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Modos normales en sistemas dipolares semiconductores
(Teorema Fluctuacion-Disipacion para ProcesosConvectivos)
Miguel Angel Olivares-Roblesa
aSeccion de Posgrado e Investigacion, Escuela Superior de IngenierıaMecanica y Electrica Culhuacan-IPN, Av. Santa Ana 1000, Col. San
Francisco Culhuacan Coyoacan 04430, Mexico D.F.
Abstract
Este proyecto se dirigio inicialmente a estudiar la respuesta fısicade sistemas de baja dimensionalidad en presencia de campos magneticosexternos. El tema fue propuesto dentro del campo de estado solido deIngenierıa en Microelectronica. Solo que dado que mi contratacion serealizo para Ingenierıa de Sistemas Energeticos era mas propio traba-jar en el area de Fluidos como lınea de investigacion de esta maestrıa.Siendo esta prioritaria en mi incorporacion a la ESIME-CU, del IPN.Ası que el proyecto se dirigio a continuar trabajo de investigacion so-bre termodinamica de procesos convectivos en Fluidos.
La investigacion se enfoco sobre dos aspectos: Teorema Fluctuacion-Disipacion para Procesos Convectivos, y el estudio de la produccionde entropıa de flujos oscilatorios entre placas paralelas.Teorema Fluctuacion-Disipacion para Procesos Convectivos:Para hacer la conexion entre la termodinamica de procesos irreversiblesy la teorıa de procesos estocasticos es necesario invocar un postu-lado de tipo Einstein-Boltzmann. En este trabajo nosotros obtuvimosel teorema fluctuacion-disipacion, el cual mostro algunas diferenciascomparadas con el caso no-convectivo por ejemplo que d2S es unafuncion de Liapunov cuando se incluyen fluctuaciones en la velocidad.Produccion de Entropıa: Con las soluciones analıticas para loscampos de velocidad y de temperatura a la mano, calculamos la pro-duccion de entropıa promediada en el tiempo tanto local como globalpara un flujo oscilatorio de un fluido newtoniano y un fluido de Maxwell.
1 INTRODUCCION
Este trabajo Teorema Fluctuacion-Disipacion para Procesos Convectivos seenmarca dentro de la termodinamica de procesos irreversibles. El propositodel trabajo es abordar problemas no resueltos satisfactoriamente en esta ramade investigacion. El trabajo se esboza como sigue:
• Termodinamica Irreversible y Procesos Estocasticos
• El Teorema de fluctuaciondisipacion para Sistemas Convectivos
• Fluctuaciones en la velocidad y la Funcion de Liapunov
• Conclusiones
2 METODOS Y MATERIALES
Teorema Fluctuacion-Disipacion para Procesos Convectivos :Revisar las generalizaciones de la relacion de Einstein-Boltzmann por difer-entes autores. Descargar estas versiones electronicas via internet a partir desubscripciones a revistas cientıficas. Modificar o re-enunciar las propuestasmas viables para sistemas convectivos donde las variaciones de la veloci-dad juegan un papel importante. Realizamos los calculos pertinentes parala obtencion del teorema fluctuacion- disipacion, usando recursos como elHandbook of Mathematical-Functions de Abramowitz y Stegun de la edito-rial DOVER, el libro de thermodynamics de Callen, Segunda Edicion y ellibro Table of Integrals, series and Products de Gradshteyn y Rizhik Sextaedicion editorial Academic Press. Estos libros fueron adquiridos con los re-cursos del proyecto.La Produccion de Entropıa de Flujos Oscilatorioslos recursos del proyectos se usaron para actualizar el equipo de computo conlas tarjetas madre optimas para el calculo numerico de la produccion de en-tropıa. Adquirimos software y las subscripciones a revistas de investigacion.En esta fase del proyecto realizamos los calculos en colaboracion del Centrode Investigaciones en Energıa de la UNAM en Temixco, Morelos.El problema de transferencia de calor de un flujo oscilatorio para un fluidoNewtoniano y un fluido de Maxwell se aborda via las ecuaciones diferencialespara los campos de temperatura y velocidades con las condiciones a la fron-tera correspondientes a las condiciones del problema en nuestro caso placasparalelas. Despues aplicamos el metodo de la produccion de mınima entropıapara encontrar los parametros que minimizan la produccion de entropıa.
2
3 RESULTADOS
La investigacion sobre el Teorema Fluctuacion-Disipacion para Procesos Con-vectivos dio lugar a la publicacion de un artıculo de investigacion en la re-vista cientıfica internacional, JOURNAL OF NON-EQUILIBRIUM THER-MODYNAMICS. En los productos esperados del proyecto originalse propuso la publicacion de un articulo a nivel internacional y secumplio.Para procesos convectivos, las fluctuaciones hidrodinamicas deben ser in-cluıdas la velocidad es una variable dinamica y aunque la entropıa no puededepender directamente de la velocidad, d2S depende de las variaciones dela velocidad. Algunos autores no incluyen las variaciones de la velocidad end2S, y asi tienen que introducir una funcion no termodinamica que reem-plaza la entropıa y depende d ela velocidad. A primera vista parece quela introduccion de tal funcion requiere la generalizacion de la relacion deEinstein-Boltzmann. Nosotros revisamos porque no es necesario generalizarla relacion de Einstein-Boltzmann en esta manera.
