Exponential peak and scaling of work fluctuations in modulated systems

Abstract

We extend the stationary-state work fluctuation theorem to periodically modulated nonlinear systems. Such systems often have coexisting stable periodic states. We show that work fluctuations sharply increase near a kinetic phase transition where the state populations are close to each other. The work variance is proportional here to the reciprocal rate of interstate switching. We also show that the variance displays scaling with the distance to a bifurcation point and find the critical exponent for a saddle-node bifurcation.PACS numbers: 05.70.Ln, 74.50.+r, 85.85.+j Since their discovery in the early 90s [1,2,3], fluctuation theorems have been attracting increasing interest. They establish general features of fluctuating systems away from thermal equilibrium, see Refs. 4, 5 for reviews. A major "test bed" for fluctuation theorems is provided by dynamical systems with a few degrees of freedom coupled to a thermal bath, a Brownian particle being an example. Much of the corresponding theoretical and experimental work refers to (i) modulated linear systems, where fluctuations have been studied both in transient and stationary regimes [6,7,8,9,10,11,12], and (ii) nonlinear systems, initially at thermal equilibrium, driven to a different, generally nonequilibrium state [13,14,15,16,17].Fluctuations in nonequilibrium dynamical systems have been attracting attention also in a different context. They play an important role in various types of mesoscopic vibrational systems of current interest. Because damping of the vibrations is typically weak, even a moderately strong resonant force can excite them to comparatively large amplitudes, where the nonlinearity becomes substantial. As a result, the system may have two or more coexisting stable states of forced vibrations [18]. Fluctuations can cause switching between these states [19] and thus significantly affect the overall behavior of the system even where they are small on average. Many features of the switching behavior and a range of phenomena and applications related to the switching, from quantum measurements to resonant frequency mixing and to high-frequency stochastic resonance have been studied experimentally [20,21,22,23,24,25,26,27,28].In this paper we analyze work fluctuations in periodically modulated nonlinear dynamical systems coupled to a bath. We derive the stationary state work fluctuation theorem and show that, under fairly general assumptions, the distribution of fluctuations of work done by the modulating force over a long time τ is Gaussian. In common with systems close to thermal equilibrium, the work variance σ 2 is proportional to the average work W , but the proportionality coefficient is not universal and depends on system parameters. It becomes exponentially large in bistable systems in the range of a kinetic phase transition where the stationary populations of the vibrational states are close to each other. This parameter range has similarity with the region of a first-order phase transition where molar fractions of the coexisting phases ar...

Work fluctuations in a nonlinear micromechanical oscillator driven far from thermal equilibrium

“…Moreover, instead of a linear dependence, σ 2 / W varies exponentially with the inverse fluctuation intensity due to the activated nature of switching. These observations are shown to be consistent with predicted generic behaviors for work fluctuations in periodically modulated nonlinear systems [24].…”

“…We compare our experimental results on work fluctuations with theoretical predictions [24]. For τ ≫ 1/ν in a bistable system, the work variance is given by:…”

“…3c). However, due to the activated nature of switching, it has been predicted that the dependence of the work fluctuations on the noise intensity and driving frequency differs considerably from the conventional fluctuation theorem [24].…”

“…In these systems, the relation between work variance and mean work generally cannot be obtained from the linear response theory and the fluctuation dissipation theorem. Theoretical analysis has identified generic features in the work fluctuations of these modulated nonlinear systems that differ from their linear counterparts [24]. To our knowledge, fluctuations in nonlinear systems that are periodically driven into multistability have not been experimentally studied in the context of fluctuation theorems.…”

Steady-state work fluctuations of a dragged particle under external and thermal noise

“…A genuinely nonlinear system modulated by a periodic field has been investigated in [39] and has also been shown to violate the SSFT.…”