RECENT PAPERS (since 2008) (Christian Maes): ( BACK TO HOME PAGE )

. Urna Basu, Matthias Kruger, Alexandre Lazarescu and Christian Maes, Frenetic aspects of second order response. arXiv:1410.7450

. Christian Maes and Stefano Steffenoni, Friction and noise for a probe in a nonequilibrium fluid . arXiv:1410.5933

We investigate the fluctuation dynamics of a probe around a deterministic motion induced by interactions with driven particles. The latter constitute the nonequilibrium medium in which the probe is immersed and is modelled as overdamped Langevin particle dynamics driven by nonconservative forces. The expansion that yields the friction and noise expressions for the reduced probe dynamics is based on linear response around a time-dependent nonequilibrium condition of the medium. The result contains an extension of the second fluctuationâ€“dissipation relation between friction and noise for probe motion in a nonequilibrium fluid.

. Christian Maes and Karel Netocny, Revisiting the Glansdorff-Prigogine criterion for stability within irreversible thermodynamics. . arXiv:1410.2183

Glansdorff and Prigogine (1970) proposed a decomposition of the entropy production rate, which today is mostly known for Markov processes as the Hatano-Sasa approach. Their context was irreversible thermodynamics which, while ignoring fluctuations, still allows a somewhat broader treatment than the one based on the Master or Fokker-Planck equation. Glansdorff and Prigogine were the first to introduce a notion of excess entropy production rate dEP and they suggested as sufficient stability criterion for a nonequilibrium macroscopic condition that dEP be positive. We find for nonlinear diffusions that their excess entropy production rate is itself the time-derivative of a local free energy which is the close-to-equilibrium functional governing macroscopic fluctuations. The positivity of the excess dEP, for which we state a simple sufficient condition, is therefore equivalent with the monotonicity in time of that functional in the relaxation to steady nonequilibrium. There also appears a relation with recent extensions of the Clausius heat theorem close-to-equilibrium. The positivity of Î´EP immediately implies a Clausius (in)equality for the excess heat. A final and related question concerns the operational meaning of fluctuation functionals, nonequilibrium free energies, and how they make their entrÃ©e in irreversible thermodynamics.

. Marco Baiesi, Urna Basu and Christian Maes, Thermal response in driven diffusive systems. arXiv:1409.3369

Evaluating the linear response of a driven system to a change in environment temperature(s) is essential for understanding thermal properties of nonequilibrium systems. The system is kept in weak contact with possibly different fast relaxing mechanical, chemical or thermal equilibrium reservoirs. Modifying one of the temperatures creates both entropy fluxes and changes in dynamical activity. That is not unlike mechanical response of nonequilibrium systems but the extra difficulty for perturbation theory via path-integration is that for a Langevin dynamics temperature also affects the noise amplitude and not only the drift part. Using a discrete-time mesh adapted to the numerical integration one avoids that ultraviolet problem and we arrive at a fluctuation expression for its thermal susceptibility. The algorithm appears stable under taking even finer resolution.

.
Urna Basu and Christian Maes, Mobility transition in a dynamic environment, arXiv:1402.5253v1 [cond-mat.stat-mech], J. Phys. A: Math. Theor. **47**, 255003 (2014).

Depending on how the dynamical activity of a particle in a random environment is influenced by an external field E, its differential mobility at intermediate E can turn negative. We discuss the case where for slowly changing random environment the driven particle shows negative differential mobility while that mobility turns positive for faster environment changes. We illustrate this transition using a 2D-lattice Lorentz model where a particle moves in a background of simple exclusion walkers. The effective escape rate of the particle (or minus its collision frequency) which is essential for its mobility-behavior depends both on E and on the kinetic rate Î³ of the exclusion walkers. Large Î³, i.e., fast obstacle motion, amounts to merely rescaling the particle's free motion with the obstacle density, while slow obstacle dynamics results in particle motion that is more singularly related to its free motion and preserves the negative differential mobility already seen at Î³ = 0. In more general terms that we also illustrate using one-dimensional random walkers, the mobility transition is between the time-scales of the quasi-stationary regime and that of the fluid limit.

