The fluctuation-dissipation theorem for contracted descriptions of Markov processes

It is shown that if the Onsager-Casimir relations and the fluctuationdissipation theorem are valid for a stationary, Gaussian, Markov process in anN-dimensional space, then these relations are valid when the process is projected into a subspace of the original space. Both time-reversal-even and time-reversal-odd variables are allowed. Previous derivations of the fluctuation-dissipation theorem for Brownian motion from fluctuating hydrodynamics are special cases of the present result. For the Brownian motion problem, the fluctuation-dissipation theorem is proven for the case of a compressible, thermally conducting fluid with a nonlocal equation of state. Arbitrary slip boundary conditions are considered as well.     

On the local central limit theorem for Gibbs processes

We derive a sufficient condition for the validity of the local central limit theorem for Gibbs processes and their isomorphism with a Bernoulli shift.     

Central limit theorem for the Lorentz process via perturbation theory

The Markov partition of the Sinai billiard allows the following heuristic interpretation for the Lorentz process with a ²-periodic configuration of scatterers: while executing a (non-Markovian) random walk on ², and particle changes its internal state according to the symbolic dynamics defined by the Markov partition. This picture can be formalized and then the Lorentz process appears as the limit of a sequence of (Markovian!) random walks with a finite but increasing number of internal states and the central limit theorem can be proved for it by perturbational expansions with uniformly bounded — in a sence related to the Perron-Frobenius theorem — coefficients and uniform remainder terms.     

Central limit theorem for mixing quantum systems and the CCR-algebra of fluctuations

We analyse macroscopic fluctuations of an infinite quantum system and introduce the CCR-C*-algebra of normal fluctuations. A non-commutative central limit theorem for mixing quantum systems is proved.     

An invariance principle for reversible Markov processes. Applications to random motions in random environments

We present an invariance principle for antisymmetric functions of a reversible Markov process which immediately implies convergence to Brownian motion for a wide class of random motions in random environments. We apply it to establish convergence to Brownian motion (i) for a walker moving in the infinite cluster of the two-dimensional bond percolation model, (ii) for ad-dimensional walker moving in a symmetric random environment under very mild assumptions on the distribution of the environment, (iii) for a tagged particle in ad-dimensional symmetric lattice gas which allows interchanges, (iv) for a tagged particle in ad-dimensional system of interacting Brownian particles. Our formulation also leads naturally to bounds on the diffusion constant.     

Memory functions of the additive Markov chains: applications to complex dynamic systems

A new approach to describing correlation properties of complex dynamic systems with long-range memory based on a concept of additive Markov chains (Phys. Rev. E 68 (2003) 061107) is developed. An equation connecting the memory and correlation function of the system under study is presented. This equation allows reconstructing a memory function using a correlation function of the system. Effectiveness and robustness of the proposed method is demonstrated by simple model examples. Memory functions of concrete coarse-grained literary texts are found and their universal power-law behavior at long distances is revealed.     

Central limit theorem in quantum field theory,generalized partons and application to deep-inelastic scattering

We prove the following elementary theorem. Ifφ ¹,...,φ ^N is a sequence of fields having identical, thougharbitrary, interactions but not interacting with each other and 〈φ _n 〉0,i=1,...,N then the generating functional of the «average» field φ^(N) may be explicitly obtained and may be written in terms of the two-point function of any of the fields φ _i . The theorem is then applied to define generalized parton fields \(\psi _j = \sum\limits_{i = 1}^N {\psi _{ij} } /\sqrt N \) as «averages» of basic fieldsψ _ij havingarbitrary interactions but not interacting with each other. We show that in the limitN→∞ Bjorken scaling, as observed at energies not too high, may be obtained if only quanta associated with generalized parton fields are excited in the hadron by the virtual photonwith no reference to the details of the underlying dynamics. ForN<∞, and the excitation of other quanta as well lead to a systematic breaking of scale invariance and the details of the dynamics are necessarily recovered which are expected to be applicable at higher energy regimes.     

The diffusion limit for reversible jump processes onZ d with ergodic random bond conductivities

We consider a reversible jump process on ? ^d whose jump rates themselves are random. We show mean square convergence of this process under diffusion scaling to a limiting Brownian motion with a certain diffusion matrix, characterizing effective conductivity.     

