Frenesy is a concept in

statistical physics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...

that measures the dynamical activity of a system's microscopic trajectories under nonequilibrium conditions. It complements the notion of

entropy production Entropy production (or generation) is the amount of entropy which is produced during heat process to evaluate the efficiency of the process. Short history Entropy is produced in irreversible processes. The importance of avoiding irreversible p ...

, which measures time-antisymmetric aspects associated with

irreversibility In thermodynamics, an irreversible process is a process that cannot be undone. All complex natural processes are irreversible, although a phase transition at the coexistence temperature (e.g. melting of ice cubes in water) is well approximated a ...

. Frenesy reflects how frequently states are visited or how many transitions occur over time and can be thought of as the "busyness" of the system's trajectories. 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 nonequilibrium processes by Christian Maes and collaborators, and has been discussed in various studies since then. In systems described by trajectory ensembles or path-space measures (e.g. originating in ''

Markov process In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, ...

es'' or ''

Langevin dynamics In physics, Langevin dynamics is an approach to the mathematical modeling of the dynamics of molecular systems using the Langevin equation. It was originally developed by French physicist Paul Langevin. The approach is characterized by the use o ...

''), 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 nonequilibrium response theory. The role of dynamical activity in trajectory ensembles was explored in the study of

large deviations In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. While some basic ideas of the theory can be traced to Laplace, the formalization started with insura ...

. The specific need for dealing with the time-symmetric fluctuation sector was explained in an early influential paper. For some time, it was discussed under the name of "traffic", for example, in several studies on macroscopic fluctuations. A year later, in the context of response theory, the term "frenetic" appeared. 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.

Role in fluctuation-response

Frenesy is used in the generalization of fluctuation-dissipation relations beyond equilibrium. In nonequilibrium 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 has been proposed to describe response phenomena when systems are driven far from equilibrium. As an extension of Kubo and Green-Kubo formulas, nonequilibrium 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. This is evident in nonequilibrium modifications of the Sutherland-Einstein relation, where mobility is no longer determined solely by the

diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...

matrix of the unperturbed system but also includes force–current correlations. 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" is supported by similar theoretical considerations. Frenetic effects also appear in second order and higher-order nonlinear response expansions around equilibrium. A frenetic contribution also appears in corrections to the fluctuation-dissipation relation of the second kind, known as the Einstein relation. The (linear)

friction Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. Types of friction include dry, fluid, lubricated, skin, and internal -- an incomplete list. The study of t ...

has an entropic and frenetic part, where the entropic part connects with the

noise Noise is sound, chiefly unwanted, unintentional, or harmful sound considered unpleasant, loud, or disruptive to mental or hearing faculties. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrat ...

in the usual equilibrium way. The frenetic part may be negative and dominating to the extent of rendering the friction negative.

Applications

The concept of frenesy is used in various areas of modern statistical physics. In the presence of

dissipation In thermodynamics, dissipation is the result of an irreversible process that affects a thermodynamic system. In a dissipative process, energy ( internal, bulk flow kinetic, or system potential) transforms from an initial form to a final form, wh ...

, kinetic aspects in the form of increased or decreased dynamical activity and reactivities can determine a system's behavior. For example, jamming, localization, or glassy behavior are induced by dynamical heterogeneities where traps become important under driving or relaxation. Relaxation behavior is indeed another instance where kinetic aspects matter and where frenesy influences and shapes the landscape of possible pathways. Kinetic

phase transition In physics, chemistry, and other related fields like biology, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic Sta ...

s are governed by large deviations in frenesy that signal transitions between dynamical phases. In

active matter Active matter is matter composed of large numbers of active "agents", each of which consumes energy in order to move or to exert mechanical forces. Such systems are intrinsically out of thermal equilibrium. Unlike thermal systems relaxing toward ...

, persistent motion is triggered by switches (discrete or continuous) introducing high frenetic activity. Other applications concern selection and steering.

Kinetic proofreading Kinetic proofreading (or kinetic amplification) is a mechanism for error correction in biochemical reactions, proposed independently by John Hopfield (1974) and Jacques Ninio (1975). Kinetic proofreading allows enzymes to discriminate between two ...

and biological error correction are examples. The presence of driving allows changes in parameters governing dynamical activity to promote certain conditions of occupation and current. When those parameters depend on and receive feedback about the actual state, the system may evolve into a different phase or develop a dynamical pattern, as witnessed in

. As frenesy captures kinetic fluctuations and dynamical activity, its potential role has also been assessed in explaining nonequilibrium processes within the

climate system Earth's climate system is a complex system with five interacting components: the Atmosphere of Earth, atmosphere (air), the hydrosphere (water), the cryosphere (ice and permafrost), the lithosphere (earth's upper rocky layer) and the biosphere ( ...

. The concept of time-symmetric dynamical activity in nonequilibrium statistical mechanics has also been explored in the study of dynamical fluctuation symmetries. It deviates from stochastic thermodynamics by stressing kinetic aspects.

Origin and context

Role in fluctuation-response

Applications

See also

References

Further reading