[Formalisme thermodynamique et grandes déviations dans les systèmes à dynamique markovienne]
Le formalisme thermodynamique, qui a d'abord été développé dans le cadre des systèmes dynamiques puis appliqué aux processus de Markov, s'avère également pertinent pour les dynamiques de Markov en temps continu, à condition toutefois d'interpréter les définitions en jeu de façon appropriée. Ce formalisme peut être reformulé en termes de fonction génératrice d'une observable, puis étendu à d'autres observables. En particulier, l'observable K donnant le nombre d'événements ayant lieu dans un intervalle de temps donné, bien que très simple, contient déjà la signature de transitions de phases dynamiques.
Pour les modèles de champ moyen à l'équilibre, et dans la limite des grands systèmes, le formalisme peut s'appliquer simplement et montre comment les transitions de phase thermodynamiques peuvent affecter les propriétés dynamiques de ces systèmes. Cela est illustré sur le cas du modèle de Potts en champ moyen, et il s'avère que le cas d'Ising diffère qualitativement des autres cas.
The thermodynamic formalism, which was first developed for dynamical systems and then applied to discrete Markov processes, turns out to be well suited for continuous time Markov processes as well, provided the definitions are interpreted in an appropriate way. Moreover, it can be reformulated in terms of the generating function of an observable, and then extended to other observables. In particular, the simple observable K giving the number of events occurring over a given time interval turns out to contain already the signature of dynamical phase transitions.
For mean-field models in equilibrium, and in the limit of large systems, the formalism is rather simple to apply and shows how thermodynamic phase transitions may modify the dynamical properties of the systems. This is exemplified with the q-state mean-field Potts model, for which the Ising limit
Publié le :
Mots-clés : Formalisme thermodynamique, Grandes déviations, Chaos
Vivien Lecomte 1 ; Cécile Appert-Rolland 2 ; Frédéric van Wijland 1
@article{CRPHYS_2007__8_5-6_609_0,
author = {Vivien Lecomte and C\'ecile Appert-Rolland and Fr\'ed\'eric van Wijland},
title = {Thermodynamic formalism and large deviation functions in continuous time {Markov} dynamics},
journal = {Comptes Rendus. Physique},
pages = {609--619},
publisher = {Elsevier},
volume = {8},
number = {5-6},
year = {2007},
doi = {10.1016/j.crhy.2007.05.005},
language = {en},
}
TY - JOUR AU - Vivien Lecomte AU - Cécile Appert-Rolland AU - Frédéric van Wijland TI - Thermodynamic formalism and large deviation functions in continuous time Markov dynamics JO - Comptes Rendus. Physique PY - 2007 SP - 609 EP - 619 VL - 8 IS - 5-6 PB - Elsevier DO - 10.1016/j.crhy.2007.05.005 LA - en ID - CRPHYS_2007__8_5-6_609_0 ER -
%0 Journal Article %A Vivien Lecomte %A Cécile Appert-Rolland %A Frédéric van Wijland %T Thermodynamic formalism and large deviation functions in continuous time Markov dynamics %J Comptes Rendus. Physique %D 2007 %P 609-619 %V 8 %N 5-6 %I Elsevier %R 10.1016/j.crhy.2007.05.005 %G en %F CRPHYS_2007__8_5-6_609_0
Vivien Lecomte; Cécile Appert-Rolland; Frédéric van Wijland. Thermodynamic formalism and large deviation functions in continuous time Markov dynamics. Comptes Rendus. Physique, Work, dissipation, and fluctuations in nonequilibrium physics, Volume 8 (2007) no. 5-6, pp. 609-619. doi : 10.1016/j.crhy.2007.05.005. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2007.05.005/
[1] Thermodynamic Formalism, Addison Wesley Publ. Co., Reading, MA, 1978
[2] Time-reversed dynamical entropy and irreversibility in Markovian random processes, J. Stat. Phys., Volume 117 (2004), pp. 599-615
[3] A note on the Ruelle pressure for a dilute disordered Sinai billiard, J. Stat. Phys., Volume 108 (2002), p. 516
[4] Lyapunov spectrum and the conjugate pairing rule for a thermostated random Lorentz gas: kinetic theory, Phys. Rev. Lett., Volume 78 (1997), pp. 207-210
[5] Chaotic properties of systems with Markov dynamics, Phys. Rev. Lett., Volume 95 (2006), p. 010601
[6] Thermodynamic formalism for systems with Markov dynamics, J. Stat. Phys., Volume 127 (2007), pp. 51-106
[7] Chaos, Scattering and Statistical Mechanics, Cambridge Nonlinear Science Series, vol. 9, Cambridge Univ. Press, 1998
