1. Non-linear, Topologically Coherent, and Compact Flows Far from Equilibrium R. M. Kiehn University of Houston www.cartan.pair.com R.M.K. EGU Vienna 6.06 April 20, 2007 From the point of view of Continuous Topological Evolution

2. R.M.K. EGU Vienna 6.06 April 20, 2007 PART III Applications

3. 1.Topological structure of Physical Systems is encoded in an Action differential 1-form A. Review of the Basic Ideas

4. 1.Topological structure of Physical Systems is encoded in an Action differential 1-form A. 2. Physical Processes can be defined in terms of contravariant vector/spinor direction fields, V. Review of the Basic Ideas

5. 1.Topological structure of Physical Systems is encoded in an Action differential 1-form A. 2. Physical Processes can be defined in terms of contravariant vector/spinor direction fields, V. 3. Continuous Topological Evolution is encoded by Cartan's magic formula (the First Law of Thermo): L (V) A = i(V)dA+di(V)A = W + dU = Q (L (V) A is the Lie differential with respect to the direction field V acting on the 1-form A) Review of the Basic Ideas

6. Topological evolution is a necessary condition for both time asymmetry and thermodynamic irreversibility Review of the Basic Ideas

7. Topological evolution is a necessary condition for both time asymmetry and thermodynamic irreversibility A unique extremal direction field which represents a conservative reversible Hamiltonian process always exists on subspaces of topological dimension 2n+1. Review of the Basic Ideas

8. Topological evolution is a necessary condition for both time asymmetry and thermodynamic irreversibility A unique extremal direction field which represents a conservative reversible Hamiltonian process always exists on subspaces of topological dimension 2n+1. A unique torsional direction field which represents a thermodynamically irreversible process always exists on subspaces of even topological dimension 2n+2. Review of the Basic Ideas

9. 1. Must choose the functional form for a 1-form of Action, A, to define a physical system. Application Requirements

10. 1. Must choose the functional form for a 1-form of Action, A, to define a physical system. 2. Possibly constrain the PTD of the Work 1-form, W, to generate the desired constrained dynamical PDE’s. Application Requirements

11. 1. Must choose the functional form for a 1-form of Action, A, to define a physical system. 2. Possibly constrain the PTD of the Work 1-form, W, to generate the desired constrained dynamical PDE’s. 3. Evaluate the non-equilibrium properties encoded as 3-form of Topological Torsion, A^F 4-form of Topological Parity. Application Requirements

12. Use the 4D electromagnetic 1-form of Action: A =A k ( x,y,z,t )dx k -  ( x,y,z,t )dt. A Universal Example

13. Use the 4D electromagnetic 1-form of Action: A =A k ( x,y,z,t )dx k -  ( x,y,z,t )dt. Define: E = - ∂(A k (x)/∂t - grad  ; B = curl A k A Universal Example

14. Use the 4D electromagnetic 1-form of Action: A =A k ( x,y,z,t )dx k -  ( x,y,z,t )dt. Define: E = - ∂(A k (x)/∂t - grad  ; B = curl A k Construct the Work 1-form: W = i(V)dA = -{E+V×B}+{VE}dt A Universal Example

15. Use the 4D electromagnetic 1-form of Action: A =A k ( x,y,z,t )dx k -  ( x,y,z,t )dt. Define: E = - ∂(A k (x)/∂t - grad  ; B = curl A k Construct the Work 1-form: W = i(V)dA = -{E+V×B}+{VE}dt Construct the 2-form F = dA, F = dA = B z dx^dy…- E z dz^dt… Then, the coefficients of the 3-form dF = ddA = 0, generate a nested set of Maxwell-Faraday PDE’s. Curl E + ∂B/∂t = 0 div B = 0. Faraday Induction A Universal Example

16. Construct the 3-form A^F = A^dA (of Topological Torsion) A ^ dA = i(T 4 )(dx^dy^dz^dt) with the 4 component Torsion Direction Field T 4 = - [ E x A +  B, A  B]. A Universal Example

