RECENT PAPERS (Christian Maes):                                                    ( BACK TO HOME PAGE )

· Christian Maes and Winny O'Kelly de Galway, A low temperature analysis of the boundary driven Kawasaki Process. arXiv:1306.1775v1 [cond-mat.stat-mech].

Low temperature analysis of nonequilibrium systems requires finding the states with the longest lifetime and that are most accessible

from other states. We determine these {\it dominant} states for a one-dimensional diffusive lattice gas subject to exclusion and with nearest neighbor interaction. They do not correspond to lowest energy configurations even though the particle current tends to zero as the temperature reaches zero. That is because the dynamical activity that sets the effective time scale, also goes to zero with temperature.

The result is a non-trivial asymptotic phase diagram, which crucially depends on the interaction coupling and the relative chemical potentials of the reservoirs.

· Christian Maes and Karel Netočný, Minimum entropy production principle, to be published on Scholarpedia.

The MINimum Entropy Production principle (MINEP) is an approximate variational characterization of steady states for thermodynamically open systems maintained out of equilibrium. Originally formulated within the framework of linear irreversible thermodynamics, it was extended to stochastic kinetics, e.g., for close-to-equilibrium systems described by a Master equation. The MINEP is consistent yet different from other nonequilibrium variational principles like the Maximum entropy production principle or the Least dissipation principle. Recent dynamical fluctuation theories provide a framework for their precise formulation, unification and systematic improvement.

· Christian Maes and Alberto Salazar, Linear response in the nonequilibrium zero range process. arXiv:1305.4157v1 [cond-mat.stat-mech].

We explore a number of explicit response formulae around the boundary driven zero range process to changes in the exit and entrance rates. In such a nonequilibrium regime kinetic (and not only thermodynamic) aspects make a difference in the response. Apart from a number of formal approaches, we illustrate a general decomposition of the linear response into entropic and frenetic contributions, the latter being realized from changes in the dynamical activity at the boundaries. In particular we so obtain nonlinear modifications to the Green-Kubo relation. We end by bringing some general remarks about the situation where that nonequilibrium response remains given by the (equilibrium) Kubo formula such as for the density profile in the boundary driven Lorentz gas.

· Christian Maes and Alberto Salazar, Active Fluctuation Symmetries. arXiv:1305.0736.

We elaborate on an observation of Maes-vanWieren (2006) to obtain fluctuation symmetries also for time-symmetric quantities. Examples are given, analytic and numerical, yielding time-symmetric path-observables with fluctuations satisfying a Gallavotti-Cohen type symmetry. From these results one is actually introduced to stationary nonequilibrium by a different phenomenology. It deals with a complementary class of what we may call active fluctuation symmetries, again general non-perturbative nonequilibrium relations but not expressed in terms of the traditional dissipative variables; they rather involve the notion of dynamical activity. In particular, we derive Green-Kubo like relations for differences in dynamical activity. The illustrations include boundary driven Kawasaki and zero range models and the spinning Lorentz gas.

· Christian Maes and Simi R. Thomas, From Langevin to generalized Langevin equations for the nonequilibrium Rouse model. Physical Review E 87, 022145 (2013). arXiv:1210.5068.

We investigate the nature of the effective dynamics and statistical forces obtained after integrating out nonequilibrium degrees of freedom. To be explicit, we consider the Rouse model for the conformational dynamics of an ideal polymer chain subject to steady driving. We compute the effective dynamics for one of the many monomers by integrating out the rest of the chain. The result is a generalized Langevin dynamics for which we give the memory and noise kernels and the effective force, and we discuss the inherited nonequilibrium aspects.

· C. Maes, S. Safaverdi, P. Visco, F. van Wijland: Fluctuation-response relations for nonequilibrium diffusions with memory, arXiv:1211.2181v2, Physical Review E 87, 022125 (2013).

Strong interaction with other particles or feedback from the medium on a Brownian particle entail memory effects in the effective dynamics. We discuss the extension of the fluctuation-dissipation theorem to nonequilibrium Langevin systems with memory. An important application is to the extension of the Sutherland-Einstein relation between diffusion and mobility. Nonequilibrium corrections include the time-correlation between the dynamical activity and the velocity of the particle, which in turn leads to information about the correlations between the driving force and the particle's displacement.

· Christian Maes and Karel Netočný: Heat bounds and the blowtorch theorem, Annales Henri Poincaré 14, 1193-1202 (2013).

We study driven systems with possible population inversion and we give optimal bounds on the relative occupations in terms of released heat. A precise meaning to Landauer's blowtorch theorem (1975) is obtained stating that nonequilibrium occupations are essentially modified by kinetic effects. Towards very low temperatures we apply a Freidlin-Wentzel type analysis for continuous time Markov jump processes.

· Christian Maes and Karel Netočný: A nonequilibrium extension of the Clausius heat theorem arXiv:1206.3423v2 [cond-mat.stat-mech].

We generalize the Clausius (in)equality to overdamped mesoscopic and macroscopic diffusions in the presence of nonconservative forces. In contrast to previous frameworks, we use a decomposition scheme for heat which is based on an exact variant of the Minimum Entropy Production Principle as obtained from dynamical fluctuation theory. This new extended heat theorem holds true for arbitrary driving and does not require assumptions of local or close to equilibrium. The argument remains exactly intact for diffusing fields where the fields correspond to macroscopic profiles of interacting particles under hydrodynamic fluctuations. We also show that the change of Shannon entropy is related to the antisymmetric part under a modified time-reversal of the time-integrated entropy flux.

· Marco Baiesi and Christian Maes, An update on nonequilibrium linear response. New Journal of Physics 15, 013004 (2013).

The unique fluctuation-dissipation theorem for equilibrium stands in contrast with the wide variety of nonequilibrium linear response formulae. Their most traditional approach is "analytic", which, in the absence of detailed balance, introduces the logarithm of the stationary probability density as observable. The theory of dynamical systems offers an alternative with a formula that continues to work when the stationary distribution is not smooth. We show that this method works equally well for stochastic dynamics, and we illustrate it with a numerical example for the perturbation of circadian cycles. A second "probabilistic" approach starts from dynamical ensembles and expands the probability weights on path space. This line suggests new physical questions, as we meet the frenetic contribution to linear response, and the relevance of the change in dynamical activity in the relaxation to a (new) nonequilibrium condition.

· Christian Maes: Nonequilibrium entropries, Physica Scripta 86, 058509 (2012).

In contrast to the quite unique entropy concept useful for systems in (local) thermodynamic equilibrium, there is a variety of quite distinct nonequilibrium entropies, reflecting different physical

points. We disentangle these entropies as they relate to heat, fluctuations, response, time-asymmetry,variational principles, monotonicity, volume contraction or statistical forces. However, not all of those extensions yield state quantities as understood thermodynamically. At the end we sketch how aspects of dynamical activity can take over for obtaining an extended Clausius relation.

· Pierre Bohec, François Gallet, Christian Maes, Soghra Safaverdi, Paolo Visco, Frédéric Van Wijland, Probing active forces via a fluctuation-dissipation relation: Application to living cells. arXiv:1203.3571v1 [cond-mat.soft], to appear in Europhysics Letters (2013).

We derive a new fluctuation-dissipation relation for non-equilibrium systems with long term memory. We show how this relation allows one to access new experimental information regarding active forces in living cells that cannot otherwise be accessed. For a silica bead attached to the wall of a living cell, we identify a cross-over time between thermally controlled fluctuations and those produced by the active forces. We show that the probe position is eventually slaved to the underlying random drive produced by the so-called active forces.

· Eliran Boksenbojm, Christian Maes, Karel Netočný and Jirka Pešek: Heat capacity in nonequilibrium steady states, Europhysics Letters 96, 40001 (2011).

We show how to extend the concept of heat capacity to nonequilibrium systems. The main idea is to consider the excess heat released by an already dissipative system when slowly changing the environment temperature. We take the framework of Markov jump processes to embed the specific physics of small driven systems and we demonstrate that heat capacities can be consistently defined in the quasistatic limit. Away from thermal equilibrium, an additional term appears to the usual energy-temperature response at constant volume, explicitly in terms of the excess work. In linear order around an equilibrium dynamics that extra term is an energy-driving response and it is entirely determined from local detailed balance. Examples illustrate how the steady heat capacity can become negative when far from equilibrium.

· Christian Maes and Senya Shlosman: Rotating states in driven clock- and XY-models, J. Stat Phys. 144, 1238–1246 (2011).

