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Chabane, L., Lazarescu, A. and Verley, G. Effective Hamiltonians and Lagrangians for Conditioned Markov Processes at Large Volume 2022 Journal of Statistical Physics
Vol. 187(1) 
article DOI URL 
Abstract: When analysing statistical systems or stochastic processes, it is often interesting to ask how they behave given that some observable takes some prescribed value. This conditioning problem is well understood within the linear operator formalism based on rate matrices or Fokker-Planck operators, which describes the dynamics of many independent random walkers. Relying on certain spectral properties of the biased linear operators, guaranteed by the Perron-Frobenius theorem, an effective process can be found such that its path probability is equivalent to the conditional path probability. In this paper, we extend those results for nonlinear Markov processes that appear when the many random walkers are no longer independent, and which can be described naturally through a Lagrangian-Hamiltonian formalism within the theory of large deviations at large volume. We identify the appropriate spectral problem as being a Hamilton-Jacobi equation for a biased Hamiltonian, for which we conjecture that two special global solutions exist, replacing the Perron-Frobenius theorem concerning the positivity of the (left and right) dominant eigenvector. We then devise a rectification procedure based on a canonical gauge transformation of the biased Hamiltonian, yielding an effective dynamics in agreement with the original conditioning. Along the way, we present simple examples in support of our conjecture, we examine its consequences on important physical objects such as the fluctuation symmetries of the biased and rectified processes as well as the dual dynamics obtained through time-reversal. We apply all those results to simple independent and interacting models, including a stochastic chemical reaction network and a population process called the Brownian Donkey.
BibTeX:
@article{Chabane_2022,
  author = {Lydia Chabane and Alexandre Lazarescu and Gatien Verley},
  title = {Effective Hamiltonians and Lagrangians for Conditioned Markov Processes at Large Volume},
  journal = {Journal of Statistical Physics},
  publisher = {Springer Science and Business Media LLC},
  year = {2022},
  volume = {187},
  number = {1},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Chabane_2022.pdf},
  doi = {https://doi.org/10.1007/s10955-022-02898-3}
}
Verley, G. Dynamical equivalence classes for Markov jump processes 2022 Journal of Statistical Mechanics: Theory and Experiment
Vol. 2022(2), pp. 023211 
article DOI URL 
Abstract: Two different Markov jump processes driven out of equilibrium by constant thermodynamic forces may have identical current fluctuations in the stationary state. The concept of dynamical equivalence classes emerges from this statement as proposed by Andrieux for discrete-time Markov chains on simple graphs. We define dynamical equivalence classes in the context of continuous-time Markov chains on multigraphs using the symmetric part of the rate matrices that define the dynamics. The freedom on the skew-symmetric part is at the core of the freedom inside a dynamical equivalence class. It arises from different splittings of the thermodynamic forces onto the system's transitions.
BibTeX:
@article{Verley2022vol2022,
  author = {Gatien Verley},
  title = {Dynamical equivalence classes for Markov jump processes},
  journal = {Journal of Statistical Mechanics: Theory and Experiment},
  publisher = {IOP Publishing},
  year = {2022},
  volume = {2022},
  number = {2},
  pages = {023211},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Verley2022vol2022.pdf},
  doi = {https://doi.org/10.1088/1742-5468/ac4981}
}
Chabane, L., Chétrite, R. and Verley, G. Periodically driven jump processes conditioned on large deviations 2020 Journal of Statistical Mechanics: Theory and Experiment
Vol. 2020(3), pp. 033208 
article DOI URL 
Abstract: We study the fluctuations of systems modeled by time periodically driven Markov jump processes. We focus on observables defined through time-periodic functions of the system's states or transitions. Using large deviation theory, canonical biasing and Doob transform, we characterize the asymptotic fluctuations of such observables after a large number of periods by obtaining the Markov process that produces them. We show that this process, called driven process, is the optimizer under constraint of the large deviation function for occupation and jumps.
BibTeX:
@article{Chabane_2020,
  author = {Lydia Chabane and Raphaël Chétrite and Gatien Verley},
  title = {Periodically driven jump processes conditioned on large deviations},
  journal = {Journal of Statistical Mechanics: Theory and Experiment},
  publisher = {IOP Publishing},
  year = {2020},
  volume = {2020},
  number = {3},
  pages = {033208},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Chabane_2020vol.pdf},
  doi = {https://doi.org/10.1088/1742-5468/ab74c4}
}
Vroylandt, H., Esposito, M. and Verley, G. Efficiency Fluctuations of Stochastic Machines Undergoing a Phase Transition 2020 Phys. Rev. Lett.
