# Timetable (PDSW02)

## Nonequilibrium Dynamics of Interacting Particle Systems

Monday 27th March 2006 to Friday 7th April 2006

 08:30 to 09:55 Registration 10:00 to 11:00 Fluctuations and large deviations in non-equilibrium systems: Lecture I The exact solutions of simple models allow us to obtain the large deviation functions of density profiles and of the current through simple systems in contact with two reservoirs at different densities. These simple models show that non-equilibrium systems have a number of properties which contrast with equilibrium systems: phase transitions in one dimension, non local free energy functional, violation of the Einstein relation between the compressibility and the density fluctuation, non-Gaussian density fluctuations. They also lead to a general expression for the current fluctuations through a diffusive system in contact with two reservoirs. B Derrida, J L Lebowitz, E R Speer Free Energy Functional for Nonequilibrium Systems: An Exactly Solvable Case Phys. Rev. Lett. 87, 150601 (2001) B Derrida, B Doucot, P-E Roche, Current fluctuations in the one dimensional Symmetric Exclusion Process with open boundaries J. Stat. Phys. 115, 717-748 (2004) T. Bodineau, B Derrida Current fluctuations in non-equilibrium diffusive systems: an additivity principle Phys. Rev. Lett. 92, 180601 (2004) INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 The fluctuation and nonequilibrium free energy theorems, Theory and experiment: Lecture I 1. We give a brief summary of the derivations of the Evans-Searles Fluctuation Theorems (FTs) and the NonEquilibrium Free Energy Theorems (Crooks and Jarzynski). The discussion is given for time reversible Newtonian dynamics. We emphasize the role played by thermostatting. We also highlight the common themes inherent in the Fluctuation and Free Energy Theorems. We discuss a number of simple consequences of the Fluctuation Theorems including the Second Law Inequality, the Kawasaki Identity and the fact that the dissipation function which is the subject of the FT, is a nonlinear generalization of the spontaneous entropy production, that is so central to linear irreversible thermodyanamics. Lastly we give a brief update on the latest experimental tests of the FTs (both steady state and transient) and the NonEquilibrium Free Energy Theorem, using optical tweezer apparatus. 2 The Fluctutation Theorem: In 1993 we discovered a relation, subsequently known as the Fluctuation Theorem (FT), which gives an analytical expression for the probability of observing Second Law violating dynamical fluctuations in small thermostatted nonequilibrium systems which are observed for a short period of time. This Theorem places quantitative restrictions on the operation of small (nano) machines and devices. These constraints cannot be circumvented. Quantitative predictions made by the Fluctuation Theorem regarding the probability of Second Law violations' have been confirmed experimentally, both using molecular dynamics computer simulation and very recently in two laboratory experiments[1] which employed optical tweezers. In this talk we give a brief summary of the theory [2] and a description of the experiments. References [1] Experimental demonstration of violations of the Second Law of Thermodynamics for small systems and short time scales, by Wang, G.M., Sevick, E.M., Mittag, E., Searles, D.J. and Evans, D.J., Phys. Rev. Lett., 89 (5), 050601/1?4 (2002). [2] The Fluctuation Theorem by Denis J Evans and Debra J Searles, Advances in Physics, 51 , 1529-1585(2002). INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 Poster session I 15:00 to 15:30 Tea 15:30 to 16:30 Hydrodynamic limit for driven diffusive systems: Lecture I The large scale behaviour of microscopic stochastic particle systems can often be described in tems of nonlinear partial differential equations which can be predicted phenomenologically or sometimes derived rigorously using probabilistic tools. For one-component systems this allows not only for computing the (deterministic) space-time evolution of the coarse-grained local order parameter, but also for the derivation of the stationary phase diagram in bulk-driven finite systems with open boundaries. The Bethe ansatz provides the means to study fluctuations on finer scales. Systems with two or more components exhibit richer physics, but the theory is far less developed, both mathematically from a probabilistic and PDE point of view and from a statistical physics perspective. Focussing on paradigmatic one-dimensional lattice gas models for driven diffusive systems far from thermal equilibrium the lecture aims at giving a non-technical overview of some well-known rigorous and some more recent numerically established results for one-component systems with conserved particle dynamics or with slow reaction kinetics and at highlighting some aspects of the present incomplete state of art for two-component systems which deserve further investigation. INI 1 16:30 to 17:30 H Hinrichsen ([Wuerzburg])Absorbing state phase transitions: Lecture I The purpose of these lectures is to give a basic introduction to the physics of phase transitions far from equilibrium. Starting with a general introduction to non-equilibrium statistical mechanics four different topics will be addressed. At first the universality class of directed percolation will be discussed, which plays a fundamental role in non-equilibrium statistical physics. The second part concerns the properties of other universality classes which have been of interest in recent years. The third part deals with phase transitions in models with long-range interactions, including memory effects and so-called Levy-flights. Finally, deposition-evaporation phenomena leading to wetting transitions out of equilibrium will be reviewed. INI 1 17:30 to 18:30 Wine Reception 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)
 09:00 to 10:00 Fluctuations and large deviations in non equilibrium systems: Lecture II The exact solutions of simple models allow us to obtain the large deviation functions of density profiles and of the current through simple systems in contact with two reservoirs at different densities. These simple models show that non-equilibrium systems have a number of properties which contrast with equilibrium systems: phase transitions in one dimension, non local free energy functional, violation of the Einstein relation between the compressibility and the density fluctuation, non-Gaussian density fluctuations. They also lead to a general expression for the current fluctuations through a diffusive system in contact with two reservoirs. B Derrida, J L Lebowitz, E R Speer Free Energy Functional for Nonequilibrium Systems: An Exactly Solvable Case Phys. Rev. Lett. 87, 150601 (2001) B Derrida, B Doucot, P-E Roche, Current fluctuations in the one dimensional Symmetric Exclusion Process with open boundaries J. Stat. Phys. 115, 717-748 (2004) T. Bodineau, B Derrida Current fluctuations in non-equilibrium diffusive systems: an additivity principle Phys. Rev. Lett. 92, 180601 (2004) INI 1 10:00 to 11:00 JL Cardy ([Oxford])Stochastic Loewner Evolution and other growth processes in two dimensions: Lecture I Random objects such as clusters in the plane can often be described in terms of the conformal mappings which take their boundaries into some standard shape. As the clusters grow, the mapping function changes in a well-defined