Sobre la investigacion de La Produccion de Entropıa de Flujos Oscila-torios entre placas paralelas: Nuestros resultados de los calculos sobre laproduccion de entropıa para estos flujos se publicaron en el 4th Interna-tional workshop on nonequilibrium thermodynamics and complex fluids, 3-7september 2006, Rhodes, Greece.En los productos esperados del proyecto original se propuso la pub-licacion de un articulo en congreso internacional y se presentaronDOS TRABAJOS.Finalmente con respecto a los estudiantes de la maestrıa de sistemas en-ergeticos, por ser esta maestrıa de nueva creacion, no hubo suficientes estu-diantes para incluirlos en la investigacion de cada profesor de la maestrıa.En mi caso no fue asignado ninguno.
4 IMPACTO
Nuestros resultados sobre Flujos Oscilatorios entre Placas Paralelas con-tribuyen a la minimizacion de las irreversibilidades en el flujo de fluidos entreplacas ayudando a entender el mecanismo optimo para el aprovechamientode la energıa en fluidos confimados.Con respecto al Teorema Fluctuacion-Disipacion para Procesos Convectivos,segun los comentarios de los arbitros: ”Reviewer #1: This is an excellent,concise and well written paper. It attacks a very fundamental issue in ir-reversible thermodynamics: In which form does the fluctuation-dissipationtheorem hold for convective processes? Old masters have attacked this prob-
3
lem before without satisfactory results. This paper seems to resolve theissue.”Lo que contribuye a la formacion de investigadores en el estudio de la ter-modinamica de procesos irreversibles y la teorıa de procesos estocasticos.
AGRADECIMIENTOS Deseamos agradecer a Y. Oono y a M. Lopez deHaro por sus utiles discusiones.
4
4th International workshop on nonequilibrium thermodynamics and complex fluids
IWNET
2006
4th International workshop on nonequilibrium thermodynamics and complex fluids 3-7 september 2006, Rhodes, Greece
Home Scope
Location Speakers Program
Abstracts Author index
Organizing Committee Scientific Committee
Venue & Accomodation Registration
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Medieval City of Rhodes
Program starts on Sunday, Sep. 3rd, 6.30pm, and ends on Thursday, Sep. 7th, at 4.00pm The registration desk will be available: on Sunday, Sept. the 3rd, in the afternoon between 5.00pm-10.30pm, and all day on Monday, Sept. the 4th.
In order to find your way to the conference site or hotel », to browse through the program », the abstracts » or author index » etc. please use the navigation bar on the left.
News
● August, 26th, 2006 Extended versions of some of the papers that will be presented in the Workshop will be submitted for publication in a special issue of the Journal of Non-Newtonian Fluid Mechanics (JNNFM), with Guest Editors: Vlasis Mavrantzas, Thanos Tzavaras and Antony Beris. The deadline for paper submission for this Special Volume is November 30th, 2006.
● July 24th, 2006 Program available online.
http://www.complexfluids.ethz.ch/cgi-bin/CONF/c (1 de 2)28/08/2006 07:56:13 p.m.
4th International workshop on nonequilibrium thermodynamics and complex fluids
● July 10th, 2006 Book of abstracts & corresponding author index available online.
● March 6th, 2006 Prof. Masao Doi has cancelled his participation due to a conflict of the conference dates with some important duties of his as a chairman of his department. The organizing committee contacted Professor Akira Onuki to replace him. Prof. Onuki has kindly accepted the invitation and he will be one of the three invited speakers.
© and Kleanthi for IWNET 2006
http://www.complexfluids.ethz.ch/cgi-bin/CONF/c (2 de 2)28/08/2006 07:56:13 p.m.
4th International workshop on nonequilibrium thermodynamics and complex fluids
14:25 2-Fluid Viscoelasticity _» H. Pleiner, J.L. Harden
14:50 Selected nonlinear physical properties of liquid crystalline elastomers _» A.M. Menzel, H.R. Brand
15:15 Suspensions of rodlike molecules: phase transition and equlilibration time scale for a shear flow _» F. Otto, C. Löschke, J. Wachsmuth
15:40 Coffee break
15:45 Poster Session »
16:35 Free time
17:00 Excursion
20:30 End of workshop day 3/5
Day 4: Wednesday morning, September 6, 2006
Session 5 Non-equilibrium thermodynamics: Approaches and formalisms Chair: A.N. Beris
08:00 Towards a thermodynamics of complex systems_» G. Nicolis
08:50 Non-Equilibrium Thermodynamics of Boundary Conditions _» H.C. Öttinger
09:15 Extended thermodynamics of polymers and superfluids_» D. Jou, J. Casas-Vazquez, M. Criado-Sancho, M.S. Mongiovi
09:40 Kinematics of turbulence in simple and polymeric fluids _» M. Grmela
http://www.complexfluids.ethz.ch/cgi-bin/CONF/c?program=on (5 de 8)28/08/2006 07:57:21 p.m.
4th International workshop on nonequilibrium thermodynamics and complex fluids
10:05 Coffee break
10:20 Nonequilibrium thermodynamics of elasto-viscoplastic deformation _» M. Hütter, T.A. Tervoort, H.C. Öttinger
10:45 On a possible difference between the barycentric velocity and the velocity that gives translational momentum in fluids _» D. Bedeaux, S. Kjelstrup, H.C. Öttinger
11:10 Non-Equilibrium Thermodynamic Fluctuations within the Framework of Path Integrals _» A. McKane, F. Vazquez, M.A. Olivares-Robles
11:35 Discussion A.N. Beris
12:05 Lunch
Day 4: Wednesday afternoon, September 6, 2006
Session 6 Coarse-graining and mesoscopic dynamics - some mathematical aspects Chair: B.J. Edwards
14:00 Dissipation and Stress _» P. Constantin
14:25 Self-similarity in Smoluchowski's coagulation equation _» G. Menon, R.L. Pego
14:50 Stress relaxation theories in the approximation of polyconvex elastodynamics by viscoelasticity _» T. Tzavaras
15:15 Numerical analysis of coarse-graining for stochastic systems _» P. Plechac
15:40 Coffee break
http://www.complexfluids.ethz.ch/cgi-bin/CONF/c?program=on (6 de 8)28/08/2006 07:57:21 p.m.