Vol. 124(25) 
article DOI URL 
Abstract: We study the efficiency fluctuations of a stochastic heat engine made of N interacting unicyclic machines and undergoing a phase transition in the macroscopic limit. Depending on N and on the observation time, the machine can explore its whole phase space or not. This affects the engine efficiency that either strongly fluctuates on a large interval of equiprobable efficiencies (ergodic case) or fluctuates close to several most likely values (nonergodic case). We also provide a proof that despite the phase transition, the decay rate of the efficiency distribution at the reversible efficiency remains the largest one although other efficiencies can now decay equally fast.
BibTeX:
@article{Vroylandt2020vol124,
  author = {Hadrien Vroylandt and Massimiliano Esposito and Gatien Verley},
  title = {Efficiency Fluctuations of Stochastic Machines Undergoing a Phase Transition},
  journal = {Phys. Rev. Lett.},
  publisher = {American Physical Society (APS)},
  year = {2020},
  volume = {124},
  number = {25},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Vroylandt2020vol124.pdf},
  doi = {https://doi.org/10.1103/physrevlett.124.250603}
}
Vroylandt, H., Lacoste, D. and Verley, G. An ordered set of power-efficiency trade-offs 2019 J. Stat. Mech: Theory Exp.
Vol. 2019(5), pp. 054002 
article DOI URL 
Abstract: In this paper, we derive a number of inequalities which express power-efficiency trade-offs that hold generally for thermodynamic machines operating in non-equilibrium stationary states. One of these inequalities concerns the output power, which is bounded by a quadratic function of the thermodynamic efficiency multiplied by a factor. Different factors can be obtained according to the level of knowledge one has about the underlying dynamics of the machine, they can depend for instance on the covariance of the input flux, the dynamical activity, or the non-equilibrium conductance.
BibTeX:
@article{Vroylandt2019_vol2019,
  author = {Hadrien Vroylandt and David Lacoste and Gatien Verley},
  title = {An ordered set of power-efficiency trade-offs},
  journal = {J. Stat. Mech: Theory Exp.},
  publisher = {IOP Publishing},
  year = {2019},
  volume = {2019},
  number = {5},
  pages = {054002},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Vroylandt2019_vol2019.pdf},
  doi = {https://doi.org/10.1088/1742-5468/ab14d7}
}
Vroylandt, H. and Verley, G. Non-equivalence of Dynamical Ensembles and Emergent Non-ergodicity 2019 J. Stat. Phys.
Vol. 174(2), pp. 404-432 
article DOI URL 
Abstract: Dynamical ensembles have been introduced to study constrained stochastic processes. In the microcanonical ensemble, the value of a dynamical observable is constrained to a given value. In the canonical ensemble a bias is introduced in the process to move the mean value of this observable. The equivalence between the two ensembles means that calculations in one or the other ensemble lead to the same result. In this paper, we study the physical conditions associated with ensemble equivalence and the consequences of non-equivalence. For continuous time Markov jump processes, we show that ergodicity guarantees ensemble equivalence. For non-ergodic systems or systems with emergent ergodicity breaking, we adapt a method developed for equilibrium ensembles to compute asymptotic probabilities while caring about the initial condition. We illustrate our results on the infinite range Ising model by characterizing the fluctuations of magnetization and activity. We discuss the emergence of non-ergodicity by showing that the initial condition can only be forgotten after a time that scales exponentially with the number of spins.
BibTeX:
@article{Vroylandt2019_vol174,
  author = {Vroylandt, Hadrien and Verley, Gatien},
  title = {Non-equivalence of Dynamical Ensembles and Emergent Non-ergodicity},
  journal = {J. Stat. Phys.},
  year = {2019},
  volume = {174},
  number = {2},
  pages = {404--432},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Vroylandt2018_Vol174.pdf},
  doi = {https://doi.org/10.1007/s10955-018-2186-7}
}
Vroylandt, H., Lacoste, D. and Verley, G. Degree of coupling and efficiency of energy converters far-from-equilibrium 2018 J. Stat. Mech: Theory Exp.  article DOI  
Abstract: In this paper, we introduce a real symmetric and positive semi-definite matrix, which we call the non-equilibrium conductance matrix, and which generalizes the Onsager response matrix for a system in a non-equilibrium stationary state. We then express the thermodynamic efficiency in terms of the coefficients of this matrix using a parametrization similar to the one used near equilibrium. This framework, then valid arbitrarily far from equilibrium allows to set bounds on the thermodynamic efficiency by a universal function depending only on the degree of coupling between input and output currents. It also leads to new general power-efficiency trade-offs valid for macroscopic machines that are compared to trade-offs previously obtained from uncertainty relations. We illustrate our results on an unicycle heat to heat converter and on a discrete model of a molecular motor.