17. Construct the 3-form A^F = A^dA (of Topological Torsion) A ^ dA = i(T 4 )(dx^dy^dz^dt) with the 4 component Torsion Direction Field T 4 = - [ E x A +  B, A  B]. ( The 4 divergence of the Torsion vector T 4 is equal to -2(E  B) = -2 . Then, L(T 4 ) A = (-E  B ) A =  A  is a Conformality Factor, and a Homogenity (including fractals) index of self similarity. (Suggested as a dissipative extension to Hamilton’s principle by RMK in 1974) A Universal Example

18. Construct the 4-form F^F = dA^dA of Topological Parity: dA ^ dA = -2(E  B)(dx^dy^dz^dt). On regions where  = (E  B)  0, the PTD(A) is 4. The thermodynamic system is not in equilibrium. A Universal Example

19. Construct the 4-form F^F = dA^dA of Topological Parity: dA ^ dA = -2(E  B)(dx^dy^dz^dt). On regions where  = (E  B)  0, the PTD(A) is 4. The thermodynamic system is not in equilibrium. And as L(T 4 ) A^ L(T 4 ) dA = Q^dQ = (-E  B) 2 A^dA  0. A Universal Example The process in the direction of the Torsion vector T4 is thermodynamically irreversible

20. The Action 1-form for a thermodynamic fluid and for an electromagnetic system are similar except for notation. A fluid = V k dx k + (V 2 /2)dt A em = A k dx k – (  )dt A Hydrodynamic Example.

21. The Action 1-form for a thermodynamic fluid and for an electromagnetic system are similar except for notation. A fluid = V k dx k + (V 2 /2)dt A em = A k dx k – (  )dt Hence the resulting theories are similar except for a change of notation A k ↔V k,  ↔-V 2 /2, a ↔ -E, ω = curlV↔B A Hydrodynamic Example.

22. The Action 1-form for a thermodynamic fluid and for an electromagnetic system are similar except for notation. A fluid = V k dx k - (V 2 /2)dt A em = A k dx k – (  )dt Hence the resulting theories are similar except for a change of notation A k ↔V k,  ↔V 2 /2, a ↔ -E, ω = curlV↔B A Hydrodynamic Example. Both the fluid and the EM systems should have a Maxwell Faraday induction law.

23. use EM notation. a process  V 4 =  [V,1], construct the Work 1-form W: W = i(  V 4 )dA = {ρ(- a - V  ω)}dx k + {-ρVa}dt A Hydrodynamic Example.

24. use EM notation. a process  V 4 =  [V,1], construct the Work 1-form W: W = i(  V 4 )dA = {ρ(- a - V  ω)}dx k + {-ρVa}dt   V/  t + grad V 2 /2 - V  curl V} k dx k + {ρV   V/  t + grad V 2 /2}dt A Hydrodynamic Example.

25. use EM notation. a process  V 4 =  [V,1], construct the Work 1-form W: W = i(  V 4 )dA = {ρ(- a - V  ω)}dx k + {-ρVa}dt   V/  t + grad V 2 /2 - V  curl V} k dx k + {ρV   V/  t + grad V 2 /2}dt  f the Work 1-form is of Pfaff dimension 1, and is exact, then W = i(  V 4 )dA = dP. A Hydrodynamic Example.

26. use EM notation. a process  V 4 =  [V,1], construct the Work 1-form W: W = i(  V 4 )dA = {ρ(- a - V  ω)}dx k + {-ρVa}dt   V/  t + grad V 2 /2 - V  curl V} k dx k + {ρV   V/  t + grad V 2 /2}dt  f the Work 1-form is of Pfaff dimension 1, and is exact, then W = i(  V 4 )dA = dP. By comparing coefficients of the differentials, it follows that {  V/  t + grad V 2 /2 - V  curl V } = dP/ρ Which is the Equation for an Eulerian fluid. A Hydrodynamic Example.

