## Steady state thermodynamics (2004)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Sasa04steadystate,

author = {Shin-ichi Sasa and Hal Tasaki},

title = {Steady state thermodynamics},

year = {2004}

}

### OpenURL

### Abstract

We propose a thermodynamic formalism that is expected to apply to a large class of nonequilibrium steady states including a heat conducting fluid, a sheared fluid, and an electrically conducting fluid. We call our theory steady state thermodynamics (SST) after Oono and Paniconi’s original proposal. The construction of SST is based on a careful examination of how the basic notions in thermodynamics should be modified in nonequilibrium steady states. We define all thermodynamic quantities through operational procedures, which can be (in principle) realized experimentally. Based on SST thus constructed, we make some nontrivial predictions, including an extension of Einstein’s formula on density fluctuation, an extension of the minimum work principle, the existence of a new osmotic pressure of a purely nonequilibrium origin, and a shift of coexistence temperature. All these predictions may be checked experimentally to test SST for its quantitative validity. Contents

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Citation Context ...conduction, i.e., obeys the Fourier law. Mathematically, basic results including the existence, uniqueness and mixing property of the nonequilibrium steady states are proved under suitable conditions =-=[28, 29]-=-, but no concrete information about the structure of the heat conducting state is available. Recently a new perturbative method for the nonequilibrium steady state of this model was developed [30]. 1.... |

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Citation Context ...mple, [37]. The simplicity of these models made it possible to resolve some delicate issues rigorously, a notable examples being the long-range correlations [38] and the anomalous current fluctuation =-=[39]-=-. A standard nontrivial model is the driven lattice gas [40], in which particles on lattice are subject to hard core on-site repulsion, nearest neighbor interaction, and a constant driving force. Many... |

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Citation Context ...uilibrium states were studied. See, for example, [37]. The simplicity of these models made it possible to resolve some delicate issues rigorously, a notable examples being the long-range correlations =-=[38]-=- and the anomalous current fluctuation [39]. A standard nontrivial model is the driven lattice gas [40], in which particles on lattice are subject to hard core on-site repulsion, nearest neighbor inte... |

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Citation Context ...It is therefore difficult to judge its physical validity and usefulness. (c) Weak coupling limit: In the weak coupling limit of quantum systems, the procedure of projection can be executed rigorously =-=[25]-=-. Relaxation to the steady state, the reciprocal relations in linear transport, and the principle of minimum entropy production are established. In this study, however, explicit forms of nonequilibriu... |

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Citation Context ...[18]. These results were obtained by employing the projection operator method pioneered by Zwanzig [19] and Mori [20]. Furthermore, through a formal argument based on the Liouville equation, McLennan =-=[21]-=- and Zubarev [22] proposed a measure that describes (or is claimed to describe) nonequilibrium states. Although the derivations of these results involve (often uncontrolled) assumptions, the nonlinear... |

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Citation Context ...awasaki-Gunton formula, and the power-low decay of spatial correlations are believed to be physically sound, since they can also be derived in simple manners from phenomenological considerations. See =-=[23]-=- for the nonlinear response relation, [24] for the Kawasaki-Gunton formula, and [18] for the long range correlations. The measure proposed by McLennan and Zubarev is supported by neither a controlled ... |

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Citation Context ... and if statistical mechanics for nonequilibrium steady states is constructed. 14was found [46, 47]. The fluctuation theorem was then extended to nonequilibrium steady states in various systems. See =-=[48, 49, 50]-=-. Now it is understood that the essence of the fluctuation theorem lies in the fact that the relevant nonequilibrium steady states are described by Gibbs measures for space-time configurations [50]. I... |

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Citation Context ...nce of long range spatial correlations of fluctuations in nonequilibrium steady states [18]. These results were obtained by employing the projection operator method pioneered by Zwanzig [19] and Mori =-=[20]-=-. Furthermore, through a formal argument based on the Liouville equation, McLennan [21] and Zubarev [22] proposed a measure that describes (or is claimed to describe) nonequilibrium states. Although t... |

33 |
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Citation Context ... Lebowitz, and Speer obtained exact large deviation functionals for the density profiles in the nonequilibrium steady states of the one dimensional lattice gas models (the symmetric exclusion process =-=[52, 53]-=- and the asymmetric exclusion process [54, 55]) attached to two particle baths with different chemical potentials. In the equilibrium states, the corresponding large deviation functional coincides wit... |

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Citation Context ...eady states is called the principle of minimum entropy production [12]. It is understood that all the correct variational principles in linear transport regime are based on the Onsager-Machlup theory =-=[13]-=- which concerns a large deviation functional for the history of fluctuations. See, for example, [14]. (d) Nonequilibrium thermodynamics: Flux-force relations with the reciprocity constitute fundamenta... |