BibTeX:
@article{Vroylandt2018vol2018,
  author = {Hadrien Vroylandt and David Lacoste and Gatien Verley},
  title = {Degree of coupling and efficiency of energy converters far-from-equilibrium},
  journal = {J. Stat. Mech: Theory Exp.},
  year = {2018},
  doi = {https://doi.org/10.1088/1742-5468/aaa8fe}
}
Vroylandt, H., Esposito, M. and Verley, G. Collective effects enhancing power and efficiency 2017 Europhys. Lett.
Vol. 120(3), pp. 30009 
article DOI URL 
Abstract: Energy conversion is most efficient for micro or nano machines with tight
coupling between input and output power. To reach meaningful amounts of
power, ensembles of N such machines must be considered. We use a model system
to demonstrate that interactions between N tightly coupled nanomachines can
enhance the power output per machine. Furthermore, while interactions break
tight coupling and thus lower efficiency in finite ensembles, the macroscopic
limit restores it and enhances both the efficiency and the output power per
nanomachine.
BibTeX:
@article{Vroylandt2017_vol120,
  author = {Hadrien Vroylandt and Massimiliano Esposito and Gatien Verley},
  title = {Collective effects enhancing power and efficiency},
  journal = {Europhys. Lett.},
  publisher = {IOP Publishing},
  year = {2017},
  volume = {120},
  number = {3},
  pages = {30009},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Vroylandt2017_vol120.pdf},
  doi = {https://doi.org/10.1209/0295-5075/120/30009}
}
Verley, G. Nonequilibrium thermodynamic potentials for continuous-time Markov chains 2016 Phys. Rev. E
Vol. 93, pp. 012111 
article DOI URL 
Abstract: We connect the rare fluctuations of an equilibrium (EQ) process and the
typical fluctuations of a nonequilibrium (NE) stationary process. In the
framework of large deviation theory, this observation allows us to introduce
NE thermodynamic potentials. For continuous-time Markov chains, we identify
the relevant pairs of conjugated variables and propose two NE ensembles: one
with fixed dynamics and fluctuating time-averaged variables, and another with
fixed time-averaged variables, but a fluctuating dynamics. Accordingly, we
show that NE processes are equivalent to conditioned EQ processes ensuring
that NE potentials are Legendre dual. We find a variational principle
satisfied by the NE potentials that reach their maximum in the NE stationary
state and whose first derivatives produce the NE equations of state and
second derivatives produce the NE Maxwell relations generalizing the Onsager
reciprocity relations.
BibTeX:
@article{Verley2016_vol93,
  author = {Verley, Gatien},
  title = {Nonequilibrium thermodynamic potentials for continuous-time Markov chains},
  journal = {Phys. Rev. E},
  publisher = {American Physical Society},
  year = {2016},
  volume = {93},
  pages = {012111},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Verley2016_vol93.pdf},
  doi = {https://doi.org/10.1103/PhysRevE.93.012111}
}
Vroylandt, H., Bonfils, A. and Verley, G. Efficiency fluctuations of small machines with unknown losses 2016 Phys. Rev. E
Vol. 93, pp. 052123 
article DOI URL 
Abstract: The efficiency statistics of a small thermodynamic machine has been recently
investigated assuming that the total dissipation is a linear combination of
two currents: the input and output currents. Here, we relax this standard
assumption and consider the question of the efficiency fluctuations for a
machine involving three different currents, first in full generality and
second for two different examples. Since the third current may not be
measurable and/or may decrease the machine efficiency, our motivation is to
study the effect of unknown losses in small machines.