manner, which is often easier to understand than the original problem. One of the simplest examples is Stochastic Loewner Evolution (SLE), which turns out to describe random curves in equilibrium statistical mechanics models. These lectures will give an introduction to the use of such conformal mappings, and to SLE in particular, from the physicist's point of view. INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 SN Majumdar (Université Paris-Sud)A class of mass transport models: Factorised steady states and condensation in real space: Lecture I Lecture-I The traditional Bose-Einstein condensation in an ideal quantum Bose gas occurs in momentum space, when a macroscopically large number of bosons condense onto the ground state. It is becoming increasingly clear over the last decade that condensation can also happen in real space (and even in one dimension) in the steady state of a broad class of physical systems. These are classical systems, generally lack a Hamiltonian and are defined by their microscopic kinetic processes. Examples include traffic jams on a highway, island formation on growing crystals and many other systems. In this lecture, I'll discuss in detail two simple models namely the Zero-range process and the Chipping model that exhbits condensation in real space. Lecture-II I'll introduce a generalized mass transport model that includes in iteself, as specail cases, the Zero-range process, the Chipping model and the Random Average process. We will derive a necessary and sufficient condition, in one dimension, for the model to have a factorised steady state. Generalization to arbitrary graphs will be mentioned also. Lecture-III We will discuss, in the context of the mass transport model, the phenomenon of condensation. In particular we will address three basic isuues: (1) WHEN does such a condensation occur (the criterion) (2) HOW does the condensation happen (the mechanism) and (3) WHAT does the condensate look like (the nature of fluctuations and lifetime of the condensate etc.)? We will see how these issues can be resolved analytically in the mass transport model. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 Poster session II 15:00 to 15:30 Tea 15:30 to 16:30 The fluctuation and nonequilibrium free energy theorems: Theory and experiment: lecture II 1. We give a brief summary of the derivations of the Evans-Searles Fluctuation Theorems (FTs) and the NonEquilibrium Free Energy Theorems (Crooks and Jarzynski). The discussion is given for time reversible Newtonian dynamics. We emphasize the role played by thermostatting. We also highlight the common themes inherent in the Fluctuation and Free Energy Theorems. We discuss a number of simple consequences of the Fluctuation Theorems including the Second Law Inequality, the Kawasaki Identity and the fact that the dissipation function which is the subject of the FT, is a nonlinear generalization of the spontaneous entropy production, that is so central to linear irreversible thermodyanamics. Lastly we give a brief update on the latest experimental tests of the FTs (both steady state and transient) and the NonEquilibrium Free Energy Theorem, using optical tweezer apparatus. 2 The Fluctutation Theorem: In 1993 we discovered a relation, subsequently known as the Fluctuation Theorem (FT), which gives an analytical expression for the probability of observing Second Law violating dynamical fluctuations in small thermostatted nonequilibrium systems which are observed for a short period of time. This Theorem places quantitative restrictions on the operation of small (nano) machines and devices. These constraints cannot be circumvented. Quantitative predictions made by the Fluctuation Theorem regarding the probability of Second Law violations' have been confirmed experimentally, both using molecular dynamics computer simulation and very recently in two laboratory experiments[1] which employed optical tweezers. In this talk we give a brief summary of the theory [2] and a description of the experiments. References [1] Experimental demonstration of violations of the Second Law of Thermodynamics for small systems and short time scales, by Wang, G.M., Sevick, E.M., Mittag, E., Searles, D.J. and Evans, D.J., Phys. Rev. Lett., 89 (5), 050601/1?4 (2002). [2] The Fluctuation Theorem by Denis J Evans and Debra J Searles, Advances in Physics, 51 , 1529-1585(2002). INI 1 16:30 to 17:30 Hydrodynamic limit for driven diffusive systems: Lecture II The large scale behaviour of microscopic stochastic particle systems can often be described in tems of nonlinear partial differential equations which can be predicted phenomenologically or sometimes derived rigorously using probabilistic tools. For one-component systems this allows not only for computing the (deterministic) space-time evolution of the coarse-grained local order parameter, but also for the derivation of the stationary phase diagram in bulk-driven finite systems with open boundaries. The Bethe ansatz provides the means to study fluctuations on finer scales. Systems with two or more components exhibit richer physics, but the theory is far less developed, both mathematically from a probabilistic and PDE point of view and from a statistical physics perspective. Focussing on paradigmatic one-dimensional lattice gas models for driven diffusive systems far from thermal equilibrium the lecture aims at giving a non-technical overview of some well-known rigorous and some more recent numerically established results for one-component systems with conserved particle dynamics or with slow reaction kinetics and at highlighting some aspects of the present incomplete state of art for two-component systems which deserve further investigation. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)
 09:00 to 10:00 Some results in the dynamical large deviation approach to macroscopic fluctuations in stochastic lattice gases: Lecture I 1. Long range space correlations as a signature of violation of time reversal invariance (TRI) We introduce the notions of strong and weak violation of time reversal invariance (TRI) according to whether the violation of TRI in the microscopic dynamics shows up or not at the hydrodynamical level. We then argue that long range space correlations seem to be a generic feature of dynamics strongly violating TRI. In particular, on the basis af a recently established Hamilton-Jacobi equation for the free energy, we show that equilibrium states of Glauber-Kawasaki type dynamics under strong violation of TRI have space correlations over a macroscopic scale. This result indicates that long range correlations are not specific to non equilibrium stationary states. 