4th International workshop on nonequilibrium thermodynamics and complex fluids
IWNET
2006
4th International workshop on nonequilibrium thermodynamics and complex fluids 3-7 september 2006, Rhodes, Greece
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ORAL PRESENTATION
Session: 5 Non-equilibrium thermodynamics: Approaches and formalisms (scheduled: Wednesday, 11:10 )
Non-Equilibrium Thermodynamic Fluctuations within the Framework of Path
Integrals
A. McKane1, F. Vazquez2, M.A. Olivares-Robles3 1 Theory Group, School of Physics and Astronomy, University of Manchester M13 9PL, United
Kingdom
2 Facultad de Ciencias, UAEM, Av. Universidad 1001, Cuernavaca, Morelos 62209, Mexico
3 Seccion de Investigacion y Posgrado, ESIME-Culhuacan, IPN, Mexico D.F.
Fluctuational non-equilibrium thermodynamics is formulated in terms of path integrals. The theory is presented in such a way that it will be applicable to a wide class of stochastic processes, including non-Markovian processes. In particular, we show how to construct the path-integral scheme when the noise-correlation matrix is singular, which is the case for fluctuations in non-equilibrium thermodynamics, since the continuity equation has no stochastic term associated with it. The theory is illustrated by calculating the light-scattering spectrum in fluids. Non-linear contributions to this quantity are also computed. © IWNET 2006
© and Kleanthi for IWNET 2006
http://www.complexfluids.ethz.ch/cgi-bin/CONF/c?search=Olivares-Robles&COUNT=2828/08/2006 08:00:57 p.m.
4th International workshop on nonequilibrium thermodynamics and complex fluids
IWNET
2006
4th International workshop on nonequilibrium thermodynamics and complex fluids 3-7 september 2006, Rhodes, Greece
Home Scope
Location Speakers Program
Abstracts Author index
Organizing Committee Scientific Committee
Venue & Accomodation Registration
News Contact
Medieval City of Rhodes
Program starts on Sunday, Sep. 3rd, 6.30pm, and ends on Thursday, Sep. 7th, at 4.00pm The registration desk will be available: on Sunday, Sept. the 3rd, in the afternoon between 5.00pm-10.30pm, and all day on Monday, Sept. the 4th.
In order to find your way to the conference site or hotel », to browse through the program », the abstracts » or author index » etc. please use the navigation bar on the left.
News
● August, 26th, 2006 Extended versions of some of the papers that will be presented in the Workshop will be submitted for publication in a special issue of the Journal of Non-Newtonian Fluid Mechanics (JNNFM), with Guest Editors: Vlasis Mavrantzas, Thanos Tzavaras and Antony Beris. The deadline for paper submission for this Special Volume is November 30th, 2006.
● July 24th, 2006 Program available online.
http://www.complexfluids.ethz.ch/cgi-bin/CONF/c (1 de 2)28/08/2006 07:56:13 p.m.
4th International workshop on nonequilibrium thermodynamics and complex fluids
● July 10th, 2006 Book of abstracts & corresponding author index available online.
● March 6th, 2006 Prof. Masao Doi has cancelled his participation due to a conflict of the conference dates with some important duties of his as a chairman of his department. The organizing committee contacted Professor Akira Onuki to replace him. Prof. Onuki has kindly accepted the invitation and he will be one of the three invited speakers.
© and Kleanthi for IWNET 2006
http://www.complexfluids.ethz.ch/cgi-bin/CONF/c (2 de 2)28/08/2006 07:56:13 p.m.
4th International workshop on nonequilibrium thermodynamics and complex fluids
15:55 Mathematical and computational methods for coarse-graining _» M.A. Katsoulakis
16:20 Some mathematical issues arising in the multiscale modelling of complex fluids_» C. Le Bris
16:45 Discussion B.J. Edwards
17:00 Free time
20:30 Gala dinner
22:30 End of workshop day 4/5
Day 5: Thursday morning, September 7, 2006
Session 7 Applications to complex materials: glasses, micelles, colloids, blends, interfaces Chair: V.G. Mavrantzas
08:00 Ergodicity-breaking in glassforming liquids (and related systems), and relaxation processes below the glass temperature, Tg_» C.A. Angell
08:50 Entropy production of oscillatory flows between parallel plates _» M. Lopez de Haro, S. Cuevas, M.A. Olivares-Robles, F. Vazquez
09:15 A thermodynamically consistent model for the thixotropic rheological behavior of concentrated colloidal star polymer solutions_» A.N. Beris, D. Vlassopoulos
09:40 Extrudate swell control by balancing short and long polyethylene chains using multi-scale modeling _» C.F.J. den Doelder
10:05 Coffee break
http://www.complexfluids.ethz.ch/cgi-bin/CONF/c?program=on (7 de 8)28/08/2006 07:57:21 p.m.
4th International workshop on nonequilibrium thermodynamics and complex fluids
10:20 Flow of Polymer blends between Concentric Cylinders _» M. Dressler, B.J. Edwards, E.J. Windhab
10:45 On the rheology of a dilute suspension of vesicles _» C. Misbah, G. Danker
11:10 Atomistic molecular dynamics simulation of the temperature and pressure dependences of local and terminal relaxations in cis-1,4-polybutadiene_» G. Tsolou, V.G. Mavrantzas
11:35 Discussion V.G. Mavrantzas
12:00 Lunch
14:00 Discussion - Closing remarks V.G. Mavrantzas
15:00 End of workshop day 5/5
© and Kleanthi for IWNET 2006
http://www.complexfluids.ethz.ch/cgi-bin/CONF/c?program=on (8 de 8)28/08/2006 07:57:21 p.m.