[8] A Gallavotti–Cohen type symmetry in the large deviation functional for stochastic dynamics, J. Stat. Phys., Volume 95 (1999), pp. 333-365
[9] Universal large deviation function of the Kardar–Parisi–Zhang equation in one dimension, J. Stat. Phys., Volume 94 (1999), pp. 1-30
[10] Dynamics of the infinite ranged Potts model, J. Stat. Phys., Volume 64 (1991), p. 653
[11] Ill Condensed Matter (R. Maynard; R. Balian; G. Toulouse, eds.), Les Houches, vol. XXXI, North-Holland, Amsterdam, 1979
- Markov spin models for image generation: explicit large deviations with respect to the number of pixels, Journal of Statistical Mechanics: Theory and Experiment, Volume 2025 (2025) no. 1 | DOI:10.1088/1742-5468/ada256
- Explicit dynamical properties of the Pelikan random map in the chaotic region and at the intermittent critical point towards the non-chaotic region, Journal of Statistical Mechanics: Theory and Experiment, Volume 2025 (2025) no. 1 | DOI:10.1088/1742-5468/adaa6c
- Large deviations at level 2.5 and for trajectories observables of diffusion processes: the missing parts with respect to their random-walks counterparts, Journal of Physics A: Mathematical and Theoretical, Volume 57 (2024) no. 9 | DOI:10.1088/1751-8121/ad26ae
- Large deviations for trajectory observables of diffusion processes in dimension d > 1 in the double limit of large time and small diffusion coefficient, Journal of Statistical Mechanics: Theory and Experiment, Volume 2024 (2024) no. 1 | DOI:10.1088/1742-5468/ad1390
- Large deviations and conditioning for chaotic non-invertible deterministic maps: analysis via the forward deterministic dynamics and the backward stochastic dynamics, Journal of Statistical Mechanics: Theory and Experiment, Volume 2024 (2024) no. 1 | DOI:10.1088/1742-5468/ad1bdc
- Inverse problem in the conditioning of Markov processes on trajectory observables: what canonical conditionings can connect two given Markov generators?, Journal of Statistical Mechanics: Theory and Experiment, Volume 2024 (2024) no. 1 | DOI:10.1088/1742-5468/ad17b7
- A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains, Journal of Statistical Mechanics: Theory and Experiment, Volume 2024 (2024) no. 7 | DOI:10.1088/1742-5468/ad526d
- Joint distribution of two local times for diffusion processes with the application to the construction of various conditioned processes, Journal of Physics A: Mathematical and Theoretical, Volume 56 (2023) no. 20 | DOI:10.1088/1751-8121/acccbd
- Revisiting boundary-driven non-equilibrium Markov dynamics in arbitrary potentials via supersymmetric quantum mechanics and via explicit large deviations at various levels, Journal of Statistical Mechanics: Theory and Experiment, Volume 2023 (2023) no. 6 | DOI:10.1088/1742-5468/acdcea
- On the Kemeny time for continuous-time reversible and irreversible Markov processes with applications to stochastic resetting and to conditioning towards forever-survival, Journal of Statistical Mechanics: Theory and Experiment, Volume 2023 (2023) no. 6 | DOI:10.1088/1742-5468/acd695
- Large deviations for the Pearson family of ergodic diffusion processes involving a quadratic diffusion coefficient and a linear force, Journal of Statistical Mechanics: Theory and Experiment, Volume 2023 (2023) no. 8 | DOI:10.1088/1742-5468/ace431
- Nonequilibrium diffusion processes via non-Hermitian electromagnetic quantum mechanics with application to the statistics of entropy production in the Brownian gyrator, Physical Review E, Volume 107 (2023) no. 1 | DOI:10.1103/physreve.107.014101
- Conditioning two diffusion processes with respect to their first-encounter properties, Journal of Physics A: Mathematical and Theoretical, Volume 55 (2022) no. 30 | DOI:10.1088/1751-8121/ac7af3
- Large deviations for metastable states of Markov processes with absorbing states with applications to population models in stable or randomly switching environment, Journal of Statistical Mechanics: Theory and Experiment, Volume 2022 (2022) no. 1 | DOI:10.1088/1742-5468/ac4519
- Conditioning diffusion processes with respect to the local time at the origin, Journal of Statistical Mechanics: Theory and Experiment, Volume 2022 (2022) no. 10 | DOI:10.1088/1742-5468/ac9618
- Microcanonical conditioning of Markov processes on time-additive observables, Journal of Statistical Mechanics: Theory and Experiment, Volume 2022 (2022) no. 2 | DOI:10.1088/1742-5468/ac4e81
