Fluctuation-Response Relations
Last update: 12 Apr 2025 10:57First version: 14 May 2015
A key result in statistical mechanics is what's known as the "fluctuation-dissipation" or "fluctuation-response" relation. Assemblages of particles in thermodynamic equilibrium actually fluctuate all the time. (Typically, the fluctuations are relatively small.) These fluctuations generally evolve according to well-behaved Markov processes that tend back towards the equilibrium values of the macroscopic variables on average. The fluctuation-dissipation theorem says that the assemblage's response to small external perturbations or forcings looks just like the dynamics of these spontaneous fluctuations. There are various ways of proving this, making use of more or less detailed physical considerations; many of them boil down to the assumption that equilibrium has to correspond to a stationary stochastic process. This in turn suggests that non-equilibrium assemblages in stationary states would have their own fluctuation-response relations, which is of special interest to me.
- See also:
- Mori-Zwanzig Formalism
- Recommended:
- Dieter Forster, Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Function [As I said, fluctuation-dissipation is a standard topic in statistical mechanics, but I found Foster's treatment particularly clarifying]
- Philip Pechukas, "Generalized Langevin Equation of Mori and Kubo", Physical Review 164 (1967): 174--175 [Deriving one of the fluctuation-dissipation relations from stationarity and (mean-square) differentiabilty alone.]
- Lars Onsager and S. Machlup, "Fluctuations and Irreversible Processes", Physical Review 91 (1953): 1505--1512
- To read:
- Kiryl Asheichyk, Matthias Krüger, "Using the fluctuation-dissipation theorem for nonconservative forces", arxiv:1908.11287
- G. Baez, H. Larralde, F. Leyvraz and Rafael A. Mendez-Sanchez, "Fluctuation-Dissipation Theorem for Metastable Systems," cond-mat/0303281
- Marco Baiesi, Christian Maes, Bram Wynants, "Fluctuations and response of nonequilibrium states", arxiv:0902.3955
- Marco Baiesi, Eliran Boksenbojm, Christian Maes and Bram Wynants, "Nonequilibrium Linear Response for Markov Dynamics, II: Inertial Dynamics", Journal of Statistical Physics 139 (2010): 492--505
- Marco Baldovin, Lorenzo Caprini, Angelo Vulpiani, "A handy fluctuation-dissipation relation to approach generic noisy systems and chaotic dynamics", arxiv:2105.12717
- M. M. Bandi, J. R. Cressman Jr., W. I. Goldburg, "Test of the Fluctuation Relation in compressible turbulence on a free surface", nlin.CD/0607037
- L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio and C. Landim
- "Fluctuations in Stationary non Equilibrium States," cond-mat/0104153
- "Macroscopic fluctuation theory for stationary non equilibrium states," cond-mat/0108040
- G. Boffetta, G. Lacorata, S. Musacchio and A. Vulpiani, "Relaxation of finite perturbations: Beyond the fluctuation-response relation", Chaos 13 (2003): 806--811
- Salvatore Calabrese, Lamberto Rondoni, Amilcare Porporato, "Thermodynamics of fluctuations in small systems interacting with the environment", arxiv:1909.09479
- Xian Chen, Chen Jia, "Mathematical foundation of nonequilibrium fluctuation-dissipation theorems for inhomogeneous diffusion processes with unbounded coefficients", arxiv:1708.09744
- Vladimir Y. Chernyak, Mcihael Chertkov and Christopher Jarzynski, "Path-integral analysis of fluctuation theorems for general Langevin processes", cond-mat/0605471
- Hyun-Myung Chun, Qi Gao, Jordan M. Horowitz, "Nonequilibrium Green-Kubo relations for hydrodynamic transport from an equilibrium-like fluctuation-response equality", arxiv:2103.09288
- Matteo Colangeli, Valerio Lucarini, "Elements of a unified framework for response formulae", arxiv:1310.1747
- Andreas Dechant, Shin-ichi Sasa, "Fluctuation-response inequality out of equilibrium", arxiv:1804.08250
- Amir Dembo, Jean-Dominique Deuschel, "Markovian perturbation, response and fluctuation dissipation theorem", arxiv:0710.4394
- Gregor Diezemann, "Fluctuation-dissipation relations for Markov processes", Physical Review E 72 (2005): 0111104
- Denis J. Evans and Debra J. Searles, "On Irreversibility, Dissipation and Response Theory", cond-mat/0612105
- Denis J. Evans, Debra J. Searles, Stephen R. Williams, "Dissipation and the Relaxation to Equilibrium", arxiv:0811.2248
- Gregory Eyink, "Fluctuation-response relations for multitime correlations," Physical Review E 62 (2000): 210--220
- Suzanne Fielding and Peter Sollich, "Observable-dependence of fluctuation-dissipation relations and effective temperatures," cond-mat/0107627
- C. H. Fleming, B. L. Hu, and Albert Roura, "Nonequilibrium fluctuation-dissipation inequality and nonequilibrium uncertainty principle", Physical Review E 88 (2013): 012102
- étienne Fodor, Cesare Nardini, Michael E. Cates, Julien Tailleur, Paolo Visco, and Frédéric van Wijland, "How Far from Equilibrium Is Active Matter?", Physical Review Letters 117 (2016): 038103
- Reinaldo García-García, Vivien Lecomte, A. B. Kolton, D. Domínguez, "Joint probability distributions and fluctuation theorems", arxiv:1111.5369
