Nonlinear response and fluctuation dissipation relations

A unified derivation of the off equilibrium fluctuation dissipation relations (FDR) is given for Ising and continous spins to arbitrary order, within the framework of Markovian stochastic dynamics. Knowledge of the FDR allows to develop zero field algorithms for the efficient numerical computation of the response functions. Two applications are presented. In the first one, the problem of probing for the existence of a growing cooperative length scale is considered in those cases, like in glassy systems, where the linear FDR is of no use. The effectiveness of an appropriate second order FDR is illustrated in the test case of the Edwards-Anderson spin glass in one and two dimensions. In the second one, the important problem of the definition of an off equilibrium effective temperature through the nonlinear FDR is considered. It is shown that, in the case of coarsening systems, the effective temperature derived from the second order FDR is consistent with the one obtained from the linear FDR.Comment: 24 pages, 6 figure

Modeling of Complex Systems II: A minimalist and unified semantics for heterogeneous integrated systems

International audienceThe purpose of this paper is to contribute to a unified formal framework for complex systems modeling. To this aim, we define a unified semantics for systems including integration operators. We consider complex systems as functional blackboxes (with internal states), whose structure and behaviors can be constructed through a recursive integration of heterogeneous components. We first introduce formal definitions of time (allowing to deal uniformly with both continuous and discrete times) and data (allowing to handle heterogeneous data), and introduce a generic synchronization mechanism for dataflows. We then define a system as a mathematical object characterized by coupled functional and states behaviors. This definition is expressive enough to capture the functional behavior of any real system with sequential transitions. We finally provide formal operators for integrating systems and show that they are consistent with the classical definitions of those operators on transfer functions which model real systems

A unified approach to time consistency of dynamic risk measures and dynamic performance measures in discrete time

In this paper we provide a flexible framework allowing for a unified study of time consistency of risk measures and performance measures (also known as acceptability indices). The proposed framework not only integrates existing forms of time consistency, but also provides a comprehensive toolbox for analysis and synthesis of the concept of time consistency in decision making. In particular, it allows for in depth comparative analysis of (most of) the existing types of time consistency -- a feat that has not be possible before and which is done in the companion paper [BCP2016] to this one. In our approach the time consistency is studied for a large class of maps that are postulated to satisfy only two properties -- monotonicity and locality. The time consistency is defined in terms of an update rule. The form of the update rule introduced here is novel, and is perfectly suited for developing the unifying framework that is worked out in this paper. As an illustration of the applicability of our approach, we show how to recover almost all concepts of weak time consistency by means of constructing appropriate update rules

Causal Fermion Systems as a Candidate for a Unified Physical Theory

The theory of causal fermion systems is an approach to describe fundamental physics. Giving quantum mechanics, general relativity and quantum field theory as limiting cases, it is a candidate for a unified physical theory. We here give a non-technical introduction.Comment: 19 pages, LaTeX, minor improvements (published version

Which quantum theory must be reconciled with gravity? (And what does it mean for black holes?)

We consider the nature of quantum properties in non-relativistic quantum mechanics (QM) and relativistic QFTs, and examine the connection between formal quantization schemes and intuitive notions of wave-particle duality. Based on the map between classical Poisson brackets and their associated commutators, such schemes give rise to quantum states obeying canonical dispersion relations, obtained by substituting the de Broglie relations into the relevant (classical) energy-momentum relation. In canonical QM, this yields a dispersion relation involving $\hbar$ but not $c$, whereas the canonical relativistic dispersion relation involves both. Extending this logic to the canonical quantization of the gravitational field gives rise to loop quantum gravity, and a map between classical variables containing $G$ and $c$, and associated commutators involving $\hbar$. This naturally defines a "wave-gravity duality", suggesting that a quantum wave packet describing {\it self-gravitating matter} obeys a dispersion relation involving $G$, $c$ and $\hbar$. We propose an ansatz for this relation, which is valid in the semi-Newtonian regime of both QM and general relativity. In this limit, space and time are absolute, but imposing $v_{\rm max} = c$ allows us to recover the standard expressions for the Compton wavelength $\lambda_C$ and the Schwarzschild radius $r_S$ within the same ontological framework. The new dispersion relation is based on "extended" de Broglie relations, which remain valid for slow-moving bodies of {\it any} mass $m$. These reduce to canonical form for $m \ll m_P$, yielding $\lambda_C$ from the standard uncertainty principle, whereas, for $m \gg m_P$, we obtain $r_S$ as the natural radius of a self-gravitating quantum object. Thus, the extended de Broglie theory naturally gives rise to a unified description of black holes and fundamental particles in the semi-Newtonian regime.Comment: 38 pages, 5 figures. Invited contribution to the Universe special issue "Open questions in black hole physics" (Gonzalo J. Olmo, Ed.). Matches published versio

Anisotropic Continuum-Molecular Models: A Unified Framework Based on Pair Potentials for Elasticity, Fracture and Diffusion-Type Problems

This paper presents a unified framework for continuum-molecular modeling of anisotropic elasticity, fracture and diffusion-based problems within a generalized two-dimensional peridynamic theory. A variational procedure is proposed to derive the governing equations of the model, that postulates oriented material points interacting through pair potentials from which pairwise generalized actions are computed as energy conjugates to properly defined pairwise measures of primary field variables. While mass is considered as continuous function of volume, we define constitutive laws for long-range interactions such that the overall anisotropic behavior of the material is the result of the assigned elastic, conductive and failure micro-interaction properties. The non-central force assumption in elasticity, together with the definition of specific orientation-dependent micromoduli functions respecting material symmetries, allow to obtain a fully anisotropic non-local continuum using a purely pairwise description of deformation and constitutive properties. A general and consistent micro-macro moduli correspondence principle is also established, based on the formal analogy with the classic elastic and conductivity tensors. The main concepts presented in this work can be used for further developments of anisotropic continuum-molecular formulations to include other mechanical behaviors and coupled phenomena involving different physics