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Phil Attard - One of the best experts on this subject based on the ideXlab platform.

  • non Equilibrium thermodynamics and Statistical Mechanics foundations and applications
    2012
    Co-Authors: Phil Attard
    Abstract:

    Preface 1. Prologue 2. Fluctuation Theory 3. Brownian Motion 4. Heat Conduction 5. Second Entropy For Fluctuating Hydrodynamics 6. Heat Convection and Non-Equilibrium Phase Transitions 7. Equilibrium Statistical Mechanics 8. Non-Equilibrium Statistical Mechanics 9. Statistical Mechanics of Steady Flow: Heat and Shear 10. Generalised Langevin Equation 11. Non-Equilibrium Computer Simulation Algorithms References Index

  • non Equilibrium thermodynamics and Statistical Mechanics foundations and applications
    2012
    Co-Authors: Phil Attard
    Abstract:

    Preface 1. Prologue 2. Fluctuation Theory 3. Brownian Motion 4. Heat Conduction 5. Second Entropy For Fluctuating Hydrodynamics 6. Heat Convection and Non-Equilibrium Phase Transitions 7. Equilibrium Statistical Mechanics 8. Non-Equilibrium Statistical Mechanics 9. Statistical Mechanics of Steady Flow: Heat and Shear 10. Generalised Langevin Equation 11. Non-Equilibrium Computer Simulation Algorithms References Index

  • theory for non Equilibrium Statistical Mechanics
    2006
    Co-Authors: Phil Attard
    Abstract:

    This paper reviews a new theory for non-Equilibrium Statistical Mechanics. This gives the non-Equilibrium analogue of the Boltzmann probability distribution, and the generalization of entropy to dynamic states. It is shown that this so-called second entropy is maximized in the steady state, in contrast to the rate of production of the conventional entropy, which is not an extremum. The relationships of the new theory to Onsager’s regression hypothesis, Prigogine’s minimal entropy production theorem, the Langevin equation, the formula of Green and Kubo, the Kawasaki distribution, and the non-Equilibrium fluctuation and work theorems, are discussed. The theory is worked through in full detail for the case of steady heat flow down an imposed temperature gradient. A Monte Carlo algorithm based upon the steady state probability density is summarized, and results for the thermal conductivity of a Lennard-Jones fluid are shown to be in agreement with known values. Also discussed is the generalization to non-Equilibrium mechanical work, and to non-Equilibrium quantum Statistical Mechanics. As examples of the new theory two general applications are briefly explored: a non-Equilibrium version of the second law of thermodynamics, and the origin and evolution of life.

Claudealain Pillet - One of the best experts on this subject based on the ideXlab platform.

  • quantum hypothesis testing and non Equilibrium Statistical Mechanics
    2012
    Co-Authors: Vojkan Jaksic, Yoshiko Ogata, Claudealain Pillet, Robert Seiringer
    Abstract:

    We extend the mathematical theory of quantum hypothesis testing to the general W*-algebraic setting and explore its relation with recent developments in non-Equilibrium quantum Statistical Mechanics. In particular, we relate the large deviation principle for the full counting statistics of entropy flow to quantum hypothesis testing of the arrow of time.

  • Topics in nonEquilibrium quantum Statistical Mechanics
    2006
    Co-Authors: Walter H. Aschbacher, Vojkan Jaksic, Yan Pautrat, Claudealain Pillet
    Abstract:

    These notes are an expanded and revised version of the lectures given by the second and fourth autor in the summer school "Open Quantum System" held in Grenoble, June 16-July 4, 2003. They provide an introduction to recent developments in non-Equilibrium Statistical Mechanics of open quantum systems, including a completely worked out (simple) example. We discuss non-Equilibrium steady states (NESS) and their structural properties, entropy production, linear response theory and weak coupling limit. The emphasis is on Ruelle's scattering approach to the construction of NESS.

  • non Equilibrium steady states of finite quantum systems coupled to thermal reservoirs
    2002
    Co-Authors: Vojkan Jaksic, Claudealain Pillet
    Abstract:

    We study the non-Equilibrium Statistical Mechanics of a 2-level quantum system, ?, coupled to two independent free Fermi reservoirs ?1, ?2, which are in thermal Equilibrium at inverse temperatures β1≠β2. We prove that, at small coupling, the combined quantum system ?+?1+?2 has a unique non-Equilibrium steady state (NESS) and that the approach to this NESS is exponentially fast. We show that the entropy production of the coupled system is strictly positive and relate this entropy production to the heat fluxes through the system.

  • Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures
    1999
    Co-Authors: Jeanpierre Eckmann, Claudealain Pillet, Luc Rey-bellet
    Abstract:

    We study the Statistical Mechanics of a finite-dimensional non-linear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a non-compact phase space. These techniques are based on an extension of the commutator method of H\\örmander used in the study of hypoelliptic differential operators.

Elena Agliari - One of the best experts on this subject based on the ideXlab platform.