Subcritical asymptotic behavior in the thermodynamic limit of reversible random polymerization processes

We consider a reversible Markov process as a chemical polymerization model and study the asymptotic behavior (in the thermodynamic limit asN+) of a particular probability distribution on the set ofN-dimensional vectors, thekth component of which is the number ofk-mers. The study establishes the existence of three stages (subcritical, near-critical, and supercritical stages) of polymerization, depending on the value of the strength of the fragmentation reaction. The present paper concentrates on the analysis of the subcritical stage. In the subcritical stages we show that the size of the largest length of polymers of sizeN is of the order logN asN+.     

An application of a representation theorem for entropy functionals

The vibrational predissociation of HD₂ ⁺ is modelled in terms of quantum-mechanical tunnelling through a minimal centrifugal barrier at given total angular momentum, J, and with statistical intermode coupling behind the barrier. It is shown that the observed strong preference for the H⁺ + D ₂ predissociation channel (over D⁺ + HD) is consistent with an experimental preference for J values in the range 0 < J < 25, a range which is also shown to be consistent with the observed H₃ ⁺ preferred range of kinetic energy release. A correlation between the total angular momentum and the kinetic energy release is also predicted.     

Entropy functionals of Kolmogorov-Sinai type and their limit theorems

Two functionals and are introduced forC ^*-dynamical systems with invariant states and stationary channels. It is shown that the Kolmogorov-Sinai-type theorems hold for these functionals and . Our functionals and are set within the framework of quantum information theory and generalize a quantum KS entropy by CNT and the mutual entropy by Ohya.     

On a simple derivation of master equations for diffusion processes driven by white noise and dichotomic Markov noise

A very simple way is presented of deriving the partial differential equations (the master equations) satisfied by the probability density for certain kinds of diffusion processes in one dimension, in which the driving term is a Gaussian white noise, or a dichotomic noise, or a combination of the two. The method involves the use of certain ‘formulas of differentiation’ to derive the equations obeyed by the characteristic functions of the processes concerned, and thence the corresponding master equations. The examples presented cover a substantial number of diffusion processes that occur in physical modelling, including some master equations derived recently in the literature for generalizations of persistent diffusion.     

Unbounded functionals on a group algebra and applications to quantum theory

A certain class of positive functionals on a group algebra is examined that is pertinent to the induced representations of Frobenius and Mackey. Though these functionals are not bounded in theL ¹ norm, continuity still persists to an extent that secures the existence of a continuous group representation obtained from Gelfand's construction. The theory thus developed provides a new aspect of both the improper states in quantum theory and the induced representations of groups. The method is applied to the Poincaré group and it is shown that the representations, in which particles can be accommodated, are determined up to unitary equivalence by unbounded functionals of a simple structure. It is stressed that representations describing an infinitely degenerate vacuum emerge from mass nonzero representations as the mass tends to zero.     

Dynamical suppression of dephasing for Markov processes

It is shown that the dephasing of a qubit caused by a Markov process can be suppressed by a successive application of π-pulses if two-time conditional probability of the stochastic variable depends only on the time-difference. The several types of the π-pulse sequence are compared in the case of the two-state jump Markov process.     

Boundary behavior of diffusion approximations to Markov jump processes

We show that diffusion approximations, including modified diffusion approximations, can be problematic since the proper choice of local boundary conditions (if any exist) is not obvious. For a class of Markov processes in one dimension, we show that to leading order it is proper to use a diffusion (Fokker-Planck) approximation to compute mean exit times with a simple absorbing boundary condition. However, this is only true for the leading term in the asymptotic expansion of the mean exit time. Higher order correction terms do not, in general, satisfy simple absorbing boundary conditions. In addition, the diffusion approximation for the calculation of mean exit times is shown to break down as the initial point approaches the boundary, and leads to an increasing relative error. By introducing a boundary layer, we show how to correct the diffusion approximation to obtain a uniform approximation of the mean exit time. We illustrate these considerations with a number of examples, including a jump process which leads to Kramers' diffusion model. This example represents an extension to a multivariate process.     

A singular perturbation approach to first passage times for Markov jump processes

We introduce singular perturbation methods for constructing asymptotic approximations to the mean first passage time for Markov jump processes. Our methods are applied directly to the integrai equation for the mean first passage time and do not involve the use of diffusion approximations. An absorbing interval condition is used to properly account for the possible jumps of the process over the boundary which leads to a Wiener-Hopf problem in the neighborhood of the boundary. A model of unimolecular dissociation is considered to illustrate our methods.