We consider 3D active plane rotators, where the interaction between the spins is of XY-type and where each spin is driven to rotate. For the clock-model, when the spins take N >>1 possible values, we conjecture that there are two low-temperature regimes. At very low temperatures and for small enough drift the phase diagram is a small perturbation of the equilibrium case. At larger temperatures the massless modes appear and the spins start to rotate synchronously for arbitrary small drift. For the driven XY-model we prove that there is essentially a unique translation-invariant and stationary distribution despite the fact that the dynamics is not ergodic.

· Jeremy Clark, Wojciech De Roeck and Christian Maes: Diffusive behavior from a quantum master equation, Journal of Mathematical Physics 52, 083303 (2011).

We study a general class of translation invariant quantum Markov evolutions for a particle on the d-dimensional regular lattice. We assume locality of the spatial jumps and exponentially fast relaxation in momentum space. It is shown that the particle position diffuses in the long time limit. We employ a fiber decomposition in momentum space of the evolution made possible by the translation invariance of the dynamics. A central limit theorem follows from perturbation theory around a single fiber whose evolution in the momentum representation is described by a Markov jump process on the d-dimensional torus.

· Christian Maes and Simi R. Thomas: Archimedes' law and its corrections for an active particle in a granular sea, J. Phys. A: Math. Theor. 44, 285001 (2011).

We study the origin of buoyancy forces acting on a larger particle moving in a granular medium subject to horizontal shaking and its corrections before fluidization. In the fluid limit Archimedes' law is verified; before the limit memory effects counteract buoyancy, as also found experimentally. The origin of the friction is an excluded volume effect between active particles, which we study more exactly for a random walker in a random environment.

The same excluded volume effect is also responsible for the mutual attraction between bodies moving in the granular medium. Our theoretical modeling proceeds via an asymmetric exclusion process, i.e., via a dissipative lattice gas dynamics simulating the position degrees of freedom of a low density granular sea.

· Soghra Safaverdi, Gerard T. Barkema, Eddy Kunnen, Adam M. Urbanowicz and Christian Maes: Saturation of front-propagation in a reaction-diffusion process describing plasma damage in porous low-k materials, Phys. Rev. B. 83, 245320 (2011).

We study a three-component reaction-diffusion system yielding an asymptotic logarithmic time-dependence for a moving interface. This Stefan-problem is modeled by coupled reaction-diffusion equations for which both one-sided Dirichlet-type and von Neumann-type boundary conditions are considered. We integrate the dependence of the interface motion on diffusion and reaction parameters and we observe a change from transport behavior and interface motion t^1/2 to logarithmic behavior log t as a function of time-scales and of the reaction-diffusion rates. We apply it to the description of the propagation of carbon depletion in porous dielectrics exposed to a low temperature plasma. This diffusion saturation is reached after about 1 minute in typical experimental situations of plasma damage in microelectronic fabrication. We predict the general dependencies on porosity and reaction rates.

· C. Maes, K. Netočný and B.Wynants: Monotone return to steady nonequilibrium, Phys. Rev. Lett. 107, 010601 (2011).

We propose and analyze a new candidate Lyapunov function for relaxation towards general nonequilibrium steady states. The proposed functional is obtained from the large time asymptotics of time-symmetric fluctuations. For driven Markov jump or diffusion processes it measures an excess in dynamical activity rates. We present numerical evidence and we report on a rigorous argument for its monotonous time-dependence close to the steady nonequilibrium; this is in contrast with the behavior of approximate Lyapunov functions based on entropy production that when driven far from equilibrium often keep exhibiting temporal oscillations even close to stationarity.

· C. Maes, K. Netočný and B.Wynants: Monotonicity of the dynamical activity, arXiv:1102.2690v2 [math-ph], J.Phys.A 45, 455001 (2012).

The Donsker-Varadhan rate function for occupation-time fluctuations has been seen numerically to exhibit monotone return to stationary nonequilibrium [Phys. Rev. Lett. 107, 010601 (2011)]. That rate function is related to dynamical activity and, except under detailed balance, it does not derive from the relative entropy for which the monotonicity in time is well understood. We give a rigorous argument that the Donsker-Varadhan function is indeed monotone under the Markov evolution at large enough times with respect to the relaxation time, provided that a ``normal linear-response'' condition is satisfied..

· Marco Baiesi, Christian Maes and Bram Wynants: The modified Sutherland-Einstein relation for diffusive nonequilibria, Proceedings of the Royal Society A 467, 2792-2809 (2011). (arXiv:1101.3227v2 [cond-mat.stat-mech].

There remains a useful relation between diffusion and mobility for a Langevin particle in a periodic medium subject to nonconservative forces. The usual fluctuation-dissipation relation easily gets modified and the mobility matrix is no longer proportional to the diffusion matrix, with a correction term depending explicitly on the (nonequilibrium) forces. We discuss this correction by considering various simple examples and we visualize the various dependencies on the applied forcing and on the time by means of simulations. For example, in all cases the diffusion depends on the external forcing more strongly than does the mobility. We also give an explicit decomposition of the symmetrized mobility matrix as the difference between two positive matrices, one involving the diffusion matrix, the other force-force correlations.

· Matteo Colangeli, Christian Maes and Bram Wynants: A meaningful expansion around detailed balance (2011), J. Phys. A: Math. Theor. 44, 095001 (13p.) (2011).

We consider Markovian dynamics modeling open mesoscopic systems which are driven away from detailed balance by a nonconservative

force. A systematic expansion is obtained of the stationary distribution around an equilibrium reference, in orders of the nonequilibrium forcing.

The first order around equilibrium has been known since the work of McLennan (1959), and involves the transient irreversible entropy flux. The expansion generalizes the McLennan formula to higher orders, complementing the entropy flux with the dynamical activity. The latter is more kinetic than thermodynamic and is a possible realization of Landauer's insight (1975) that, for nonequilibrium, the relative occupation of states also depends on the noise along possible escape routes. In that way nonlinear response around equilibrium can be meaningfully discussed in terms of two main quantities only, the entropy flux and the dynamical activity. The expansion makes mathematical sense as shown in the simplest cases from exponential ergodicity.

· Juan Ruben Gomez-Solano, Artyom Petrosyan, Sergio Ciliberto and Christian Maes: Non-equilibrium linear response of micron-sized systems, Journal of Statistical Mechanics, P01008 (2011).

The linear response of non-equilibrium systems with Markovian dynamics satisfies a generalized fluctuation-dissipation relation derived from time symmetry and antisymmetry properties of the fluctuations. The relation involves the sum of two correlation functions of the observable of interest: one with the entropy excess and the second with the excess of dynamical activity with respect to the unperturbed process. We illustrate this approach in the experimental determination of the linear response of the potential energy of a Brownian particle in a toroidal optical trap. The overdamped particle motion is effectively confined to a circle, undergoing a periodic potential and driven out of equilibrium by a non-conservative force. Independent direct and indirect measurement of the linear response around a non-equilibrium steady state are performed in this simple experimental system. The same ideas are applicable to the non-equilibrium linear response of more general micron-sized systems immersed in Newtonian fluids either in stationary or non-stationary states and possibly including inertial degrees of freedom.

· Christian Maes, Karel Netočný and Simi R. Thomas: General no-go condition for stochastic pumping, arXiv:1002.3811, J. Chem. Phys. 132, 234116 (2010). Copyright Abstract J.Chem.Phys.

The control of chemical dynamics requires understanding the effect of time-dependent transition rates between chemo-mechanical molecular configurations. Pumping means generating a net current, e.g. per period in the time-dependence, through a cycle of consecutive states. The working of artificial machines or synthesized molecular motors depends on it. In this paper we give short and simple proofs of no-go theorems, some of which appeared before but here with essential extensions to non-Markovian dynamics, including the study of the diffusion limit. It allows to exclude certain protocols in the working of chemical motors where only the depth of the energy well is changed in time and not the barrier height between pairs of states. We also show how pre-existing steady state currents are in general modified with a multiplicative factor when this time-dependence is turned on.

· M. Baiesi, E. Boksenbojm, C. Maes and B.Wynants: Nonequilibrium Linear Response for Markov Dynamics, II: Inertial Dynamics. arXiv:0912.0694, Journal of Statistical Physics 139, 492–505 (2010).

We continue our study of the linear response of a nonequilibrium system. This Part~II concentrates on models of open and driven inertial dynamics but the structure and the interpretation of the result remain unchanged: the response can be expressed as a sum of two temporal correlations in the unperturbed system, one entropic, the other frenetic. The decomposition arises from the (anti)symmetry under time-reversal on the level of the nonequilibrium action. The response formula involves a statistical averaging over explicitly known observables but, in contrast with the equilibrium situation, they depend on the model dynamics in terms of an excess in dynamical activity. As an example, the Einstein relation between mobility and diffusion constant is modified by a correlation term between the position and the momentum of the particle.