  • Equilibrium Statistical Mechanics on correlated random graphs
    2011
    Co-Authors: Adriano Barra, Elena Agliari
    Abstract:

    Biological and social networks have recently attracted great attention from physicists. Among several aspects, two main ones may be stressed: a non-trivial topology of the graph describing the mutual interactions between agents and, typically, imitative, weighted, interactions. Despite such aspects being widely accepted and empirically confirmed, the schemes currently exploited in order to generate the expected topology are based on ap rioriassumptions and, in most cases, implement constant intensities for links. Here we propose a simple shift (−1, +1) → (0, +1) in the definition of patterns in a Hopfield model: a straightforward effect is the conversion of frustration into dilution. In fact, we show that by varying the bias of pattern distribution, the network topology (generated by the reciprocal affinities among agents, i.e. the Hebbian rule) crosses various well-known regimes, ranging from fully connected, to an extreme dilution scenario, then to completely disconnected. These features, as well as small-world properties, are, in this context, emergent and no longer imposed ap riori. The model is throughout investigated also from a thermodynamics perspective: the Ising model defined on the resulting graph is analytically solved (at a replica symmetric level) by extending the double stochastic stability technique, and presented together with its fluctuation theory for a picture of criticality. Overall, our findings show that, at least at Equilibrium, dilution (of whatever kind) simply decreases the strength of the coupling felt by the spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The main difference with respect to previous investigations is that, within our approach, replicas

  • Equilibrium Statistical Mechanics on correlated random graphs
    2010
    Co-Authors: Adriano Barra, Elena Agliari
    Abstract:

    Biological and social networks have recently attracted enormous attention between physicists. Among several, two main aspects may be stressed: A non trivial topology of the graph describing the mutual interactions between agents exists and/or, typically, such interactions are essentially (weighted) imitative. Despite such aspects are widely accepted and empirically confirmed, the schemes currently exploited in order to generate the expected topology are based on a-priori assumptions and in most cases still implement constant intensities for links. Here we propose a simple shift in the definition of patterns in an Hopfield model to convert frustration into dilution: By varying the bias of the pattern distribution, the network topology -which is generated by the reciprocal affinities among agents - crosses various well known regimes (fully connected, linearly diverging connectivity, extreme dilution scenario, no network), coupled with small world properties, which, in this context, are emergent and no longer imposed a-priori. The model is investigated at first focusing on these topological properties of the emergent network, then its thermodynamics is analytically solved (at a replica symmetric level) by extending the double stochastic stability technique, and presented together with its fluctuation theory for a picture of criticality. At least at Equilibrium, dilution simply decreases the strength of the coupling felt by the spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The main difference with respect to previous investigations and a naive picture is that within our approach replicas do not appear: instead of (multi)-overlaps as order parameters, we introduce a class of magnetizations on all the possible sub-graphs belonging to the main one investigated: As a consequence, for these objects a closure for a self-consistent relation is achieved.

  • new perspectives in the Equilibrium Statistical Mechanics approach to social and economic sciences
    2010
    Co-Authors: Elena Agliari, Adriano Barra, Raffaella Burioni, Pierluigi Contucci
    Abstract:

    In this chapter we review some recent development in the mathematical modeling of quantitative sociology by means of Statistical Mechanics. After a short pedagogical introduction to static and dynamic properties of many body systems, we develop a theory for particle (agents) interactions on random graph.

Adriano Barra - One of the best experts on this subject based on the ideXlab platform.

  • Equilibrium Statistical Mechanics of bipartite spin systems
    2011
    Co-Authors: Adriano Barra, Giuseppe Genovese, Francesco Guerra
    Abstract:

    The aim of this paper is to give an extensive treatment of bipartite mean field spin systems, pure and disordered. At first, bipartite ferromagnets are investigated, and an explicit expression for the free energy is achieved through a new minimax variational principle. Then, via the Hamilton?Jacobi technique, the same structure of the free energy is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one. The same is investigated for bipartite spin-glasses. By the Borel?Cantelli lemma we obtain the control of the high temperature regime, while via the double stochastic stability technique we also obtain the explicit expression of the free energy in the replica symmetric approximation, uniquely defined by a minimax variational principle again. We also obtain a general result that states that the free energies of these systems are convex linear combinations of their independent one-party model counterparts. For the sake of completeness, we show further that at zero temperature the replica symmetric entropy becomes negative and, consequently, such a symmetry must be broken. The treatment of the fully broken replica symmetry case is deferred to a forthcoming paper. As a first step in this direction, we start deriving the linear and quadratic constraints to overlap fluctuations.

