## Stochastic measures and modular evolution in non-equilibrium thermodynamics (908)

### BibTeX

@MISC{Hernández-lemus908stochasticmeasures,

author = {Enrique Hernández-lemus and Jesús K. Estrada-gil},

title = {Stochastic measures and modular evolution in non-equilibrium thermodynamics},

year = {908}

}

### OpenURL

### Abstract

We present an application of the theory of stochastic processes to model and categorize non-equilibrium physical phenomena. The concepts of uniformly continuous probability measures and modular evolution lead to a systematic hierarchical structure for (physical) correlation functions and non-equilibrium thermodynamical potentials. It is proposed that macroscopic evolution equations (such as dynamic correlation functions) may be obtained from a non-equilibrium thermodynamical description, by using the fact that extended thermodynamical potentials belongs to a certain class of statistical systems whose probability distribution functions are defined by a stationary measure; although a measure which is, in general, different from the equilibrium Gibbs measure. These probability measures obey a certain hierarchy on its stochastic evolution towards the most probable (stationary) measure. This in turns defines a convergence sequence. We propose a formalism which considers the mesoscopic stage (typical of nonlocal dissipative processes such as the ones described by extended irreversible thermodynamics) as being governed by stochastic dynamics due to the effect of non-equilibrium fluctuations. Some applications of the formalism are described. Scope The paper is outlined as follows: Section 1 is a brief introduction to the problem of applying measure theoretical tools to the study of many-particle physical systems, also some recent developments in the field are mentioned. We sketch how the probability measures approach has been applied to equilibrium systems (states). In section 2 we present an extension of such method for the case of non-equilibrium systems (processes) by means of the Choquet-Meyer Theorem on continuous measures. In this section two propositions (2.1 and 2.3) are made in terms of non-equilibrium stochastic measures as how to deal with systems out of

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Citation Context ... and entropic-measure when referring to the latter. 5 Of course it is possible to replace this modular evolution by the action of a Liouvillian super-operator or propagator. See for example [1], also =-=[52]-=- For a modern introduction to modular operators see [20]6 The cyclical behavior of the trace (i.e. Trρη(τ) = Trρ ∗ η (τ + ıβ⋆ )) gives the following condition on the modular evolution between two sta... |

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Citation Context ...riables, when the latter are considered (as is the case in extended irreversible thermodynamics [13]) as coming from stochastic processes (in general non-markovian noises) in a mesoscopic lengthscale =-=[18, 12]-=-. This statement has only been proved formally for dilute gaseous systems by means of quantum kinetic theory [13, 18, 12], but there is strong evidence supporting its validity for several other physic... |

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Citation Context ... that, due to their intrinsic dissipative nature, such systems do not have a unique definition of entropy for each state [50, 13, 51], because of the non-volume preserving property of its phase space =-=[25, 11, 27]-=-. Since phase space volume is not preserved it is, in general impossible to define a unique global measure [59], derive from this measure a probability distribution function and characterize from it a... |

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Citation Context ... that, due to their intrinsic dissipative nature, such systems do not have a unique definition of entropy for each state [50, 13, 51], because of the non-volume preserving property of its phase space =-=[25, 11, 27]-=-. Since phase space volume is not preserved it is, in general impossible to define a unique global measure [59], derive from this measure a probability distribution function and characterize from it a... |

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Citation Context ...ription of a highly complex non-equilibrium property. For the case of non-equilibrium couplings and its relation to the observed phenomena of hysteresis (cf. section 3.3) it has been possible to show =-=[35]-=- that if the relaxation times of the phenomena are compared to the stochastic time given by the associated measure, then a criteria is given as to when will we experimentally observe hysteretic phenom... |

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Citation Context ...defined through a KMS-type condition. We can think of local thermodynamic equilibrium as a state that cannot be distinguished from a global equilibrium state by infinitesimally localized measurements =-=[36]-=-. The quantitative description of the term infinitesimally localized measurements could be given in terms of a one-parameter scaling procedure [36] over the observable. We introduce a parametric diffe... |

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Citation Context ...ty conservation, etc.) usually taken into account by some kind of Lagrange multiplier formalism (in the case of extended thermodynamics these multipliers are introduced, for example, by Liu’s Method) =-=[19, 38, 44]-=-. Nevertheless by applying a novel irreversible thermodynamics formalism [14] it has been stated that they also represent Gibbsian ensemble averages of a collection of macroscopic field variables, whe... |

2 |
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Citation Context ...cally relevant problems could be better understood in terms of uniformly continuous measures of parameter state spaces. The problem of extending the notion of Gibbs states to quasi-local interactions =-=[45]-=-, coupled quantum systems [37], the hierarchical structure of physical descriptions in terms of the coarse-graining [30] exemplifies such problems. The recent arise of the so-called non-extensive stat... |