· Christian Maes and Karel Netočný: Rigorous meaning of McLennan ensembles, arXiv:0911.1032, Journal of Mathematical Physics 51, 015219 (2010).

We analyze the exact meaning of expressions for nonequilibrium stationary distributions in terms of entropy changes. They were originally introduced by McLennan for mechanical systems close to equilibrium and more recent work by Komatsu and Nakagawa has shown their intimate relation to the transient fluctuation symmetry. Here we derive these distributions for jump and diffusion Markov processes and we clarify the order of the limits that take the system both to its stationary regime and to the close-to-equilibrium regime. In particular, we prove that it is exactly the (finite) transient component of the irreversible part of the entropy flux that corrects the Boltzmann distribution to first order in the driving. We add further connections with the notion of local equilibrium, with the Green-Kubo relation and with a generalized expression for the stationary distribution in terms of a reference equilibrium process.

· Christian Maes and Bram Wynants: On a response formula and its interpretation, arXiv:0910.2320, Markov Processes and Related Fields 16, 45-58 (2010).

We present a physically inspired generalization of equilibrium response formulae, the fluctuation-dissipation theorem, to Markov jump processes possibly describing interacting particle systems out-of-equilibrium, following the recent work of Baiesi, Maes and Wynants. Here, the time-dependent perturbation adding a potential V with small amplitude h(t) changes the rates W(x,y) for the transition x à y into W(t;x,y) = W(x,y) exp (h(t) [bV(y)-aV(x)]) as first considered by Diezemann; a,b are constants. We observe that the linear response relation shows a reciprocity symmetry in the nonequilibrium stationary regime and we interpret the connection with dynamical fluctuation theory. Dedicated to the 15th birthday of Markov Processes and Related Fields.

· Jeremy Clark and Christian Maes: Diffusive behavior for randomly kicked Newtonian particles in a spatially periodic medium, Communications in Mathematical Physics 301, 229–283 (2011).

We prove a central limit theorem for the momentum distribution of a particle undergoing an unbiased spatially periodic random forcing at exponentially distributed times without friction. The start is a linear Boltzmann equation for the phase space density, where the average energy of the particle grows linearly in time. Rescaling time, the momentum converges to a Brownian motion, and the position is its time-integral showing superdiffusive scaling with time t^3/2 . The analysis has two parts: (1) to show that the particle spends most of its time at high energy, where the spatial environment is practically invisible; (2) to treat the low energy incursions where the motion is dominated by the deterministic force, with potential drift but where symmetry arguments cancel the ballistic behavior. The last problem is most prominent in one dimension, on which we concentrate.

· E. Kunnen, G.T. Barkema, C. Maes, D. Shamiryan, A. Urbanowicz, H. Struyf and M.R. Baklanov: Integrated diffusion - recombination model for describing the logarithmic time-dependence of plasma damage in porous low-k materials, Microelectronic Engineering 88, 631-634 (2011).

This work proposes an extended model that describes the propagation of damage in porous low-k material exposed to a plasma. Recent work has indicated that recombination and diffusion play a more dominant role than VUV light in oxygen plasma induced damage. Especially at low depths, the radical concentration is determined by the number of radicals that disappear back into the plasma while the final depth of damage is defined by recombination of oxygen atoms. A logarithmic equation has been proposed to describe the behavior as a function of time. In this work this equation is extended to take, diffusion into account, next to recombination. The results are in agreement with one-dimensional random walk theory calculations.

· M. Baiesi, C. Maes and B. Wynants: Nonequilibrium linear response for Markov dynamics, I: jump processes and overdamped diffusions ( PS ) , ONLINE J.Stat.Phys. 137, 1094-1116 (2009); DOI 10.1007/s10955-009-9852-8.

Systems out of equilibrium, in stationary as well as in nonstationary regimes, display a linear response to energy impulses simply expressed as the sum of two specific temporal correlation functions. There is a natural interpretation of these quantities. The first term corresponds to the correlation between observable and excess entropy flux yielding a relation with energy dissipation like in equilibrium. The second term comes with a new meaning: it is the correlation between the observable and the frenesy, the linear order of excess in dynamical activity or reactivity, playing an important role in dynamical fluctuation theory out-of-equilibrium. It appears as a generalized escape rate in the occupation statistics. The resulting response formula holds for all observables and allows direct numerical or experimental evaluation, for example in the discussion of effective temperatures, as it only involves the statistical averaging of explicit quantities, e.g. without needing an expression for the nonequilibrium distribution. The physical interpretation and the mathematical derivation are independent of many details of the dynamics, but in this first part they are restricted to Markov jump processes and overdamped diffusions.

· Christian Maes: Fluctuations and response out-of-equilibrium, Progress of Theoretical Physics, supplement 184, 318-328 (2010). Yukawa International Workshop 2009, Kyoto: Frontiers in Nonequilibrium Physics.

We discuss some recently visited positions towards dealing with nonequilibria from the mathematical point of view of Markov networks.

· C. Maes, K. Netočný and B.Wynants: Dynamical fluctuations for semi-Markov processes, arXiv:0905.4897v2, J. Phys. A: Math. Theor. 42, 365002 (2009).

We develop an Onsager-Machlup-type theory for nonequilibrium semi-Markov processes. Our main result is an exact large time asymptotics for the joint probability of the occupation times and the currents in the system, establishing some generic large deviation structures. We discuss in detail how the nonequilibrium driving and the non-exponential waiting time distribution influence the occupation-current statistics. The violation of the Markov condition is reflected in the emergence of a new type of nonlocality in the fluctuations. Explicit solutions are obtained for some examples of driven random walks on the ring.

• M. Baiesi, C. Maes and B. Wynants: Fluctuations and response of nonequilibrium states, arXiv:0902.3955v3, Phys. Rev. Lett. 103, 010602 (2009).

A generalized fluctuation-response relation is found for thermal systems driven out of equilibrium. Its derivation is independent of many details of the dynamics, which is only required to be first-order. The result gives a correction to the equilibrium fluctuation-dissipation theorem, in terms of the correlation between observable and the excess in dynamical activity caused by the perturbation. Previous approaches to this problem are recovered and extended in a unifying scheme.

• C. Maes, K. Netočný and B. Shergelashvili: On the nonequilibrium relation between potential and stationary distribution for driven diffusion, ( PS ), Phys. Rev.E 80, 011121 (2009).

We investigate the relation between an applied potential and the corresponding stationary state occupation for nonequilibrium and overdamped diffusion processes. This relation typically becomes long ranged resulting in global changes for the relative density when the potential is locally perturbed, and inversely, we find that the potential needs to be wholly rearranged for the purpose of creating a locally changed density. The direct question, determining the density as a function of the potential, comes under the response theory out of equilibrium. The inverse problem of determining the potential that produces a given stationary distribution naturally arises in the study of dynamical fluctuations. This link to the fluctuation theory results in a variational characterization of the stationary density upon a given potential and vice versa.

• W. De Roeck, C. Maes, K. Netočný and L. Rey-Bellet: A note on the non-commutative Laplace-Varadhan integral lemma, Reviews of Mathematical Physics 22, 839-858 (2010).

We continue the study of the free energy of quantum lattice spin systems where to the local Hamiltonian H an arbitrary mean field term is added, an analytic function of the arithmetic mean of some local observables X and Y that do not mutually commute. By slightly extending a recent paper by Hiai, Mosonyi, Ohno and Petz, we prove in general that the free energy is given by a variational principle over the range of the operators X and Y. As there, the result is a noncommutative extension of the Laplace-Varadhan asymptotic formula, with a rate function that is in general different from the rate function for the large deviations of X.

• M. Baiesi, C. Maes and K. Netočný: Computation of current cumulants for small nonequilibrium systems, ( PS ). cond-mat arXiv:0807.0145, Journal of Statistical Physics 135, 57-75 (2009).

We analyze a systematic algorithm for the exact computation of the current cumulants in stochastic nonequilibrium systems, recently discussed in the framework of full counting statistics for mesoscopic systems. This method is based on identifying the current cumulants from a Rayleigh-Schrödinger perturbation expansion for the generating function. Here it is derived from a simple path-distribution identity and extended to the joint statistics of multiple currents. For a possible thermodynamical interpretation we compare this approach to a generalized Onsager-Machlup formalism. We present calculations for the boundary driven Kawasaki dynamics on a one-dimensional chain, both for attractive and repulsive particle interactions.

• Stochastic Models in Biological Sciences, Eds. R. Bürger, C. Maes and J. Miekisz, Banach Center Publications 80, Warszawa (2008).