4th International workshop on nonequilibrium thermodynamics and complex fluids
IWNET
2006
4th International workshop on nonequilibrium thermodynamics and complex fluids 3-7 september 2006, Rhodes, Greece
Home Scope
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ORAL PRESENTATION
Session: 7 Applications to complex materials: glasses, micelles, colloids, blends, interfaces (scheduled: Thursday, 08:50 )
Entropy production of oscillatory flows between parallel plates
M. Lopez de Haro1, S. Cuevas1, M.A. Olivares-Robles2, F. Vazquez3 1 Centro de Investigacion en Energia, UNAM, Temixco, Morelos 62580, Mexico
2 Seccion de Investigacion y Posgrado, ESIME-Culhuacan, IPN, Mexico D.F.
3 Facultad de Ciencias, UAEM, Cuernavaca, Morelos 62209, Mexico
The heat transfer problem of a zero-mean oscillatory flow of both a Newtonian and a Maxwell fluid between infinite parallel plates with boundary conditions of the third kind is considered. With the analytic solutions for the velocity and temperature fields at hand, the local and global time-averaged entropy production are computed. The consequences of convective cooling of the plates are assessed for this problem. © IWNET 2006
© and Kleanthi for IWNET 2006
http://www.complexfluids.ethz.ch/cgi-bin/CONF/c?search=Olivares-Robles&COUNT=2728/08/2006 08:00:48 p.m.
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Date: Jul 03, 2006
To: "Miguel A Olivares-Robles" [email protected]
From: "Journal of Non-Equilibrium Thermodynamics" [email protected]
Subject: JNE Acceptance JNE-D-06-00018R1
Dear Dr Olivares-Robles, We are glad to inform that your paper ON THE FLUCTUATION-DISSIPATION THEOREM FOR CONVECTIVE PROCESSES reg.no. JNE-D-06-00018R1 has now been accepted for publication in the Journal of Non-Equilibrium Thermodynamics. With the release of the manuscript for publication, the author transfers the copyright to Walter de Gruyter GmbH & Co. KG, Berlin - New York, including the rights to produce offprints, photocopies or translations. The manusript will soon be sent to the typesetters. You will receive galley proofs together with your manuscript for proof reading as soon as they are available. Yours sincerely, Torsten Krueger for Professor Juergen U. Keller Editor-in-Chief __________________________________ Torsten Krueger Managing Editor, Journal of Non-Equilibrium Thermodynamics Walter de Gruyter GmbH & Co. KG Genthiner Strasse 13 10785 Berlin, Germany Tel. ++49-30-26005-176 Fax. ++49-30-26005-298 E-Mail: [email protected]
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J. Non-Equilib. Thermodyn.2007 �Vol. 32 � pp. 1–12
J. Non-Equilib. Thermodyn. � 2007 �Vol. 32 �No. 16 Copyright 2007 Walter de Gruyter �Berlin �New York. DOI 10.1515/JNETDY.2007.aaa
On the Fluctuation-Dissipation Theoremfor Convective Processes
Alan J. McKane1, Federico Vazquez2, and Miguel A. Olivares-Robles3,*1 Theory Group, School of Physics and Astronomy, University of Manchester,Manchester M13 9PL, UK2 Facultad de Ciencias, Universidad Autonoma del Estado de Morelos, AvenidaUniversidad 1001, Chamilpa, Cuernavaca, Morelos 62209, Mexico3 Seccion de Posgrado e Investigacion, Escuela Superior de Ingenierıa Mecanicay Electrica Culhuacan-IPN, Av. Santa Ana 1000, Col. San Francisco CulhuacanCoyoacan 04430, Mexico D.F.
*Corresponding author ([email protected])
Abstract
When making the connection between the thermodynamics of irreversibleprocesses and the theory of stochastic processes through the fluctuation–dissipation theorem, it is necessary to invoke a postulate of the Einstein–Boltzmann type. For convective processes hydrodynamic fluctuations must beincluded, the velocity is a dynamical variable and although the entropy cannotdepend directly on the velocity, d2S will depend on velocity variations. Someauthors do not include velocity variations in d2S, and so have to introduce anon-thermodynamic function which replaces the entropy and does depend onthe velocity. At first sight, it seems that the introduction of such a functionrequires 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 re-lation in this way. We then obtain the fluctuation–dissipation theorem, whichshows some di¤erences as compared with the non-convective case. We alsoshow that d2S is a Liapunov function when it includes velocity fluctuations.
1. Introduction
Velocity fluctuations play an important role in a variety of non-equilibriumphenomena. Mention can be made, for instance, of time-dependent di¤usion
(AutoPDF V7 28/11/06 12:12) WDG (170�240mm) Tmath J-1657 JNET, 32:1 PMU: H(A1) 17/11/2006 pp. 1–12 1657_32-1_06-18 (p. 1)
(06-18)
processes in binary liquid mixtures, where they are the principal mechanismleading to anomalously large fluctuations in concentration [1]. Also, the cou-pling between temperature and transverse-velocity fluctuations in the well-known case of a horizontal fluid layer heated from below may be associatedwith a small convective heat transfer below the Rayleigh–Benard instability[2]. It is natural to consider these kind of problems from the point of view ofirreversible thermodynamics. However, there is no prescription for how to in-troduce the velocity fluctuations into the formalism.