2. Dynamical phase transitions in large current fluctuations of stochastic lattice gases In works in collaboration with L. Bertini, A. De Sole, D. Gabrielli and C. Landim we have shown that in large fluctuations of the current, averaged over long intervals of time, transitions to different dynamical regimes can take place. These are revealed by the time dependence of the thermodynamic variables associated to the fluctuations. In this case time shift invariance is spontaneously broken. So far two examples are known, the weakly asymmetric simple exclusion process with periodic boundary conditions discussed by Bodineau and Derrida and the Kipnis-Marchioro-Presutti model for which we have provided a rigorous proof of the transition. INI 1 10:00 to 11:00 Dynamics of growing and equilibrium networks: Lecture I The lectures will present an overview of the large-scale topological and dynamical properties of complex networks. First the methodology used to obtain large scale maps of several real networked systems in the social, biological and technology domains will be reviewed. Then the statistical features and regularities observed in the large scale structure of complex networks will be discussed along with the formulation of adequate dynamical models. Finally the effect of network complexity in spreading and percolation processes will be analyzed. INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 SN Majumdar (Université Paris-Sud)A class of mass transport models: Factorised steady states and condensation in real space: Lecture II Lecture-I The traditional Bose-Einstein condensation in an ideal quantum Bose gas occurs in momentum space, when a macroscopically large number of bosons condense onto the ground state. It is becoming increasingly clear over the last decade that condensation can also happen in real space (and even in one dimension) in the steady state of a broad class of physical systems. These are classical systems, generally lack a Hamiltonian and are defined by their microscopic kinetic processes. Examples include traffic jams on a highway, island formation on growing crystals and many other systems. In this lecture, I'll discuss in detail two simple models namely the Zero-range process and the Chipping model that exhbits condensation in real space. Lecture-II I'll introduce a generalized mass transport model that includes in iteself, as specail cases, the Zero-range process, the Chipping model and the Random Average process. We will derive a necessary and sufficient condition, in one dimension, for the model to have a factorised steady state. Generalization to arbitrary graphs will be mentioned also. Lecture-III We will discuss, in the context of the mass transport model, the phenomenon of condensation. In particular we will address three basic isuues: (1) WHEN does such a condensation occur (the criterion) (2) HOW does the condensation happen (the mechanism) and (3) WHAT does the condensate look like (the nature of fluctuations and lifetime of the condensate etc.)? We will see how these issues can be resolved analytically in the mass transport model. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 Contributed Seminar: Stochastic thermodynamics: Energy Conservation and entropy production along a single trajectory For stochastic dynamics of driven non-equilibrium systems, entropy production can be defined along a single trajectory [1]. It consists of two parts, entropy change of the system itself and entropy change of the surrounding medium. Total entropy production fulfills an integral fluctuation theorem for arbitrary initial state and arbitrary driving. For steady states, the total entropy production obeys the detailed fluctuation theorem even for finite times. These theorems can be derived without the notion of a surrounding heat bath of constant temperature. In the presence of such a bath as it is typical for many colloidal and biomolecular systems, however, a first law-like energy balance along the trajectory allows to identify dissipated heat and equate it with the entropy change of the medium. I will sketch the derivation of these results both for a Langevin type dynamics of continuous degrees of freedom and for a master equation dynamics on a discrete set of states. Illustrative examples for the first type include our recent experiments on a colloidal particle in a time-dependent non-harmonic potential [2]. Examples for discrete dynamics include enzym models [3,4] and our experiments on an athermal optically driven two-level system [5]. [1] U. Seifert, Phys. Rev. Lett. 95: 040602/1-4, 2005. [2] V. Blickle et al, Phys. Rev. Lett. 96: 070603/1-4, 2006. [3] U. Seifert, Europhys. Lett. 70: 36-41, 2005. [4] T. Schmiedl, T. Speck and U. Seifert, cond-mat 0601636, 2006. [5] S. Schuler et al, Phys. Rev. Lett. 94: 180602/1-4, 2005. INI 1 15:00 to 15:30 Tea 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)
 09:00 to 10:00 H Hinrichsen ([Wuerzburg])Absorbing state phase transitions: Lecture II The purpose of these lectures is to give a basic introduction to the physics of phase transitions far from equilibrium. Starting with a general introduction to non-equilibrium statistical mechanics four different topics will be addressed. At first the universality class of directed percolation will be discussed, which plays a fundamental role in non-equilibrium statistical physics. The second part concerns the properties of other universality classes which have been of interest in recent years. The third part deals with phase transitions in models with long-range interactions, including memory effects and so-called Levy-flights. Finally, deposition-evaporation phenomena leading to wetting transitions out of equilibrium will be reviewed. INI 1 10:00 to 11:00 Fluctuations and large deviations in non equilibrium systems: Lecture III The exact solutions of simple models allow us to obtain the large deviation functions of density profiles and of the current through simple systems in contact with two reservoirs at different densities. These simple models show that non-equilibrium systems have a number of properties which contrast with equilibrium systems: phase transitions in one dimension, non local free energy functional, violation of the Einstein relation between the compressibility and the density fluctuation, non-Gaussian density fluctuations. They also lead to a general expression for the current fluctuations through a diffusive system in contact with two reservoirs. B Derrida, J L Lebowitz, E R Speer Free Energy Functional for Nonequilibrium Systems: An Exactly Solvable Case Phys. Rev. Lett. 87, 150601 (2001) B Derrida, B Doucot, P-E Roche, Current fluctuations in the one dimensional Symmetric Exclusion Process with open boundaries J. Stat. Phys. 115, 717-748 (2004) T. Bodineau, B Derrida Current fluctuations in non-equilibrium diffusive systems: an additivity principle Phys. Rev. Lett. 92, 180601 (2004) INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 SN Majumdar (Université Paris-Sud)A class of mass transport models: Factorised steady states and condensation in real space: Lecture III Lecture-I The traditional Bose-Einstein condensation in an ideal quantum Bose gas occurs in momentum space, when a macroscopically large number of bosons condense onto the ground state. It is becoming increasingly clear over the last decade that condensation can also happen in real space (and even in one dimension) in the steady state of a broad class of physical systems. These are classical systems, generally lack a Hamiltonian and are defined by their microscopic kinetic processes. Examples include traffic jams on a highway, island formation on growing crystals and many other systems. In this lecture, I'll discuss in detail two simple models namely the Zero-range process and the Chipping model that exhbits condensation in real space. Lecture-II I'll introduce a generalized mass transport model that includes in iteself, as specail cases, the Zero-range process, the Chipping model and the Random Average process. We will derive a necessary and sufficient condition, in one dimension, for the model to have a factorised steady state. Generalization to arbitrary graphs will be mentioned also. Lecture-III We will discuss, in the context of the mass transport model, the phenomenon of condensation. In particular we will address three basic isuues: (1) WHEN does such a condensation occur (the criterion) (2) HOW does the condensation happen (the mechanism) and (3) WHAT does the condensate look like (the nature of fluctuations and lifetime of the condensate etc.)