Hans Christian Öttinger. Department of Materials, ETH Zürich, Switzerland

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1 Lessons from NonequilibriumThermodynamics: Flow-Driven Glasses Hans Christian Öttinger Department of Materials, ETH Zürich, Switzerland

2 Summary & Conclusions Geometry suggests thermodynamics (at equilbrium & far beyond)

3 Summary & Conclusions Geometry suggests thermodynamics (at equilbrium & far beyond) Deriving thermodynamics is difficult, formulating it is easy

4 Summary & Conclusions Geometry suggests thermodynamics (at equilbrium & far beyond) Deriving thermodynamics is difficult, formulating it is easy Poisson structures (used to generate reversible dynamics) contain only static information

5 Summary & Conclusions Geometry suggests thermodynamics (at equilbrium & far beyond) Deriving thermodynamics is difficult, formulating it is easy Poisson structures (used to generate reversible dynamics) contain only static information The geometric structure of isolated systems has direct consequences for open systems (Poisson brackets, Lie groups & Lie algebras, semidirect products)

6 Summary & Conclusions Geometry suggests thermodynamics (at equilbrium & far beyond) Deriving thermodynamics is difficult, formulating it is easy Poisson structures (used to generate reversible dynamics) contain only static information The geometric structure of isolated systems has direct consequences for open systems (Poisson brackets, Lie groups & Lie algebras, semidirect products) Poisson brackets suggest a static indicator for the onset of the glass transition

9 Summary & Conclusions Geometry suggests thermodynamics (at equilbrium & far beyond) Deriving thermodynamics is difficult, formulating it is easy Poisson structures (used to generate reversible dynamics) contain only static information The geometric structure of isolated systems has direct consequences for open systems (Poisson brackets, Lie groups & Lie algebras, semidirect products) Poisson brackets suggest a static indicator for the onset of the glass transition

10 Approach to Nonequilibrium Thermodynamics Step Equilibrium Nonequilibrium 1 work vs heat reversible vs irreversible dynamics 2 3 4

11 Approach to Nonequilibrium Thermodynamics Step Equilibrium Nonequilibrium 1 work vs heat reversible vs irreversible dynamics 2 3 de = δw + δq dx = v dt reversible + v irreversible 4

12 Approach to Nonequilibrium Thermodynamics Step Equilibrium Nonequilibrium 1 work vs heat reversible vs irreversible dynamics de = δw + δq dx = v dt reversible + v irreversible de = pdv + μdn + δq dx = L δh dt δx v irreversible

13 Approach to Nonequilibrium Thermodynamics Step Equilibrium Nonequilibrium 1 work vs heat reversible vs irreversible dynamics de = δw + δq dx = v dt reversible + v irreversible de de = pdv + μdn + δq dx = L δh dt δx v irreversible = pdv + μdn + TdS dx = L δh M δs dt δx δx

14 GENERIC Structure General equation for the nonequilibrium reversible-irreversible coupling metriplectic structure (P. J. Morrison, 1986) Lx ( ) dx = Lx ( ) dt δe ( x) δx + Mx ( ) δs( x) = 0 Mx ( ) δx L antisymmetric, Jacobi identity δs ( x) δx δe( x) = 0 δx M Onsager/Casimir symm., positive-semidefinite

15 Summary & Conclusions Geometry suggests thermodynamics (at equilbrium & far beyond) Deriving thermodynamics is difficult, formulating it is easy Poisson structures (used to generate reversible dynamics) contain only static information The geometric structure of isolated systems has direct consequences for open systems (Poisson brackets, Lie groups & Lie algebras, semidirect products) Poisson brackets suggest a static indicator for the onset of the glass transition

16 Statistical Mechanics: Energy & Entropy E = E 0 x relative entropy

17 Think About Entropy Has this messy room a large or small entropy?

18 Statistical Mechanics: Friction Matrix τ 1 M jk = Π k( 0)Π j () xdt k B 0 f f t Green-Kubo Π j x = x j 1 M jk = Δ 2k B τ τ Π j Δ τ Π k x Einstein cf. D = ( Δx) 2 2Δt τ 1 τ2

19 Poisson Operator L jk = { Π j, Π k } x L contains only static information Selection of the structural variables is needed L describes convection behavior (deformation)

20 Summary & Conclusions Geometry suggests thermodynamics (at equilbrium & far beyond) Deriving thermodynamics is difficult, formulating it is easy Poisson structures (used to generate reversible dynamics) contain only static information The geometric structure of isolated systems has direct consequences for open systems (Poisson brackets, Lie groups & Lie algebras, semidirect products) Poisson brackets suggest a static indicator for the onset of the glass transition

21 Poisson Structure Lie group: space transformations Semidirect products: action of the group of space transformations on structural variables Lie algebra: velocity fields Dual of the Lie algebra: momentum density fields

22 Poisson Structure Lie group: space transformations Lie algebra: velocity fields Semidirect products: action of the group of space transformations on structural variables open system Dual of the Lie algebra: momentum density fields driving device

23 Summary & Conclusions Geometry suggests thermodynamics (at equilbrium & far beyond) Deriving thermodynamics is difficult, formulating it is easy Poisson structures (used to generate reversible dynamics) contain only static information The geometric structure of isolated systems has direct consequences for open systems (Poisson brackets, Lie groups & Lie algebras, semidirect products) Poisson brackets suggest a static indicator for the onset of the glass transition

24 A Vision of Glasses inherent structures Debenedetti & Stillinger, Nature 410 (2001) 259

25 Nonequilibrium Thermodynamics Variable Structural variable: inherent structure pair correlation function g is ( rr, ). Study rearrangements for energy minimization after deformation!

Study rearrangements for energy minimization after deformation!

26 Nonequilibrium Thermodynamics Variable Structural variable: inherent structure pair correlation function g is ( rr, ). Study rearrangements for energy minimization after deformation! at onset of glassy dynamics melt [enhancement factor 600] [enhancement factor 14] 2d E. Del Gado, P. Ilg, M. Kröger, hco, Phys Rev Lett 2008 γ = 0.001

27 Nonequilibrium Thermodynamics Order Parameter? normalized mean magnitude of rearrangements E. Del Gado, P. Ilg, M. Kröger, hco, Phys Rev Lett d spatially modulated

28 Summary & Conclusions Geometry suggests thermodynamics (at equilbrium & far beyond) Deriving thermodynamics is difficult, formulating it is easy Poisson structures (used to generate reversible dynamics) contain only static information The geometric structure of isolated systems has direct consequences for open systems (Poisson brackets, Lie groups & Lie algebras, semidirect products) Poisson brackets suggest a static indicator for the onset of the glass transition

Hydrodynamics, Thermodynamics, and Mathematics

Hydrodynamics, Thermodynamics, and Mathematics Hans Christian Öttinger Department of Mat..., ETH Zürich, Switzerland Thermodynamic admissibility and mathematical well-posedness 1. structure of equations