BibTeX:
@article{Vroylandt2016_vol93,
  author = {Vroylandt, H. and Bonfils, A. and Verley, G.},
  title = {Efficiency fluctuations of small machines with unknown losses},
  journal = {Phys. Rev. E},
  publisher = {American Physical Society},
  year = {2016},
  volume = {93},
  pages = {052123},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Vroylandt2016_vol93.pdf},
  doi = {https://doi.org/10.1103/PhysRevE.93.052123}
}
Polettini, M., Verley, G. and Esposito, M. Efficiency Statistics at All Times: Carnot Limit at Finite Power 2015 Phys. Rev. Lett.
Vol. 114, pp. 050601 
article DOI URL 
Abstract: We derive the statistics of the efficiency under the assumption that
thermodynamic fluxes fluctuate with normal law, parametrizing it in terms of
time, macroscopic efficiency, and a coupling parameter ζ . It has a
peculiar behavior: no moments, one sub-, and one super-Carnot maxima
corresponding to reverse operating regimes (engine or pump), the most
probable efficiency decreasing in time. The limit ζ → 0
where the Carnot bound can be saturated gives rise to two extreme situations,
one where the machine works at its macroscopic efficiency, with Carnot limit
corresponding to no entropy production, and one where for a transient time
scaling like 1 / ζ microscopic fluctuations are enhanced in such a way
that the most probable efficiency approaches the Carnot limit at finite
entropy production.
BibTeX:
@article{Polettini2015_vol114,
  author = {Polettini, M. and Verley, G. and Esposito, M.},
  title = {Efficiency Statistics at All Times: Carnot Limit at Finite Power},
  journal = {Phys. Rev. Lett.},
  publisher = {American Physical Society},
  year = {2015},
  volume = {114},
  pages = {050601},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Polettini2015_vol114.pdf},
  doi = {https://doi.org/10.1103/PhysRevLett.114.050601}
}
Tusch, S., Kundu, A., Verley, G., Blondel, T., Miralles, V., Démoulin, D., Lacoste, D. and Baudry, J. Energy versus Information Based Estimations of Dissipation Using a Pair of Magnetic Colloidal Particles 2014 Phys. Rev. Lett.
Vol. 112, pp. 180604 
article DOI URL 
Abstract: Using the framework of stochastic thermodynamics, we present an experimental
study of a doublet of magnetic colloidal particles that is manipulated by a
time-dependent magnetic field. Because of hydrodynamic interactions, each
bead experiences a state-dependent friction, which we characterize using a
hydrodynamic model. In this work, we compare two estimates of the dissipation
in this system: the first one is energy based since it relies on the measured
interaction potential, while the second one is information based since it
uses only the information content of the trajectories. While the latter only
offers a lower bound of the former, we find it to be simple to implement and
of general applicability to more complex systems.
BibTeX:
@article{Tusch2014_vol112,
  author = {Tusch, S. and Kundu, A. and Verley, G. and Blondel, T. and Miralles, V. and Démoulin, D. and Lacoste, D. and Baudry, J.},
  title = {Energy versus Information Based Estimations of Dissipation Using a Pair of Magnetic Colloidal Particles},
  journal = {Phys. Rev. Lett.},
  publisher = {American Physical Society},
  year = {2014},
  volume = {112},
  pages = {180604},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Tusch2014_vol112.pdf},
  doi = {https://doi.org/10.1103/PhysRevLett.112.180604}
}
Verley, G., Van den Broeck, C. and Esposito, M. Work statistics in stochastically driven systems 2014 New J. Phys.
Vol. 16(9), pp. 095001 
article DOI URL 
Abstract: We identify the conditions under which a stochastic driving that induces
energy changes into a system coupled with a thermal bath can be treated as a
work source. When these conditions are met, the work statistics satisfy the
Crooks fluctuation theorem traditionally derived for deterministic drivings.
We illustrate this fact by calculating and comparing the work statistics for
a two-level system driven respectively by a stochastic and a deterministic
piecewise constant protocol.
BibTeX:
@article{Verley2014_vol16,
  author = {Verley, G. and Van den Broeck, C. and Esposito, M.},
  title = {Work statistics in stochastically driven systems},
  journal = {New J. Phys.},
  year = {2014},
  volume = {16},
  number = {9},
  pages = {095001},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Verley2014_vol16.pdf},
  doi = {https://doi.org/10.1088/1367-2630/16/9/095001}
}
Verley, G., Willaert, T., Van den Broeck, C. and Esposito, M. The unlikely Carnot efficiency 2014 Nat. Commun.
