Kinetic theory of information -- the dynamics of information

A kinetic approach to the notion of information is proposed, based on Liouville kinetic theory. The general kinetic equation for the evolution of the N-particle information $\mathcal{I}_N$ in a Hamiltonian system of large particle number $N\gg 1$ is obtained. It is shown that the $N$-particle information is strictly conserved. Defining reduced particle number information densities in phase space should be possible to obtain a kinetic equation for the ordinary one-particle information $\mathcal{I}_1\equiv \mathcal{I}$ following the Bogoliubov prescription. The kinetic equation for $\mathcal{I}$ is a kind of generalized Boltzmann equation with interaction term depending on the hierarchy of reduced informations. This term in its general form is the most general expression for the Kolmogorov entropy rate of evolution of the information.Comment: 5 pages, no figures, accepted in Front. Phy

Dynamics of the Fisher Information Metric

We present a method to generate probability distributions that correspond to metrics obeying partial differential equations generated by extremizing a functional $J[g^{\mu\nu}(\theta^i)]$, where $g^{\mu\nu}(\theta^i)$ is the Fisher metric. We postulate that this functional of the dynamical variable $g^{\mu\nu}(\theta^i)$ is stationary with respect to small variations of these variables. Our approach enables a dynamical approach to Fisher information metric. It allows to impose symmetries on a statistical system in a systematic way. This work is mainly motivated by the entropy approach to nonmonotonic reasoning.Comment: 11 page

Information Percolation with Equilibrium Search Dynamics

We solve for the equilibrium dynamics of information sharing in a large population. Each agent is endowed with signals regarding the likely outcome of a random variable of common concern. Individuals choose the effort with which they search for others from whom they can gather additional information. When two agents meet, they share their information. The information gathered is further shared at subsequent meetings, and so on. Equilibria exist in which agents search maximally until they acquire sufficient information precision, and then minimally. A tax whose proceeds are used to subsidize the costs of search improves information sharing and can in some cases increase welfare. On the other hand, endowing agents with public signals reduces information sharing and can in some cases decrease welfare

Information dynamics: Temporal behavior of uncertainty measures

We carry out a systematic study of uncertainty measures that are generic to dynamical processes of varied origins, provided they induce suitable continuous probability distributions. The major technical tool are the information theory methods and inequalities satisfied by Fisher and Shannon information measures. We focus on a compatibility of these inequalities with the prescribed (deterministic, random or quantum) temporal behavior of pertinent probability densities.Comment: Incorporates cond-mat/0604538, title, abstract changed, text modified, to appear in Cent. Eur. J. Phy

Quantum Information Dynamics and Open World Science

One of the fundamental insights of quantum mechanics is that complete knowledge of the state of a quantum system is not possible. Such incomplete knowledge of a physical system is the norm rather than the exception. This is becoming increasingly apparent as we apply scientific methods to increasingly complex situations. Empirically intensive disciplines in the biological, human, and geosciences all operate in situations where valid conclusions must be drawn, but deductive completeness is impossible. This paper argues that such situations are emerging examples of {it Open World} Science. In this paradigm, scientific models are known to be acting with incomplete information. Open World models acknowledge their incompleteness, and respond positively when new information becomes available. Many methods for creating Open World models have been explored analytically in quantitative disciplines such as statistics, and the increasingly mature area of machine learning. This paper examines the role of quantum theory and quantum logic in the underpinnings of Open World models, examining the importance of structural features of such as non-commutativity, degrees of similarity, induction, and the impact of observation. Quantum mechanics is not a problem around the edges of classical theory, but is rather a secure bridgehead in the world of science to come

From Information Geometry to Newtonian Dynamics

Newtonian dynamics is derived from prior information codified into an appropriate statistical model. The basic assumption is that there is an irreducible uncertainty in the location of particles so that the state of a particle is defined by a probability distribution. The corresponding configuration space is a statistical manifold the geometry of which is defined by the information metric. The trajectory follows from a principle of inference, the method of Maximum Entropy. No additional "physical" postulates such as an equation of motion, or an action principle, nor the concepts of momentum and of phase space, not even the notion of time, need to be postulated. The resulting entropic dynamics reproduces the Newtonian dynamics of any number of particles interacting among themselves and with external fields. Both the mass of the particles and their interactions are explained as a consequence of the underlying statistical manifold.Comment: Presented at MaxEnt 2007, the 27th International Workshop on Bayesian Inference and Maximum Entropy Methods (July 8-13, 2007, Saratoga Springs, New York, USA

Testing Quantum Dynamics in Genetic Information Processing

Does quantum dynamics play a role in DNA replication? What type of tests would reveal that? Some statistical checks that distinguish classical and quantum dynamics in DNA replication are proposed.Comment: 4 pages, latex. (v2) Several points elaborated. Published version, formatted according to the journal styl

Information geometry, dynamics and discrete quantum mechanics

We consider a system with a discrete configuration space. We show that the geometrical structures associated with such a system provide the tools necessary for a reconstruction of discrete quantum mechanics once dynamics is brought into the picture. We do this in three steps. Our starting point is information geometry, the natural geometry of the space of probability distributions. Dynamics requires additional structure. To evolve the probabilities $P^k$, we introduce coordinates $S^k$ canonically conjugate to the $P^k$ and a symplectic structure. We then seek to extend the metric structure of information geometry, to define a geometry over the full space of the $P^k$ and $S^k$. Consistency between the metric tensor and the symplectic form forces us to introduce a K\"ahler geometry. The construction has notable features. A complex structure is obtained in a natural way. The canonical coordinates of the K\"ahler space are precisely the wave functions of quantum mechanics. The full group of unitary transformations is obtained. Finally, one may associate a Hilbert space with the K\"ahler space, which leads to the standard version of quantum theory. We also show that the metric that we derive here using purely geometrical arguments is precisely the one that leads to Wootters' expression for the statistical distance for quantum systems.Comment: 12 pages. Presented at MaxEnt 2012, the 32nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, July 15-20, 2012, Garching near Munich, Germany. Updated version includes corrections of typos and minor revision