Fluctuation theorem for Hamiltonian Systems: Le Chatelier’s principle

Abstract

For thermostated dissipative systems, the fluctuation theorem gives an analytical expression for the ratio of probabilities that the time-averaged entropy production in a finite system observed for a finite time takes on a specified value compared to the negative of that value. In the past, it has been generally thought that the presence of some thermostating mechanism was an essential component of any system that satisfies a fluctuation theorem. In the present paper, we point out that a fluctuation theorem can be derived for purely Hamiltonian systems, with or without applied dissipative fields. DOI: 10.1103/PhysRevE.63.051105 PACS number͑s͒: 05.20.Ϫy, 47.10.ϩg The fluctuation theorem ͓1-3͔ ͑FT͒ gives a general formula for the logarithm of the probability ratio that in a thermostated dissipative system, the time-averaged entropy production ⌺ t takes a value A to minus the value, ϪA,From this equation it is obvious that as the averaging time or system size increases, it becomes exponentially likely that the entropy production will be positive. The fluctuation theorem is important for at least three reasons: first, it gives an expression for the probability that in a finite system observed for a finite time, the Second Law will be violated; secondly, it gives one of the very few exact fluctuation relations that are known for nonequilibrium steady states, even far from equilibrium; thirdly, it can be derived using some of the standard results of the mathematical theory of dynamical systems theory ͓3͔. The theorem was initially proposed ͓1͔ for nonequilibrium steady states that are thermostated in such a way that the total energy of the system is constant. Subsequently, it was shown by Gallavotti and Cohen ͓3͔ that the theorem could be proved for sufficiently chaotic, isoenergetic nonequilibrium systems using the Sinai-Ruelle-Bowen measure ͓3͔. For transient trajectory segments that start at tϭ0 from an ensemble of initial phase-space vectors ⌫͑0͒, and evolve in time under the influence of a reversible deterministic thermostat and an applied dissipative field towards a unique nonequilibrium steady state, a transient fluctuation theorem can be derived using the Liouville measure ͓2,4͔. It has also been shown that the theorem is valid for a wide class of stochastic nonequilibrium systems ͓5͔.It has recently be shown ͓4͔ that if initial phases are sampled from a known N-particle phase-space distribution function, f (⌫,0), and if we define a dissipation function, ⍀͑⌫͒, bywhere ‫⌫•⌫ץ/ץ‪⌳(⌫)ϵ‬‬ is the phase-space compression factor, then one can derive a transient fluctuation theorem,Thermostats lead to nonzero expressions for the phase-space compression factor. Equation ͑2͒ is consistent with all known deterministic transient fluctuation theorems covering a wide variety of initial ensemble types and thermostating mechanisms ͓4͔.This relationship ͑3͒ has been tested using computer simulations for a range of nonequilibrium steady-state systems in which the phase-space contraction is nonzero ͓2,4,5͑b͒,6͔. It predicts t...

Numerical study of the steady state fluctuation relations far from equilibrium

“…[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.…”

Time-dependent fluctuation theorem

“…The validity of the FT has been confirmed for systems in the absence of a thermostat [11] and most recently, the FT was verified in the isobaric-isothermal ensemble [7]. Recently the TFT has been confirmed in a laboratory experiment using optical tweezers applied to a single colloid particle in solution [12].…”

“…Many numerical simulations have been performed verifying the FT in various ensembles and with various dynamics [1][2]4,[7][8][9][10][11]. The validity of the FT has been confirmed for systems in the absence of a thermostat [11] and most recently, the FT was verified in the isobaric-isothermal ensemble [7].…”

“…Equation (3) has been tested via computer simulations for a range of ensembles with a large range of dynamics [1][2]4,[7][8][9][10][11].…”

Isobaric–isothermal fluctuation theorem

“…͑3͒ was applied to an adiabatic system, where ⌳(t)ϵ0 and the phase volumetherefore does not change. 6 In the present note we show that a FT can be derived for a nonequilibrium system where the initial phase space distribution is isothermal-isobaric and where the dynamics constrains the hydrostatic pressure and kinetic temperature to constant values.…”