## Fluctuation-Response Relations

*27 Feb 2017 16:30*

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.

- 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]
- Lars Onsager and S. Machlup, "Fluctuations and Irreversible
Processes", Physical Review
**91**(1953): 1505--1512

- To read:
- 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 - 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 - Matteo Colangeli, Valerio Lucarini, "Elements of a unified framework for response formulae", arxiv:1310.1747
- 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 - Reinaldo García-García, Vivien Lecomte, A. B. Kolton, D. Domínguez, "Joint probability distributions and fluctuation theorems", arxiv:1111.5369
- 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
- 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

- "Dependence of the Fluctuation-Dissipation Temperature on the Choice of Observable", Physical Review Letters
- 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 - David Ruelle, "A review of linear response theory for general differentiable dynamical systems", arxiv:0901.0484
- 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