Frenesy is a concept in statistical physics that measures the dynamical activity of a system's microscopic trajectories under non-equilibrium conditions.[1] It complements the notion of entropy production, which measures time-antisymmetric aspects associated with irreversibility. Frenesy represents how frequently states are visited or how many transitions occur over time, and how busy the system's trajectories are. It relates to reactivities, escape rates, and residence times of a physical state.
The concept of frenesy was introduced in 2006 in the study of non-equilibrium processes by Christian Maes and collaborators.[1][2][3] In systems described by trajectory ensembles or path-space measures (e.g. originating in Markov processes or Langevin dynamics), frenesy is associated with the time-symmetric part of the action functional, containing trajectory-dependent terms such as escape rates, undirected traffic, and the total number of configuration changes. As with many physical observables, it is the change in frenesy that makes the relevant quantity, particularly in the context of non-equilibrium response theory.[1][citation needed]
The role of dynamical activity in trajectory ensembles was explored in the study of large deviations.[4][5] The specific need for dealing with the time-symmetric fluctuation sector was explained in another 2006 paper by Christian Maes et al.[6] Previous the concept had been was discussed under the name "traffic".[7][8] A year later, in the context of response theory, the term "frenetic" appeared.[9]
Mathematically, in a stochastic trajectory under local detailed balance, entropy production is tied to the asymmetry between forward and time-reversed paths, whereas frenesy quantifies the symmetric part invariant under time reversal. As such, it measures changes in dynamical activity or quiescence depending on the reference process and the level of description.[citation needed]
Role in fluctuation-response
Frenesy is used in the generalization of fluctuation-dissipation relations beyond equilibrium. In non-equilibrium steady states, the linear response of an observable depends on both the correlation with entropy production and correlations with frenesy. This correction was proposed to describe response phenomena when systems are driven far from equilibrium.[citation needed]
As an extension of Kubo and Green-Kubo formulas, non-equilibrium linear response theory allows the response to be decomposed into an "entropic" term and a "frenetic" term. The frenetic component is absent in equilibrium but becomes significant under external driving forces.[citation needed] This is evident in non-equilibrium modifications of the Sutherland-Einstein relation, where mobility is no longer determined solely by the diffusion matrix of the unperturbed system but also includes force–current correlations.[10] The frenetic contribution can lead to negative responses, such as for differential mobility or non-equilibrium specific heats. This phenomenon, described as "pushing more for getting less",[11] is supported by similar theoretical considerations.[12] Frenetic effects also appear in second order and higher-order nonlinear response expansions around equilibrium.[13][14]
A frenetic contribution also appears in corrections to the fluctuation-dissipation relation of the second kind, known as the Einstein relation. The linear friction has an entropic and frenetic part, where the entropic part is connected with the thermal noise. The frenetic part may be negative and dominating to the extent of rendering the friction negative.[15]
Applications
A van der Pol oscillator models oscillations with nonlinear amplification and damping; it has been used to model biological heartbeat. Frenetic control may play a role in analysis of these systems.[16]
↑ Garrahan, Juan P.; Jack, Robert L.; Lecomte, Vivien; Pitard, Estelle; Van Duijvendijk, Kristina; Van Wijland, Frédéric (2009). "First-order dynamical phase transition in models of glasses: An approach based on ensembles of histories". Journal of Physics A: Mathematical and Theoretical. 42 (7). arXiv:0810.5298. Bibcode:2009JPhA...42g5007G. doi:10.1088/1751-8113/42/7/075007.
↑ Baiesi, Marco; Maes, Christian; Wynants, Bram (2011). "The modified Sutherland–Einstein relation for diffusive non-equilibria". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 467 (2134): 2792–2809. arXiv:1101.3227. Bibcode:2011RSPSA.467.2792B. doi:10.1098/rspa.2011.0046.
↑ Zia, R. K. P.; Praestgaard, E. L.; Mouritsen, O. G. (2002). "Getting more from pushing less: Negative specific heat and conductivity in nonequilibrium steady states". American Journal of Physics. 70 (4): 384–392. arXiv:cond-mat/0108502. Bibcode:2002AmJPh..70..384Z. doi:10.1119/1.1427088.
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