? We will see how these issues can be resolved analytically in the mass transport model. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 Poster session III 15:00 to 15:30 Tea 15:30 to 16:30 Dynamics of growing and equilibrium networks: Lecture II The lectures will present an overview of the large-scale topological and dynamical properties of complex networks. First the methodology used to obtain large scale maps of several real networked systems in the social, biological and technology domains will be reviewed. Then the statistical features and regularities observed in the large scale structure of complex networks will be discussed along with the formulation of adequate dynamical models. Finally the effect of network complexity in spreading and percolation processes will be analyzed. INI 1 16:30 to 17:30 JL Cardy ([Oxford])Stochastic Loewner Evolution and other growth processes in two dimensions: Lecture II Random objects such as clusters in the plane can often be described in terms of the conformal mappings which take their boundaries into some standard shape. As the clusters grow, the mapping function changes in a well-defined manner, which is often easier to understand than the original problem. One of the simplest examples is Stochastic Loewner Evolution (SLE), which turns out to describe random curves in equilibrium statistical mechanics models. These lectures will give an introduction to the use of such conformal mappings, and to SLE in particular, from the physicist's point of view. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)
 09:00 to 10:00 JL Cardy ([Oxford])Stochastic Loewner Evolution and other growth processes in two dimensions: Lecture III Random objects such as clusters in the plane can often be described in terms of the conformal mappings which take their boundaries into some standard shape. As the clusters grow, the mapping function changes in a well-defined manner, which is often easier to understand than the original problem. One of the simplest examples is Stochastic Loewner Evolution (SLE), which turns out to describe random curves in equilibrium statistical mechanics models. These lectures will give an introduction to the use of such conformal mappings, and to SLE in particular, from the physicist's point of view. INI 1 10:00 to 11:00 Some results in the dynamical large deviation approach to macroscopic fluctuations in stochastic lattice gases: Lecture II 1. Long range space correlations as a signature of violation of time reversal invariance (TRI) We introduce the notions of strong and weak violation of time reversal invariance (TRI) according to whether the violation of TRI in the microscopic dynamics shows up or not at the hydrodynamical level. We then argue that long range space correlations seem to be a generic feature of dynamics strongly violating TRI. In particular, on the basis af a recently established Hamilton-Jacobi equation for the free energy, we show that equilibrium states of Glauber-Kawasaki type dynamics under strong violation of TRI have space correlations over a macroscopic scale. This result indicates that long range correlations are not specific to non equilibrium stationary states. 2. Dynamical phase transitions in large current fluctuations of stochastic lattice gases In works in collaboration with L. Bertini, A. De Sole, D. Gabrielli and C. Landim we have shown that in large fluctuations of the current, averaged over long intervals of time, transitions to different dynamical regimes can take place. These are revealed by the time dependence of the thermodynamic variables associated to the fluctuations. In this case time shift invariance is spontaneously broken. So far two examples are known, the weakly asymmetric simple exclusion process with periodic boundary conditions discussed by Bodineau and Derrida and the Kipnis-Marchioro-Presutti model for which we have provided a rigorous proof of the transition. INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 Hydrodynamic limit for driven diffusive systems: Lecture III The large scale behaviour of microscopic stochastic particle systems can often be described in tems of nonlinear partial differential equations which can be predicted phenomenologically or sometimes derived rigorously using probabilistic tools. For one-component systems this allows not only for computing the (deterministic) space-time evolution of the coarse-grained local order parameter, but also for the derivation of the stationary phase diagram in bulk-driven finite systems with open boundaries. The Bethe ansatz provides the means to study fluctuations on finer scales. Systems with two or more components exhibit richer physics, but the theory is far less developed, both mathematically from a probabilistic and PDE point of view and from a statistical physics perspective. Focussing on paradigmatic one-dimensional lattice gas models for driven diffusive systems far from thermal equilibrium the lecture aims at giving a non-technical overview of some well-known rigorous and some more recent numerically established results for one-component systems with conserved particle dynamics or with slow reaction kinetics and at highlighting some aspects of the present incomplete state of art for two-component systems which deserve further investigation. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 J de Gier ([Melbourne])Contributed Seminar: Exact solution of the dynamics of the PASEP with open boundaries The dynamics of the asymmetric exclusion process is governed by the spectrum of its transition matrix. In particular its lowest excited state describes the approach to stationarity at large times. I will discuss the exact diagonalisation of the transition matrix of the partially asymmetric exclusion process with the most general open boundary conditions. The resulting Bethe ansatz equations describe the {\em complete} spectrum of the transition matrix. For totally asymmetric diffusion I will present exact results for the spectral gap and derive the dynamical phase diagram. We observe boundary induced crossovers in and between massive, diffusive and KPZ scaling regimes. INI 1 15:00 to 15:30 Tea 15:30 to 16:30 H Hinrichsen ([Wuerzburg])Absorbing state phase transitions: Lecture III The purpose of these lectures is to give a basic introduction to the physics of phase transitions far from equilibrium. Starting with a general introduction to non-equilibrium statistical mechanics four different topics will be addressed. At first the universality class of directed percolation will be discussed, which plays a fundamental role in non-equilibrium statistical physics. The second part concerns the properties of other universality classes which have been of interest in recent years. The third part deals with phase transitions in models with long-range interactions, including memory effects and so-called Levy-flights. Finally, deposition-evaporation phenomena leading to wetting transitions out of equilibrium will be reviewed. INI 1 16:30 to 17:30 Dynamics of growing and equilibrium networks: Lecture III The lectures will present an overview of the large-scale topological and dynamical properties of complex networks. First the methodology used to obtain large scale maps of several real networked systems in the social, biological and technology domains will be reviewed. Then the statistical features and regularities observed in the large scale structure of complex networks will be discussed along with the formulation of adequate dynamical models. Finally the effect of network complexity in spreading and percolation processes will be analyzed. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)