- Conditioned diffusion processes with an absorbing boundary condition for finite or infinite horizon, Physical Review E, Volume 106 (2022) no. 4 | DOI:10.1103/physreve.106.044117
- Anomalous dynamical large deviations of local empirical densities and activities in the pure and in the random kinetically constrained East model, The European Physical Journal B, Volume 95 (2022) no. 2 | DOI:10.1140/epjb/s10051-022-00281-5
- Markov trajectories: Microcanonical Ensembles based on empirical observables as compared to Canonical Ensembles based on Markov generators, The European Physical Journal B, Volume 95 (2022) no. 8 | DOI:10.1140/epjb/s10051-022-00386-x
- Large deviations for the skew-detailed-balance lifted-Markov processes to sample the equilibrium distribution of the Curie-Weiss model, Journal of Statistical Mechanics: Theory and Experiment, Volume 2021 (2021) no. 10 | DOI:10.1088/1742-5468/ac22f9
- Inhomogeneous asymmetric exclusion processes between two reservoirs: large deviations for the local empirical observables in the mean-field approximation, Journal of Statistical Mechanics: Theory and Experiment, Volume 2021 (2021) no. 12 | DOI:10.1088/1742-5468/ac4044
- Large deviations for Markov processes with stochastic resetting: analysis via the empirical density and flows or via excursions between resets, Journal of Statistical Mechanics: Theory and Experiment, Volume 2021 (2021) no. 3 | DOI:10.1088/1742-5468/abdeaf
- Large deviations of the Lyapunov exponent in 2D matrix Langevin dynamics with applications to one-dimensional Anderson localization models, Journal of Statistical Mechanics: Theory and Experiment, Volume 2021 (2021) no. 3 | DOI:10.1088/1742-5468/abe408
- Revisiting the Ruelle thermodynamic formalism for Markov trajectories with application to the glassy phase of random trap models, Journal of Statistical Mechanics: Theory and Experiment, Volume 2021 (2021) no. 6 | DOI:10.1088/1742-5468/ac06c1
- Large deviations at various levels for run-and-tumble processes with space-dependent velocities and space-dependent switching rates, Journal of Statistical Mechanics: Theory and Experiment, Volume 2021 (2021) no. 8 | DOI:10.1088/1742-5468/ac0edf
- Jump-drift and jump-diffusion processes: large deviations for the density, the current and the jump-flow and for the excursions between jumps, Journal of Statistical Mechanics: Theory and Experiment, Volume 2021 (2021) no. 8 | DOI:10.1088/1742-5468/ac12c5
- Microscopic fluctuation theory (mFT) for interacting Poisson processes, Journal of Physics A: Mathematical and Theoretical, Volume 52 (2019) no. 13 | DOI:10.1088/1751-8121/ab0978
- Large deviations for dynamical fluctuations of open Markov processes, with application to random cascades on trees, Journal of Physics A: Mathematical and Theoretical, Volume 52 (2019) no. 2 | DOI:10.1088/1751-8121/aaf141
- A comparison of dynamical fluctuations of biased diffusion and run-and-tumble dynamics in one dimension, Journal of Physics A: Mathematical and Theoretical, Volume 52 (2019) no. 42 | DOI:10.1088/1751-8121/ab4349
- Statistical physics of long dynamical trajectories for a system in contact with several thermal reservoirs, Journal of Physics A: Mathematical and Theoretical, Volume 52 (2019) no. 48 | DOI:10.1088/1751-8121/ab4f1a
- Large deviations for the density and current in non-equilibrium-steady-states on disordered rings, Journal of Statistical Mechanics: Theory and Experiment, Volume 2 (2019) no. 2, p. 023206 | DOI:10.1088/1742-5468/ab02dc
- Introduction to dynamical large deviations of Markov processes, Physica A: Statistical Mechanics and its Applications, Volume 504 (2018), pp. 5-19 | DOI:10.1016/j.physa.2017.10.046
- Simulating Rare Events in Dynamical Processes, Journal of Statistical Physics, Volume 145 (2011) no. 4, pp. 787-811 | DOI:10.1007/s10955-011-0350-4
- Computation of Current Cumulants for Small Nonequilibrium Systems, Journal of Statistical Physics, Volume 135 (2009) no. 1, pp. 57-75 | DOI:10.1007/s10955-009-9723-3
- Thermodynamic Formalism for Systems with Markov Dynamics, Journal of Statistical Physics, Volume 127 (2007) no. 1, p. 51 | DOI:10.1007/s10955-006-9254-0
Cité par 35 documents. Sources : Crossref, NASA ADS
Commentaires - Politique
Vous devez vous connecter pour continuer.
S'authentifier