• M. Baiesi, C. Maes and K. Netočný: Exact computation of current cumulants in small Markovian systems, In: Modeling and Simulation of New Materials, Tenth Granada Lectures, Eds. P.L.Garrido, P.I.Hurtado and J.Marro, AIP Conference Proceedings Volume 1091, pp.220-224 (2009)., ( PS ), arXiv:0811.0469 (November 2008).

We describe an algorithm computing the exact value of the mean current, its variance, and higher order cumulants for stochastic driven systems. The method uses a Rayleigh-Schrodinger perturbation expansion of the generating function of the current, and can be extended to compute covariances of multiple currents. As an example of application of the method, we give numerical evidence for a simple relation between the second and the fourth cumulants of the current in a symmetric exclusion process.

• C. Maes, K. Netočný and B. Wynants: On and beyond entropy production; the case of Markov jump processes, ( PS ), Markov Processes and Related Fields 14, 445-464 (2008).

How is it that entropy derivatives almost in their own are characterizing the state of a system close to equilibrium, and what happens further away from it? We explain within the framework of Markov jump processes why fluctuation theory can be based on considerations involving entropy production alone when perturbing around the detailed balance condition. Variational principles such as that of minimum entropy production are understood in that way. Yet, further away from equilibrium, dynamical fluctuations reveal a structure where the time-symmetric sector crucially enters. The fluctuations of densities and currents get coupled and a time-symmetric notion of dynamical activity becomes the counterpart and equal player to the entropy production. The results are summarized in an extended Onsager-Machlup Lagrangian, which in its quadratic approximation is expected to be quite general in governing the small fluctuations of nonequilibrium systems whose macroscopic behavior can be written in terms of a Master equation autonomously describing the time-dependence of densities and currents.

• C. Maes, K. Netočný and B. Wynants: Steady state statistics of driven diffusions. ( PS ), Physica A 387, 2675-2689 (2008); arXiv:0708.0489.

We consider overdamped diffusion processes driven out of thermal equilibrium and we analyze their dynamical steady fluctuations. We mainly consider the joint fluctuations of occupation times and currents because they incorporate respectively the time-symmetric and the time-antisymmetric sector of the fluctuations. An explicit expression is given for the large time asymptotics of this joint distribution, and we explain how the occupation and current fluctuations get mutually coupled out of equilibrium. We highlight the canonical structure of the joint fluctuations.

• J. Derezinski, C. Maes and W. De Roeck: Fluctuations of Quantum Currents and Unravelings of Master Equations. arXiv:cond-mat/0703594, J. Stat. Phys. 131, 341-356 (2008).

The very notion of a current fluctuation is problematic in the quantum context. We study that problem in the context of nonequilibriumstatistical mechanics, both in a microscopic setup and in a Markovian model. Our answer is based on a rigorous result that relates the weak coupling limit of fluctuations of reservoir observables under a global unitary evolution with the statistics of the so-called quantum trajectories. These quantum trajectories are frequently considered in the context of quantum optics, but they remain useful for more general nonequilibrium systems In contrast with the approaches found in the literature, we do not assume that the system is continuously monitored. Instead, our starting point is a relatively realistic unitary dynamics of the full system.

• C. Maes and K. Netočný: The canonical structure of dynamical fluctuations in mesoscopic nonequilibrium steady states. ( PS ) arXiv:0705.2344 Europhysics Letters 82, 30003 (2008).

We give the explicit structure of the functional governing the dynamical density and current fluctuations for a mesoscopic system in a nonequilibrium steady state. Its canonical form determines a generalised Onsager-Machlup theory. We assume that the system is described as a Markov jump process satisfying a local detailed balance condition such as typical for stochastic lattice gases and for chemical networks. We identify the entropy current and the traffic between the mesoscopic states as extra terms in the fluctuation functional with respect to the equilibrium dynamics. The density and current fluctuations are coupled in general, except close to equilibrium where their decoupling explains the validity of entropy production principles.

• Bidzina M.Shergelashvili, G. Gogoberidze, C. Maes and S.Poedts: Magnetic helicity driven MHD turbulence: a mechanism for the creation of post-flare magnetic structures.

The inverse turbulent cascade of magnetic helicity is studied in the context of the dynamics of solar flares. In particular, its role in the formation of post-flare loop structures is investigated. An updated model of structure formation is developed, based on the standard model of decaying MHD turbulence. A quantitative study of the problem reveals characteristic timescales of the process that are in good agreement with the available observational evidence.

• S. Bruers, C. Maes and K. Netočný: On the validity of entropy production principles for linear electrical circuits ( PDF ): J. Stat. Phys. 129, 725-740 (2007). arXiv:cond-mat/0701035

We explain the (non-)validity of close-to-equilibrium entropy production principles in the context of linear electrical circuits. Both the minimum and the maximum entropy production principles are understood within dynamical fluctuation theory. The starting point are Langevin equations obtained by combining Kirchoff's laws with a Johnson-Nyquist noise at each dissipative element in the circuit. The main observation is that the fluctuation functional for time averages, that can be read off from the path-space action, is in first order around equilibrium given by an entropy production rate. That allows to understand beyond the schemes of irreversible thermodynamics (1) the validity of the least dissipation, the minimum entropy production, and the maximum entropy production principles close to equilibrium; (2) the role of the observables' parity under time-reversal and, in particular, the origin of Landauer's counterexample (1975) from the fact that the fluctuating observable there is odd under time-reversal; (3) the critical remark of Jaynes (1980) concerning the apparent inappropriateness of entropy production principles in temperature-inhomogeneous circuits.

• C. Maes and K. Netočný: Minimum entropy production principle from a dynamical fluctuation law ( PDF ): J. Math. Phys. 48, 053306 (2007), arXiv:math-ph/0612063.

The minimum entropy production principle provides an approximative variational characterization of close-to-equilibrium stationary states, both for macroscopic systems and for stochastic models. Analyzing the fluctuations of the empirical distribution of occupation times for a class of Markov processes, we identify the entropy production as the large deviation rate function, up to leading order when expanding around a detailed balance dynamics. In that way, the minimum entropy production principle is recognized as a consequence of the structure of dynamical fluctuations, and its approximate character gets an explanation. We also discuss the subtlety emerging when applying the principle to systems whose degrees of freedom change sign under kinematical time-reversal.

• M. Baiesi, C. Maes and B.M. Shergelashvili: Correlated flares in models of a magnetized 'canopy', Physica A 387, 167-176 (2008).

A model of the Lu-Hamilton kind is applied to the study of critical behavior of the magnetized solar atmosphere. The main novelty is that its driving is done via sources undergoing a diffusion. This mimics the effect of a virtual turbulent substrate forcing the system. The system exhibits power-law statistics not only in the size of the flares, but also in the distribution of the waiting times.

• B.M. Shergelashvili, C. Maes, S. Poedts and T.V. Zaqarashvili: Amplification of compressional MHD waves in systems with forced entropy oscillations. Phys. Rev. E 76, 046404 (2007).

The propagation of compressional MHD waves is studied for an externally driven system. It is assumed that the combined action of the external sources and sinks of the entropy results in the harmonic oscillation of the entropy (and temperature) in the system. It is found that with the appropriate resonant conditions fast and slow waves get amplified due to the phenomenon of parametric resonance. Besides, it is shown that the considered waves are mutually coupled as a consequence of the nonequilibrium state of the background medium. The coupling is strongest when the plasma-beta is about equal to 1. The proposed formalism is sufficiently general and can be applied for many dynamical systems, both under terrestrial and astrophysical conditions.

• C. Maes, K. Netočný and B. Shergelashvili: A selection of nonequilibrium issues ( PDF ): In: Methods of Contemporary Mathematical Statistical Physics, Ed. Roman Kotecký, Lecture Notes in Mathematics 1970, pp. 247-306, Springer, 2009.

We give a pedagogical introduction to a selection of recently discussed topics in nonequilibrium statistical mechanics, concentrating mostly on formal structures and on general principles. Part I contains an overview of the formalism of lattice gases that we use to explain various symmetries and inequalities generally valid for nonequilibrium systems, including the fluctuation symmetry, Jarzynski equality, and the direction of currents. That mostly concerns the time-antisymmetric part of dynamical fluctuation theory. We also briefly comment on recent attempts to combine that with the time-symmetric sector in a Langrangian or extended Onsager-Machlup approach. In Part II we concentrate on the macroscopic state and how entropy provides a bridge between microscopic dynamics and macroscopic irreversibility; included is a construction of quantum macroscopic states and a result on the equivalence of ensembles.

• C. Maes and K. Netočný: Static and Dynamical Nonequilibrium Fluctuations ( PDF ): cond-mat/0612525, Comptes Rendus –Physique 8, 591-597 (2007).

