Fluctuation formula in the Nosé-Hoover thermostated Lorentz gas

Dolowschiák, M.; Kovács, Zoltán

doi:10.1103/physreve.71.025202

Cited by 23 publications

(56 citation statements)

References 24 publications

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“…The numerical results of [34][35][36] agree with the prediction that FR for the rate function of a 0 is valid even beyond a 0 = σ + . The prediction that (at least near equilibrium) the rate function of a should satisfy FR only up to a = σ + and that should become linear for a ≥ a + at the moment has been experimentally confirmed only in Gaussian cases [4,24,25].…”

Section: How To Remove Singularitiessupporting

confidence: 78%

Chaotic Hypothesis, Fluctuation Theorem and Singularities

et al. 2006

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The chaotic hypothesis has several implications which have generated interest in the literature because of their generality and because a few exact predictions are among them. However its application to Physics problems requires attention and can lead to apparent inconsistencies. In particular there are several cases that have been considered in the literature in which singularities are built in the models: for instance when among the forces there are Lennard-Jones potentials (which are infinite in the origin) and the constraints imposed on the system do not forbid arbitrarily close approach to the singularity even though the average kinetic energy is bounded. The situation is well understood in certain special cases in which the system is subject to Gaussian noise; here the treatment of rather general singular systems is considered and the predictions of the chaotic hypothesis for such situations are derived. The main conclusion is that the chaotic hypothesis is perfectly adequate to describe the singular physical systems we consider, i.e. deterministic systems with thermostat forces acting according to Gauss' principle for the constraint of constant total kinetic energy ("isokinetic Gaussian thermostats"), close and far from equilibrium. Near equilibrium it even predicts a fluctuation relation which, in deterministic cases with more general thermostat forces (i.e. not necessarily of Gaussian isokinetic nature), extends recent relations obtained in situations in which the thermostatting forces satisfy Gauss' principle. This relation agrees, where expected, with the fluctuation theorem for perfectly chaotic systems. The results are compared with some recent works in the literature.

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Section: How To Remove Singularitiessupporting

confidence: 78%

Chaotic Hypothesis, Fluctuation Theorem and Singularities

et al. 2006

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“…We also test equation ⌳-FR ͑2͒ for this system, although it is not expected to hold at small fields. 19,20,36 The dynamics for this system are not symplectic or symplectic when the system is out of equilibrium, 39 so we do not expect conjugate pairing of Lyapunov exponents; however, we find that it is possible to drive the system so that the numbers of positive and negative exponents are unequal but negative fluctuations in the dissipation can still be observed. The system has five degrees of freedom, and therefore fiveLyapunov exponents.…”

Section: Introductionmentioning

confidence: 87%

“…For constant temperature dynamics it has proved impossible to confirm ͑2͒ numerically, particularly for weak fields. 6,19,20 However, because ͑2͒ is an asymptotic relation it is always possible that the empirical data have not been considered at sufficiently long times for convergence to occur. The status of ͑2͒ for nonisoenergetic dynamics has recently been considered in detail by Evans et al 36 To further complicate the issue, the formal derivation of ͑2͒ ͑Refs.…”

Section: Introductionmentioning

confidence: 99%

“…[1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] Equation ͑2͒ has been shown to apply in both the linear and nonlinear regimes to isoenergetic systems, and Eq. ͑3͒ has been shown to apply in both the linear and nonlinear regimes to a range of systems including isoenergetic, isokinetic, and Nosé-Hoover thermostatted systems, [1][2][3][4][5][6][7][8][9][10]17,19,20,34 and has recently been verified experimentally. 24 More recently Searles et al have presented a detailed mathematical proof of ͑3͒ for chaotic systems.…”

Section: Introductionmentioning

confidence: 99%

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Numerical study of the steady state fluctuation relations far from equilibrium