 09:00 to 10:00 Soluble models of self-organized criticality: Lecture I In these lectures, I will discuss the abelian sandpile model of self-organized criticality, and its related models. The abelian group structure of the model, the burning test for recurrent states, equivalence to the spanning trees problem will be described. The exact solution of the directed version of the model in any dimension will be explained, and its relation to Scheidegger's model of river basins, Takayasu's aggregation model and the voter model will be discussed. I will summarize the known results about the undirected models. Generalization to the abelian distributed processors model, and time-dependent properties and the universality of critical behavior in sandpiles will be briefly discussed. INI 1 10:00 to 11:00 Introduction to nonequilibrium work theorems: Lecture I When we drive a physical system away from a state of thermal equilibrium by forcing a change in one of its variables -- e.g. when we push a piston into a gas, or when we stretch a single molecule using optical tweezers -- then we perform a certain amount of work, W, on the system. Over the past decade, a number of results -- collectively known as nonequilibrium work theorems (NWT's) -- have revealed that equilibrium information is subtly encoded in the statistics of W, even when the system is driven significantly far from equilibrium. In my three lectures I will present an introduction to these theoretical results, as well as to their applications in the context of experiments and numerical simulations aimed at estimating thermodynamic properties of complex systems. The first lecture will present a general overview of these results. In the remaining two lectures I plan to cover a number of topics, including: useful mathematical tools for deriving and analyzing NWT's; practical issues regarding the applications of these results; generalizations of NWT's, and their relation to Fluctuation Theorems; and the connection of NWT's to deeper issues of macroscopic irreversibility. INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 Field-theoretic approaches to interacting particle systems: Lecture I It is explained how field-theoretic methods and the dynamic renormalization group (RG) can be applied to study the universal scaling properties of interacting particle systems far from thermal equilibrium that either undergo a continuous phase transition or display generic scale invariance It is described how the master equation for stochastic particle reaction processes can be mapped onto a field theory action. The RG is then employed to analyze the ensuing power laws in simple diffusion-limited annihilation reactions as well as generic continuous transitions from active to inactive, absorbing states, which are characterized by the power laws of (critical) directed percolation. Certain other important universality classes including dynamic percolation and parity-conserving branching and annihilating random walks are discussed, and some open issues are mentioned. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 Contributed Seminar: Membrane nanotubes pulled cooperatively by molecular motors Long and very dynamic tubular structures growing along the cytoskeleton have been observed in vivo. Similar tubular nano-structures have been obtained in vitro with a minimal system composed of kinesins grafted on the membranes of giant vesicles, and moving along immobilized microtubule. When the kinesins are individually grafted to single lipids, dynamical clusters of motors pulling the tubes have been observed at the tip of the tubes, in agreement with theory. Different dynamical regimes have been observed : below a threshold depending on membrane tension and motor concentration, no tube is formed but above this threshold, we observe either stable tubes or oscillating tubes depending on the relative vesicle size and tube length INI 1 15:00 to 15:30 Tea 15:30 to 16:30 Modelling of traffic flow and related transport systems: Lecture I In this series of lectures the basic empirical properties and theoretical modelling approaches for various traffic systems will be reviewed. The list of topics includes: 1) Empirical traffic data and their interpretation 2) Modelling approaches for highway traffic (hydrodynamic models, car-following models, etc.) 3) Cellular automata models of highway traffic (ASEP, Nagel-Schreckenberg model and their extensions) 4) Pedestrian dynamics 5) Transport in biological systems (ant trails, intracellular transport, etc) INI 1 16:30 to 17:30 Clustering, coarsening and directed transport in a granular gas: Lecture I 1. Clustering and Coarsening in a Granular Gas Granular gases are of great scientific and economic relevance. Scientific, because of their tendency to spontaneously separate into dense and dilute regions, which makes them fundamentally different from any textbook molecular gas. Economic, because no less than 5 per cent of the global energy budget is wasted due to problems with granular matter in conveyor belts, sorting machines, mixers, and other industrial machinery. Here we study - experimentally, numerically, and theoretically - the clustering of particles in a vertically vibrated array of N connected compartments. For strong shaking, the particles spread evenly over the compartments, but if the shaking strength is lowered beneath a critical level this uniform distribution gives way to a clustered state, consisting of a few well-filled compartments and a lot of diluted ones. In the course of time, this state coarsens: The smaller clusters are eaten by the larger ones, until finally only one big cluster remains. This coarsening process is exceptionally slow, with the mass of the surviving cluster growing only as the square root of log t. 2. Clustering and Directed Transport In this second lecture we turn to the wonderful world of ratchets, which have become a hot topic in recent years. In order to extract mechanical work on a molecular scale (e.g., to make a muscle move), nature uses the concept of a Brownian ratchet: The stochastic forces from a noisy environment are converted into a directed motion. Here we create a "granular ratchet", exploiting the clustering phenomenon from the previous lecture in a slightly adapted array of N connected compartments: The stochastically colliding particles spontaneously generate a particle current perpendicular to the direction of energy input. This is the first practical realization of the theoretically predicted concept of a stochastic ratchet as a collective effect in a symmetric geometry. A related problem of prime importance in modern society is the clustering of cars on the highway. We show how the formation of traffic jams on the Dutch highway A58 is well described - and predicted! - by a flux model similar to the one we use for the clustering of granular particles 3. Granular Impact Jets A steel ball dropped onto loose, very fine sand ("dry quicksand") creates an upward jet exceeding the release height of the ball. There is a striking similarity with the impact of an object in a liquid: The jet is generated by the gravity-driven collapse of the void created by the ball, and the focused pressure pushes the sand straight up into the air. Using a 2-dimensional experimental setup and high-speed imaging, the collapse of the void is visualized. For high impact velocities the void collapse is seen to entrain air. The entrained air bubble slowly rises through the sand, and upon reaching the surface causes a granular eruption. The experimental observations are quantitatively explained by a Rayleigh-type model. Parallels are drawn with impacts on a planetary scale. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)
