Invariant Observables and the Individual Ergodic Theorem

The notion of the almost everywhere equality of observables is introduced. The limit of Cesaro means is an invariant observable with respect to this notion.     

Consistency of Microcanonical and Canonical Finite-Size Scaling

Typically, in order to obtain finite-size scaling laws for quantities in the microcanonical ensemble, an assumption is taken as a starting point. In this paper, consistency of such a Microcanonical Finite-Size Scaling Assumption with its commonly accepted canonical counterpart is shown, which puts Microcanonical Finite-Size Scaling on a firmer footing.     

Microcanonical Finite-Size Scaling

In the microcanonical ensemble, suitably defined observables show nonanalyticities and power-law behavior even for finite systems. For these observables, a microcanonical finite-size scaling theory is established and combined with the experimentally observed power-law behavior. Scaling laws are obtained which relate exponents of the finite system and critical exponents of the infinite system to the system-size dependence of the affiliated microcanonical observables.     

Microcanonical and Canonical Ensembles for fMRI Brain Networks in Alzheimer’s Disease

This paper seeks to advance the state-of-the-art in analysing fMRI data to detect onset of Alzheimer’s disease and identify stages in the disease progression. We employ methods of network neuroscience to represent correlation across fMRI data arrays, and introduce novel techniques for network construction and analysis. In network construction, we vary thresholds in establishing BOLD time series correlation between nodes, yielding variations in topological and other network characteristics. For network analysis, we employ methods developed for modelling statistical ensembles of virtual particles in thermal systems. The microcanonical ensemble and the canonical ensemble are analogous to two different fMRI network representations. In the former case, there is zero variance in the number of edges in each network, while in the latter case the set of networks have a variance in the number of edges. Ensemble methods describe the macroscopic properties of a network by considering the underlying microscopic characterisations which are in turn closely related to the degree configuration and network entropy. When applied to fMRI data in populations of Alzheimer’s patients and controls, our methods demonstrated levels of sensitivity adequate for clinical purposes in both identifying brain regions undergoing pathological changes and in revealing the dynamics of such changes.     

The one-dimensional Boltzmann gas: The ergodic hypothesis and the phase portrait of small systems

The concept of ergodicity and its application to microcanonical systems composed of few particles of different mases are clarified. The distribution functions in position and velocity are theoretically derived and numerically verified. Moreover, we deal with a one-dimensional Boltzmann gas where the order relation (connected to the one dimensionality) brings constraints depending on the two classes of boundary conditions enforced (reflecting, periodic). The numerical simulations on a one-dimensional Boltzmann gas act as real experiments and allow us to play on the constraints to which the system is subjected.     

Tricritical and Critical Exponents in Microcanonical Ensemble of Systems with Long-Range Interactions

We explore the tricritical points and the critical lines of both Blume-Emery-Griffiths and Ising model within long-range interactions in the microcanonical ensemble. For K=K_MTP, the tricritical exponents take the values β=1/4, 1=γ^-≠γ⁺=1/2 and 0=α^-≠α⁺=-1/2, which disagree with classical (mean field) values. When K>K_MTP, the phase transition becomes second order and the critical exponents have classical values except close to the canonical tricritical parameters (K_CTP), where the values of the critical expoents become β=1/2, 1=γ^-≠γ⁺=2 and 0=α⁺≠α⁺=1.     

ALADIN热曲线的微正则系综分析

用微正则瞬时多碎裂统计模型分析了ALADIN热曲线.结果表明落在ALADIN热曲线温度平台(温度扭曲)附近的弹剩余核(热核)的理论代表点,并不落在相应弹剩余核热曲线的温度平台(扭曲)范围内,从而对ALADIN热曲线是否可作为核液气相变的证据提出质疑.     

