Optimal Linear Responses for Markov Chains and Stochastically Perturbed Dynamical Systems

The linear response of a dynamical system refers to changes to properties of the system when small external perturbations are applied. We consider the little-studied question of selecting an optimal perturbation so as to (i) maximise the linear response of the equilibrium distribution of the system, (ii) maximise the linear response of the expectation of a specified observable, and (iii) maximise the linear response of the rate of convergence of the system to the equilibrium distribution. We also consider the inhomogeneous, sequential, or time-dependent situation where the governing dynamics is not stationary and one wishes to select a sequence of small perturbations so as to maximise the overall linear response at some terminal time. We develop the theory for finite-state Markov chains, provide explicit solutions for some illustrative examples, and numerically apply our theory to stochastically perturbed dynamical systems, where the Markov chain is replaced by a matrix representation of an approximate annealed transfer operator for the random dynamical system.     

Kinetic theory of hydrodynamic flows. II. The drag on a sphere and on a cylinder

In this paper we continue the study of solutions of the extended Boltzmann equation started previously. In particular, we study an iterated solution of the equation that can be used to describe the flow of a rarefied gas around a macroscopic object. We discuss the rarefied flow and then show how the iterated solution can be extended into the hydrodynamic regime. The results for the drag force and for the distribution function of the gas molecules are shown to be identical to the results obtained in a previous paper by a generalization of the normal solution method. We also discuss the special properties of both rarefied and continuum flows around a cylinder and show that in both regions one must take into account Oseen-like terms which naturally appear in the extended Boltzmann equation. In the hydrodynamic regime we obtain Lamb's formula for the force on the cylinder. By relating the terms in the iterated expression to dynamical events taking place in the fluid, we are able to discuss the dynamical origin of the results obtained here.A preliminary report on the work described here and in Part I was given in Ref. 2.     

Time-dependent shell-model theory of dissipative heavy-ion collisions

A transport theory is formulated within a time-dependent shell-model approach. Time averaging of the equations for macroscopic quantities lead to irreversibility and justifies weak-coupling limit and Markov approximation for the (energy-conserving) one- and two-body collision terms. Two coupled equations for the occupation probabilities of dynamical single-particle states and for the collective variable are derived and explicit formulas for transition rates, dynamical forces, mass parameters and friction coefficients are given. The applicability of the formulation in terms of characteristic quantities of nuclear systems is considered in detail and some peculiarities due to memory effects in the initial equilibration process of heavy-ion collisions are discussed.     

Kinetic Ising model in a time-dependent oscillating external magnetic field: effective-field theory

Recently,Shi et al.[2008 Phys.Lett.A 372 5922] have studied the dynamical response of the kinetic Ising model in the presence of a sinusoidal oscillating field and presented the dynamic phase diagrams by using an effective-field theory(EFT) and a mean-field theory(MFT).The MFT results are in conflict with those of the earlier work of Tom’e and de Oliveira,[1990 Phys.Rev.A 41 4251].We calculate the dynamic phase diagrams and find that our results are similar to those of the earlier work of Tom’e and de Oliveira;hence the dynamic phase diagrams calculated by Shi et al.are incomplete within both theories,except the low values of frequencies for the MFT calculation.We also investigate the influence of external field frequency(ω) and static external field amplitude(h0) for both MFT and EFT calculations.We find that the behaviour of the system strongly depends on the values of ω and h0.     

Characterizing the dynamical importance of network nodes and links

The largest eigenvalue of the adjacency matrix of networks is a key quantity determining several important dynamical processes on complex networks. Based on this fact, we present a quantitative, objective characterization of the dynamical importance of network nodes and links in terms of their effect on the largest eigenvalue. We show how our characterization of the dynamical importance of nodes can be affected by degree-degree correlations and network community structure. We discuss how our characterization can be used to optimize techniques for controlling certain network dynamical processes and apply our results to real networks.     

Bose-Einstein condensates beyond mean field theory: quantum backreaction as decoherence

We propose an experiment to measure the slow log(N) convergence to mean field theory (MFT) around a dynamical instability. Using a density matrix formalism instead of the standard macroscopic wave function approach, we derive equations of motion which go beyond MFT and provide accurate predictions for the quantum break time. The leading quantum corrections appear as decoherence of the reduced single-particle quantum state.     

Induced maps,Markov extensions and invariant measures in one-dimensional dynamics

A way to study ergodic and measure theoretic aspects of interval maps is by means of the Markov extension. This tool, which ties interval maps to the theory of Markov chains, was introduced by Hofbauer and Keller. More generally known are induced maps, i.e. maps that, restricted to an element of an interval partition, coincide with an iterate of the original map.We will discuss the relation between the Markov extension and induced maps. The main idea is that an induced map of an interval map often appears as a first return map in the Markov extension. For S-unimodal maps, we derive a necessary condition for the existence of invariant probability measures which are absolutely continuous with respect to Lebesgue measure. Two corollaries are given.     