 09:00 to 10:00 Fourier Law in low-dimensional systems: Lecture I The study of transport properties in low-dimensional models has attracted much attention in the scientific community. In fact, it has pointed out new basic for the theory of non-equilibrium stationary processes, while providing interesting perspectives of applications to new materials. These lectures will be mainly devoted to survey the achievements on the problem of heat transport in 1D and 2D systems. A pedagogically suitable approach will be proposed to the students, going through a historical pathway. Numerical results will be presented together with the main rigorous approaches and analytical results. These lectures aim at providing also an overview of the state of the art on this research topic, with reference to still open, challenging problems. Department of Physics, LE Fermi 2, 50125 Firenze INI 1 10:00 to 11:00 Introduction to nonequilibrium work theorems: Lecture II When we drive a physical system away from a state of thermal equilibrium by forcing a change in one of its variables -- e.g. when we push a piston into a gas, or when we stretch a single molecule using optical tweezers -- then we perform a certain amount of work, W, on the system. Over the past decade, a number of results -- collectively known as nonequilibrium work theorems (NWT's) -- have revealed that equilibrium information is subtly encoded in the statistics of W, even when the system is driven significantly far from equilibrium. In my three lectures I will present an introduction to these theoretical results, as well as to their applications in the context of experiments and numerical simulations aimed at estimating thermodynamic properties of complex systems. The first lecture will present a general overview of these results. In the remaining two lectures I plan to cover a number of topics, including: useful mathematical tools for deriving and analyzing NWT's; practical issues regarding the applications of these results; generalizations of NWT's, and their relation to Fluctuation Theorems; and the connection of NWT's to deeper issues of macroscopic irreversibility. INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 Clustering, coarsening and directed transport in a Granular gas: Lecture II 1. Clustering and Coarsening in a Granular Gas Granular gases are of great scientific and economic relevance. Scientific, because of their tendency to spontaneously separate into dense and dilute regions, which makes them fundamentally different from any textbook molecular gas. Economic, because no less than 5 per cent of the global energy budget is wasted due to problems with granular matter in conveyor belts, sorting machines, mixers, and other industrial machinery. Here we study - experimentally, numerically, and theoretically - the clustering of particles in a vertically vibrated array of N connected compartments. For strong shaking, the particles spread evenly over the compartments, but if the shaking strength is lowered beneath a critical level this uniform distribution gives way to a clustered state, consisting of a few well-filled compartments and a lot of diluted ones. In the course of time, this state coarsens: The smaller clusters are eaten by the larger ones, until finally only one big cluster remains. This coarsening process is exceptionally slow, with the mass of the surviving cluster growing only as the square root of log t. 2. Clustering and Directed Transport In this second lecture we turn to the wonderful world of ratchets, which have become a hot topic in recent years. In order to extract mechanical work on a molecular scale (e.g., to make a muscle move), nature uses the concept of a Brownian ratchet: The stochastic forces from a noisy environment are converted into a directed motion. Here we create a "granular ratchet", exploiting the clustering phenomenon from the previous lecture in a slightly adapted array of N connected compartments: The stochastically colliding particles spontaneously generate a particle current perpendicular to the direction of energy input. This is the first practical realization of the theoretically predicted concept of a stochastic ratchet as a collective effect in a symmetric geometry. A related problem of prime importance in modern society is the clustering of cars on the highway. We show how the formation of traffic jams on the Dutch highway A58 is well described - and predicted! - by a flux model similar to the one we use for the clustering of granular particles 3. Granular Impact Jets A steel ball dropped onto loose, very fine sand ("dry quicksand") creates an upward jet exceeding the release height of the ball. There is a striking similarity with the impact of an object in a liquid: The jet is generated by the gravity-driven collapse of the void created by the ball, and the focused pressure pushes the sand straight up into the air. Using a 2-dimensional experimental setup and high-speed imaging, the collapse of the void is visualized. For high impact velocities the void collapse is seen to entrain air. The entrained air bubble slowly rises through the sand, and upon reaching the surface causes a granular eruption. The experimental observations are quantitatively explained by a Rayleigh-type model. Parallels are drawn with impacts on a planetary scale. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 Poster session IV INI 1 15:00 to 15:30 Tea 15:30 to 16:30 Soluble models of self-organized criticality: Lecture II In these lectures, I will discuss the abelian sandpile model of self-organized criticality, and its related models. The abelian group structure of the model, the burning test for recurrent states, equivalence to the spanning trees problem will be described. The exact solution of the directed version of the model in any dimension will be explained, and its relation to Scheidegger's model of river basins, Takayasu's aggregation model and the voter model will be discussed. I will summarize the known results about the undirected models. Generalization to the abelian distributed processors model, and time-dependent properties and the universality of critical behavior in sandpiles will be briefly discussed. INI 1 16:30 to 17:30 Field-theoretic approaches to interacting particle systems: Lecture II It is explained how field-theoretic methods and the dynamic renormalization group (RG) can be applied to study the universal scaling properties of interacting particle systems far from thermal equilibrium that either undergo a continuous phase transition or display generic scale invariance It is described how the master equation for stochastic particle reaction processes can be mapped onto a field theory action. The RG is then employed to analyze the ensuing power laws in simple diffusion-limited annihilation reactions as well as generic continuous transitions from active to inactive, absorbing states, which are characterized by the power laws of (critical) directed percolation. Certain other important universality classes including dynamic percolation and parity-conserving branching and annihilating random walks are discussed, and some open issues are mentioned. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)