Various notions of fluctuations exist depending on the way one chooses to measure them. We discuss two extreme cases (continuous measurement versus long inter-measurement times) and we see their relation with entropy production and with escape rates. A simple explanation of why the relative entropy satisfies a Hamilton-Jacobi equation is added.

• W. De Roeck and C. Maes: Symmetries of the ratchet current ( PS ), Phys. Rev. E 76 , 051117 (2007).

Recent advances in nonequilibrium statistical mechanics shed new light on the ratchet effect. The ratchet motion can thus be understood in terms of symmetry (breaking) considerations. We introduce an additional symmetry operation besides time-reversal, that effectively reverses the nonequilibrium driving. That operation of field-reversal combined with time-reversal decomposes the nonequilibrium action so to clarify under what circumstances the ratchet current is a second order effect around equilibrium, what is the direction of the ratchet current and what are possibly the symmetries in its fluctuations.

• C. Maes, F. Redig and E. Saada: Freezing transitions in non-Fellerian particle systems ( PS ), J. Stat. Phys. 127, 171-189 (2007).

Non-Fellerian processes show phenomena that are unseen in standard interacting particle systems. We consider freezing transitions in one-dimensional non-Fellerian processes which are built from the abelian sandpile additions to which in one case, spin flips are added, and in another case, the so called anti-sandpile subtractions. In the first case and as a function of the sandpile addition rate, there is a sharp transition from a non-trivial invariant measure to the invariant measure of the sandpile process. For the combination sandpile plus anti-sandpile, there is a sharp transition from one frozen state to the other anti-state.

• C. Maes and Hal Tasaki: Second law of thermodynamics for macroscopic mechanics coupled to thermodynamic degrees of freedom ( PDF ), Letters in Mathematical Physics 79, 251-261 (2007) .

Starting from and only using classical Hamiltonian dynamics, we prove for the first time the maximum work principle in a system where macroscopic dynamical degrees of freedom are intrinsically coupled to microscopic degrees of freedom. Unlike in many of the standard and recent works on the second law, the macroscopic dynamics is not governed by an external action but undergoes the back reaction of the microscopic degrees of freedom. Our theorems cover such physical situations as impact between macroscopic bodies, thermodynamic machines, and molecular motors. Our work identifies and quantifies the physical limitations on the applicability of the second law for small systems.

· J. Depoorter and C. Maes: Stavskaya's measure is weakly Gibbsian ( PDF ), Markov Processes and Related Fields 12, 791-804 (2006).

Stavskaya's model is a one-dimensional Boolean probabilistic cellular automaton very similar to the contact and directed percolation processes. There is always an absorbing measure but close to the deterministic limit the model also shows a non-trivial invariant measure. We show that the latter ``Stavskaya's'' measure is weakly Gibbsian with an exponentially decaying interaction potential.

· C.Maes: Challenges and trends in interacting particle systems, in the Proceedings to the IMA conference on Mathematics in the Science of Complex Systems, 18-21 September, 2006.

Some meditations are presented on the status of Interacting Particle Systems in the light of studies in complex systems. Probabilistic reasoning and a rigorous understanding of large scale dynamics contribute significantly in clarifying various complex issues. (Talk given in Warwick at the IMA conference on Mathematics in the Science of Complex Systems}, 18-21 September 2006.)

· W. De Roeck and C. Maes:  Steady state fluctuations of the dissipated heat for a quantum stochastic model   ( PDF  ), Reviews in Mathematical Physics 18, 619-653 (2006).

We introduce a quantum stochastic dynamics for heat conduction. A multi-level subsystem is coupled to reservoirs at different temperatures.  Energy quanta are detected in the reservoirs allowing the study of steady state fluctuations of the entropy production. Our main result states a symmetry in its large deviation rate function.

• M. Baiesi, T. Jacobs, C. Maes and Nikos S. Skantzos: Fluctuation symmetries for work and heat ( PDF ) Phys. Rev. E 74, 021111 (2006).

We consider a particle dragged through a medium at constant temperature as described by a Langevin equation with a time-dependent potential. The time-dependence is specified by an external protocol. We give conditions on potential and protocol under which the dissipative work satisfies an exact symmetry in its fluctuations for all times. We also present counter examples to that exact fluctuation symmetry when our conditions are not satisfied. Finally, we consider the dissipated heat which differs from the work by a temporal boundary term. We explain when and why there can be a correction to the standard fluctuation theorem due to the unboundedness of that temporal boundary. However, the corrected fluctuation symmetry has again a general validity.

• M. Baiesi and C. Maes: Realistic time correlations in sandpiles, Europhysics Letters 75, 413-419 (2006).

A sandpile cellular automaton achieves complex temporal correlations, like a 1/f spectrum, if the position where it is perturbed diffuses slowly rather than changing completely at random, showing that the spatial correlations of the driving are deeply related to the intermittent activity. Hence, recent arguments excluding the relevance of self-organized criticality in seismicity and in other contexts are inaccurate. As a toy model of single fault evolution, and despite of its simplicity, our automaton uniquely reproduces the scaling form of the broad distributions of waiting times between earthquakes.

• C. Maes and M.H. van Wieren: Time-symmetric fluctuations in nonequilibrium systems ( PDF ) Phys. Rev. Letters 96, 240601 (2006).

For nonequilibrium steady states, we identify observables whose fluctuations satisfy a general symmetry and for which a new reciprocity relation can be shown. Unlike the situation in recently discussed fluctuation theorems, these observables are time-reversal symmetric. That is essential for exploiting the fluctuation symmetry beyond linear response theory. Besides time-reversal, a crucial role is played by the reversal of the driving fields, that further resolves the space-time action. In particular, the time-symmetric part in the space-time action determines second order effects of the nonequilibrium driving.

• W. De Roeck, C. Maes and K. Netočný: Quantum Macrostates, Equivalence of Ensembles and an H-theorem (PDF ) J. Math. Phys. 47, 073303 (2006).

Before the thermodynamic limit, macroscopic averages need not commute for a quantum system. As a consequence, aspects of macroscopic fluctuations or of constrained equilibrium require a careful analysis, when dealing with several observables. We propose an implementation of ideas that go back to John von Neumann's writing about the macroscopic measurement. We apply our scheme to the relation between macroscopic autonomy and an H-theorem, and to the problem of equivalence of ensembles. In particular, we prove a quantum version of the asymptotic equipartition theorem. The main point of departure is an expression of a law of large numbers for a sequence of states that start to concentrate, as the size of the system gets larger, on the macroscopic values for the different macroscopic observables. Deviations from that law are governed by the entropy.

• C. Maes: Elements of nonequilibrium statistical mechanics ( PDF ), Summer school Les Houches, Session LXXXIII, 2005, A. Bovier, F. Dunlop, F. den Hollander, A. van Enter and J. Dalibard, eds. (Elsevier), Course 13, pp. 607-655 (2006).
• W. De Roeck, C. Maes and K. Netočný: H-Theorems from Macroscopic Autonomous Equations ( PS ) J. Stat. Phys. 123, 571-584 (2006).

The H-theorem is an extension of the Second Law to a time-sequence of states that need not be equilibrium ones. In this paper we review and we rigorously establish the connection with macroscopic autonomy. If for a Hamiltonian dynamics for many particles, at all times the present macrostate determines the future macrostate, then its entropy is non-decreasing as a consequence of Liouville's theorem. That observation, made since long, is here rigorously analyzed with special care to reconcile the application of Liouville's theorem (for a finite number of particles) with the condition of autonomous macroscopic evolution (sharp only in the limit of infinite scale separation); and to evaluate the presumed necessity of a semigroup property for the macroscopic evolution.

· C. Maes: New Trends in Interacting Particle Systems ( PDF ) Markov Processes and Related Fields 11, 283-288 (2005).

Since about a decade some new trends in the study of interacting particle systems can be identified. New is meant here with respect to some standard references and plays both in new mathematical challenges and in new models or phenomena. Examples are collected in the present volume. I give them here a short introduction.

• M. Baiesi and C. Maes: On the enstrophy dissipation in two-dimensional turbulence ( PDF ) Phys. Rev. E 72, 056314 (2005).

Insight into the problem of two-dimensional turbulence can be obtained by an analogy with a heat conduction network. It allows the identification of an entropy function associated to the enstrophy dissipation and that fluctuates around a positive (mean) value. While the corresponding enstrophy network is highly nonlocal, the direction of the enstrophy current follows from the Second Law of Thermodynamics. An essential parameter is the ratio T(k) = γ(k)/(ν k²) of the intensity of driving γ(k)>0 as a function of wavenumber k, to the dissipation strength ν k², where ν is the viscosity. The enstrophy current flows from higher to lower values of T(k), similar to a heat current from higher to lower temperature. Our probabilistic analysis of the enstrophy dissipation and the analogy with heat conduction thus complements and visualizes the more traditional spectral arguments for the direct enstrophy cascade. We also show a fluctuation symmetry in the distribution of the total entropy production which relates the probabilities of direct and inverse enstrophy cascades.