Non-equilibrium phenomena and molecular reaction dynamics: mode space,energy space and conformer space

The ability to characterise and control matter far away from equilibrium is a frontier challenge facing modern science. In this article, we sketch out a heuristic structure for thinking about the different ways in which non-equilibrium phenomena can impact molecular reaction dynamics. Our analytical schema includes three different regimes, organised according to increasing dynamical resolution: at the lowest resolution, we have conformer phase space, at an intermediate resolution, we have energy space; and at the highest resolution, we have mode space. Within each regime, we discuss practical definitions of non-equilibrium phenomena, mostly in terms of the corresponding relaxation timescales. Using this analytical framework, we discuss some recent non-equilibrium reaction dynamics studies spanning isolated small-molecule ensembles, gas-phase ensembles and solution-phase ensembles. This includes new results that provide insight into how non-equilibrium phenomena impact the solution-phase alkene–hydroboration reaction. We emphasise that interesting non-equilibrium dynamical phenomena often occur when the relaxation timescales characterising each regime are similar. In closing, we reflect on outstanding challenges and future research directions to guide our understanding of how non-equilibrium phenomena impact reaction dynamics.     

What Can One Learn About Self-Organized Criticality from Dynamical Systems Theory?

We develop a dynamical system approach for the Zhang model of self-organized criticality, for which the dynamics can be described either in terms of iterated function systems or as a piecewise hyperbolic dynamical system of skew-product type. In this setting we describe the SOC attractor, and discuss its fractal structure. We show how the Lyapunov exponents, the Haussdorf dimensions, and the system size are related to the probability distribution of the avalanche size via the Ledrappier–Young formula.     

Stochastic perturbation theory: a low-scaling approach to correlated electronic energies

We discuss a stochastic implementation of M?ller-Plesset (MP) theory based upon the concept of a "graph," a set of connected Slater determinants. We show how contributions from an arbitrary level, MPn, of perturbation theory can be expressed diagrammatically in terms of graphs, and that these may be stochastically sampled to give a good estimate of the energy. We show this to be the case for Ne, Ar, N2, and H2O molecules. N-molecule chains of He atoms and H2 molecules at equilibrium and stretched geometries show an effective scaling of O[N(2.6)] and O[N(5.6)] for MP2 and MP3 theories.     

How hidden are hidden processes? A primer on crypticity and entropy convergence

We investigate a stationary process's crypticity--a measure of the difference between its hidden state information and its observed information--using the causal states of computational mechanics. Here, we motivate crypticity and cryptic order as physically meaningful quantities that monitor how hidden a hidden process is. This is done by recasting previous results on the convergence of block entropy and block-state entropy in a geometric setting, one that is more intuitive and that leads to a number of new results. For example, we connect crypticity to how an observer synchronizes to a process. We show that the block-causal-state entropy is a convex function of block length. We give a complete analysis of spin chains. We present a classification scheme that surveys stationary processes in terms of their possible cryptic and Markov orders. We illustrate related entropy convergence behaviors using a new form of foliated information diagram. Finally, along the way, we provide a variety of interpretations of crypticity and cryptic order to establish their naturalness and pervasiveness. This is also a first step in developing applications in spatially extended and network dynamical systems.     

The gravitational interaction: Spin,rotation, and quantum effects-a review

Previous work on spin, rotation, and quantum effects in gravitation is surveyed, with particular emphasis on the gravitational two-body interaction, both for elementary particles and for macroscopic bodies. Applications considered include (a) the precession of a gyroscope, (b) rotational effects on the equations of motion for the orbit, (c) binary systems, particularly the binary pulsar PSR 1913+16, and (d) the prospects of measuring spin-orbit and spin-spin forces in the laboratory. In addition, we discuss quantum effects that arise in the interaction between elementary particles. In particular, we point out the potentially decisive role of these forces in high-density matter, with emphasis on the fact that repulsive forces arise that may prevent gravitational collapse. All of the above considerations are within the framework of Einstein's theory of general relativity, albeit extended to treat spin-dependent and quantum forces. Finally, we consider the additional quantum terms that are present if one works with a generalization of Einstein's theory, the Einstein-Cartan-Sciama-Kibble theory of gravitation, in which the spin of matter, as well as its mass, plays a dynamical role.     

