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2014 | OriginalPaper | Buchkapitel

2. The Dissipation Function: Its Relationship to Entropy Production, Theorems for Nonequilibrium Systems and Observations on Its Extrema

verfasst von : James C. Reid, Sarah J. Brookes, Denis J. Evans, Debra J. Searles

Erschienen in: Beyond the Second Law

Verlag: Springer Berlin Heidelberg

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Abstract

In this chapter we introduce the dissipation function, and discuss the behaviour of its extrema. The dissipation function allows the reversibility of a nonequilibrium process to be quantified for systems arbitrarily close to or far from equilibrium. For a system out of equilibrium, the average dissipation over a period, t, will be positive. For field driven flow in the thermodynamic and small field limits, the dissipation function becomes proportional to the rate of entropy production from linear irreversible thermodynamics. It can therefore be considered as an entropy-like quantity that remains useful far from equilibrium and for relaxation processes. The dissipation function also appears in three important theorems in nonequilibrium statistical mechanics: the fluctuation theorem, the dissipation theorem and the relaxation theorem. In this chapter we introduce the dissipation function and the theorems, and show how they quantify the emergence of irreversible behaviour in perturbed, steady state, and relaxing nonequilibrium systems. We also examine the behaviour of the dissipation function in terms of the extrema of the function using numerical and analytical approaches.

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Fußnoten
1
Note that this is often trivially satisfied in a stochastic system.
 
2
Outside of equilibrium, microscopic temperature expressions are ill defined. Often expressions such as the kinetic temperature (the equipartition expression in momenta) or configurational temperature (a similar expression in position) are used, however these expressions only correspond to the temperature of the system, and each other, at equilibrium [15].
 
3
Equation (2.7) applies to systems with no field or a constant field. For the case of a time-dependent field see [19].
 
4
Note that a special case of this relation was derived much earlier, see [21] for details.
 
5
We note that \( - \int_{0}^{ - t} \varOmega (\varvec{\varGamma }(s))ds = \int_{0}^{t} \varOmega (\varvec{\varGamma }^{*} (s))ds \) if the dynamics are time reversible and that \( \int \varOmega (\varvec{\varGamma }^{*} ,s)f(\varvec{\varGamma })d\varvec{\varGamma } = \int \varOmega (\varvec{\varGamma }^{*} ,s)f(\varvec{\varGamma }^{*} )d\varvec{\varGamma }^{*} = \left\langle {\varOmega (s)} \right\rangle \) since the probability of observing ensemble members is constant (\( f(\varvec{\varGamma })d\varvec{\varGamma } = f(\varvec{\varGamma }^{*} )d\varvec{\varGamma }^{*} \), see [1]). We assume that the system eventually relaxes to an equilibrium state, \( f_{2} (\varvec{\varGamma }) = \mathop {\lim }\nolimits_{t \to \infty } \ln f(\varvec{\varGamma },t) \). From Eq. (2.7), \( \mathop {\lim }\nolimits_{t \to \infty } \ln f(\varvec{\Upgamma},t) = \mathop {\lim }\nolimits_{t \to \infty } ( - \int_{0}^{ - t} \Upomega (\varvec{\varGamma }(s))ds + \ln f(\varvec{\varGamma },0)) \), which can be expressed \( \ln f_{2} (\varvec{\varGamma }) = \int_{0}^{\infty } \varOmega (\varvec{\varGamma }^{*} (s))ds + \ln f_{1} (\varvec{\varGamma }) \). Then taking the ensemble average with respect to the initial distribution function we obtain \( \left\langle {\ln f_{2} } \right\rangle_{{f_{1} }} = \left\langle {\varOmega_{\infty } } \right\rangle_{{f_{1} }} + \left\langle {\ln f_{1} } \right\rangle_{{f_{1} }}. \)
 
6
A conformal system relaxes such that the nonequilibrium distribution is of the form (\( f(\varvec{\varGamma },t) = \exp ( - \beta H(\varvec{\varGamma }) + \lambda (t)g(\varvec{\varGamma }))/Z,\forall t \)) and the deviation function, \( g \), is a constant over the relaxation.
 
