Fluctuation theorem applied to the Nos\'e-Hoover thermostated Lorentz gas

Thomas Gilbert

REVIEW

Fluctuation theorem applied to the Nos\'e-Hoover thermostated Lorentz gas

Thomas Gilbert

Statistical Mechanics · Submitted 20 Feb 2006 · 16 references

Pith's reading: For the Nosé-Hoover thermostated Lorentz gas, phase space contraction fluctuations follow the modified Gallavotti-Cohen theorem, with linear small fluctuations and saturated large fluctuations.

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arXiv IDcond-mat/0602458 (version 1)

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CitationThomas Gilbert. "Fluctuation theorem applied to the Nos\'e-Hoover thermostated Lorentz gas." arXiv:cond-mat/0602458 (2006). https://pith.science/paper/RYRKQKTU

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Claim paper

The paper

The paper claims

For the Nosé-Hoover thermostated Lorentz gas, phase space contraction fluctuations follow the modified Gallavotti-Cohen theorem, with linear small fluctuations and saturated large fluctuations.

The reading

The paper studies a simple mechanical model: a point particle moves through a periodic array of hard disks, pushed by a constant electric field, and kept at roughly constant kinetic energy by a Nosé-Hoover thermostat, which is an extra variable that adjusts a friction-like force. In such nonequilibrium systems, the Gallavotti-Cohen fluctuation theorem predicts a specific symmetry between the probability of seeing a positive fluctuation of the phase space contraction rate and the probability of seeing the corresponding negative fluctuation. An earlier paper by Dolowschiák and Kovács reported that this symmetry fails for the Nosé-Hoover thermostated Lorentz gas.

Gilbert re-examines the same model with longer simulations and with a version of the theorem, due to Bonetto, Gallavotti, Giuliani and Zamponi, that allows for unbounded fluctuations. He compares his data with the formula of van Zon and Cohen, which describes how the fluctuation ratio should behave for small fluctuations (linearly) and for large fluctuations (saturating). The simulations use two field strengths, E = 0.1 and E = 0.5, and are consistent with the predicted curve once finite-time corrections are extrapolated away. The author therefore concludes that the earlier negative result came from not running long enough and from ignoring the saturation at large fluctuations.

The paper is a numerical case study, not a new derivation. Its main limitations are that no error bars, code, or data files are provided, and the agreement is judged visually. The author also states that even weaker fields require much longer simulations, so the claimed independence from field strength rests on only a few driving amplitudes.

Extended reading notes

Core claim

The paper claims that "the Lorentz gas with Nosé-Hoover thermostating has fluctuations of the phase space contraction rate which are entirely consistent with [Eq. (2)], showing both saturation for large fluctuations and linear behavior for small fluctuations." If true, the modified Gallavotti-Cohen fluctuation theorem, with the van Zon-Cohen shape f(p), correctly describes this deterministic thermostated system, and the earlier contrary conclusion of Dolowschiák and Kovács is wrong.

Load-bearing premise

The use of the van Zon-Cohen prediction, Eqs. (4)-(5), rests on the assumption that the long-time fluctuations of σ0 are Gaussian. The paper cites [5] for this and checks kurtosis and skewness on the same trajectory data, but it does not prove or independently establish Gaussianity. If σ0 had non-Gaussian tails, the predicted f(p) would not apply and the comparison in Figs. 1-2 would lose its meaning. This assumption enters after Eq. (5), where the paper states "According to [5], the fluctuations of the time averages of σ0 for the Nosé-Hoover Lorentz gas are Gaussian."

Original abstract

We present numerical evidence supporting the validity of the Gallavotti-Cohen Fluctuation Theorem applied to the driven Lorentz gas with Nos\'e-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.

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The abstract, sentence by sentence

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We present numerical evidence supporting the validity of the Gallavotti-Cohen Fluctuation Theorem applied to the driven Lorentz gas with Nos\'e-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.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present numerical evidence supporting the validity of the Gallavotti-Cohen Fluctuation Theorem applied to the driven Lorentz gas with Nos\'e-Hoover thermostating.	contribution 0.52	0.95	0.88	asserted 0.79
s1	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.	result 0.59	0.64	0.16	asserted 0.80

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Figures

**Figure 1.** FIG. 1: Probability density function (above) and verificati

**Figure 2.** FIG. 2: Same as Fig. 1 for

**Figure 3.** FIG. 3: Height of saturation measured at

Figures reproduced from arXiv:cond-mat/0602458 by the authors.

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References

Works cited (16)

[1]

D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. Lett. 71, 2401 (1993)

View on Pith 1993
[2]

D. J. Evans and D. J. Searles, Phys. Rev. E 50, 1645 (1994)

View on Pith 1994
[3]

G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett. 74, 2694 (1995); J. Stat. Phys. 80 931 (1995)

View on Pith 1995
[4]

B. Moran and W. G. Hoover, J. Stat. Phys. 48, 709 (1987)

View on Pith 1987
[5]

(similar claims can be found in [7]) that the ﬂuctua- tion theorem does not hold if the driving is small can be attributed to two reasons; (i) these authors did not con- sider properly the saturation of the ﬂuctuation at larger p > p ∗ , and (ii) they did not run the simulation long enough to observe the small but clear linear region at p < p ∗. The small...