The Generalized Canonical Ensemble and Its Universal Equivalence with the Microcanonical Ensemble

This paper shows for a general class of statistical mechanical models that when the microcanonical and canonical ensembles are nonequivalent on a subset of values of the energy, there often exists a generalized canonical ensemble that satisfies a strong form of equivalence with the microcanonical ensemble that we call universal equivalence. The generalized canonical ensemble that we consider is obtained from the standard canonical ensemble by adding an exponential factor involving a continuous function g of the Hamiltonian. For example, if the microcanonical entropy is C², then universal equivalence of ensembles holds with g taken from a class of quadratic functions, giving rise to a generalized canonical ensemble known in the literature as the Gaussian ensemble. This use of functions g to obtain ensemble equivalence is a counterpart to the use of penalty functions and augmented Lagrangians in global optimization. linebreak Generalizing the paper by Ellis et al. [J. Stat. Phys. 101:999–1064 (2000)], we analyze the equivalence of the microcanonical and generalized canonical ensembles both at the level of equilibrium macrostates and at the thermodynamic level. A neat but not quite precise statement of one of our main results is that the microcanonical and generalized canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the generalized microcanonical entropy s–g is concave. This generalizes the work of Ellis et al., who basically proved that the microcanonical and canonical ensembles are equivalent at the level of equilibrium macrostates if and only if they are equivalent at the thermodynamic level, which is the case if and only if the microcanonical entropy s is concave.     

Ergodic Properties of the Quantum Geodesic Flow on Tori

We study ergodic averages for a class of pseudodifferential operators on the flatN-dimensional torus with respect to the Schrödinger evolution. The later can be consider a quantization of the geodesic flow on . We prove that, up to semi-classically negligible corrections, such ergodic averages are translationally invariant operators.Mathematics Subject Classifications (2000) 58J50, 58J40, 81S10.     

Ergodic Properties of the Non-Markovian Langevin Equation

We discuss the dissipative dynamics of a classical particle coupled to an infinitely extended heat reservoir. We announce a number of results concerning the ergodic properties of this model. The novelty of our approach is that it extends beyond Markovian dynamics to the case where the Langevin equation is driven by colored noise. Our method works in arbitrary space dimension, and for fully nonlinear systems.     

Ergodic Properties for a Quantum Nonlinear Dynamics

We present a quantum system composed of infinitely many particles, subject to a nonquadratic Hamiltonian, for which it is possible to investigate the long time behavior of the dynamics and its ergodic properties. We do so both for the KMS states and for a large class of locally normal invariant states, whose very existence is already of some interest.     

Probes of equipartition in nonlinear Hamiltonian systems

The time scales for equipartition to be reached are studied using a generalization of the recently introduced measure of ergodicity in liquids. The-Fermi-Pasta-Ulam model is chosen as an illustration. The measures are constructed by following the evolution of the systems using two independent initial conditions. The time-averaged property of an observable is calculated using the two dynamical trajectories. The measure is essentially the norm in the space of the observable obtained from the two trajectories. We show that the time-dependent behavior of the measure is a good indicator of the equipartition in large nonlinear systems. The numerical results show that equipartitioning critically depends on the initial conditions, and even when adequate mode mixing occurs the time scales appear to be extremely long.     

Microcanonical ensemble extensive thermodynamics of Tsallis statistics

The microscopic foundation of the generalized equilibrium statistical mechanics based on the Tsallis entropy is given by using the Gibbs idea of statistical ensembles of the classical and quantum mechanics. The equilibrium distribution functions are derived by the thermodynamic method based upon the use of the fundamental equation of thermodynamics and the statistical definition of the functions of the state of the system. It is shown that if the entropic index ξ=1/(q−1) in the microcanonical ensemble is an extensive variable of the state of the system, then in the thermodynamic limit the principle of additivity and the zero law of thermodynamics are satisfied. In particular, the Tsallis entropy of the system is extensive and the temperature is intensive. Thus, the Tsallis statistics completely satisfies all the postulates of the equilibrium thermodynamics. Moreover, evaluation of the thermodynamic identities in the microcanonical ensemble is provided by the Euler theorem. The principle of additivity and the Euler theorem are explicitly proved by using the illustration of the classical microcanonical ideal gas in the thermodynamic limit.     

A Note on Invariant Observables

The ergodic theory and particularly the individual ergodic theorem were studied in many structures. Recently the individual ergodic theorem has been proved for MV-algebras of fuzzy sets (Riečan, 2000; Riečan and Neubrunn, 1997) and even in general MV-algebras (Jurečková, 2000). The notion of almost everywhere equality of observables was introduced by B. Riečan and M. Jurečková in Riečan and Jurečková (2005). They proved that the limit of Cesaro means is an invariant observable for P-observables. In this paper show that the assumption of P-observable can be omitted.     

近独立子系系统的统计规律(二)——微正则分布函数

从微正则系统理论出发,导出了任意数目粒子构成的近独立子系系统的分布函数,在粒子数N趋于无限大的极限情况下,得到了麦克斯韦分布律.     