Alternative interpretations of power-law distributions found in nature

We investigate two inherently different classes of probability density functions (pdfs) that share the common property of power law tails: the α-stable Le?vy process and the linear Markov diffusion process with additive and multiplicative Gaussian noise. Dynamical processes described by these distributions cannot be uniquely identified as belonging to one or the other class either by diverging variance due to power-law tails in the pdf or by the possible existence of skew. However, there are distinguishing features that may be found in sufficiently well sampled time series. We examine these features and discuss how they may guide the development of proper approximations to equations of motion underlying dynamical systems. An additional result of this research was the identification of a variable describing the relative importance of the multiplicative and independent additive noise forcing in our linear Markov process. The distribution of this variable is generally skewed, depending on the level of correlation between the additive and multiplicative noise.     

QCD at finite density and color superconductivity

The paper contains a brief review of recent applications of many-body theory to quark matter. We discuss the progress in theory of dense quark matter during the last two years, especially color superconductivity. We emphasize that there are two basic dynamical reasons for it: short-range forces induced by instantons and long-range ones mediated by exchanges of magnetic gluons. For quark matter which is supposed to be found in neutron stars, both lead to superconducting gaps on the order of 100 MeV. The most surprising facts are the rather impressive richness of different phases and their robustness in respect to variation of the fundamental interaction.     

Factorisation scale independence,the connection between alternative explanations of the EMC effect and QCD predictions for nuclear properties

We examine different models for deep inelastic scattering in nuclei in the context of the operator product expansion where the operator matrix elements involve a factorisation scale characterising the separation of short or long distance physics. By exploiting the independence of physical quantities upon this scale we can connect seemingly different models for the nucleus such as dynamical rescaling and the standard convolution models of nuclear physics, allowing nuclear properties to be simply expressed in terms of the anomalous dimensions of QCD. We discuss how non-convolution contributions may also be described by dynamical rescaling and we show how to extend dynamical rescaling to describe spin dependent quantities.     

Controlling chaotic solitons in Frenkel-Kontorova chains by disordered driving forces

We discuss a general mechanism explaining the taming effect of phase disorder in external forces on chaotic solitons in damped, driven, Frenkel-Kontorova chains. We deduce analytically an effective random equation of motion governing the dynamics of the soliton center of mass for which we obtain numerically the regions in the control parameter space where chaotic solitons are suppressed. We find that such predictions are in excellent agreement with results of computer simulations of the original Frenkel-Kontorova chains. We show theoretically how such a fundamental mechanism explains recent numerical results concerning extended chaos in arrays of coupled pendula [S. F. Brandt, Phys. Rev. Lett. 96, 034104 (2006)10.1103/PhysRevLett.96.034104].     

Markov chains theory for scale-free networks

This paper proposes a Markov chain method to predict the growth dynamics of the individual nodes in scale-free networks, and uses this to calculate numerically the degree distribution. We first find that the degree evolution of a node in the BA model is a nonhomogeneous Markov chain. An efficient algorithm to calculate the degree distribution is developed by the theory of Markov chains. The numerical results for the BA model are consistent with those of the analytical approach. A directed network with the logarithmic growth is introduced. The algorithm is applied to calculate the degree distribution for the model. The numerical results show that the system self-organizes into a scale-free network.     

Nature computes: information processing in quantum dynamical systems

Nature intrinsically computes. It has been suggested that the entire universe is a computer, in particular, a quantum computer. To corroborate this idea we require tools to quantify the information processing. Here we review a theoretical framework for quantifying information processing in a quantum dynamical system. So-called intrinsic quantum computation combines tools from dynamical systems theory, information theory, quantum mechanics, and computation theory. We will review how far the framework has been developed and what some of the main open questions are. On the basis of this framework we discuss upper and lower bounds for intrinsic information storage in a quantum dynamical system.     

Constrained Markovian Dynamics of Random Graphs

We introduce a statistical mechanics formalism for the study of constrained graph evolution as a Markovian stochastic process, in analogy with that available for spin systems, deriving its basic properties and highlighting the role of the ‘mobility’ (the number of allowed moves for any given graph). As an application of the general theory we analyze the properties of degree-preserving Markov chains based on elementary edge switchings. We give an exact yet simple formula for the mobility in terms of the graph’s adjacency matrix and its spectrum. This formula allows us to define acceptance probabilities for edge switchings, such that the Markov chains become controlled Glauber-type detailed balance processes, designed to evolve to any required invariant measure (representing the asymptotic frequencies with which the allowed graphs are visited during the process). As a corollary we also derive a condition in terms of simple degree statistics, sufficient to guarantee that, in the limit where the number of nodes diverges, even for state-independent acceptance probabilities of proposed moves the invariant measure of the process will be uniform. We test our theory on synthetic graphs and on realistic larger graphs as studied in cellular biology, showing explicitly that, for instances where the simple edge swap dynamics fails to converge to the uniform measure, a suitably modified Markov chain instead generates the correct phase space sampling.