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Citation Context ... Lebowitz, and Speer obtained exact large deviation functionals for the density profiles in the nonequilibrium steady states of the one dimensional lattice gas models (the symmetric exclusion process =-=[52, 53]-=- and the asymmetric exclusion process [54, 55]) attached to two particle baths with different chemical potentials. In the equilibrium states, the corresponding large deviation functional coincides wit... |

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Citation Context ...to resolve some delicate issues rigorously, a notable examples being the long-range correlations [38] and the anomalous current fluctuation [39]. A standard nontrivial model is the driven lattice gas =-=[40]-=-, in which particles on lattice are subject to hard core on-site repulsion, nearest neighbor interaction, and a constant driving force. Many results, both theoretical and numerical, have been obtained... |

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Citation Context ...ave been obtained [37, 41], but the structure of the nonequilibrium steady state is still not very well understood except for some partial results including the recent perturbation expansion [42]. In =-=[43]-=-, hydrodynamic limit and fluctuation was studied for the nonequilibrium steady state in the driven lattice gas. Possibility of thermodynamics of driven lattice gas being “shapedependent” was pointed o... |

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Citation Context ...ation functionals for the density profiles in the nonequilibrium steady states of the one dimensional lattice gas models (the symmetric exclusion process [52, 53] and the asymmetric exclusion process =-=[54, 55]-=-) attached to two particle baths with different chemical potentials. In the equilibrium states, the corresponding large deviation functional coincides with the thermodynamic free energy. Moreover thei... |

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Citation Context ...matical structure consisting of mathematical relations among certain quantities in a physical system. The mathematical structure of thermodynamics is clearly and abstractly explained, for example, in =-=[1, 2, 3]-=-. The conventional thermodynamics for equilibrium systems is a typical and no doubt the most important example of thermodynamics, but it is not the only example (see, for example, section 3 of [2] and... |

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Citation Context ...mula [17] for a nonlinear shear viscosity and normal stresses. Another example is the establishment of the existence of long range spatial correlations of fluctuations in nonequilibrium steady states =-=[18]-=-. These results were obtained by employing the projection operator method pioneered by Zwanzig [19] and Mori [20]. Furthermore, through a formal argument based on the Liouville equation, McLennan [21]... |

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Citation Context .... The shift of the critical temperature under the influence of shear flow as well as the corresponding critical exponents were calculated by analyzing the so called model H with the steady shear flow =-=[36]-=-. 6 The Boltzmann equation can be derived from the BBGKY hierarchy in a low density limit around the (spatially uniform) equilibrium state. (See [32] for the mathematical justification of the derivati... |

6 |
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Citation Context ...nduction in classical mechanics is the chain of coupled anharmonic oscillators whose two ends are attached to two heat baths with different temperatures. From numerical simulations (see, for example, =-=[27]-=-) it is expected that the model exhibits a “healthy” heat conduction, i.e., obeys the Fourier law. Mathematically, basic results including the existence, uniqueness and mixing property of the nonequil... |

6 |
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Citation Context ...[28, 29], but no concrete information about the structure of the heat conducting state is available. Recently a new perturbative method for the nonequilibrium steady state of this model was developed =-=[30]-=-. 1.3.3 Approaches from meso-scale models We turn to approaches to nonequilibrium steady states that employ a class of models which are neither microscopic (as in mechanical treatments) nor macroscopi... |

5 |
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Citation Context ...matical structure consisting of mathematical relations among certain quantities in a physical system. The mathematical structure of thermodynamics is clearly and abstractly explained, for example, in =-=[1, 2, 3]-=-. The conventional thermodynamics for equilibrium systems is a typical and no doubt the most important example of thermodynamics, but it is not the only example (see, for example, section 3 of [2] and... |

5 |
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Citation Context ...cs, the Liouville equation can be a starting point for microscopic considerations. An example is the derivation of the non-linear response relation of [16], which leads to the Kawasaki-Gunton formula =-=[17]-=- for a nonlinear shear viscosity and normal stresses. Another example is the establishment of the existence of long range spatial correlations of fluctuations in nonequilibrium steady states [18]. The... |

4 |
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Citation Context ...d on the Liouville equation: In classical mechanics, the Liouville equation can be a starting point for microscopic considerations. An example is the derivation of the non-linear response relation of =-=[16]-=-, which leads to the Kawasaki-Gunton formula [17] for a nonlinear shear viscosity and normal stresses. Another example is the establishment of the existence of long range spatial correlations of fluct... |