27. By adding potential energy terms such that V 2 /2  V 2 /2 + , and adding constraints on the Work 1-form, the reversible Extremal, Bernoulli and Stokes flows can be emulated as special cases.. A Hydrodynamic Example.

28. By adding potential energy terms such that V 2 /2  V 2 /2 + , and adding constraints on the Work 1-form, the reversible Extremal, Bernoulli and Stokes flows can be emulated as special cases. The PDE’s describing fluid flow depend upon the non-zero contributions to the Work 1-form which are Spinors. A Hydrodynamic Example.

29. By adding potential energy terms such that V 2 /2  V 2 /2 + , and adding constraints on the Work 1-form, the reversible Extremal, Bernoulli and Stokes flows can be emulated as special cases. The PDE’s describing fluid flow depend upon the non-zero contributions to the Work 1-form which are Spinors. Any non zero contribution to the work 1-form, W, must be transversal to the Process V 4 = [V, 1], as i(V 4 )W= i(V 4 )i(V 4 )dA = 0, hence: A Hydrodynamic Example.

30. By adding potential energy terms such that V 2 /2  V 2 /2 + , and adding constraints on the Work 1-form, the reversible Extremal, Bernoulli and Stokes flows can be emulated as special cases. The PDE’s describing fluid flow depend upon the non-zero contributions to the Work 1-form which are Spinors. Any non zero contribution to the work 1-form, W, must be transversal to the Process V 4 = [V, 1], as i(V 4 )W= i(V 4 )i(V 4 )dA = 0, hence: W= i(V 4 )dA =  k (dx k – V k dt) It is the Lagrange multiplier  k that is of interest, for it includes topological fluctuations (Spinors) to the Work 1-form. A Hydrodynamic Example.

31. Next, assume that the 3 vector  k is represented by ν  2 V  k = ν  2 V = ν(curl curl V – grad div V) and dΘ = dP/   grad div V    Bulk viscosity ν = shear viscosity A Hydrodynamic Example.

32. Next, assume that the 3 vector  k is represented by ν  2 V  k = ν  2 V = ν(curl curl V – grad div V) and dΘ = dP/   grad div V    Bulk viscosity ν = shear viscosity Substitute these expressions into the work 1-form to yield   V/  t + grad V 2 /2  V  curl V =  dP/ρ   grad div V  ν  2 V The Navier-Stokes Equations A Hydrodynamic Example.

33. Next, the purpose is to show that the Navier-Stokes flows can admit Topological Torsion, and thermodynamically irreversible processes, a key feature of TURBULENCE. Note that from the Navier-Stokes equations, an expression for a (the acceleration of expansion) can be obtained. (The acceleration due to rotation is encoded in the vorticity, ω). a = - V  curl V + dP/ρ -   grad div V  ν  2 V And this result can be used in the Universal Formulas for the Navier-Stokes Topological Torsion Vector and the N-S Topological Parity dissipation coefficent. A Hydrodynamic Example.

34. Compute the Topological Torsion 4 vector from the universal EM expression T 4 = [E  A +  B, A B] ↔ [a  V + V 2 /2 ω, V ω] = [T, h] T = hV- V 2 /2 ω - V  {gradP/ρ +   grad div V- ν V   2 V } h = V ω A Hydrodynamic Example. Similarly the Topological Parity Dissipation coefficient becomes K = -2(a ω) = 2 {gradP/ρ +  B grad div V- ν V   2 V}ω

35. CONCLUSIONS For the Navier-Stokes fluid, if the vorticity is zero, ω = 0, the flow produces NO irreversible dissipation, and the flow is not turbulent. If the acceleration is zero, a = 0, or is orthogonal to the vorticity, there is NO irreversible dissipation and no Turbulence. A Hydrodynamic Example.

36. CONCLUSIONS For the Navier-Stokes fluid, if the vorticity is zero, ω = 0, the flow produces NO irreversible dissipation, and the flow is not turbulent. If the acceleration, a, is zero, or is orthogonal to the vorticity, there is NO irreversible dissipation and no Turbulence. Only when the component of fixed point acceleration of expansion, a, is parallel to the fixed point acceleration of rotation, ω, such that aω  0, does the Navier-Stokes flow admit irreversible dissipation. A Hydrodynamic Example.