Vol. 5, pp. 4721 
article DOI URL 
Abstract: The efficiency of an heat engine is traditionally defined as the ratio of its
average output work over its average input heat. Its highest possible value
was discovered by Carnot in 1824 and is a cornerstone concept in
thermodynamics. It led to the discovery of the second law and to the
definition of the Kelvin temperature scale. Small-scale engines operate in
the presence of highly fluctuating input and output energy fluxes. They are
therefore much better characterized by fluctuating efficiencies. In this
study, using the fluctuation theorem, we identify universal features of
efficiency fluctuations. While the standard thermodynamic efficiency is, as
expected, the most likely value, we find that the Carnot efficiency is,
surprisingly, the least likely in the long time limit. Furthermore, the
probability distribution for the efficiency assumes a universal scaling form
when operating close-to-equilibrium. We illustrate our results analytically
and numerically on two model systems.
BibTeX:
@article{Verley2014_vol5,
  author = {Verley, G. and Willaert, T. and Van den Broeck, C. and Esposito, M.},
  title = {The unlikely Carnot efficiency},
  journal = {Nat. Commun.},
  year = {2014},
  volume = {5},
  pages = {4721},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Verley2014_vol5_Arxiv.pdf},
  doi = {https://doi.org/10.1038/ncomms5721}
}
Verley, G., Willaert, T., Van den Broeck, C. and Esposito, M. Universal theory of efficiency fluctuations 2014 Phys. Rev. E
Vol. 90, pp. 052145 
article DOI URL 
Abstract: Using the fluctuation theorem supplemented with geometric arguments, we
derive universal features of the (long-time) efficiency fluctuations for
thermal and isothermal machines operating under steady or periodic driving,
close or far from equilibrium. In particular, the probabilities for observing
the reversible efficiency and the least likely efficiency are identical to
those of the same machine working under the time-reversed driving. For
time-symmetric drivings, this reversible and the least probable efficiency
coincide.
BibTeX:
@article{Verley2014_vol90,
  author = {Verley, G. and Willaert, T. and Van den Broeck, C. and Esposito, M.},
  title = {Universal theory of efficiency fluctuations},
  journal = {Phys. Rev. E},
  publisher = {American Physical Society},
  year = {2014},
  volume = {90},
  pages = {052145},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Verley2014_vol90.pdf},
  doi = {https://doi.org/10.1103/PhysRevE.90.052145}
}
Verley, G., Van den Broeck, C. and Esposito, M. Modulated two-level system: Exact work statistics 2013 Phys. Rev. E
Vol. 88, pp. 032137 
article DOI URL 
Abstract: We consider an open two-level system driven by a piecewise constant periodic
field and described by a rate equation with Fermi, Bose, and Arrhenius rates,
respectively. We derive an analytical expression for the generating function
and large deviation function of the work performed by the field and show that
a work fluctuation theorem holds.
BibTeX:
@article{Verley2013_vol88,
  author = {Verley, G. and Van den Broeck, C. and Esposito, M.},
  title = {Modulated two-level system: Exact work statistics},
  journal = {Phys. Rev. E},
  publisher = {American Physical Society},
  year = {2013},
  volume = {88},
  pages = {032137},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Verley2013_vol88.pdf},
  doi = {https://doi.org/10.1103/PhysRevE.88.032137}
}
Verley, G. Fluctuations et réponse des systèmes hors d'équillibre 2012 School: Université Pierre et Marie Curie  phdthesis URL 
Abstract: In recent years, a broad number of works have shed light on the nature of
irreversibility of nonequilibrium phenomena, have permitted to redefine
quantities such as work or entropy production at the level of trajectories,
and to enlighten their symmetries through the fluctuation theorems. During
this thesis, we focused on these results in a purely classical and Markovian
framework. We have unified the various formulations of the generalized
fluctuation dissipation theorem around steady and unsteady nonequilibrium
state by identifying the various sources of irreversibility and by making
them appear in the system response. We have tried to link the fluctuation
theorems with the theory of linear response which was possible using a
generalization of the relation of Hatano and Sasa. In doing so, we found that
different functionals verified symmetry properties similar to the fluctuation
theorem for total entropy production. The study of the generalization of
Hatano and Sasa relation led us to an inequality of the same type as the
Clausius inequality. This new inequality gives a lower bound for the system
entropy production during transition between non-stationary states, as the
Clausius inequality do for transitions between equilibrium states. We have
illustrated these ideas on a few simple cases analytical or numerical and in
an experiment where we manipulate magnetic colloids. In these experiments, a
Crooks relation is expected for a system with space dependent diffusion
coefficient, allowing to consider new tests of stochastic thermodynamics.