The standard method of introducing fluctuations into irreversible thermo-dynamics is through the Einstein–Boltzmann relation, PS P expfd2S=2kBg,where PS is the stationary probability distribution and d2S is the second vari-ation of the local entropy [3]. In this paper we will be interested in convectiveprocesses where the velocity is included as a dynamical variable, and in theexplicit form for d2S in this case. It should be noted, and is widely appreci-ated, that the entropy does not depend directly on the velocity of the system:velocity is a hydrodynamic, but not a thermodynamic variable. Thereforesome authors, notably Glansdor¤ and Prigogine [4], do not include velocityvariations in the expression for d2S.
A solution to this problem could be to introduce a new function which is es-sentially a generalisation of the entropy, which does depend on the velocity.This would not be a thermodynamic function, but it would then be necessaryto generalise the Einstein–Boltzmann relation in such a way that entropywould be replaced by this new function. Such a function has been introducedsome time ago by Glansdor¤ and Prigogine, but in the context of thermody-namic and hydrodynamic stability [4]. They suggested defining a new func-tion zC s� v2=2T0, where s is the entropy per unit mass, v is the barycentricvelocity, and T0 is the temperature in the reference state (for example, thetemperature in equilibrium). The analogous quantity for the system as awhole will be denoted by Z and is given by Z ¼
Ðrz dV , just as S ¼
Ðrs dV .
This function has not been utilised a great deal, perhaps in part becauseamong those who explicitly use the Z-function [4–7], most do not consistentlyuse the definition given above, sometimes using the (varying) temperature T inplace of the (non-varying) reference temperature T0.
The main reason why the function Z has not been widely used is no doubtthe demonstration by Oono [6] that d2S does in fact contain velocity varia-tions, even though the entropy does not depend on the velocity. In fact, theentropy may be written in terms of the velocity if other variables are intro-duced that exactly cancel out the velocity dependence [8]. To see this let uswrite [6]
dU ¼ T dS � p dV þ mg dNg; ð1Þ
(AutoPDF V7 28/11/06 12:12) WDG (170�240mm) Tmath J-1657 JNET, 32:1 PMU: H(A1) 17/11/2006 pp. 1–12 1657_32-1_06-18 (p. 2)
2 A.J. McKane et al.
J. Non-Equilib. Thermodyn. � 2007 �Vol. 32 �No. 1
where U is the internal energy, V the volume, Ng the number of moles of theg-chemical species, p the pressure, and mg the chemical potential of the g-species. In addition, let ET be the total energy:
ET ¼ U þmvmvm=2; ð2Þ
vm being the barycentric velocity and m the mass. We assume that the changesin potential energy due to altitude, for instance, are negligible. Therefore weomit a potential term in this definition. Then Eq. (1) can be written as
dET ¼ T dS � p dV þ mg dNg þmvm dvm: ð3Þ
Now note that in Eq. (3) the term dET �mvm dvm does not depend on velocityin accordance with the definition of the total energy, Eq. (2). So the entropyin Eq. (3) does not depend on the velocity and the thermodynamic consis-tency of this form of Gibbs relation, Eq. (1), is ensured.
Oono also showed that d2Z is nothing else but d2S. However, mention mustbe made of the fact that d2S and d2Z are only equal within approximationschemes where T can be replaced by T0. There is also a lack of consensus asto whether d2S is a Liapunov function in systems where velocity is a dynam-ical variable: some authors believe it is [9], others believe it is not [4]. Some ofthis confusion involves matters of principle, some involves matters of nota-tion (for instance, d2S meaning two entirely di¤erent things), and some in-volves inconsistencies in definitions of key quantities. Our objective in thispaper is to clarify many of these points, by examining their consequences inthe context of linear theories of irreversible thermodynamics, and to obtainthe explicit form of the fluctuation–dissipation theorem for convective pro-cesses. We remark in passing that there are a whole set of di¤erent subtletiesand controversies in extending these ideas to the non-linear regime [5, 10–12],but we do not explore these here.
2. Irreversible thermodynamics and stochastic processes
A fluid being described within linear irreversible thermodynamics (LIT) re-quires five local variables: the volume per unit mass v, the barycentric velocityvm, and the temperature T [3], but our conclusions will be more widely appli-cable, for example applying also to a fluid in extended irreversible thermody-namics (EIT), which requires 14 dynamic variables [13–17]. To keep the no-tation general, we will denote the fluctuations in the independent dynamicvariables as abðr; tÞ, where b ¼ 1; . . . ;N and assume that they satisfy a set ofLangevin-type equations:
(AutoPDF V7 28/11/06 12:12) WDG (170�240mm) Tmath J-1657 JNET, 32:1 PMU: H(A1) 17/11/2006 pp. 1–12 1657_32-1_06-18 (p. 3)
On the Fluctuation-Dissipation Theorem for Convective Processes 3
J. Non-Equilib. Thermodyn. � 2007 �Vol. 32 �No. 1
qabðr; tÞqt
¼ �Xc
ðdr 0Gbcðr; r 0Þacðr 0; tÞ þ ~ffbðr; tÞ: ð4Þ
Here the first term on the right-hand side is a result of the linearisation of themacroscopic equation about the stationary state and ~ffbðr; tÞ is a stochasticterm that represents fluctuations in the system. For the particular case of afluid within LIT N ¼ 5 and the five local variables a1; . . . ; a5 are the scaledversions of fluctuations in fv; vm;Tg. Specifically, if the equilibrium state isdenoted by fv0; 0;T0g, and fluctuations away from this state by fv1; vm;T1g,then we define the ab by [9, 18]:
a1 ¼ �r3=20 v1; amþ1 ¼
r1=20
cTvm; a5 ¼
r0Cv
T0c2T
� �1=2
T1; ð5Þ
with m ¼ 1; 2; 3. Here r0 is the mass density, cT the isothermal speed ofsound, and Cv the specific heat at constant volume, all in equilibrium. Theserescalings simplify the algebraic structure of the results. We use the same no-tation for the velocity and the velocity fluctuations, since no confusion shouldarise.
