and cutwidth.If there exists a graph G∈F such thatφ(G)=x for any integer x in the interval[a,b],where a and b are the minimum and maximum ofφon F,respectively,thenφis said to have ergodicity on F.Let Gnbe the set of simple connected graphs with order n and Tnthe set of trees with order n.In this paper,we investigate the ergodicity of bandwidth,cutwidth,cyclic bandwidth,the bandwidth sum and cyclic bandwidth sum on Tn and Gn.
文摘In this paper, we investigate the flow of customers through queuing systems with randomly varying intensities. The analysis of the Kolmogorov-Chapman system of stationary equations for this model showed that it is not possible to construct a convenient symbolic solution. In this paper an attempt is made to circumvent this requirement by referring to the ergodicity theorems, which gives the conditions for the existence of the limit distribution in the service processes, but do not require knowledge of them.
文摘In this paper, we give a survey on the PhD thesis of the first author. There theexistence and ergodicity on invariant measures of set-valued mappings are discused.
基金supported by the Natural Science Foundation of Jiangsu Province,P.R.China(No.BK20220553)the China Postdoctoral Science Foundation(No.2023M742955)Z.Qiu's work was supported by the National Natural Science Foundation of China(No.12071217).
文摘In this paper,a stochastic framework with a general nonmonotonic response function is formulated to investigate the competition dynamics between two species in a chemostat environment.The model incorporates both white noise and telegraph noise,the latter being described by Markov process.The existence of a unique global positive solution for the stochastic chemostat model is established.Subsequently,by using the ergodic theory of Markov process and utilizing techniques of stochastic analysis,the critical value differentiating between persistence in mean and extinction for the microorganism species is explored.Moreover,the existence of a unique stationary distribution is proved by using stochastic Lyapunov analysis.Finally,numerical simulations are introduced to support the obtained results.
文摘针对智能反射面(reconfigurable intelligent surface,RIS)不具备复杂的感知和信号处理能力,使得信道状态信息(channel state information,CSI)估计存在误差的情况,构建了不完美Nakagami-m衰落信道下的RIS辅助端到端的无线通信系统模型,分析了此模型下系统的中断概率(outage probability,OP)和各态历经容量(ergodic capacity,EC).首先基于矩匹配法推导了端到端信干噪比(signal to interference plus noise ratio,SINR)的累积分布函数(cumulative distribution function,CDF),然后基于此CDF导出了OP与EC的解析闭合表达式,最后推导了高SINR条件下OP与EC的近似表达式.仿真实验结果表明:解析结果与蒙特卡罗仿真数据高度吻合,且在高SINR情况下与近似曲线逼近;完美CSI的OP值会随着SINR增大而持续降低,而不完美CSI会出现地板现象,且相关系数ρ越小,OP值越大;通过增加RIS的反射元件数量可有效减少信道估计误差带来的性能损失.
文摘对D2D(Device to Device)通信系统的隐蔽通信问题进行研究,提出了一种基于智能反射面(Intelligent Reflecting Surfaces,IRS)的D2D通信系统,该系统采用双IRS模式。首先分析了各信号的信干噪比(SINR)和各自的期望值,并据此推导出系统遍历总容量的解形式,接着针对隐蔽通信建立了二元假设问题,推导出误检测率的解形式,并据此计算出平均最小误检测率。除此之外,还分析了无IRS系统的遍历总容量和平均最小误检测率,与双IRS系统进行对比。仿真结果表明,双IRS系统的遍历总容量和平均最小误检测率优于无IRS系统,双IRS系统能够比传统的无IRS系统获得更高的容量和更低的误检测率。