7
Strictly a system relaxes as time tends towards infinity, but in practice at \( t_{eq} \) the system has relaxed.
 
8
Simulation parameters: 50 Fluid particles, 22 Wall particles, T = 1, \( \rho = 0.3 \), 100,000 trajectories.
 
9
Simulation parameters: 64 Fluid particles, 64 Wall particles, T = 1, \( \rho = 0.8 \), 100,000 Trajectories.
 
Literatur
1.
Zurück zum Zitat Evans, D.J., Williams, S.R., Searles, D.J.: J. Chem. Phys. 135, 194107 (2011)CrossRef Evans, D.J., Williams, S.R., Searles, D.J.: J. Chem. Phys. 135, 194107 (2011)CrossRef
2.
Zurück zum Zitat de Groot, S., Mazur, P.: Non-Equilibrium Thermodynamics. Dover Books on Physics. Dover Publications, NY (1984) de Groot, S., Mazur, P.: Non-Equilibrium Thermodynamics. Dover Books on Physics. Dover Publications, NY (1984)
10.
Zurück zum Zitat Broda, E., Gay, L.: Ludwig Boltzmann: man, physicist, philosopher. Ox Bow Press, Woodbridge (1983). Translating: L. Boltzmann, Re-joinder to the Heat Theoretical Considerations of Mr E. Zermelo (1896) Broda, E., Gay, L.: Ludwig Boltzmann: man, physicist, philosopher. Ox Bow Press, Woodbridge (1983). Translating: L. Boltzmann, Re-joinder to the Heat Theoretical Considerations of Mr E. Zermelo (1896)
11.
12.
Zurück zum Zitat Carberry, D.M., Reid, J.C., Wang, G.M., Sevick, E.M., Searles, D.J., Evans, D.J.: Phys. Rev. Lett. 92, 140601 (2004)CrossRef Carberry, D.M., Reid, J.C., Wang, G.M., Sevick, E.M., Searles, D.J., Evans, D.J.: Phys. Rev. Lett. 92, 140601 (2004)CrossRef
13.
Zurück zum Zitat Joubaud, S., Garnier, N.B., Ciliberto, S.: J. Stat. Mech: Theory Exp. 2007, P09018 (2007)CrossRef Joubaud, S., Garnier, N.B., Ciliberto, S.: J. Stat. Mech: Theory Exp. 2007, P09018 (2007)CrossRef
14.
15.
16.
Zurück zum Zitat Evans, D.J., Searles, D.J., Williams, S.R.: J. Chem. Phys. 128, 014504 (2008)CrossRef Evans, D.J., Searles, D.J., Williams, S.R.: J. Chem. Phys. 128, 014504 (2008)CrossRef
17.
Zurück zum Zitat Evans, D.J., Searles, D.J., Williams, S.R.: J. Stat. Mech: Theory Exp. 2009, P07029 (2009)CrossRef Evans, D.J., Searles, D.J., Williams, S.R.: J. Stat. Mech: Theory Exp. 2009, P07029 (2009)CrossRef
18.
Zurück zum Zitat Evans, D.J., Searles, D.J., Williams, S.R.: Diffusion fundamentals III. In: Chmelik, C., Kanellopoulos, N., Karger, J., Theodorou, D. (eds.), pp. 367–374. Leipziger Universitats Verlag, Leipzig (2009) Evans, D.J., Searles, D.J., Williams, S.R.: Diffusion fundamentals III. In: Chmelik, C., Kanellopoulos, N., Karger, J., Theodorou, D. (eds.), pp. 367–374. Leipziger Universitats Verlag, Leipzig (2009)
19.
20.