· C. Maes and  M.H. van Wieren:   Thermoelectric phenomena via an interacting particle system   (  PDF  ), J. Phys. A: Math. Gen. 38, 005-1020 (2005).

We present a mesoscopic model for thermoelectric phenomena in terms of interacting particle system, a lattice electron gas dynamics that is a suitable extension of the standard simple exclusion process. We concentrate on electronic heat and charge transport in different but connected metallic substances. The electrons hop  between energy-cells located alongside the spatial extension of the metal wire. When changing energy level, the system exchanges energy with the environment. At equilibrium the distribution satisfies the Fermi-Dirac occupation-law. Installing different temperatures at two connections induces an electromotive    force (Seebeck effect) and upon forcing an electric current, an additional heat flow is produced at the junctions (Peltier heat).
We derive the linear response behavior relating the Seebeck and Peltier coefficients as an application of Onsager reciprocity.
We also indicate the higher order corrections.  The entropy production is characterized as the anti-symmetric part under  time-reversal of the space-time Lagrangian.

• T . Jacobs and C. Maes:  Reversibility and Irreversibility within the Quantum Formalism    (   PDF  ), Physicalia Magazine 27, 119-130 (2005).

We state and review a number of general points that are perhaps less emphasized in the existing literature. We first explain what is meant by mechanical reversibility and how it applies in classical mechanics and, somewhat differently, for the free evolution in quantum mechanics. If the evolution is not free, i.e., it is interrupted by measurements, the quantum formalism is challenged by the problem of retrodiction. Related to that is the notion of statistical reversibility which is very similar to what is more commonly known as the condition of detailed balance, at least for stochastic processes describing the spatio- temporal fluctuations in equilibrium. Finally, we describe the emergence of thermodynamic irreversibility. Here there is little difference between the classical and the quantum set-up even though one could have thought that the presence of a discrete energy spectrum prohibits dissipation for Hamiltonian evolutions.

• C. Maes, F. Redig and E. Saada:  Abelian SandPile Models in Infinite Volume Sankhyā, the Indian Journal of Statistics 67, 634-661 (2005).

Since its introduction by Bak,Tang and Wiesenfeld, the abelian sandpile dynamics has been studied extensively in finite volume. There are many problems posed by the existence of a sandpile dynamics in an infinite volume S: its invariant distribution should be the thermodynamic limit (does the latter exist?) of
the invariant measure for the finite volume dynamics; the extension of the sand grains addition operator to infinite volume is related to the boundary effects of the dynamics in finite volume; finally, the crucial difficulty of the definition of a Markov process in infinite volume is that, due to sand avalanches, the interaction is long range,  so that no use of the Hille-Yosida theorem is possible.
In this review paper, we recall the needed results in finite volume, then explain how to deal with infinite volume  when S=Z, S=T is an infinite tree, S=Z^d with d large, and when the dynamics is dissipative (i.e. sand grains may disappear at each toppling).

· E. Dinaburg, C. Maes, S. Pirogov, F. Redig and A. Rybko:   The Potts model built on sand, J. Stat. Phys. 117, 179-198 (2004).

We consider the q=4 Potts model on the square lattice with an additional hard-core nonlocal interaction.  That interaction arises from the choice of the reference measure taken to be the uniform measure on the recurrent configurations for the abelian sandpile model. In that reference measure some correlation functions have a power-law decay. We investigate the low-temperature phase diagram and we prove the existence of a single stable phase with exponential decay of correlations. For all boundary conditions the density of 4 in the infinite volume limit goes to one as the temperature tends to zero.

• W. De Roeck and C. Maes:    A quantum version of free energy - irreversible work relations, Phys. Rev. E 69, 26115 (2004). 

We give a quantum version of the Jarzynski relation between the distribution of work done over a certain time-interval on a system and the difference of equilibrium free energies.  The main new ingredient is the identification of work depending on the quantum history of the system and the proper definition of various quantum ensembles over which the averages should be made.  We also discuss a number of different regimes that have been considered by other authors and which are unified in the present set-up. In all cases, and quantum or classical, it is a general relation between heat and time-reversal that makes the Jarzynski relation so universally valid.

• C. Maes: Fluctuation relations and positivity of the entropy production in irreversible dynamical systems, Nonlinearity 17, 1305 - 1316 (2004).

We give a generalization of fluctuation identities and inequalities for the entropy production when the microscopic dynamics preserves the phase space volume but is not assumed to be dynamically reversible.  We find that general first properties such as the strict positivity of the entropy production or the validity of an H-theorem are unaffected by the irreversibility of the dynamics.  Fluctuation identities describing the relation between entropy production and its time-reversal must however be changed and now involve the time-reversed dynamics.

• I. Callens, W. De Roeck, T. Jacobs, C. Maes and K.  Netočný:   Quantum entropy production as a measure of irreversibility  ( PDF  )  Physica D: Nonlinear phenomena 187, 383-391 (2004).

We consider conservative quantum evolutions possibly interrupted by macroscopic measurements. When started in a nonequilibrium state, the resulting path-space measure is not time-reversal invariant and the weight of time-reversal breaking equals the exponential of the entropy production. The mean entropy production can then be expressed via a relative entropy on the level of histories. That gives a partial extension of the result for classical systems, that the entropy production is given by the source term of time-reversal breaking in the path-space measure.

• C. Maes, F. Redig and E. Saada:   The Infinite Volume Limit of Dissipative Abelian Sandpiles    (  PDF  )  Commun. Math. Phys. 244, 395-417 (2004).

We construct the thermodynamic limit of the stationary measures of the Bak-Tang-Wiesenfeld sandpile model with a dissipative toppling matrix (sand grains may disappear at each toppling). We prove uniqueness and mixing properties of this measure and we obtain an infinite volume ergodic Markov process leaving it invariant. We show how to extend the Dhar formalism of the abelian group of toppling operators' to infinite volume in order to obtain a compact abelian group with a unique Haar measure representing the uniform distribution over the recurrent configurations that create finite avalanches.

•  C. Maes:   On the origin and the use of fluctuation relations for the entropy    (   PDF  )   Séminaire Poincaré 2,  29-62 (2003).

        Since about a decade various fluctuation relations for the entropy production have been derived and analyzed.   These relations deal with
        symmetries of the entropy production under time-reversal and have been proposed as a non-perturbative generalization of
        fluctuation-dissipation relations. I describe a unifying framework for understanding these relations and I present an algorithm to derive them. The fluctuation relations all follow from the main observation that in great generality the path-dependent entropy production is the source-term of time-reversal breaking in the Lagrangian over space-time histories. That is illustrated via a number of examples as well as via a general theoretical argument. I move these relations away from the strict dynamical background in which they originated and take them back to the context of statistical mechanics where entropy is understood in the sense of Boltzmann, as measuring the typicality of a manifest condition.
I discuss how a relation between work and free energy is naturally put in that framework and how the transient and steady state fluctuation theorems are simple consequences. Part of the interest in the resulting fluctuation symmetries is that they are so universally valid, a rare occasion in nonequilibrium statistical mechanics.  However they do not provide a systematic perturbation expansion for response functions. For that one needs to go back to the full Lagrangian and also consider the nonequilibrium modifications to its time-symmetric part.

• W. De Roeck, T. Jacobs, C. Maes and K. Netočný:   An extension of the Kac ring model, J. Phys. A: Math. Gen. 36, 11547 – 11559  (2003).   

       We introduce a unitary dynamics for quantum spins which is an extension of a model introduced by Mark Kac to clarify the
        phenomenon of relaxation to equilibrium.  When the number of spins gets very large, the magnetization satisfies an autonomous
        equation as function of time with exponentially fast relaxation to the equilibrium magnetization as determined by the microcanonical
        ensemble.  This is proven as a law of large numbers with respect to a class of initial data.  The corresponding Gibbs-von Neumann
        entropy is also computed and its monotonicity in time discussed.

• C. Maes and M.H. van Wieren:   A Markov Model for Kinesin,  J. Stat. Phys. 112, 329-355 (2003).    

        We investigate the validity of a Markov approach for the motility of kinesin. We show in detail how the various mechano-chemical states and reaction rates that were explored experimentally, may be used to create a Markov-chain model. We compare the performance of this model to motility data and we find global similarities in the load and ATP-concentration dependency of speed and average distance travelled. We also discuss the relation between the experimentally found stalling behavior and thermodynamic expectations. Finally, the Markov chain modelling provides a way to calculate the mean entropy production and the (power) efficiency.