The analysis of the fluctuations is made more transparent if we adopt an ab-breviated form where the continuous labels r and r 0 are replaced by the dis-crete labels j and k and where the summation convention is assumed. In thiscase, (4) becomes
_aa jbðtÞ þ G
jkbc a
kc ðtÞ ¼ ~ff j
b ðtÞ; b; c ¼ 1; . . . ;N: ð6Þ
To complete the specification of the stochastic dynamics, the statistics of the
stochastic terms ~ff jb ðtÞ need to be given. We will take them to have a Gaussian
distribution with mean zero and correlator
3 ~ff jb ðtÞ ~ff k
c ðt 0Þ4 ¼ 2Qjkbcdðt� t 0Þ: ð7Þ
The requirement that they have zero mean follows from the fact that we askthat the ab have zero mean: 3a j
b4 ¼ 0. The matrix Q is real, symmetric, andpositive semidefinite. We will not give an explicit form for the matrix G here:it may be straightforwardly derived by a linearisation of the macroscopicequations [9]. As will be discussed below, the matrix Q may be given in termsof the matrix G and another matrix E, which is the covariant matrix of the ak
b
in the stationary state:
3alea
mf 4S ¼ ðE�1Þ lmef : ð8Þ
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Therefore, the stochastic dynamics will be completely specified if we can de-termine the matrix E. Clearly we need some new information from which tofind it. This is the Einstein–Boltzmann relation.
The Gaussian assumption determines the class of phenomena to be dealt with.In general, the Gaussian assumption is valid for a wide range of conditions inwhich the physical variables do not change too fast with time [3]. It may besaid that the su‰cient condition for the validity of this assumption is the localequilibrium hypothesis. Nevertheless, the system may be in a non-equilibriumnon-stationary state in which such a hypothesis is not satisfied and yet will bewell described throughout using the Gaussian assumption.
We now introduce the fluctuation–dissipation theorem by recalling that an-other way of specifying the stochastic process defined by Eqs. (6) and (7) isthrough the Fokker–Planck equation [19, 20],
qPða; tÞqt
¼ q
qajb
½G jkbc a
kc Pða; tÞ� þ
q2
qajbqa
kc
½QjkbcPða; tÞ�; ð9Þ
where Pða; tÞ is the probability distribution function of the local variables a.This is a linear Fokker–Planck equation and so the solution is a Gaussian,which may be written down explicitly as [21]
Pða; tÞ ¼ Nðdet XðtÞÞ�1=2 � exp � 1
2aTXðtÞ�1
a
� �; ð10Þ
where N is a normalisation constant and where the matrix XðtÞ is givenby
Xðt� t0Þ ¼ 2
ð t
t0
e�ðt�t 0ÞGQeðt�t 0ÞG dt 0: ð11Þ
Here initial conditions have been set at t ¼ t0 and we have made use of thefact that 3a j
b4 ¼ 0. By letting t0 ! �l, we find the stationary distribution.It has the form (10), but with XðtÞ replaced by
XðlÞ ¼ 2
ð t
�le�ðt�t 0ÞGQeðt�t 0ÞG dt 0
¼ 2
ðl0
e�rGQerG dr: ð12Þ
To make use of the Einstein–Boltzmann relation, let us observe that since theakb have zero mean, and since they are linearly related to the f k
b , which are
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On the Fluctuation-Dissipation Theorem for Convective Processes 5
J. Non-Equilib. Thermodyn. � 2007 �Vol. 32 �No. 1
Gaussian, they also have a Gaussian distribution with a stationary probabilitydistribution of the form
PSðaÞ ¼ N exp � 1
2ajbE
jkbc a
kc
� �: ð13Þ
Here a ¼ ða1; a2; . . .Þ where ai ¼ ðai1; . . . ; a
iNÞ and N is a normalisation con-
stant. By comparing Eq. (13) with Eq. (10) when t0 ! �l, we can make theidentification
E�1 ¼ XðlÞ ¼ 2
ðl0
e�rGQerG dr: ð14Þ
Performing the integral in Eq. (14) gives the result [21]
2Qijab ¼ Gik
acðE�1Þkjcb þ ðE�1Þ ikacGTkjcb ; ð15Þ
where T denotes transpose. This is the fluctuation–dissipation theorem of thetheory. It is the required relationship that gives the matrix Q in terms of thematrices G and E.
