Bjerre, Peter

Buch-Madsen, Tommy

Olsen, Andreas Elmerdahl

Samsøe-Petersen, Anders

Formålet med dette projekt er at undersøge, dels hvorvidt gruppeteori kan bruges som model for isomorfe kurver i IPL-væsker, og dels hvorvidt forklaringen på isomorf opførsel kan findes som symmetri i løsninger til de, for dette projekt ukendte, differentialligninger, der ligger til grund for dynamikken i simulering af viskøse væsker. Ved at analysere IPL-modellen med gruppeteori bliver der opstillet en transformationsgruppe GIPL til beskrivelse af ækvivalensrelationen n/3 1 /T1 = n/3 2 /T2. Der bliver udfra dette vist, at isomorfe kurver kan beskrives som banekurver i (T, )-planet, hvor enhver banekurve kan identificeres som et element i kvotientrummet ved hjælp af isochorer og isotermer. Ud fra dette konkluderes det, at banekurver kan bruges som model for isomorfer, da disse indfanger visse isomorfe karakteristika. Til sammenligning med de ukendte underliggende differentialligninger anvendes varmeledningsligningen som et simpelt eksempel på et differentialigningssystem. Udfra Olver [1993] bliver der opstillet en symmetrigruppe, Gi, for hvilken løsningen T(x, t) = je−(x−Kt) er gruppeinvariant. Ud fra parametrene (k, c, ), der udgør konstanten K, opstilles en ækvivalensrelation og hertil en gruppe, Geq, der beskriver ækvivalensrelationen. Det vises således, at løsninger til varmeledningsligningen er invariante overfor transformation af denne ved Geq. Udfra analysen af varmeledningsligningen med hensyn til symmetrigrupper, gruppeinvariante løsninger og invarians overfor transformation af parametrene i varmeledningsligningen postuleres det, at eksistensen af isomorfer antyder tilstedeværelsen af løsninger, som er invariante overfor transformation af det underliggende differentialligningssystem. En sådan invarians burde derfor kunne findes i alle væsker, der udviser isomorf opførsel. For IPL-væsker formodes det, at ækvivalensrelationen, givet ved det konstante forhold mellem n/3 og T langs de isomorfe kurver, har relation til denne invarians. The purpose of this project is to investigate, in part whether group theory, can be used as a model for isomorphic curves in IPL-liquids, and in part whether the explanation of isomorphic behaviour can be found as symmetry in solutions to the, for this project unknown, differential equations that controls the dynamics in simulations of vicous liquids By analysing the IPL-model with group theory a transformation group GIPL was suggested as a way of describing the equivalence relation n/3 1 /T1 = n/3 2 /T2. From this it was shown, that isomorphic curves can be described as orbits in the (T, )-plane, where each of the orbits can be identified as elements in the quotient space by means of isochors and isotherms. From this it is concluded, that orbits can be used as a model for isomorphs, since these capture some of the isomorphic characteristics. As a frame of reference to the unknown differential equations, the heat equation is used as a simple example of a system of differential equations. Based on Olver [1993] a symmetry group Gi is established, for which the solution T(x, t) = je−(x−Kt) is group invariant. From the parameters (k, c, ), which define the contant K, an equivalence relation is created along with a group, Geq, that describes this relation. From this it is shown, that solutions to the heat equation are invariant to transformation by Geq. From the analysis of the heat equaiton in respect to symmetry groups, group invariant solutions and invariance towards transformations of the parameters in the heat equation it is postulatet, that the existence of isomorphs imply the presence of solutions that are invariant to the transformation of the underlying system of differential equations. Such an invariance should thus be pressent in all liquids that exhibit isomorphic behaviour. For IPL-liquids it is surmised that the equivalence relation given by a constant ratio between n/3 and T along isomophic curves has a relation to this invariance.Formålet med dette projekt er at undersøge, dels hvorvidt gruppeteori kan bruges som model for isomorfe kurver i IPL-væsker, og dels hvorvidt forklaringen på isomorf opførsel kan findes som symmetri i løsninger til de, for dette projekt ukendte, differentialligninger, der ligger til grund for dynamikken i simulering af viskøse væsker. Ved at analysere IPL-modellen med gruppeteori bliver der opstillet en transformationsgruppe GIPL til beskrivelse af ækvivalensrelationen n/3 1 /T1 = n/3 2 /T2. Der bliver udfra dette vist, at isomorfe kurver kan beskrives som banekurver i (T, )-planet, hvor enhver banekurve kan identificeres som et element i kvotientrummet ved hjælp af isochorer og isotermer. Ud fra dette konkluderes det, at banekurver kan bruges som model for isomorfer, da disse indfanger visse isomorfe karakteristika. Til sammenligning med de ukendte underliggende differentialligninger anvendes varmeledningsligningen som et simpelt eksempel på et differentialigningssystem. Udfra Olver [1993] bliver der opstillet en symmetrigruppe, Gi, for hvilken løsningen T(x, t) = je−(x−Kt) er gruppeinvariant. Ud fra parametrene (k, c, ), der udgør konstanten K, opstilles en ækvivalensrelation og hertil en gruppe, Geq, der beskriver ækvivalensrelationen. Det vises således, at løsninger til varmeledningsligningen er invariante overfor transformation af denne ved Geq. Udfra analysen af varmeledningsligningen med hensyn til symmetrigrupper, gruppeinvariante løsninger og invarians overfor transformation af parametrene i varmeledningsligningen postuleres det, at eksistensen af isomorfer antyder tilstedeværelsen af løsninger, som er invariante overfor transformation af det underliggende differentialligningssystem. En sådan invarians burde derfor kunne findes i alle væsker, der udviser isomorf opførsel. For IPL-væsker formodes det, at ækvivalensrelationen, givet ved det konstante forhold mellem n/3 og T langs de isomorfe kurver, har relation til denne invarians. The purpose of this project is to investigate, in part whether group theory, can be used as a model for isomorphic curves in IPL-liquids, and in part whether the explanation of isomorphic behaviour can be found as symmetry in solutions to the, for this project unknown, differential equations that controls the dynamics in simulations of vicous liquids By analysing the IPL-model with group theory a transformation group GIPL was suggested as a way of describing the equivalence relation n/3 1 /T1 = n/3 2 /T2. From this it was shown, that isomorphic curves can be described as orbits in the (T, )-plane, where each of the orbits can be identified as elements in the quotient space by means of isochors and isotherms. From this it is concluded, that orbits can be used as a model for isomorphs, since these capture some of the isomorphic characteristics. As a frame of reference to the unknown differential equations, the heat equation is used as a simple example of a system of differential equations. Based on Olver [1993] a symmetry group Gi is established, for which the solution T(x, t) = je−(x−Kt) is group invariant. From the parameters (k, c, ), which define the contant K, an equivalence relation is created along with a group, Geq, that describes this relation. From this it is shown, that solutions to the heat equation are invariant to transformation by Geq. From the analysis of the heat equaiton in respect to symmetry groups, group invariant solutions and invariance towards transformations of the parameters in the heat equation it is postulatet, that the existence of isomorphs imply the presence of solutions that are invariant to the transformation of the underlying system of differential equations. Such an invariance should thus be pressent in all liquids that exhibit isomorphic behaviour. For IPL-liquids it is surmised that the equivalence relation given by a constant ratio between n/3 and T along isomophic curves has a relation to this invariance

Matematisk modellering af isomorfer i IPL-væsker: Mathematical modelling of isomorphs in IPL-liquids

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