On the fluctuation-dissipation theorem for convective processes

Alan J. McKane, Federico Vázquez, Miguel A. Olivares-Robles

Research output: Contribution to journalArticleResearchpeer-review

Abstract

When making the connection between the thermodynamics of irreversible processes and the theory of stochastic processes through the uctuation-dissipation theorem, it is necessary to invoke a postulate of the Einstein-Boltzmann type. For convective processes hydrodynamic uctuations must be included; the velocity is a dynamical variable and although the entropy cannot depend directly on the velocity, δ2S will depend on velocity variations. Some authors do not include velocity variations in δ2S, and so have to introduce a non-thermodynamic function which replaces the entropy and does depend on the velocity. At rst sight, it seems that the introduction of such a function requires a generalisation of the Einstein-Boltzmann relation to be invoked. We review the reason why it is not necessary to introduce such a function, and therefore why there is no need to generalise the Einstein-Boltzmann relation in this way. We then obtain the uctuation-dissipation theorem, which shows some differences as compared with the non-convective case. We also show that δ2S is a Liapunov function when it includes velocity uctuations. © © Walter de Gruyter 2007.
Original languageAmerican English
Pages (from-to)29-40
Number of pages12
JournalJournal of Non-Equilibrium Thermodynamics
DOIs
StatePublished - 20 Feb 2007

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dissipation
theorems
Entropy
entropy
Liapunov functions
irreversible processes
visual perception
stochastic processes
axioms
Random processes
Hydrodynamics
hydrodynamics
Thermodynamics
thermodynamics

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title = "On the fluctuation-dissipation theorem for convective processes",
abstract = "When making the connection between the thermodynamics of irreversible processes and the theory of stochastic processes through the uctuation-dissipation theorem, it is necessary to invoke a postulate of the Einstein-Boltzmann type. For convective processes hydrodynamic uctuations must be included; the velocity is a dynamical variable and although the entropy cannot depend directly on the velocity, δ2S will depend on velocity variations. Some authors do not include velocity variations in δ2S, and so have to introduce a non-thermodynamic function which replaces the entropy and does depend on the velocity. At rst sight, it seems that the introduction of such a function requires a generalisation of the Einstein-Boltzmann relation to be invoked. We review the reason why it is not necessary to introduce such a function, and therefore why there is no need to generalise the Einstein-Boltzmann relation in this way. We then obtain the uctuation-dissipation theorem, which shows some differences as compared with the non-convective case. We also show that δ2S is a Liapunov function when it includes velocity uctuations. {\circledC} {\circledC} Walter de Gruyter 2007.",
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On the fluctuation-dissipation theorem for convective processes. / McKane, Alan J.; Vázquez, Federico; Olivares-Robles, Miguel A.

In: Journal of Non-Equilibrium Thermodynamics, 20.02.2007, p. 29-40.

Research output: Contribution to journalArticleResearchpeer-review

TY - JOUR

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AU - Vázquez, Federico

AU - Olivares-Robles, Miguel A.

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N2 - When making the connection between the thermodynamics of irreversible processes and the theory of stochastic processes through the uctuation-dissipation theorem, it is necessary to invoke a postulate of the Einstein-Boltzmann type. For convective processes hydrodynamic uctuations must be included; the velocity is a dynamical variable and although the entropy cannot depend directly on the velocity, δ2S will depend on velocity variations. Some authors do not include velocity variations in δ2S, and so have to introduce a non-thermodynamic function which replaces the entropy and does depend on the velocity. At rst sight, it seems that the introduction of such a function requires a generalisation of the Einstein-Boltzmann relation to be invoked. We review the reason why it is not necessary to introduce such a function, and therefore why there is no need to generalise the Einstein-Boltzmann relation in this way. We then obtain the uctuation-dissipation theorem, which shows some differences as compared with the non-convective case. We also show that δ2S is a Liapunov function when it includes velocity uctuations. © © Walter de Gruyter 2007.

AB - When making the connection between the thermodynamics of irreversible processes and the theory of stochastic processes through the uctuation-dissipation theorem, it is necessary to invoke a postulate of the Einstein-Boltzmann type. For convective processes hydrodynamic uctuations must be included; the velocity is a dynamical variable and although the entropy cannot depend directly on the velocity, δ2S will depend on velocity variations. Some authors do not include velocity variations in δ2S, and so have to introduce a non-thermodynamic function which replaces the entropy and does depend on the velocity. At rst sight, it seems that the introduction of such a function requires a generalisation of the Einstein-Boltzmann relation to be invoked. We review the reason why it is not necessary to introduce such a function, and therefore why there is no need to generalise the Einstein-Boltzmann relation in this way. We then obtain the uctuation-dissipation theorem, which shows some differences as compared with the non-convective case. We also show that δ2S is a Liapunov function when it includes velocity uctuations. © © Walter de Gruyter 2007.

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