3. The fluctuation–dissipation theorem for convective systems
The result (13) may be compared directly [18] with the Einstein–Boltzmannrelation
PSðaÞP expfd2S=2kBg; ð16Þ
so that
SðaÞ ¼ Seq �1
2kBa
jbE
jkbc a
kc : ð17Þ
The indices b and c in Eq. (13) or Eq. (17) run from 1 to N (from 1 to 5 inLIT) and include the velocity as a variable. However, if only specific volume(or density) and temperature are included as variables in d2S [4, 22], then itapparently seems that Eqs. (13) and (16) cannot be compared to determinethe E
jkbc matrix. Thus, it seems clear that the d2S that we need to use in the
Einstein–Boltzmann relation is the one that allows for variations in the veloc-ity. In fact, as shown by Oono [6],
d2S ¼ d1
T
� �dU þ d
p
T
� �dV �mdvmdvm
T
¼ d2Sjv �mdvmdvm
T; ð18Þ
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where d2Sjv is d2S with no variation in the velocity. Using d2S, rather thand2Sjv allows Eqs. (13) and (16) to be compared and the matrix E determined.It should be noted that (i) in [18], the additional term to be added to d2Sjv wasgiven as mdvmdð�vm=TÞ, and (ii) in [6] it was stated that d2Z ¼ d2S – whereasfrom the definition of z we see that
d2Z ¼ d2Sjv �mdvmdvm
T0: ð19Þ
Both the results (i) and (ii) are true in the linear regime, where T�1 may be re-placed by T�1
0 , but they are not true in general; the correct form for d2S isgiven in Eq. (18), and d2Z is not equal to d2S, it is given by Eq. (19). A conse-quence of this is that in the linear regime the Einstein–Boltzmann relation mayalso be written as PS P expfd2Z=2kBg. This means that if we were to use d2Sjv,as Glansdor¤ and Prigogine do, we would need to invoke this latter form ofthe Einstein–Boltzmann relation to identify the matrix E and so make theconnection between irreversible thermodynamics and the theory of stochasticprocesses, at least in the linear regime. However, as we have stressed, there isno need to introduce this extra postulate, and we may use the usual formPS P expfd2S=2kBg, as long as the correct form of d2S (18) is used.
We can now come back to the task of determining the matrix E. Let us firstwrite down the expression for d2S without velocity variations in terms of thescaled versions of v1 and T1, namely a1 and a5, to see explicitly where the pro-cess fails. After some straightforward manipulations [18] of this standard re-sult [22], we obtain, using the Einstein–Boltzmann relation,
PSðaÞP expc2T
2kBT0½�a
j1a
j1 � a
j5a
j5�
� �: ð20Þ
If this result were to be compared with Eq. (13), then it would imply that Ewould be diagonal, but with entries corresponding to the velocity fluctuationsbeing zero. This is clearly not correct since, for instance, the velocity–velocitycorrelation function in equilibrium (8) would be formally infinite. Using in-stead the form of d2S allowing for velocity variation we find
PSðaÞP expc2T
2kBT0½�a
jba
jb�
� �; ð21Þ
since vm ¼ ðc2T=r0Þ1=2
amþ1 and where b ¼ 1; . . . ; 5. A comparison with Eq. (8)gives the identification
Ejkbc ¼ c2T
kBT0djkdbc: ð22Þ
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J. Non-Equilib. Thermodyn. � 2007 �Vol. 32 �No. 1
This now gives a consistent result, which when used in conjunction with thefluctuation–dissipation theorem (15), completely specifies the stochastic dy-namics described by Eqs. (6) and (7) or by Eq. (9). An explicit expression formatrix Q is obtained by substituting Eq. (22) into Eq. (15). The result is
Qjkbc ¼
kBT0
AS
jkbc ; ð23Þ
where S jkbc represents the symmetric part of the dynamic matrix G:
Smþ1;nþ1ðr; r 0Þ ¼1
r0½2mXmrns þ zdmrdns�
q2
qxrqx 0s
dðr� r 0Þ; ð24Þ
S55ðr; r 0Þ ¼1
r0Cldmn
q2
qxmqx 0n
dðr� r 0Þ; ð25Þ
with all other Sbcðr; r 0Þ, including S11ðr; r 0Þ, equal to zero. The tensor Xmnrs isdefined by
Xmnrs ¼1
2dmrdns þ dmsdnr �
2
3dmndrs
� �: ð26Þ
In Eqs. (24) and (25), the continuum limit has been taken so that the discretespatial variables j, k have been replaced by r, r 0. As mentioned above, all thematrices in Eq. (15) are 5� 5 in the convective case, unlike in the non-convective case where they are 2� 2.
The discussion above took place within the framework of LIT, which con-tains five dynamical variables, but the idea is more general. We have alreadymentioned EIT where the dissipative fluxes are raised to the same status asthe thermodynamic variables. In this case, d2S (where S now denotes the cor-responding non-equilibrium thermodynamic potential in place of the localequilibrium entropy) contains terms involving these fluxes, as well as themore conventional thermodynamical variables, but not the velocity variables[13–17]. Written in terms of scaled variables, it has the form [18]
PSðaÞP exp
�c2T
2kBT0
��a
j1a
j1 � a
j5a
j5 �
1
2ao j
mnao j
nm
� ajmþ10a
jmþ10 � a
j14a
j14
��: ð27Þ
Here the variables ao j
mn, ajmþ10 and a
j14 are scaled versions of the traceless stress
tensor, the heat flux, and the trace of the stress tensor, respectively. The result
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(27) su¤ers from the same defect as Eq. (20), but if we now include the veloc-ity variations in d2S, then we again obtain (21), but now with b ¼ 1; . . . ; 14.Therefore, the matrix E can be consistently identified, and again is given by(22).
4. Velocity fluctuations and the Liapunov function
Finally, within the context of LIT or EIT, we can investigate the claim thatd2Z is a Liapunov function, but that d2S can no longer be adopted as a Lia-punov function when velocity is included as a dynamical variable [4]. In thelanguage we have been using in this paper, the former is d2S and the latter isd2Sjv, and this is the notation we will use in what follows. To investigatewhether these functions are Liapunov functions, we begin from the form ofd2S su‰ciently near equilibrium that LIT will apply:
d2S ¼ � c2TT0
ajbðtÞa
jbðtÞ: ð28Þ
Here the a jb are averaged variables, that is, non-fluctuating variables that obey
the hydrodynamic balance equations. From Eq. (28) we see that d2Sa0 withequality if and only if a j
bðtÞ ¼ 0. Di¤erentiating Eq. (28) with respect to timegives
d
dtðd2SÞ ¼ � 2c2T
T0_aa jbðtÞa
jbðtÞ ¼
2c2TT0
Gjkbc a
kc ðtÞa
jbðtÞ
¼ 2c2TT0
Sjkbc a
kc ðtÞa
jbðtÞ; ð29Þ
where Sjkbc is the symmetric part of G jk
bc . Using the expressions for S jkbc , Eqs.
