Causal conditioning and instantaneous coupling in causality graphs

The paper investigates the link between Granger causality graphs recently formalized by Eichler and directed information theory developed by Massey and Kramer. We particularly insist on the implication of two notions of causality that may occur in physical systems. It is well accepted that dynamical causality is assessed by the conditional transfer entropy, a measure appearing naturally as a part of directed information. Surprisingly the notion of instantaneous causality is often overlooked, even if it was clearly understood in early works. In the bivariate case, instantaneous coupling is measured adequately by the instantaneous information exchange, a measure that supplements the transfer entropy in the decomposition of directed information. In this paper, the focus is put on the multivariate case and conditional graph modeling issues. In this framework, we show that the decomposition of directed information into the sum of transfer entropy and information exchange does not hold anymore. Nevertheless, the discussion allows to put forward the two measures as pillars for the inference of causality graphs. We illustrate this on two synthetic examples which allow us to discuss not only the theoretical concepts, but also the practical estimation issues.Comment: submitte

Local equilibrium and the second law of thermodynamics for irreversible systems with thermodynamic inertia

Validity of local equilibrium has been questioned for non-equilibrium systems which are characterized by delayed response. In particular, for systems with non-zero thermodynamic inertia, the assumption of local equilibrium leads to negative values of the entropy production, which is in contradiction with the second law of thermodynamics. In this paper we address this question by suggesting a variational formulation of irreversible evolution of a system with non-zero thermodynamic inertia. We introduce the Lagrangian, which depends on the properties of the normal and the so-called "mirror-image" systems. We show that the standard evolution equations, in particular the Maxwell-Cattaneo-Vernotte equation, can be derived from the variational procedure without going beyond the assumption of local equilibrium. We also argue, that the second law of thermodynamics should be understood as a consequence of the variational procedure and the property of local equilibrium. For systems with instantaneous response this leads to the standard requirement of the local instantaneous entropy production being always positive. However, if a system is characterized by delayed response, the formulation of the second law of thermodynamics should be altered. In particular, the quantity, which is always positive, is not the instantaneous entropy production, but the entropy production averaged over the period of the heat wave.Comment: 12 pages, 7 figure

Relative Entropy of a Freely Cooling Granular Gas

The time evolution and stationary values of the entropy per particle of a homogeneous freely cooling granular gas, relative to the maximum entropy consistent with the instantaneous translational and rotational temperatures, is analyzed by means of a Sonine approximation involving fourth-degree cumulants. The results show a rich variety of dependencies of the relative entropy on time and on the coefficients of normal and tangential restitution, including a peculiar behavior in the quasi-smooth limit.Comment: 6 pages; 2 figures; contributed paper at the 28th International Symposium on Rarefied Gas Dynamics (Zaragoza, Spain, July 9-13, 2012

Dynamics of entanglement entropy and entanglement spectrum crossing a quantum phase transition

We study the time evolution of entanglement entropy and entanglement spectrum in a finite-size system which crosses a quantum phase transition at different speeds. We focus on the Ising model with a time-dependent magnetic field, which is linearly tuned on a time scale $\tau$ . The time evolution of the entanglement entropy displays different regimes depending on the value of $\tau$, showing also oscillations which depend on the instantaneous energy spectrum. The entanglement spectrum is characterized by a rich dynamics where multiple crossings take place with a gap-dependent frequency. Moreover, we investigate the Kibble-Zurek scaling of entanglement entropy and Schmidt gap.Comment: Accepted for publication in Phys. Rev.

Fluctuations of entropy production in turbulent thermal convection

We report on a numerical experiment performed to analyze fluctuations of the entropy production in turbulent thermal convection, a physical configuration that represents here a prototypical case of an out-of-equilibrium dissipative system. We estimate the entropy production from instantaneous measurements of the local temperature and velocity fields sampled along the trajectory of a large number of point-wise Lagrangian tracers. The entropy production is characterized by large fluctuations and becomes often negative. This represents a sort of "finite-time" violation of the second principle of thermodynamics, since the direction of the energy flux is opposite to that prescribed by the external gradient. We clearly show that the fluctuations of entropy production observed in the present system verify the fluctuation relation (FR), even though the system is time-irreversible

A statistical mechanical interpretation of instantaneous codes

In this paper we develop a statistical mechanical interpretation of the noiseless source coding scheme based on an absolutely optimal instantaneous code. The notions in statistical mechanics such as statistical mechanical entropy, temperature, and thermal equilibrium are translated into the context of noiseless source coding. Especially, it is discovered that the temperature 1 corresponds to the average codeword length of an instantaneous code in this statistical mechanical interpretation of noiseless source coding scheme. This correspondence is also verified by the investigation using box-counting dimension. Using the notion of temperature and statistical mechanical arguments, some information-theoretic relations can be derived in the manner which appeals to intuition.Comment: 5 pages, Proceedings of the 2007 IEEE International Symposium on Information Theory, pp.1906 - 1910, Nice, France, June 24 - 29, 200