• Joel L. Lebowitz and C. Maes:  Entropy - a dialogue ,   In: Entropy, Princeton University Press,  Princeton and Oxford. Eds. A. Greven,  G. Keller and  G. Warnecke, pp. 269-276 (2003).

        A scientist who was always very curious about the relation between entropy and information but was afraid to ask, dies and is ushered into a very large room where there sits behind a desk the angel of knowledge.  On the  desk there is a shiny sphere with some special aura surrounding it.  The eye of the scientist is attracted to the sphere and he asks what is in  it.  The angel  answers.

• C. Maes:  Entropy Production in Driven Spatially Extended Systems,  In: Entropy, Princeton Uniiversity Press, Princeton and Oxford. Eds. A. Greven, G. Keller and G. Warnecke, pp. 251-267 (2003).

        This is a short review of the statistical  mechanical definition of entropy production for systems composed of a large number of interacting components.  Emphasis is on open systems  driven away from equilibrium where the entropy production can be identified with a logarithmic ratio of microstate multiplicities of the original macrostate with respect to the time-reversed state. A special
role is taken by Gibbs measures for the stationary spatio-temporal  distribution of trajectories. The mean entropy production is always  non-negative and it is zero only when the system is in equilibrium.  The fluctuations of the entropy production satisfy a symmetry first observed and then derived for the phase space contraction rate in a class of strongly chaotic dynamical systems.  Aspects of the  general framework are illustrated via a bulk driven diffusive lattice gas.

• C. Maes, K. Netočný and M. Verschuere:   Heat Conduction Networks  (  PDF  )   J. Stat. Phys. 111, 1219-1244 (2003).

         We study networks of interacting oscillators, driven at the boundary by heat baths at possibly different temperatures.   A set of first elementary questions are discussed concerning the uniqueness of a stationary possibly Gibbsian density and the nature of the entropy production and the local heat currents.  We also derive a Carnot efficiency relation for the work that can be extracted from the heat engine.

• C. Maes and E. Verbitskiy:   Large deviations and a fluctuation symmetry for chaotic homeomorphisms    ( PDF )  Commun. Math. Phys. 233, 137-151 (2003).

         We consider expansive homeomorphisms with the specification  property.   We give a new simple proof of a large deviation principle for Gibbs measures corresponding to a regular potential and we establish a general symmetry of the rate function for the large deviations of the antisymmetric part, under time-reversal,  of the potential.  This generalizes the Gallavotti-Cohen fluctuation theorem to a larger class of chaotic systems.

• C. Maes and K. Netočný:  Time-reversal and Entropy  ( PDF )    J. Stat. Phys. 110, 269-310 (2003),

          There is a relation between the irreversibility of thermodynamic processes as expressed by the breaking of time-reversal symmetry, and the entropy production in such processes. We explain on an elementary mathematical level the relations between entropy production, phase-space contraction and time-reversal starting from a deterministic dynamics. Both closed and open systems, in the transient and in the steady regime, are considered.  The main result identifies under general conditions the statistical mechanical entropy production as the source term of time-reversal breaking in the path space measure for the evolution of reduced variables.  This provides a general algorithm for computing the entropy production and to understand in a unified way a number of useful (in)equalities.  We also discuss the Markov approximation. Important are a number of old theoretical ideas for connecting the microscopic dynamics with thermodynamic behavior.

•  C. Maes and K. Netočný: Spacetime expansions for weakly coupled interacting particle systems (  PDF ), J. Phys. A. 35, 3053-3077 (2002).

       We consider classes  of both discrete time (parallel updating) and continuous time (sequential updating) interacting particle systems in the weak coupling regime.  We set up a perturbation analysis for the space-time distributions around
the uncoupled dynamics and we construct the Gibbsian potential for the time-evolved measures.

• C. Maes, F. Redig and E. Saada (2002)  The abelian sandpile model on an infinite tree , Annals of Probability  30, 2081-2107   (   PDF )

We consider the standard Abelian sandpile process on the Bethe lattice. We show the existence of the thermodynamic limit for the
finite volume stationary measures and the existence of a unique infinite volume Markov process exhibiting features of self-organized criticality.

• C. Maes, F. Redig and M. Verschuere (2002)  No current without heat (  PDF ), J. Stat.Phys. 106, 569-587

     We show for a large class of interacting particle systems
     that whenever the stationary measure is not reversible for the dynamics,
     then the mean entropy production in the steady state is strictly positive.
     This extends to the thermodynamic limit the equivalence between
     microscopic reversibility and zero mean entropy production: time-reversal
     invariance cannot be spontaneously broken.

• C. Maes (2001)  STATISTICAL MECHANICS OF ENTROPY PRODUCTION: Gibbsian hypothesis and local fluctuations (  PDF   ),  not to be published.

       It is argued that a Gibbsian formula for the space-time distribution of
       microscopic trajectories of a nonequilibrium system provides a unifying
       framework for recent results on the fluctuations of the entropy
       production.  The variable entropy production is naturally expressed as
       the time-reversal symmetry breaking part of the space-time action
       functional.  Its mean is always positive. This is both supported by a
       Boltzmann type analysis by counting the change in phase space extension
       corresponding to the macrostate as by various examples of nonequilibrium
       models.  As the Gibbsian set-up allows for non-Markovian dynamics, we
       also get a local fluctuation theorem for the entropy production in
      globally Markovian models. In order to study the response of the system to
      perturbations, we can apply the standard Gibbs formalism.

• C. Maes, F. Redig and M. Verschuere (2001)  From global to local fluctuation theorems   ( PDF   ), Moscow Mathematical Journal 1. 421-438.

       The Gallavotti-Cohen fluctuation theorem suggests a
       general symmetry in the fluctuations of the entropy production, a basic
       concept in the theory of irreversible processes, based on results in the
       theory of strongly chaotic maps.
       We study this symmetry for some standard models of nonequilibrium steady states.  We give a general strategy to
       derive a  local fluctuation theorem exploiting the Gibbsian features
       of the stationary space-time distribution.  This is applied to spin flip
       processes and to the asymmetric exclusion process. 

• C. Maes, F. Redig and M. Verschuere (2001) Entropy Production for Interacting Particle Systems , Markov  Proc. Rel. Fields 7, 119-134.

       The present article provides a mathematical
       introduction to the concept of entropy production in stochastic
       dynamics. We discuss the mean rate of entropy production for a
       class of interacting particle systems.

• C. Maes (2001) Weakly Gibbsian measures: how strong?  XIIIth International Congress on Mathematical Physics, Imperial College, London, UK (2000), Eds. .A. Fokas,  A. Grigoryan, T. Kibble, B. Zegarlinski, pp. 235-242, International Press.

       This paper presents a quick review of current research into an extension of the
       standard Gibbs formalism for lattice spin systems.  Weakly Gibbsian
       measures are presented as a Gibbsian restoration of certain non-Gibbsian measures.

• A.C.D. van Enter, C. Külske and C. Maes (2000) Comment on ``Critical Behavior of the Randomly Diluted 2D Ising Model: A Grand Ensemble Approach'' , Phys. Rev. Lett. 84, 6134.

         We point out that the grand ensemble or Morita-approach to disordered systems
         is as such ill defined.  When trying  to incorporate quenched disorder in
         an enlarged thermal (or annealed)  ensemble, one does not necessarily deal
         with Gibbs distributions.  The assumption that there exists an effective
         (regular) Hamiltonian is not innocent.  This  is important for the inference of thermodynamic results.

• P.A. Ferrari, C. Maes, L. Ramos and F. Redig (2000)  On the Hydrodynamic Equilibrium of a Rod in a Lattice Fluid , J. Phys. A: Math. Gen. 33, 4725-4740.

We model the behavior of a big (Brazil) nut in a medium of small nuts with a stochastic asymmetric
simple exclusion  dynamics of  a polymer-monomer lattice system.  The polymer or `rod' can move up or
down in an external negative field, occupying N horizontal lattice sites where the monomers cannot enter. The monomers (at most one per site) or `fluid particles' are moving symmetrically in the horizontal plane and asymmetrically in the vertical direction, also with a negative field.  For a fixed position of the rod, thislattice fluid is in equilibrium with a vertical height profile reversible for the monomers' motion.
Upon `shaking' (speeding up the monomers) the motion of the `rod' dynamically decouples from
that of the monomers resulting in a reversible random walk for the
rod around an average height proportional to log N.

• C. Maes and F. Redig (2000)  Positivity of Entropy Production, J. Stat. Phys. 101 , 3-16.

     We discuss the positivity of the mean entropy production for stochastic
     systems driven from equilibrium.
     Non-zero entropy production is closely linked with violation of the
     detailed balance condition.  This connection is rigorously obtained
     for spinflip dynamics.
     We remark that the positivity of entropy production depends on the
     choice of time-reversal transformation, hence on the choice of the
     dynamical variables in the system of interest.