  • Equilibrium Statistical Mechanics on correlated random graphs
    2011
    Co-Authors: Adriano Barra, Elena Agliari
    Abstract:

    Biological and social networks have recently attracted great attention from physicists. Among several aspects, two main ones may be stressed: a non-trivial topology of the graph describing the mutual interactions between agents and, typically, imitative, weighted, interactions. Despite such aspects being widely accepted and empirically confirmed, the schemes currently exploited in order to generate the expected topology are based on ap rioriassumptions and, in most cases, implement constant intensities for links. Here we propose a simple shift (−1, +1) → (0, +1) in the definition of patterns in a Hopfield model: a straightforward effect is the conversion of frustration into dilution. In fact, we show that by varying the bias of pattern distribution, the network topology (generated by the reciprocal affinities among agents, i.e. the Hebbian rule) crosses various well-known regimes, ranging from fully connected, to an extreme dilution scenario, then to completely disconnected. These features, as well as small-world properties, are, in this context, emergent and no longer imposed ap riori. The model is throughout investigated also from a thermodynamics perspective: the Ising model defined on the resulting graph is analytically solved (at a replica symmetric level) by extending the double stochastic stability technique, and presented together with its fluctuation theory for a picture of criticality. Overall, our findings show that, at least at Equilibrium, dilution (of whatever kind) simply decreases the strength of the coupling felt by the spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The main difference with respect to previous investigations is that, within our approach, replicas

  • Equilibrium Statistical Mechanics of bipartite spin systems
    2010
    Co-Authors: Adriano Barra, Giuseppe Genovese, Francesco Guerra
    Abstract:

    Aim of this paper is to give an extensive treatment of bipartite mean field spin systems, ordered and disordered: at first, bipartite ferromagnets are investigated, achieving an explicit expression for the free energy trough a new minimax variational principle. Furthermore via the Hamilton-Jacobi technique the same free energy structure is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one. The same is investigated for bipartite spin-glasses: By the Borel-Cantelli lemma a control of the high temperature regime is obtained, while via the double stochastic stability technique we get also the explicit expression of the free energy at the replica symmetric level, uniquely defined by a minimax variational principle again. A general results that states that the free energies of these systems are convex linear combinations of their independent one party model counterparts is achieved too. For the sake of completeness we show further that at zero temperature the replica symmetric entropy becomes negative and, consequently, such a symmetry must be broken. The treatment of the fully broken replica symmetry case is deferred to a forthcoming paper. As a first step in this direction, we start deriving the linear and quadratic constraints to overlap fluctuations.

  • Equilibrium Statistical Mechanics on correlated random graphs
    2010
    Co-Authors: Adriano Barra, Elena Agliari
    Abstract:

    Biological and social networks have recently attracted enormous attention between physicists. Among several, two main aspects may be stressed: A non trivial topology of the graph describing the mutual interactions between agents exists and/or, typically, such interactions are essentially (weighted) imitative. Despite such aspects are widely accepted and empirically confirmed, the schemes currently exploited in order to generate the expected topology are based on a-priori assumptions and in most cases still implement constant intensities for links. Here we propose a simple shift in the definition of patterns in an Hopfield model to convert frustration into dilution: By varying the bias of the pattern distribution, the network topology -which is generated by the reciprocal affinities among agents - crosses various well known regimes (fully connected, linearly diverging connectivity, extreme dilution scenario, no network), coupled with small world properties, which, in this context, are emergent and no longer imposed a-priori. The model is investigated at first focusing on these topological properties of the emergent network, then its thermodynamics is analytically solved (at a replica symmetric level) by extending the double stochastic stability technique, and presented together with its fluctuation theory for a picture of criticality. At least at Equilibrium, dilution simply decreases the strength of the coupling felt by the spins, but leaves the paramagnetic/ferromagnetic flavors unchanged. The main difference with respect to previous investigations and a naive picture is that within our approach replicas do not appear: instead of (multi)-overlaps as order parameters, we introduce a class of magnetizations on all the possible sub-graphs belonging to the main one investigated: As a consequence, for these objects a closure for a self-consistent relation is achieved.

  • new perspectives in the Equilibrium Statistical Mechanics approach to social and economic sciences
    2010
    Co-Authors: Elena Agliari, Adriano Barra, Raffaella Burioni, Pierluigi Contucci
    Abstract:

    In this chapter we review some recent development in the mathematical modeling of quantitative sociology by means of Statistical Mechanics. After a short pedagogical introduction to static and dynamic properties of many body systems, we develop a theory for particle (agents) interactions on random graph.

Martin Hairer - One of the best experts on this subject based on the ideXlab platform.

  • non Equilibrium Statistical Mechanics of strongly anharmonic chains of oscillators
    2000
    Co-Authors: Martin Hairer
    Abstract:

    We study the model of a strongly non-linear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, Rey-Bellet [EPR99a, EPR99b] to potentials with essentially arbitrary growth at infinity. This extension is possible by introducing a stronger version of Hormander's theorem for Kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains.

  • non Equilibrium Statistical Mechanics of strongly anharmonic chains of oscillators
    1999
    Co-Authors: Jeanpierre Eckmann, Martin Hairer
    Abstract:

    We study the model of a strongly non-linear chain of particles coupled to two heat baths at different temperatures. Our main result is the existence and uniqueness of a stationary state at all temperatures. This result extends those of Eckmann, Pillet, Rey-Bellet to potentials with essentially arbitrary growth at infinity. This extension is possible by introducing a stronger version of H\"ormander's theorem for Kolmogorov equations to vector fields with polynomially bounded coefficients on unbounded domains.