· Christian Maes, Nonequilibrium physics aspects of probabilistic cellular automata. Contribution to educational book on Probabilistic Cellular Automata, eds. Wioletta Ruszel and Emilio Cirillo

Probabilistic cellular automata (PCA) are used to model a variety of discrete spatially extended systems undergoing parallel-updating. We propose an embedding of a number of classical nonequilibrium concepts in the PCA-world. We start from time-symmetric PCA, satisfying detailed balance, and we give their Kubo formula for linear response. Close-to-detailed balance we investigate the form of the McLennan distribution and the minimum entropy production principle. More generally, when time-symmetry is broken in the stationary process, there is a fluctuation symmetry for a corresponding entropy flux. For linear response around nonequilibria we also give the linear response which is now not only entropic in nature.

·
Christian Maes, Revisiting the Glansdorff-Prigogine
criterion for stability within irreversible thermodynamics, Talk given at
the Solvay Workshop (ULBruxelles, Belgium) on the *Thermodynamics of Small Systems* 2-4
December 2013.

Glansdorff and Prigogine
proposed a decomposition of the entropy production rate, which has more
recently been rediscovered in various constructions of nonequilibrium
statistical mechanics. Their context was
irreversible thermodynamics which, while ignoring fluctuations, still allows a
somewhat broader treatment than the one based on the Master or Fokker-Planck equation. Glansdorff and Prigogine were the first to introduce a
notion of excess entropy production rate δ^{2}EP and
they suggested as sufficient stability criterion for a nonequilibrium macroscopic condition that δ^{2}EP
be positive. In joint work with Karel
Netočnı we find that their excess entropy
production rate is itself the time-derivative of a deformed free energy
functional. The positivity of the excess
production rate δ^{2}EP, for which we state a simple
sufficient condition, is therefore equivalent with the monotonicity in time of
that functional in the relaxation to steady nonequilibrium.

There also appears a relation with recent extensions of
the Clausius heat theorem close-to-equilibrium. The
positivity of δ^{2}EP
immediately implies a Clausius (in)equality for the
excess heat. We have proposed a nonperturbative version using a modified excess entropy
production that we also review.

A final and related question concerns the operational meaning of fluctuation functionals, nonequilibrium free energies, and how they make their entrée in irreversible thermodynamics.

·
Urna Basu and Christian Maes, Mobility
transition in a dynamic environment, arXiv:1402.5253v1
[cond-mat.stat-mech] accepted for publication in J.
Phys. A.

Depending on how the dynamical activity of a particle in a random environment is influenced by an external field E, its differential mobility at intermediate E can turn negative. We discuss the case where for slowly changing random environment the driven particle shows negative differential mobility while that mobility turns positive for faster environment changes. We illustrate this transition using a 2D-lattice Lorentz model where a particle moves in a background of simple exclusion walkers. The effective escape rate of the particle (or minus its collision frequency) which is essential for its mobility-behavior depends both on E and on the kinetic rate γ of the exclusion walkers. Large γ, i.e., fast obstacle motion, amounts to merely rescaling the particle's free motion with the obstacle density, while slow obstacle dynamics results in particle motion that is more singularly related to its free motion and preserves the negative differential mobility already seen at γ = 0. In more general terms that we also illustrate using one-dimensional random walkers, the mobility transition is between the time-scales of the quasi-stationary regime and that of the fluid limit.

· Wojciech De Roeck, Christian Maes, Karel Netočnı and Marius Schütz, Locality and nonlocality of classical restrictions of quantum spin systems with applications to quantum large deviations and entanglement. arXiv:1312.4782v1 [math-ph].

We study the projection on classical spins starting from quantum equilibria. We show Gibbsianness or quasi-locality of the resulting classical spin system for a class of gapped quantum systems at low temperatures including quantum ground states. A consequence of Gibbsianness is the validity of a large deviation principle in the quantum system which is known and here recovered in regimes of high temperature or for thermal states in one dimension. On the other hand we give an example of a quantum ground state with strong nonlocality in the classical restriction, giving rise to what we call measurement induced entanglement, and still satisfying a large deviation principle.