Zurück zum Zitat Evans, D.J., Williams, S.R., Searles, D,J.: Nonlinear dynamics of nanosystems. In: Radons, G., Rumpf, B., Schuster, H. (eds.), pp. 84–86. Wiley-VCH, NJ (2010) Evans, D.J., Williams, S.R., Searles, D,J.: Nonlinear dynamics of nanosystems. In: Radons, G., Rumpf, B., Schuster, H. (eds.), pp. 84–86. Wiley-VCH, NJ (2010)
21.
Zurück zum Zitat Evans, D.J., Morriss, G.P.: Statistical Mechanics of Nonequilibrium Liquids. Cambridge Unviersity Press, Cambridge (2008)CrossRefMATH Evans, D.J., Morriss, G.P.: Statistical Mechanics of Nonequilibrium Liquids. Cambridge Unviersity Press, Cambridge (2008)CrossRefMATH
23.
24.
Zurück zum Zitat Brookes, S.J., Reid, J.C., Evans, D.J., Searles, D.J.: J. Phys: Conf. Ser. 297, 012017 (2011)CrossRef Brookes, S.J., Reid, J.C., Evans, D.J., Searles, D.J.: J. Phys: Conf. Ser. 297, 012017 (2011)CrossRef
25.
Zurück zum Zitat Hartkamp, R., Bernardi, S., Todd, B.D.: J. Chem. Phys. 136, 064105 (2012)CrossRef Hartkamp, R., Bernardi, S., Todd, B.D.: J. Chem. Phys. 136, 064105 (2012)CrossRef
26.
Zurück zum Zitat Evans, D.J., Searles, D.J., Williams, S.R.: J. Chem. Phys. 132, 024501 (2010)CrossRef Evans, D.J., Searles, D.J., Williams, S.R.: J. Chem. Phys. 132, 024501 (2010)CrossRef
27.
Zurück zum Zitat Reid, J.C., Evans, D.J., Searles, D.J.: J. Chem. Phys. 136, 021101 (2012)CrossRef Reid, J.C., Evans, D.J., Searles, D.J.: J. Chem. Phys. 136, 021101 (2012)CrossRef
34.
Zurück zum Zitat Weeks, J.D., Chandler, D., Andersen, H.C.: J. Chem. Phys. 54, 5237 (1971)CrossRef Weeks, J.D., Chandler, D., Andersen, H.C.: J. Chem. Phys. 54, 5237 (1971)CrossRef
36.
37.
Zurück zum Zitat Zhang, F., Isbister, D.J., Evans, D.J.: Phys. Rev. E 64, 021102 (2001)CrossRef Zhang, F., Isbister, D.J., Evans, D.J.: Phys. Rev. E 64, 021102 (2001)CrossRef
40.
Zurück zum Zitat Dzwinel, W., Alda, W., Pogoda, M., Yuen, D.: Physica D 137, 157 (2000)CrossRef Dzwinel, W., Alda, W., Pogoda, M., Yuen, D.: Physica D 137, 157 (2000)CrossRef
41.
Zurück zum Zitat Kadau, K., Germann, T.C., Hadjiconstantinou, N.G., Lomdahl, P.S., Dimonte, G., Holian, B.L., Alder, B.J.: Proc. Natl. Acad. Sci. U.S.A. 101, 5851 (2004)MathSciNetCrossRefMATH Kadau, K., Germann, T.C., Hadjiconstantinou, N.G., Lomdahl, P.S., Dimonte, G., Holian, B.L., Alder, B.J.: Proc. Natl. Acad. Sci. U.S.A. 101, 5851 (2004)MathSciNetCrossRefMATH
42.
Zurück zum Zitat Hoover, W.G.: Lecture Notes in Physics 258, Molecular Dynamics. Springer, London (1986) Hoover, W.G.: Lecture Notes in Physics 258, Molecular Dynamics. Springer, London (1986)
Metadaten
Titel
The Dissipation Function: Its Relationship to Entropy Production, Theorems for Nonequilibrium Systems and Observations on Its Extrema
verfasst von
James C. Reid
Sarah J. Brookes
Denis J. Evans
Debra J. Searles
Copyright-Jahr
2014
Verlag
Springer Berlin Heidelberg
DOI
https://doi.org/10.1007/978-3-642-40154-1_2