2 |
Kinetic Theory of a Dilute Gas System under Steady Heat Conduction
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Citation Context ...ium phenomena 6 , nonequilibrium stationary distribution functions have been calculated. Recently, for example, a systematic calculation for heat conducting nonequilibrium steady states was performed =-=[33, 34]-=-. Such a study reveals detailed properties of the nonequilibrium steady states, and may become an important guide in construction of phenomenology and statistical mechanical theory. The relation of th... |

2 |
Test of information theory on the Boltzmann equation
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Citation Context ...ium phenomena 6 , nonequilibrium stationary distribution functions have been calculated. Recently, for example, a systematic calculation for heat conducting nonequilibrium steady states was performed =-=[33, 34]-=-. Such a study reveals detailed properties of the nonequilibrium steady states, and may become an important guide in construction of phenomenology and statistical mechanical theory. The relation of th... |

1 |
Statistical mechanics: a short treatise
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Citation Context ...thermodynamics for equilibrium systems is a typical and no doubt the most important example of thermodynamics, but it is not the only example (see, for example, section 3 of [2] and Appendix 1. A1 of =-=[4]-=-). Then it makes sense to look for a thermodynamics in a physical context other than equilibrium systems. We wish to do that for nonequilibrium steady states. If it turns out that there is no sensible... |

1 |
A new determination of molecular dimensions (University of Zurich dissertation
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(Show Context)
Citation Context ...nequilibrium physics is Einstein’s celebrated work on the Brownian motion. We have no intention of going deeply into the work, but wish to mention that Einstein’s formula D = kBT µ, (1.11) derived in =-=[5, 6]-=- represents a deep fact that the transport coefficient (the mobility µ) in a driven nonequilibrium state is directly related to the diffusion constant D, which characterizes fluctuation in the equilib... |

1 |
Linear response theory — a historical perspective
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(Show Context)
Citation Context ...ions was found in various contexts, the first example being that by Nyquist [9], who precedes Onsager. These formulae are now known under the generic name linear response relations. See, for example, =-=[10, 11]-=-. We believe that the conceptual basis of these relations should be sought in a certain form of regression hypothesis, i.e., quantitative correspondence between nonequilibrium transport and equilibriu... |

1 |
Variational principles in thermodynamics and statistical mechanics of irreversible processes
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(Show Context)
Citation Context ...e correct variational principles in linear transport regime are based on the Onsager-Machlup theory [13] which concerns a large deviation functional for the history of fluctuations. See, for example, =-=[14]-=-. (d) Nonequilibrium thermodynamics: Flux-force relations with the reciprocity constitute fundamental ingredients of the standard theory known as nonequilibrium thermodynamics, which provides a macros... |

1 |
High-temperature expansion for steady nonequilibrium states in driven lattice gases, preprint cond-mat/0407552
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(Show Context)
Citation Context ...erical, have been obtained [37, 41], but the structure of the nonequilibrium steady state is still not very well understood except for some partial results including the recent perturbation expansion =-=[42]-=-. In [43], hydrodynamic limit and fluctuation was studied for the nonequilibrium steady state in the driven lattice gas. Possibility of thermodynamics of driven lattice gas being “shapedependent” was ... |

1 |
Shape-dependent thermodynamics and nonlocal hydrodynamics in a non-Gibbsian steady state of a drift-diffusion system, Phys
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(Show Context)
Citation Context ...odynamic limit and fluctuation was studied for the nonequilibrium steady state in the driven lattice gas. Possibility of thermodynamics of driven lattice gas being “shapedependent” was pointed out in =-=[44]-=-. In SST, such a shape-dependence is properly taken into account in the basic formalism. For us the driven lattice gas provides a very nice “proving ground” for various proposals and conjectures of SS... |

1 |
Thermodynamic relations in a driven lattice gas: numerical experiments
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(Show Context)
Citation Context ...attice gas provides a very nice “proving ground” for various proposals and conjectures of SST. Some of our discussions in the present paper are based on earlier numerical works by Hayashi and Sasa in =-=[45]-=-. In Appendix A of the present paper, we also discuss theoretical results about SST realized in driven lattice gases. In spite of all these interesting works, we always have to keep in mind that physi... |

1 |
Morrriss, Probability of second law violations
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Citation Context ...m, 7 Needless to say, thermodynamic phases may be in principle determined from microscopic descriptions when and if statistical mechanics for nonequilibrium steady states is constructed. 14was found =-=[46, 47]-=-. The fluctuation theorem was then extended to nonequilibrium steady states in various systems. See [48, 49, 50]. Now it is understood that the essence of the fluctuation theorem lies in the fact that... |