A new approach for the justification of ensembles in quantum statistical mechanics — I

In this and the following paper, a new approach for the justification of ensembles in statistical mechanics is given. The essential physical idea is that a measurement is an average of values arising from disjoint regions in three-space. This idea is given a mathematical basis in terms of a class of operators called local operators, and the first paper is devoted primarily to the development of the properties of local operators. In particular, a complete characterization of the bounded local operators on ₂ spaces of finite measure is given. Two results of importance for statistical mechanics are also derived. First, it is shown that the observables of quantum mechanics are local operators. Second, it is shown that the expectation value of an observable for a pure state can be written formally as an ensemble average. In the following paper, these results are used to develop a new approach for the justification of statistical ensembles.This work was supported in part by research grants from the National Science Foundation and the U.S. Public Health Service. The material of this paper is contained in a doctoral dissertation submitted by the author to the University of Oregon (1969).     

The Ergodic Theory of Traffic Jams

We introduce and analyze a simple probabilistic cellular automaton which emulates the flow of cars along a highway. Our Traffic CA captures the essential features of several more complicated algorithms, studied numerically by K. Nagel and others over the past decade as prototypes for the emergence of traffic jams. By simplifying the dynamics, we are able to identify and precisely formulate the self-organized critical evolution of our system. We focus here on the Cruise Control case, in which well-spaced cars move deterministically at maximal speed, and we obtain rigorous results for several special cases. Then we introduce a symmetry assumption that leads to a two-parameter model, described in terms of acceleration () and braking () probabilities. Based on the results of simulations, we map out the (, ) phase diagram, identifying three qualitatively distinct varieties of traffic which arise, and we derive rigorous bounds to establish the existence of a phase transition from free flow to jams. Many other results and conjectures are presented. From a mathematical perspective, Traffic CA provides local, particle-conserving, one-dimensional dynamics which cluster, and converge to a mixture of two distinct equilibria.     

Isospin dependence of nuclear multifragmentation in statistical model

The evolution of nuclear disintegration mechanisms with increasing excitation energy, from com- pound nucleus to multifragmentation, has been studied by using the Statistical Multifragmentation Model (SMM) within a micro-canonical ensemble. We discuss the observable characteristics as functions of excitation energy in multifragmentation, concentrating on the isospin dependence of the model in its decaying mechanism and break-up fragment configuration by comparing the A<,0> = 200, Z<,0> = 78 and A<,0> = 200, Z<,0> = 100 systems. The calculations indicate that the neutron-rich system (Z<,0> = 78) translates to a fission-like process from evaporation later than the symmetric nucleus at a lower excitation energy, but gets a larger average multiplicity as the excitation energy increases above 1.0 MeV/u.     

Microcanonical density functionals for critical systems: An exact one-dimensional example

Free-energy functionals suitable for describing realistic, nonuniform systems near criticality are discussed with emphasis on the advantages of a local formalism. It is proposed to investigatemicro canonical functionals in which both the usual order-parameter (or magnetization) density m(r)and the local energy density (r), which has independent critical fluctuations, are employed. This approach is tested by an exact calculation of the microcanonical functional [{m}, {}] in the continuum limit for a one-dimensional Ising model. Remarkably, the microcanonical functional is found to be local irrespective of the proximity to the critical point (located at zero temperature and zero field). Furthermore, its form relates closely to the scaling postulate advanced earlier by de Gennes and Fisher and displays features of conformal covariance.     

Ergodic properties of infinite-dimensional stochastic systems

The ergodic properties of two stochastic models _I and _II are investigated. Each model is described by a fieldx(t),t > 0, on the lattice =Z ^d,d < . For _I,x(t) evolves according to the equations wherex _s (t) R for eachs eF. Here the {w_s(t): s } are independent, one-dimensional Wiener processes, ₂ is a bounded interaction between adjacent lattice sites, and the potentials ₁ and ₂ satisfy appropriate regularity conditions. It is shown that for each model,x(t) is a Markov process on an infinite-dimensional phase spaceX. The probability measures onX that satisfy the Dobrushin-Lanford-Ruelle (DLR) conditions are stationary for this process and have a mixing property. Moreover, for _I any stationary, time-reversal-invariant probability measure that has certain regularity properties must satisfy the DLR conditions.This paper is based on a portion of the author's Ph.D. thesis.⁽²⁾