Frenesy (physics)

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 reflects 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.

Origin and context

The notion of frenesy was introduced in 2006 in the study of non-equilibrium processes by Christian Maes and collaborators, and has been discussed in various studies since then.^[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 an early influential paper.^[6] For some time, it was discussed under the name "traffic", for example, in several studies on macroscopic fluctuations.^[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 on 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 not only on the correlation with entropy production but also on 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, which is often 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 connects with the noise in the usual equilibrium way. 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 non-linear amplification and damping; it has been used to model biological heartbeat. Frenetic control may play a role in analysis of these systems.^[16]

See also

• Non-equilibrium thermodynamics
• Stochastic thermodynamics
• Fluctuation theorem
• Linear response theory