BibTeX:
@phdthesis{Verley2012_vol,
  author = {Verley, G.},
  title = {Fluctuations et réponse des systèmes hors d'équillibre},
  school = {Université Pierre et Marie Curie},
  year = {2012},
  url = {http://tel.archives-ouvertes.fr/tel-00748590}
}
Verley, G., Chétrite, R. and Lacoste, D. Inequalities Generalizing the Second Law of Thermodynamics for Transitions between Non-stationary States 2012 Phys. Rev. Lett.
Vol. 108, pp. 120601 
article DOI URL 
Abstract: We discuss the consequences of a variant of the Hatano-Sasa relation in which
a nonstationary distribution is used in place of the usual stationary one. We
first show that this nonstationary distribution is related to a difference of
traffic between the direct and dual dynamics. With this formalism, we extend
the definition of the adiabatic and nonadiabatic entropies introduced by M.
Esposito and C. Van den Broeck in Phys. Rev. Lett. 104, 090601 (2010) for the
stationary case. We also obtain interesting second- law-like inequalities for
transitions between nonstationary states.
BibTeX:
@article{Verley2012_vol108,
  author = {Verley, G. and Chétrite, R. and Lacoste, D.},
  title = {Inequalities Generalizing the Second Law of Thermodynamics for Transitions between Non-stationary States},
  journal = {Phys. Rev. Lett.},
  publisher = {American Physical Society},
  year = {2012},
  volume = {108},
  pages = {120601},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Verley2012_vol108.pdf},
  doi = {https://doi.org/10.1103/PhysRevLett.108.120601}
}
Verley, G. and Lacoste, D. Fluctuation theorems and inequalities generalizing the second law of thermodynamics out of equilibrium 2012 Phys. Rev. E
Vol. 86, pp. 051127 
article DOI URL 
Abstract: We present a general framework for systems which are prepared in a
nonstationary nonequilibrium state in the absence of any perturbation and
which are then further driven through the application of a time-dependent
perturbation. By assumption, the evolution of the system must be described by
Markovian dynamics. We distinguish two different situations depending on the
way the nonequilibrium state is prepared; either it is created by some
driving or it results from a relaxation following some initial nonstationary
conditions. Our approach is based on a recent generalization of the
Hatano-Sasa relation for nonstationary probability distributions. We also
investigate whether a form of the second law holds for separate parts of the
entropy production and for any nonstationary reference process, a question
motivated by the work of M. Esposito et al. [Phys. Rev. Lett. 104, 090601
(2010)]. We find that although the special structure of the theorems derived
in this reference is not recovered in the general case, detailed fluctuation
theorems still hold separately for parts of the entropy production. These
detailed fluctuation theorems contain interesting generalizations of the
second law of thermodynamics for nonequilibrium systems.
BibTeX:
@article{Verley2012_vol86a,
  author = {Verley, G. and Lacoste, D.},
  title = {Fluctuation theorems and inequalities generalizing the second law of thermodynamics out of equilibrium},
  journal = {Phys. Rev. E},
  publisher = {American Physical Society},
  year = {2012},
  volume = {86},
  pages = {051127},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Verley2012_vol86a.pdf},
  doi = {https://doi.org/10.1103/PhysRevE.86.051127}
}
Verley, G. and Lacoste, D. Fluctuations and response from a Hatano and Sasa approach 2012 Phys. Scr.
Vol. 86, pp. 058505 
article DOI URL 
Abstract: In this commentary paper, we present some of the main ideas on deriving a
modified fluctuation-dissipation theorem off equilibrium, which in the end
can all be related to an approach based on a generalized Hatano?Sasa
relation. This generalized Hatano?Sasa relation also contains an interesting
inequality, which can be viewed as a generalization of the second law of
thermodynamics to transitions between non-equilibrium states.
