^{1,a)}and Pierre Gaspard

^{1}

### Abstract

The fluctuationtheorems for dissipation and the currents are applied to the stochastic version of the reversible Brusselator model of nonequilibrium oscillating reactions. It is verified that the symmetry of these theorems holds far from equilibrium in the regimes of noisy oscillations. Moreover, the fluctuationtheorem for the currents is also verified for a truncated Brusselator model.

The authors thank Professor G. Nicolis for support and encouragement in this research. D.A. is grateful to the FRS-FNRS Belgium for financial support. This research is financially supported by the “Communauté française de Belgique” (Contract “Actions de Recherche Concertées” No. 04/09-312) and the FRS-FNRS Belgium (Contract FRFC No. 2.4577.04).

I. INTRODUCTION

II. THE BRUSSELATOR MODEL OF OSCILLATIONS

A. The macroscopic level

B. The mesoscopic level

III. FLUCTUATIONTHEOREM FOR DISSIPATION

IV. FLUCTUATIONTHEOREM FOR CURRENTS

V. THE TRUNCATED MODEL

VI. CONCLUSIONS

### Key Topics

- Probability theory
- 19.0
- Chemical reaction theory
- 14.0
- Cumulative distribution functions
- 12.0
- Signal generators
- 11.0
- Stochastic processes
- 11.0

## Figures

Simulation by the Gillespie algorithm of the oscillatory regime for the reversible Brusselator [Eqs. (1)–(3)]. The values of the concentrations are , , and the reaction constants , , . From the top to the bottom, the extensivity parameter takes the values: (a) , (b) , (c) , and (d) . The first column depicts the phase portrait in the plane of the numbers and of molecules. The second column shows the number as a function of time. The third one depicts the autocorrelation function [Eq. (18)] of the number , which is normalized to unity.

Simulation by the Gillespie algorithm of the oscillatory regime for the reversible Brusselator [Eqs. (1)–(3)]. The values of the concentrations are , , and the reaction constants , , . From the top to the bottom, the extensivity parameter takes the values: (a) , (b) , (c) , and (d) . The first column depicts the phase portrait in the plane of the numbers and of molecules. The second column shows the number as a function of time. The third one depicts the autocorrelation function [Eq. (18)] of the number , which is normalized to unity.

The generating function [Eq. (20)] numerically obtained for the Brusselator as the eigenvalue by the method of Refs. 5 and 12. The extensivity parameter takes the value while the control parameter takes the values . The other parameters are fixed as explained in the text.

The generating function [Eq. (20)] numerically obtained for the Brusselator as the eigenvalue by the method of Refs. 5 and 12. The extensivity parameter takes the value while the control parameter takes the values . The other parameters are fixed as explained in the text.

The generating function [Eq. (20)] numerically obtained for the Brusselator as the eigenvalue by the method of Refs. 5 and 12. The extensivity parameter takes the value while the control parameter takes the values . The other parameters are fixed as explained in the text.

Probability distribution function for the cumulated current over a time interval . The concentration [B] is fixed so that the affinity [Eq. (24)] takes the value . The other parameters take the values , , , , and .

Probability distribution function for the cumulated current over a time interval . The concentration [B] is fixed so that the affinity [Eq. (24)] takes the value . The other parameters take the values , , , , and .

Probability distribution functions for the cumulated current over the time interval . The affinity takes the values , ln 1.3, and ln 1.5. They obey the symmetry [Eq. (28)]. The parameters take the same values as in Fig. 4.

Probability distribution functions for the cumulated current over the time interval . The affinity takes the values , ln 1.3, and ln 1.5. They obey the symmetry [Eq. (28)]. The parameters take the same values as in Fig. 4.

Probabilities of the negative events (solid line) compared with the predictions (crosses) of the fluctuation relation (29). is the cumulated current and the affinity takes the value . The parameters take the same values as in Fig. 4.

Probabilities of the negative events (solid line) compared with the predictions (crosses) of the fluctuation relation (29). is the cumulated current and the affinity takes the value . The parameters take the same values as in Fig. 4.

Probability distribution functions for the cumulated current over the time interval in the truncated model [Eqs. (30) and (31)]. The affinity takes the values , ln 1.2, ln 1.4 for left to right. The concentration [B] is fixed accordingly. The other parameters are , , , and .

Probability distribution functions for the cumulated current over the time interval in the truncated model [Eqs. (30) and (31)]. The affinity takes the values , ln 1.2, ln 1.4 for left to right. The concentration [B] is fixed accordingly. The other parameters are , , , and .

Probabilities of the negative events (solid line) compared with the predictions (crosses) of the fluctuation relation (29) in the truncated model [Eqs. (30) and (31)]. is the cumulated current over the time interval and the affinity is given by . The other parameters take the same values as in Fig. 7.

Probabilities of the negative events (solid line) compared with the predictions (crosses) of the fluctuation relation (29) in the truncated model [Eqs. (30) and (31)]. is the cumulated current over the time interval and the affinity is given by . The other parameters take the same values as in Fig. 7.

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