37. CONCLUSIONS For the Navier-Stokes fluid, if the vorticity is zero, ω = 0, the flow produces NO irreversible dissipation, and the flow is not turbulent. If the acceleration, a, is zero, or is orthogonal to the vorticity, there is NO irreversible dissipation and no Turbulence. Only when the component of fixed point acceleration of expansion, a, is parallel to the fixed point acceleration of rotation, ω, does the Navier-Stokes flow admit irreversible dissipation. Bottom line: When Spinor Fluctuations are included, The Navier-Stokes equations admit Turbulence and thermodynamically Irreversible Processes. A Hydrodynamic Example.

38. Regions where PTD  2 generate a connected topology; PTD  3 generate a disconnected topology. Continuous Processes can represent the evolution from disconnected topology (  3) to a connected topology (  2). Continuous Processes can NOT represent the evolution from a connected topology (  2) to a disconnected topology (  3). Overview of the Cartan Topology

39. Regions where PTD  2 generate a connected topology; PTD  3 generate a disconnected topology. Continuous Processes can represent the evolution from disconnected topology (  3) to a connected topology (  2). Continuous Processes can NOT represent the evolution from a connected topology (  2) to a disconnected topology (  3). Overview of the Cartan Topology Therefore Continuity and  2 determine A Topological Arrow of Time.

41. R.M.K. EGU Vienna 6.06 April 20, 2007 If time allows, consider other formats for the Action 1-form and its use in Variational Principles.

42. R.M.K. EGU Vienna 6.06 April 20, 2007 Consider the Classic Lagrange Action A L = L(q,V,t)dt Suppose A L encodes a thermodynamic system The Variational Problem: Find a path such that the integral of the Action is a minimum.

43. R.M.K. EGU Vienna 6.06 April 20, 2007 Consider the Classic Lagrange Action A L = L(q,V,t)dt BUT FIRST Utilize Topological Thermodynamics and Continuous Topological Evolution

44. R.M.K. EGU Vienna 6.06 April 20, 2007 Consider the Classic Lagrange Action A L = L(q,V,t)dt Compute the PTD(A L )

45. R.M.K. EGU Vienna 6.06 April 20, 2007 Consider the Classic Lagrange Action A L = L(q,V,t)dt Compute the PTD(A L ) dA L = dL(q,V,t)^dt and A L ^ dA L = 0

46. R.M.K. EGU Vienna 6.06 April 20, 2007 Consider the Classic Lagrange Action A L = L(q,V,t)dt Compute the PTD(A L ) dA L = dL(q,V,t)^dt and A L ^ dA L = 0 Hence the Pfaff Topological Dimension of A L is PTD(A) = 2 So the Classic Lagrange Action A L describes ISOLATED EQUILIBRIUM ONLY !!!

47. R.M.K. EGU Vienna 6.06 April 20, 2007 Topological Thermodynamics tells us that the Classic Lagrangian procedure will never describe Non-Equilibrium systems

48. R.M.K. EGU Vienna 6.06 April 20, 2007 Topological Thermodynamics tells us that the Classic Lagrangian procedure will never describe Non-Equilibrium systems The basic problem is the assumption of Kinematic Perfection (a single parameter group): dq – Vdt = 0 (A very strong topological constraint.)

49. R.M.K. EGU Vienna 6.06 April 20, 2007 Topological Thermodynamics tells us that the Classic Lagrangian procedure will never describe Non-Equilibrium systems The basic problem is the assumption of Kinematic Perfection (a single parameter group): dq – Vdt = 0 (A very strong topological constraint.) Why not relax the topological constraint to include topological fluctuations: dq – Vdt = Δq  0

50. R.M.K. EGU Vienna 6.06 April 20, 2007 So Utilize the Lagrange - Hilbert Action A LH = L(q k,V k,t)dt + P k (dq k -V k dt) The P k are Lagrange multipliers The Lagrange Hilbert Action admits thermodynamic topological fluctuations about the path of Kinematic Perfection.