• C. Maes, F. Redig, F. Takens, A. Van Moffaert and E. Verbitsky (2000) Intermittency and weak Gibbs states , Nonlinearity  13, 1681-1698.

    We show that the natural invariant state for Manneville-Pomeau maps can be
     characterized as a weakly  Gibbsian  distribution.
     In this way we make a connection between the study of intermittency via
     non-uniformly expanding maps and the thermodynamic formalism for non-uniformly convergent interactions.

• H.-O. Georgii, O. Häggström and C. Maes (2001)  The random geometry of equilibrium phases , Phase Transitions and Critical Phenomena, vol 18 , Eds C. Domb and J.L. Lebowitz (Academic Press, London) pp 1-142.

This is a large survey about applications of percolation theory in equilibrium statistical mechanics. The chapters are entitled as follows:

Introduction.
2. Equilibrium phases.
3. Some models.
4. Coupling and stochastic domination.
5. Percolation.
6. Random-cluster representations.

7. Uniqueness and exponential mixing from non-percolation.
8. Phase transition and percolation.
9. Random interactions.
10. Continuum models.

• C. Maes, F. Redig and A. Van Moffaert (2000) On the definition of entropy production, via examples , J. Math. Phys. 41 , 1528-1554.

We present a definition of entropy  production rate for classes of deterministic and stochastic
dynamics.  The point of departure is a Gibbsianrepresentation of the steady state pathspace measure for which `the density' is determined with respect to the  time-reversed process.  The Gibbs formalism is used as a      unifying algorithm capable of incorporating basic properties of entropy production in nonequilibrium systems.      Our definition is motivated by recent work on the Gallavotti-Cohen (local) fluctuation theorem and it is     illustrated via a number of examples.

• C. Maes, F. Redig, S. Shlosman and A. Van Moffaert (2000) Percolation, Path Large Deviations and Weak Gibbsianity , Comm. Math. Phys. 209, 517-545.

We present a unified approach to establishing the Gibbsian character of a wide class of non-Gibbsian states, arising in the Renormalisation Group theory. Within the realm of the Pirogov-Sinai theory for lattice spin systems, we prove that RG transformations applied to low temperature phases give rise to weakly Gibbsian measures. In other words, we show that the Griffiths-Pearce-Israel scenario of RG pathologies is carried by atypical configurations. The renormalized measures are described by an effective Gibbsian interaction, with relative energies well-defined on a full measure set of configurations. In this way we complete the first part of the Dobrushin Restoration Program: to give a Gibbsian description to non-Gibbsian states. A disagreement percolation estimate is used in the proof to bound the decay of quenched correlations through which the interaction potential is constructed. The percolation is controlled via a novel type of pathwise large deviation theory.

• A.C.D. van Enter, C. Maes and S.B. Shlosman (2000) Dobrushin'sprogram: Gibbs, weakly and almost Gibbs random field , Amer. Math. Soc. Transl. 198, 59-70. In: On Dobrushin's way.  From probability to statistical mechanics , Eds. R. Minlos, Yu. Suhov, S. Shlosman.

• A.C.D. van Enter, C. Maes, R.H. Schonmann and S.B. Shlosman (2000) The Griffiths Singularity Random Field  , Amer. Math. Soc. Transl. 198, 51-58. In: On Dobrushin's way.  From probability to statistical mechanics , Eds. R. Minlos, Yu. Suhov, S. Shlosman.

•  C. Maes, F. Redig, E. Saada and A. Van Moffaert (2000) On the thermodynamic limit for a one-dimensional sandpile process , Markov Processes and  Related Fields 6 , 1-21.

    Considering the standard abelian sandpile model in one dimension,
    we construct an infinite volume Markov process corresponding to its
    thermodynamic (infinite volume) limit. The main difficulty we overcome is
    the strong non-locality of the dynamics. However, using similar ideas as in
    recent extensions of the standard Gibbs formalism for lattice spin systems,
    we can identify a set of `good' configurations on which the dynamics is
    effectively local. We prove that every configuration converges in a finite time
    to the unique invariant measure.

• C. Maes C. and S. Shlosman (1999)  Freezing Transition in the Ising Model without Internal Contours , Prob. Th. Rel. Fields 115, 479-503.

    We consider the low temperature Ising model in a uniform magnetic field h>0 with minus boundary conditions
    and  conditioned on   having no internal contours. This simple contour model defines a
    non-Gibbsian spin state. For large enough magnetic fields (h>h_c) this state is concentrated on the single
    spin configuration of all spins up.  For  smaller values (h < h_c, h=h_c)  the spin state is non-trivial.
    At the critical pointh_c > 0 the magnetization jumps discontinuously.  Freezing also provides an
    example of a translationinvariant weakly Gibbsian state which is not almost Gibbs.

• C. Maes (1999)  The Fluctuation Theorem as a Gibbs Property , J. Stat. Phys. 95, 367-392.

    Common ground to recent studies exploiting relations between
    dynamical systems and non-equilibrium statistical mechanics is, so we argue,
    the standard Gibbs formalism applied on the level of space-time histories.
    The assumptions (chaoticity principle) underlying the Gallavotti-Cohen
    fluctuation theorem make it possible, using symbolic dynamics, to employ the
    theory of one-dimensional lattice spin systems. The Kurchan and
    Lebowitz-Spohn analysis of this fluctuation theorem for stochastic dynamics
    can be restated on the level of the space-time measure which is a Gibbs
    measure for an interaction determined by the transition probabilities. In this
    note we understand the fluctuation theorem as a Gibbs property as it follows
    from the very definition of Gibbs state. We give a local version of the
    fluctuation theorem in the Gibbsian context and we derive from this a version
    also for some class of spatially extended stochastic dynamics.

• C. Maes and G. Gielis (1999)  A simple approach to time-inhomogeneous dynamics , J. Phys. A: Math. Gen.,

32, 5389-5407.

   We study  time--inhomogeneous stochastic dynamical systems with vanishing noise. Conditions implying
   strong  ergodicity are proposed; the asymptotic probability distribution is
   characterized and we estimate the    speed  of relaxation. Applying these results to  simulated annealing enables us
   to  introduce systems that converge faster than  a classical annealing dynamics determined by a Gibbs factor.

• C. Maes, F. Redig and  A. Van Moffaert (1999) The restriction of the Ising model to a layer , J. Stat. Phys. 96 , 69-107.

    We discuss the status of recent Gibbsian descriptions of the
    restriction of the Ising phases to a layer. We concentrate on the projection of
    the two-dimensional low temperature Ising phases for which we prove a
    variational principle.

• C. Maes, F. Redig and A. Van Moffaert (1999) Almost Gibbsian versus Weakly Gibbsian measures , Stoch. Proc. Appl. 79 , 1-15.

We consider various extensions of the standard definition of Gibbs states for lattice spin systems. When a random field has conditional distributions which are almost surely continuous (almost Gibbsian field), then there is a potential for that field which is almost surely summable (weakly Gibbsian field). This generalizes the standard Kozlov-Sullivan theorems. The converse is not true in general. We give (counter)examples illustrating the relation between topological and measure-theoretic aspects of generalized Gibbs definitions.

• G. Gielis, C. Maes and K. Vande Velde (1999) Randomly interacting particle systems: the uniqueness regime ,Ann. H. Poincare, Physique Theorique 70 , 445-472 .

• K. S. Alexander, F. Cesi, L. Chayes, C. Maes and F.  Martinelli (1998) Convergence to Equilibrium of Random Ising Models in the Griffiths' Phase , J. Stat. Phys. 92, 337-351.

We consider Glauber--type dynamics for disordered Ising spin systems with nearest neighbor pair interactions in the Griffiths' phase. We prove that in a nontrivial portion of the Griffiths' phase the system has exponentially decaying correlations of distant functions with probability exponentially close to 1. This condition has, in turn, been shown to imply that the convergence to equilibrium is faster than any stretched exponential, and that the average over the disorder of the time-autocorrelation function, goes to equilibrium faster than exp[- k (log t)^{d/(d-1)}]. We then show that for the diluted Ising model these upper bounds are optimal.

• C. Maes, F. Redig and  A. Van Moffaert (1998) Potentials for One-Dimensional Restrictions of Gibbs Measures p.485-500. In: Mathematical Results in Statistical Mechanics, Eds. S. Miracle-Sole, J.Ruiz, V. Zagrebnov, World Scientific (1999). Proceedings of the Marseille Conference, July 27-31, 1998.

( BACK TO HOME PAGE )

For Brian: monster-paper