 09:00 to 10:00 Applications of nonequilibrium models in biological systems: Lecture I There are many biological functions that involve movement of motors along a filament or polymeric molecule. The motors use chemical energy to propel themselves along the track. The description of these systems is in many cases done using models of driven systems. The lectures will give an introduction to molecular motors discussing their use in biological systems and the experiments which study them. Then several application of models of driven systems in the interpretation of experiments will be reviewed. INI 1 10:00 to 11:00 Field-theoretic approaches to interacting particle systems: Lecture III It is explained how field-theoretic methods and the dynamic renormalization group (RG) can be applied to study the universal scaling properties of interacting particle systems far from thermal equilibrium that either undergo a continuous phase transition or display generic scale invariance It is described how the master equation for stochastic particle reaction processes can be mapped onto a field theory action. The RG is then employed to analyze the ensuing power laws in simple diffusion-limited annihilation reactions as well as generic continuous transitions from active to inactive, absorbing states, which are characterized by the power laws of (critical) directed percolation. Certain other important universality classes including dynamic percolation and parity-conserving branching and annihilating random walks are discussed, and some open issues are mentioned. INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 Growth models in one dimension and random matrices: Lecture I The motion of an interface separating a stable from an unstable phase is a well-studied problem in nonequilibrium dynamics, in particular since the field-theoretic formulation by Kardar, Parisi, and Zhang. For a one-dimensional interface many universal quantities of physical interest can be computed exactly. Surprisingly enough, there are links to the soft edge scaling of Gaussian unitary matrices. We explain the type of growth models which can be handled, how the connection to random matrices arises, and some of the predictions for one-dimensional growth. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 Soluble models of self-organized criticality: Lecture III In these lectures, I will discuss the abelian sandpile model of self-organized criticality, and its related models. The abelian group structure of the model, the burning test for recurrent states, equivalence to the spanning trees problem will be described. The exact solution of the directed version of the model in any dimension will be explained, and its relation to Scheidegger's model of river basins, Takayasu's aggregation model and the voter model will be discussed. I will summarize the known results about the undirected models. Generalization to the abelian distributed processors model, and time-dependent properties and the universality of critical behavior in sandpiles will be briefly discussed. INI 1 15:00 to 15:30 Tea 19:30 to 18:00 Conference Dinner - Selwyn College (Dining Hall)
 09:00 to 10:00 Growth models in one dimension and random matrices: Lecture II The motion of an interface separating a stable from an unstable phase is a well-studied problem in nonequilibrium dynamics, in particular since the field-theoretic formulation by Kardar, Parisi, and Zhang. For a one-dimensional interface many universal quantities of physical interest can be computed exactly. Surprisingly enough, there are links to the soft edge scaling of Gaussian unitary matrices. We explain the type of growth models which can be handled, how the connection to random matrices arises, and some of the predictions for one-dimensional growth. INI 1 10:00 to 11:00 Fourier Law in low-dimensional systems: Lecture II The study of transport properties in low-dimensional models has attracted much attention in the scientific community. In fact, it has pointed out new basic for the theory of non-equilibrium stationary processes, while providing interesting perspectives of applications to new materials. These lectures will be mainly devoted to survey the achievements on the problem of heat transport in 1D and 2D systems. A pedagogically suitable approach will be proposed to the students, going through a historical pathway. Numerical results will be presented together with the main rigorous approaches and analytical results. These lectures aim at providing also an overview of the state of the art on this research topic, with reference to still open, challenging problems. Department of Physics, LE Fermi 2, 50125 Firenze INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 Applications of nonequilibrium models in biological systems: Lecture II There are many biological functions that involve movement of motors along a filament or polymeric molecule. The motors use chemical energy to propel themselves along the track. The description of these systems is in many cases done using models of driven systems. The lectures will give an introduction to molecular motors discussing their use in biological systems and the experiments which study them. Then several application of models of driven systems in the interpretation of experiments will be reviewed. INI 1 12:30 to 13:30 Lunch at Wolfson Court 14:00 to 15:00 New critical phenomena in complex networks Most of real-world networks are extremely compact, infinite-dimensional objects. Consequently, any cooperative model on any of these network substrates is surely in situation above the upper critical dimension. This is why critical phenomena in these models should be precisely described in the framework of a mean field approach. Nonetheless, due to specific architectures of complex networks, these mean field theories are surprisingly non-standard. We discuss the unusual critical phenomena in complex networks by using representative examples: the Ising and Potts models, the percolation and its generalizations, etc. Remarkably, the critical behaviours are very different in equilibrium and growing networks. We explain that in a class of growing networks, the percolation and Ising models may even demonstrate a critical singularity of the Berezinskii-Kosterlitz-Thouless kind. We also touch upon the bootstrap (k-core) percolation problem and the k-core organization of complex networks. S N Dorogovtsev, J F F Mendes, Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford University Press, Oxford, 2003); Adv. Phys. 51, 1079 (2002). S N Dorogovtsev, A V Goltsev, J F F Mendes, Ising model on networks with an arbitrary distribution of connections, Phys. Rev. E 66, 016104 (2002). S N Dorogovtsev, J F F Mendes, A N Samukhin, Anomalous percolation properties of growing networks, Phys. Rev. E 64, 066110 (2001). M Bauer, S Coulomb, S N Dorogovtsev, Phase transition with the Berezinskii-Kosterlitz-Thouless singularity in the Ising model on a growing network, Phys. Rev. Lett. 94, 200602 (2005). S N Dorogovtsev, A V Goltsev, J F F Mendes, k-core organization of complex networks, Phys. Rev. Lett. 96, 17 February (2006). INI 1 15:00 to 15:30 Tea 15:30 to 16:30 Modelling of traffic flow and related transport problems: Lecture II In this series of lectures the basic empirical properties and theoretical modelling approaches for various traffic systems will be reviewed. The list of topics includes: 1) Empirical traffic data and their interpretation 2) Modelling approaches for highway traffic (hydrodynamic models, car-following models, etc.) 3) Cellular automata models of highway traffic (ASEP, Nagel-Schreckenberg model and their extensions) 4) Pedestrian dynamics 5) Transport in biological systems (ant trails, intracellular transport, etc) INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)