Williams

Searles

Evans

2006

The Journal of Chemical Physics

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Nonchaotic attractor with a highly fluctuating finite-time Lyapunov exponent in a hybrid optical system AIP Conf. Proc. 501, 342 (2000); 10.1063/1.59945Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. A thermostatted dynamical model with five degrees of freedom is used to test the fluctuation relation of Evans and Searles ͑⍀-FR͒ and that of Gallavotti and Cohen ͑⌳-FR͒. In the absence of an external driving field, the model generates a time-independent ergodic equilibrium state with two conjugate pairs of Lyapunov exponents. Each conjugate pair sums to zero. The fluctuation relations are tested numerically both near and far from equilibrium. As expected from previous work, near equilibrium the ⍀-FR is verified by the simulation data while the ⌳-FR is not confirmed by the data. Far from equilibrium where a positive exponent in one of these conjugate pairs becomes negative, we test a conjecture regarding the ⌳-FR ͓Bonetto et al., Physica D 105, 226 ͑1997͒; Giuliani et al., J. Stat. Phys. 119, 909 ͑2005͔͒. It was conjectured that when the number of nontrivial Lyapunov exponents that are positive becomes less than the number of such negative exponents, then the form of the ⌳-FR needs to be corrected. We show that there is no evidence for this conjecture in the empirical data. In fact, when the correction factor differs from unity, the corrected form of ⌳-FR is less accurate than the uncorrected ⌳-FR. Also as the field increases the uncorrected ⌳-FR appears to be satisfied with increasing accuracy. The reason for this observation is likely to be that as the field increases, the argument of the ⌳-FR more and more accurately approximates the argument of the ⍀-FR. Since the ⍀-FR works for arbitrary field strengths, the uncorrected ⌳-FR appears to become ever more accurate as the field increases. The final piece of evidence against the conjecture is that when the smallest positive exponent changes sign, the conjecture predicts a discontinuous change in the "correction factor" for ⌳-FR. We see no evidence for a discontinuity at this field strength.

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“…Thus consider a timereversible dissipative system with average phase space contraction rate σ(Γ) = σ + > 0 and assume this system verifies the chaotic hypothesis. In a language similar to that used in [5], the statement of the fluctuation theorem for the dimensionless contraction amplitude p is that there exists a finite number 1 ≤ p * < ∞, so thatwhere P τ (p) denotes the probability of observing, over a time interval τ , a fluctuation of the phase space contraction rate σ τ = pσ + . In particular, thinking about the external driving parameter in the Lorentz gas, the above result should be independent of its amplitude (|E| > 0).…”

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confidence: 99%

Fluctuation theorem applied to the Nosé–Hoover thermostated Lorentz gas

Gilbert

2006

Phys. Rev. E

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We present numerical evidence supporting the validity of the Gallavotti-Cohen Fluctuation Theorem applied to the driven Lorentz gas with Nosé-Hoover thermostating. It is moreover argued that the asymptotic form of the fluctuation formula is independent of the amplitude of the driving force, in the limit where it is small. PACS numbers: 05.70. Ln, Over the last decade, different versions of fluctuation formulas have been the focus of a number of publications in the field of non-equilibrium statistical physics. In particular, dissipative deterministic dynamical systems with time-reversal symmetry have attracted some attention as potential candidates to model externally driven systems with a thermostating mechanism [1,2,3]. Two distinct results have been proposed, which, in the context of isokinetic thermostats, both characterize the fluctuations of entropy production. One, due to Evans and Searles [2], is usually referred to as transient fluctuation theorem, while the other, due to Gallavotti and Cohen [3], is simply known as the fluctuation theorem. The former addresses the fluctuations of the work done by the external forcing on the system, and the latter the fluctuations of the phase space contraction rate of non-equilibrium stationary states. It has been rigorously proved in the context of Anosov systems [3].To be definite, consider the externally driven periodic Lorentz gas with Gaussian thermostating [4]. The trajectory of a particle in between elastic collisions is described by the equationṗ = E − αp, where α = E · p/p 2 is a reversible damping mechanism that acts so as to keep the kinetic energy constant. α, the phase space contraction rate for this system is, as seen from its expression, equal to the work done on the particle divided by the constant temperature. Thus the work done on the particle is exactly compensated by the heat dissipation. In other words, work and heat dissipation statistics are identical for this system. Dolowschiák and Kovács [5] recently made the observation that work and phase space contraction rate fluctuations behave very differently for the externally driven Lorentz gas with Nosé-Hoover thermostating. On the one hand, the work fluctuations, whether large or small, obey the Evans-Searles formula, in agreement with similar observations made for other systems [6,7]. On the other hand, the authors observed that the phase space contraction rate fluctuations rapidly saturate. Moreover no observation of a linear regime of fluctuations in a limited range was reported. * thomas.gilbert@ulb.ac.beThe Nosé-Hoover thermostated Lorentz gas on a periodic lattice has phase space coordinates Γ = (q, p, ζ). Here q and p respectively denote the position and momentum of the particle, and ζ is the variable associated to the thermal reservoir. Between two elastic collisions, the dynamics is specified by the equationṡHere E denotes the external field, τ resp the relaxation time of the thermostat and T the temperature [13]. An essential difference between the Gaussian and Nosé-Hoover thermostated Lore...

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Fluctuation formula in the Nosé-Hoover thermostated Lorentz gas

Cited by 23 publications

References 24 publications

Chaotic Hypothesis, Fluctuation Theorem and Singularities

Chaotic Hypothesis, Fluctuation Theorem and Singularities

Numerical study of the steady state fluctuation relations far from equilibrium

Fluctuation theorem applied to the Nosé–Hoover thermostated Lorentz gas

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