·
Christian Maes,
Karel Netočnı and Winny O'Kelly de
Galway, Low temperature behavior of nonequilibrium multilevel systems. arXiv:1309.6184v1 [cond-mat.stat-mech], J. Phys. A.: Math Gen. **47**, 035002 (2014)

We give a low temperature formula for the stationary occupations in Markovian systems away from detailed balance. Two applications are discussed, one to determine the direction of the ratchet current and one on population inversion. Both can take advantage of low temperature to improve the gain and typical nonequilibrium features. The new formula brings to the foreground the importance of kinetic aspects in terms of reactivities for deciding the levels with highest occupation and thus gives a detailed quantitative meaning to Landauer's blowtorch theorem at low temperature.

·
Christian Maes, On the Second Fluctuation-Dissipation Theorem for Nonequilibrium Baths. arXiv:1309.3160v1
[cond-mat.stat-mech], J. Stat. Phys. **154**, 705-722 (2014).

Baths produce friction and random forcing on particles suspended in them. The relation between noise and friction in (generalized) Langevin equations is usually referred to as the second fluctuation-dissipation theorem. We show what is the proper nonequilibrium extension, to be applied when the environment is itself active and driven. In particular we determine the effective Langevin dynamics of a probe from integrating out a steady nonequilibrium environment. The friction kernel picks up a frenetic contribution, i.e., involving the environment's dynamical activity, responsible for the breaking of the standard Einstein relation.

·
Pieter Baerts, Urna Basu, Christian Maes and Soghra Safaverdi, The frenetic origin of
negative differential response, arXiv:1308.5613v2 [cond-mat.stat-mech] Phys. Rev. E **88**,
052109 (2013).

The Green-Kubo formula for linear response coefficients gets modified when dealing with nonequilibrium dynamics. In particular negative differential conductivities are allowed to exist away from equilibrium. We give a unifying framework for such negative differential response in terms of the frenetic contribution in the nonequilibrium formula. It corresponds to a negative dependence of the escape rates and reactivities on the driving forces. Partial caging in state space and reduction of dynamical activity with increased driving cause the current to drop. These are time-symmetric kinetic effects that are believed to play a major role in the study of nonequilibria. We give various simple examples treating particle and energy transport, which all follow the same pattern in the dependence of the dynamical activity on the nonequilibrium driving, made visible from recently derived nonequilibrium response theory.

·
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], J.
Stat. Phys.**153**, 991–1007 (2013).

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 *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 Netočnı, Minimum entropy production principle, Scholarpedia, 8(7):9664 (2013).

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:1305.4157v2
[cond-mat.stat-mech], Chaos, Solitons & Fractals **64**, 78-87 (2014).

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. New
Journal of Physics* ***16, **015019 (2014). arXiv:1305.0736.

In contrast with the
understanding of
fluctuation symmetries for
entropy production, similar ideas applied to the time-symmetric fluctuation
sector have been less explored. Here we
give detailed derivations of time-symmetric fluctuation symmetries in boundary
driven particle systems such as the open Kawasaki lattice gas and the zero
range model. As a measure of time-symmetric dynamical
activity we take the difference (N_{1} – N_{L})/T in the
number of particles entering or leaving the system at the left versus the right
edge of the system over time T. We show that this quantity satisfies a
fluctuation symmetry from which we derive a new Green-Kubo type relation. It will follow then that the system is more
active at the edge connected to the particle reservoir with the largest
chemical potential. We also apply these exact relations derived for stochastic
particle models to a deterministic case, the spinning Lorentz gas, where the
symmetry relation for the activity is checked numerically.

·
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.

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 Netočnı, 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 Netočnı, A nonequilibrium extension
of the Clausius heat theorem. J. Stat. Phys. **154,** 188-203 (2014).

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 Physics15,013004 (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], Europhysics
Letters **102**, 50005 (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 Netočnı 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. Netočnı
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. Netočnı
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 Netočnı 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 Netočnı: 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.

·
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. Netočnı 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. Netočnı 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. Netočnı 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. Netočnı: 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

- M. Baiesi, C. Maes and K. Netočnı: 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. Netočnı 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. Netočnı 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.
__,__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. Netočnı: 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.

- 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.