(24) and (25), and integrating by parts gives
d
dtðd2SÞ ¼ 2c2T
r0T0
ðdr 2mDmnDmn þ zD2
mm þl
Cv
qa5
qxm
qa5
qxm
� �b0; ð30Þ
where we have gone back to an explicit notation for the continuous spacevariable r. In Eq. (30), l, z, and m are the thermal conductivity, the bulk vis-cosity, and the shear viscosity, respectively, Dmn is the symmetric part of thescaled velocity gradient, and Dmn its traceless form:
Dmn ¼1
2
qamþ1
qxnþ qanþ1
qxm
� �; Dmn ¼ Dmn �
1
3Drrdmn: ð31Þ
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This shows explicitly, when d2S is defined in terms of the averaged variables,that it is a Liapunov function, as suggested by Glansdor¤ and Prigogine [4].However, this calculation is identical to one carried out in [9], where dS=dtwas evaluated and shown to be non-negative. Since all of these calculationshave been carried out in the linear regime, and dS=dt ¼ ð1=2Þ dðd2SÞ=dt ¼ð1=2Þ dðd2ZÞ=dt, this is not surprising. Note that the inequality in Eq. (30) isan equality if and only if Dmn ¼ 0, Dmm ¼ 0, and qa5=qxm ¼ 0. From the con-stitutive relations for LIT, this corresponds to the vanishing of the tracelessstress tensor and its trace and of the heat flux. This condition corresponds tothe thermodynamic equilibrium state and it is equivalent to the conditionajbðtÞ ¼ 0 found when d2S given by Eq. (28) is equal to zero.
A similar calculation may be carried out for EIT. In this case, Eqs. (28) and(29) also hold, but now with the indices b and c running from 1 to 14. Theforms of the S
jkbc are di¤erent for EIT – in some ways they are simpler, since
they do not involve derivatives, and so no integration by parts is required toobtain an explicit expression for the time derivative of d2S. Using the expres-sions for S jk
bc given in [18] for EIT, one finds that
d
dtðd2SÞ ¼ 2c2T
T0
ðdr
1
2t�12 amnanm þ t�1
0 a14a14 þ t�11 amþ10amþ10
� �b0; ð32Þ
where the ti, i ¼ 0; 1; 2 are the relaxation times of the various fluxes. Onceagain, d2S is seen to be a Liapunov function, with the inequality in Eq. (32)becoming an equality if and only if amn ¼ 0, a14 ¼ 0, and amþ10 ¼ 0. These arejust scaled versions of the traceless stress tensor and its trace, and of the heatflux, and so equality is obtained when these vanish, just as for LIT. If we usethis method to try and show that d2Sjv is a Liapunov function, we find, forexample in the case of LIT,
d2Sjv ¼ � c2TT0
ða j1ðtÞa
j1ðtÞ þ a
j5ðtÞa
j5ðtÞÞ; ð33Þ
and di¤erentiating with respect to time gives
d
dtðd2SjvÞ ¼ � 2c2T
T0ð _aa j
1ðtÞaj1ðtÞ þ _aa j
5ðtÞaj5ðtÞÞ
¼ 2c2TT0
ðG jk1c a
kc ðtÞa
j1ðtÞ þ G
jk5c a
kc ðtÞa
j5ðtÞÞ: ð34Þ
Substituting the actual expressions for G jkbc [9, 18] in Eq. (34) does not give an
expression that is manifestly positive semidefinite. This is no doubt what
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Glansdor¤ and Prigogine meant by saying that d2S loses its properties as aLiapunov function when velocity is included as a dynamical variable. How-ever, since we are assuming that v is fixed in the definition of d2S, it might bemore consistent to take v to be a constant in the balance equations. If we dothis, we find that only the third term in the parentheses in Eq. (30) is present.It now follows that dðd2SjvÞ=dtb0.
5. Conclusions
In summary, when studying fluctuations in irreversible thermodynamics usingthe formalism of Langevin or Fokker–Planck equations, velocity is includedas a variable. When making use of the Einstein–Boltzmann relation to deter-mine the exact form of the fluctuation–dissipation relation, the form of d2Swhere velocity variation is allowed must be used. Although S and dS maybe written in forms that do not involve velocity, d2S does depend on the ve-locity variation. If, as some authors do, d2S is taken not to include velocityvariations – using what we have called d2Sjv – then these velocity variationshave to be introduced by some other means, for example, by the introduc-tion of the Z function. However, in this case an added postulate of the formPS P expfd2Z=2kBg has to be introduced. Clearly, this is unnecessary sincethe usual Einstein–Boltzmann relation, with the correct use of d2S, that is,including velocity variations, may be used without contradiction to com-plete the link between thermodynamic and hydrodynamic fluctuations andthe theory of stochastic processes.
Acknowledgements
We wish to thank Y. Oono and M. Lopez de Haro for useful discussions.AJM wishes to thank the Department of Physics at the Universidad Auton-oma del Estado de Morelos for hospitality while this work was carried out.Financial support from CONACYT-Mexico under project number40454 and from PROMEP-Mexico is gratefully acknowledged.
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