- Jerome Garnier-Brun, Jean-Philippe Bouchaud, Michael Benzaquen, "Bounded Rationality and Animal Spirits: A Fluctuation-Response Approach to Slutsky Matrices", arxiv:2206.04468
- A. Giuliani, F. Zamponi and G. Gallavotti, "Fluctuation Relation beyond Linear Response Theory", cond-mat/0412455
- J. R. Gomez-Solano, A. Petrosyan, S. Ciliberto, R. Chetrite, and K. Gawedzki, "Experimental Verification of a Modified Fluctuation-Dissipation Relation for a Micron-Sized Particle in a Nonequilibrium Steady State", Physical Review Letters 103 (2009): 040601
- Giacomo Gradenigo, Andrea Puglisi, Alessandro Sarracino, Dario Villamaina, Angelo Vulpiani, "Out-of-equilibrium generalized fluctuation-dissipation relations", arxiv:1203.4941
- Martin Hairer, Andrew J Majda, "A simple framework to justify linear response theory", Nonlinearity 23 (2010): 909, arxiv:0909.4313
- Takahiro Harada and Shin-ichi Sasa
- "Energy dissipation and violation of the fluctuation-response relation in non-equilibrium Langevin systems", cond-mat/0510723
- "Fluctuations, Responses and Energetics of Molecular Motors", cond-mat/0610757
- Hisao Hayakawa and Michio Otsuki, "Nonequilibrium identities and response theory for dissipative particles", Physical Review E 88 (2013): 032117
- Kumiko Hayashi and Shin-ichi Sasa, "Linear response theory in stochastic many-body systems revisited", cond-mat/0507719
- Tristan Holsten, Matthias Krüger, "A Thermodynamic Non-Linear Response Relation", arxiv:2101.05019
- Hiroshi Iyetomi, Yasuhiro Nakayama, Hideaki Aoyama, Yoshi Fujiwara, Yuichi Ikeda, Wataru Souma, "Fluctuation-Dissipation Theory of Input-Output Interindustrial Correlations", arxiv:0912.1985
- Christopher Jarzynski, "Fluctuation relations and strong inequalities for thermally isolated systems", arxiv:1907.09604
- Gerhard Jung, Friederike Schmid, "Fluctuation-Dissipation Relations Far from Equilibrium: A Case Study", arxiv:2106.00818
- Guglielmo Lacorata, Angelo Vulpiani, "Fluctuation-Response Relation and modeling in systems with fast and slow dynamics", Nonlinear Processes in Geophysics (?) 14 (2007): 681--694, arxiv:0711.1064
- Valerio Lucarini, "Stochastic perturbations to dynamical systems: a response theory approach", Journal of Statistical Physics 146 (2012): 774--786, arxiv:1103.0237
- Valerio Lucarini, "Response Theory for Equilibrium and Non-Equilibrium Statistical Mechanics: Causality and Generalized Kramers-Kronig relations", Journal of Statistical Physics 131 (2008): 543--558, arxiv:0710.0958
- Valerio Lucarini, Matteo Colangeli, "Beyond the linear Fluctuation-Dissipation Theorem: the Role of Causality", arxiv:1202.1073
- C. Maes, S. Safaverdi, P. Visco, and F. van Wijland, "Fluctuation-response relations for nonequilibrium diffusions with memory", Physical Review E 87 (2013): 022125
- Kirsten Martens, Eric Bertin and Michel Droz
- "Dependence of the Fluctuation-Dissipation Temperature on the Choice of Observable", Physical Review Letters 103 (2009): 260602
- "Entropy-based characterizations of the observable dependence of the fluctuation-dissipation temperature", Physical Review E 81 (2010): 061107
- J. Mehl, B. Lander, C. Bechinger, V. Blickle, and U. Seifert, "Role of Hidden Slow Degrees of Freedom in the Fluctuation Theorem", Physical Review Letters 108 (2012): 220601
- Aurelio Patelli, Shamik Gupta, Cesare Nardini, and Stefano Ruffo, "Linear response theory for long-range interacting systems in quasistationary states", Physical Review E 85 (2012): 021133
- Jonas Richter, Mats H. Lamann, Christian Bartsch, Robin Steinigeweg, Jochen Gemmer, "Relaxation of dynamically prepared out-of-equilibrium initial states within and beyond linear response theory", arxiv:1905.03292
- David Ruelle, "A review of linear response theory for general differentiable dynamical systems", arxiv:0901.0484
- Manuel Santos Gutiérrez, Valerio Lucarini, "Response and Sensitivity Using Markov Chains", arxiv:1907.12881
- A. Sarracino, A. Vulpiani, "On the Fluctuation-Dissipation Relation in non-equilibrium and non-Hamiltonian systems", arxiv:1909.03726
- Geoffrey L. Sewell, "Macrostatistics and Fluctuating Hydrodynamics", arxiv:1206.2750
- Yair Shokef, Guy Bunin, and Dov Levine, "Fluctuation-dissipation relations in driven dissipative systems", Physical Review E 73 (2006): 046132, cond-mat/0511409
- M. H. Vainstein, I. V. L. Costa and F. A. Oliveira, "Mixing, Ergodicity and the Fluctuation-Dissipation Theorem in complex systems", cond-mat/0501448
- Mark S. Williamson, Chad W. Thackeray, Peter M. Cox, Alex Hall, Chris Huntingford, and Femke J. M. M. Nijsse, "Emergent constraints on climate sensitivities", Reviews of Modern Physics 93 (2021): 025004
- David H. Wolpert
- "Uncertainty relations and fluctuation theorems for Bayes nets", arxiv:1911.02700
- "Fluctuation theorems for multiple co-evolving systems", arxiv:2003.11144
- Adrianne Zhong and Michael R. DeWeese, "Beyond Linear Response: Equivalence between Thermodynamic Geometry and Optimal Transport", Physical Review Letters 133 (2024): 057102