View on Pith
[6]

M. Dolowschi´ ak and Z. Kov´ acs, Phys. Rev. E 71, 025202(R), (2005)

View on Pith 2005
[7]

D. J. Evans and D. J. Searles, J. Chem. Phys. 113, 3503 (2000)

View on Pith 2000
[8]

D. J. Evans, D. J. Searles, and L. Rondoni, Phys. Rev. E 71, 056120 (2005)

View on Pith 2005
[9]

F. Bonetto, G. Gallavotti, A. Giuliani, and F. Zamponi, Chaotic hypothesis, ﬂuctuation theorem and singularities , cond-mat/0507672, to appear in J. Stat. Phys

View on Pith Pith review arXiv
[10]

K. Rateitschak, R. Klages, and W. G. Hoover, J. Stat. Phys. 101, 61 (2000)

View on Pith 2000
[11]

R. van Zon and E. G. D. Cohen, Phys. Rev. Lett. 91, 110601 (2003)

View on Pith 2003
[12]

A. Giuliani, F. Zamponi, and G. Gallavotti, J. Stat. Phy s. 119, 909 (2005)

View on Pith 2005
[13]

N. Garnier and S. Ciliberto, Phys. Rev. E 71, 060101 (R) (2005)

View on Pith 2005
[14]

The role of T as temperature is consistent as long as the external ﬁeld is non zero. For zero ﬁeld, the system has a limit-cycle. It is therefore not ergodic and the notion of temperature is ill-deﬁned

View on Pith
[15]

The Lorentz gas or Gaussian thermostated driven Lorentz gas is strictly speaking not Anosov because of the instantaneous collisions with the scatterers

View on Pith
[16]

We could check this observation to very good accuracy. For instance, the distributions of σ 0 corresponding to the data presented in Figs. 1-2 have kurtosis (fourth moment divided by second moment squared) equal to 3, the Gaus- sian value, within one percent (and similarly for higher moments 6 and 8). Likewise the skewness (third moment squared by second ...

View on Pith

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

D. J. Evans, E. G. D. Cohen, and G. P. Morriss, Phys. Rev. Lett. 71, 2401 (1993)

View on Pith 1993

[2] [2]

D. J. Evans and D. J. Searles, Phys. Rev. E 50, 1645 (1994)

View on Pith 1994

[3] [3]

G. Gallavotti and E. G. D. Cohen, Phys. Rev. Lett. 74, 2694 (1995); J. Stat. Phys. 80 931 (1995)

View on Pith 1995

[4] [4]

B. Moran and W. G. Hoover, J. Stat. Phys. 48, 709 (1987)

View on Pith 1987

[5] [5]

(similar claims can be found in [7]) that the ﬂuctua- tion theorem does not hold if the driving is small can be attributed to two reasons; (i) these authors did not con- sider properly the saturation of the ﬂuctuation at larger p > p ∗ , and (ii) they did not run the simulation long enough to observe the small but clear linear region at p < p ∗. The small...

View on Pith

[6] [6]

M. Dolowschi´ ak and Z. Kov´ acs, Phys. Rev. E 71, 025202(R), (2005)

View on Pith 2005

[7] [7]

D. J. Evans and D. J. Searles, J. Chem. Phys. 113, 3503 (2000)

View on Pith 2000

[8] [8]

D. J. Evans, D. J. Searles, and L. Rondoni, Phys. Rev. E 71, 056120 (2005)

View on Pith 2005

[9] [9]

F. Bonetto, G. Gallavotti, A. Giuliani, and F. Zamponi, Chaotic hypothesis, ﬂuctuation theorem and singularities , cond-mat/0507672, to appear in J. Stat. Phys

View on Pith Pith review arXiv

[10] [10]

K. Rateitschak, R. Klages, and W. G. Hoover, J. Stat. Phys. 101, 61 (2000)

View on Pith 2000

[11] [11]

R. van Zon and E. G. D. Cohen, Phys. Rev. Lett. 91, 110601 (2003)

View on Pith 2003

[12] [12]

A. Giuliani, F. Zamponi, and G. Gallavotti, J. Stat. Phy s. 119, 909 (2005)

View on Pith 2005

[13] [13]

N. Garnier and S. Ciliberto, Phys. Rev. E 71, 060101 (R) (2005)

View on Pith 2005

[14] [14]

The role of T as temperature is consistent as long as the external ﬁeld is non zero. For zero ﬁeld, the system has a limit-cycle. It is therefore not ergodic and the notion of temperature is ill-deﬁned

View on Pith

[15] [15]

The Lorentz gas or Gaussian thermostated driven Lorentz gas is strictly speaking not Anosov because of the instantaneous collisions with the scatterers

View on Pith

[16] [16]

We could check this observation to very good accuracy. For instance, the distributions of σ 0 corresponding to the data presented in Figs. 1-2 have kurtosis (fourth moment divided by second moment squared) equal to 3, the Gaus- sian value, within one percent (and similarly for higher moments 6 and 8). Likewise the skewness (third moment squared by second ...

View on Pith