BibTeX:
@article{Verley2012_vol86,
  author = {Verley, G. and Lacoste, D.},
  title = {Fluctuations and response from a Hatano and Sasa approach},
  journal = {Phys. Scr.},
  year = {2012},
  volume = {86},
  pages = {058505},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Verley2012_vol86.pdf},
  doi = {https://doi.org/10.1088/0031-8949/86/05/058505}
}
Verley, G., Chétrite, R. and Lacoste, D. Modified fluctuation-dissipation theorem for general non-stationary states and application to the Glauber-Ising chain 2011 J. Stat. Mech: Theory Exp.
Vol. 10, pp. P10025 
article DOI URL 
Abstract: In this paper, we present a general derivation of a modified
fluctuation-dissipation theorem (MFDT) valid near an arbitrary non-stationary
state for a system obeying Markovian dynamics. We show that the method for
deriving modified fluctuation-dissipation theorems near non-equilibrium
stationary states used by Prost et al (2009 Phys. Rev. Lett. 103 090601) is
generalizable to non-stationary states. This result follows from both
standard linear response theory and from a transient fluctuation theorem,
analogous to the Hatano?Sasa relation. We show that this modified
fluctuation-dissipation theorem can be interpreted at the trajectory level
using the notion of stochastic trajectory entropy, in a way which is similar
to what has been done recently in the case of the MFDT near non-equilibrium
steady states (NESS). We illustrate this framework with two solvable
examples: the first example corresponds to a Brownian particle in a harmonic
trap subjected to a quench of temperature and to a time-dependent stiffness;
the second example is a classic model of coarsening systems, namely the 1D
Ising model with Glauber dynamics.
BibTeX:
@article{Verley2011_vol10,
  author = {Verley, G. and Chétrite, R. and Lacoste, D.},
  title = {Modified fluctuation-dissipation theorem for general non-stationary states and application to the Glauber-Ising chain},
  journal = {J. Stat. Mech: Theory Exp.},
  year = {2011},
  volume = {10},
  pages = {P10025},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Verley2011_vol10.pdf},
  doi = {https://doi.org/10.1088/1742-5468/2011/10/P10025}
}
Verley, G. and Lacoste, D. Fluctuation relations and fluctuation-response for molecular motors 2011
Vol. 1332AIP Conf. Proc., pp. 247-248 
inproceedings DOI URL 
Abstract: Fluctuation relations are a set of remarkable relations obeyed by a large
class of systems and arbitrarily far from equilibrium. It is interesting to
discuss the implications of these relations for molecular motors, which are
chemically driven enzymes. These enzymes operate stochastically at the
molecular level and for these reasons undergo large thermal fluctuations.
Using simple ratchet models of molecular motors, the various forms of
fluctuation relations can be illustrated in a simple way. In the linear
regime, finite time fluctuation relations imply specific modified
fluctuation-dissipation relations.
BibTeX:
@inproceedings{Verley2011_vol1332,
  author = {Verley, G. and Lacoste, D.},
  title = {Fluctuation relations and fluctuation-response for molecular motors},
  booktitle = {AIP Conf. Proc.},
  year = {2011},
  volume = {1332},
  pages = {247--248},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Verley2011_vol1332.pdf},
  doi = {https://doi.org/10.1063/1.3569518}
}
Verley, G., Mallick, K. and Lacoste, D. Modified fluctuation-dissipation theorem for non-equilibrium steady states and applications to molecular motors 2011 Europhys. Lett.
Vol. 93, pp. 10002 
article DOI URL 
Abstract: We present a theoretical framework to understand a modified
fluctuation-dissipation theorem valid for systems close to non-equilibrium
steady states and obeying Markovian dynamics. We discuss the interpretation
of this result in terms of trajectory entropy excess. The framework is
illustrated on a simple pedagogical example of a molecular motor. We also
derive in this context generalized Green-Kubo relations similar to the ones
obtained recently in Seifert U., Phys. Rev. Lett., 104 (2010) 138101 for more
general networks of biomolecular states.
BibTeX:
@article{Verley2011_vol93,
  author = {Verley, G. and Mallick, K. and Lacoste, D.},
  title = {Modified fluctuation-dissipation theorem for non-equilibrium steady states and applications to molecular motors},
  journal = {Europhys. Lett.},
  year = {2011},
  volume = {93},
  pages = {10002},
  url = {https://pperso.ijclab.in2p3.fr/page_perso/Verley/Papers/Verley2011_vol93.pdf},
  doi = {https://doi.org/10.1209/0295-5075/93/10002}
}