 09:00 to 10:00 Applications of nonequilibrium models in biological systems: Lecture III There are many biological functions that involve movement of motors along a filament or polymeric molecule. The motors use chemical energy to propel themselves along the track. The description of these systems is in many cases done using models of driven systems. The lectures will give an introduction to molecular motors discussing their use in biological systems and the experiments which study them. Then several application of models of driven systems in the interpretation of experiments will be reviewed. INI 1 10:00 to 11:00 Modelling of traffic flow and related transport problems: Lecture III In this series of lectures the basic empirical properties and theoretical modelling approaches for various traffic systems will be reviewed. The list of topics includes: 1) Empirical traffic data and their interpretation 2) Modelling approaches for highway traffic (hydrodynamic models, car-following models, etc.) 3) Cellular automata models of highway traffic (ASEP, Nagel-Schreckenberg model and their extensions) 4) Pedestrian dynamics 5) Transport in biological systems (ant trails, intracellular transport, etc) INI 1 11:00 to 11:30 Coffee 11:30 to 12:30 Growth models in one dimension and random matrices: Lecture III The motion of an interface separating a stable from an unstable phase is a well-studied problem in nonequilibrium dynamics, in particular since the field-theoretic formulation by Kardar, Parisi, and Zhang. For a one-dimensional interface many universal quantities of physical interest can be computed exactly. Surprisingly enough, there are links to the soft edge scaling of Gaussian unitary matrices. We explain the type of growth models which can be handled, how the connection to random matrices arises, and some of the predictions for one-dimensional growth. INI 1 12:30 to 13:30 Lunch at Wolfson Court 15:00 to 15:30 Tea INI 1 15:30 to 16:30 Introduction to nonequilibrium work theorems: Lecture III When we drive a physical system away from a state of thermal equilibrium by forcing a change in one of its variables -- e.g. when we push a piston into a gas, or when we stretch a single molecule using optical tweezers -- then we perform a certain amount of work, W, on the system. Over the past decade, a number of results -- collectively known as nonequilibrium work theorems (NWT's) -- have revealed that equilibrium information is subtly encoded in the statistics of W, even when the system is driven significantly far from equilibrium. In my three lectures I will present an introduction to these theoretical results, as well as to their applications in the context of experiments and numerical simulations aimed at estimating thermodynamic properties of complex systems. The first lecture will present a general overview of these results. In the remaining two lectures I plan to cover a number of topics, including: useful mathematical tools for deriving and analyzing NWT's; practical issues regarding the applications of these results; generalizations of NWT's, and their relation to Fluctuation Theorems; and the connection of NWT's to deeper issues of macroscopic irreversibility. INI 1 16:30 to 17:30 Clustering, coarsening and directed transport in a granular gas: Lecture III 1. Clustering and Coarsening in a Granular Gas Granular gases are of great scientific and economic relevance. Scientific, because of their tendency to spontaneously separate into dense and dilute regions, which makes them fundamentally different from any textbook molecular gas. Economic, because no less than 5 per cent of the global energy budget is wasted due to problems with granular matter in conveyor belts, sorting machines, mixers, and other industrial machinery. Here we study - experimentally, numerically, and theoretically - the clustering of particles in a vertically vibrated array of N connected compartments. For strong shaking, the particles spread evenly over the compartments, but if the shaking strength is lowered beneath a critical level this uniform distribution gives way to a clustered state, consisting of a few well-filled compartments and a lot of diluted ones. In the course of time, this state coarsens: The smaller clusters are eaten by the larger ones, until finally only one big cluster remains. This coarsening process is exceptionally slow, with the mass of the surviving cluster growing only as the square root of log t. 2. Clustering and Directed Transport In this second lecture we turn to the wonderful world of ratchets, which have become a hot topic in recent years. In order to extract mechanical work on a molecular scale (e.g., to make a muscle move), nature uses the concept of a Brownian ratchet: The stochastic forces from a noisy environment are converted into a directed motion. Here we create a "granular ratchet", exploiting the clustering phenomenon from the previous lecture in a slightly adapted array of N connected compartments: The stochastically colliding particles spontaneously generate a particle current perpendicular to the direction of energy input. This is the first practical realization of the theoretically predicted concept of a stochastic ratchet as a collective effect in a symmetric geometry. A related problem of prime importance in modern society is the clustering of cars on the highway. We show how the formation of traffic jams on the Dutch highway A58 is well described - and predicted! - by a flux model similar to the one we use for the clustering of granular particles 3. Granular Impact Jets A steel ball dropped onto loose, very fine sand ("dry quicksand") creates an upward jet exceeding the release height of the ball. There is a striking similarity with the impact of an object in a liquid: The jet is generated by the gravity-driven collapse of the void created by the ball, and the focused pressure pushes the sand straight up into the air. Using a 2-dimensional experimental setup and high-speed imaging, the collapse of the void is visualized. For high impact velocities the void collapse is seen to entrain air. The entrained air bubble slowly rises through the sand, and upon reaching the surface causes a granular eruption. The experimental observations are quantitatively explained by a Rayleigh-type model. Parallels are drawn with impacts on a planetary scale. INI 1 18:45 to 19:30 Dinner at Wolfson Court (Residents Only)