·         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

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, Physical Review E 87, 022145 (2013).

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  t1/2  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 t3/2 .  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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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: 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.

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.

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.

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.

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.

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.

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.

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.

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.

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