51. R.M.K. EGU Vienna 6.06 April 20, 2007 A LH = L(q k,V k,t)dt + P k (dq k -V k dt) using H(q,v,p,t) = (L-P k V k ) A LH = P k dq k + H(q,v,p,t)dt Which is in an Eulerian-Fluid format !! Note that the the Lagrange - Hilbert Action can be re-written as

52. R.M.K. EGU Vienna 6.06 April 20, 2007 A LH = L(q k,V k,t)dt + P k (dq k -V k dt) has at first glance 3n+1 geometric variables, but it turns out that the maximum PTD (A) is NOT 3n+1, but the PTD (A LH ) is 2n+2. ( Symplectic) If the 3n+1 functions are all functions of 4 thermodynamic variables, the PTD (A) is at most 4. Consider the Lagrange - Hilbert Action

53. R.M.K. EGU Vienna 6.06 April 20, 2007 Bottom Line Lagrange-Hilbert Action or the Eulerian Action can be used for Non-Equilibrium systems. Both methods are equivalent and include fluctuations about kinematic perfection. (The Lagrange Action cannot)

54. Regions where PTD  2 generate a connected topology; PTD  3 generate a disconnected topology. Continuous Processes can represent the evolution from disconnected topology (  3) to a connected topology (  2). Continuous Processes can NOT represent the evolution from a connected topology (  2) to a disconnected topology (  3). Review of the Basic Ideas

55. Example: Continuous evolution can describe the irreversible evolution on an “Open” symplectic domain of Pfaff dimension 4, with evolutionary irreversible orbits being attracted to a contact “Closed” domain of Pfaff dimension 3, with an ultimate decay to the “Isolated-Equilibrium” domain of Pfaff dimension 2 or less (integrable Caratheodory surface). Emergence of Stationary States far from equilibrium by irreversible Processes

56. dA^dA < 0 A^dA > 0 A^dA<0 Pfaff Dimension 4 Pfaff Dimension 3 A^dA = 0 Pfaff Dimension 2 LIMIT CYCLE OPEN Symplectic Isolated-Equilibrium Irreversible Process Irreversible Decay on a Symplectic Manifold to a Contact Manifold of disconnected components, then to an Isolated-Equilibrium State. CLOSED Contact Disconnected Components

57. Evolution starts on the 2k+2 symplectic manifold with orbits being attracted to 2k+1 domains where the momenta become canonical: p -  L/  v  0. Topological evolution can either continue to reduce the Pfaff topological dimension, or the process on the Contact 2k+1 manifold can become “Hamiltonian extremal”, and the topological change stops /mod fluctuations. The resulting contact manifold becomes a “deformable, or topologically stationary” state, “Far from Equilibrium”. A LaGrange Euler Scenario

58. The Sliding - Rolling Ball page 1 Consider a bowling ball with initial translational and rotational energy, thrown to the floor of the bowling alley. Initially the ball skids or slips on a 2k + 2 symplectic manifold irreversibly reducing its energy and angular momentum via “friction” forces. From arbitrary initial conditions, the evolution is attracted to a 2k + 1 contact manifold, where the ball rolls without slipping, and the anholonomic constraint vanishes. dx -  d  = 0

59. The Sliding - Rolling Ball page 2 The subsequent motion, neglecting air resistance, continues in a Hamiltonian manner without change of Kinetic Energy or Angular Momentum. The 1-form of Action can be written as: A = L(t,x, ,v,  )dt +...+ s  (dx - d  ) The analytic dynamics are described in the next picture

60. The Sliding - Rolling Ball page 3 Irreversible Evolution - Pfaff dimension 2n+2 = 6 V-  > 0  < 0 V-  > 0  = 0 V-  > 0  > 0 V-  = 0 Extremal Hamiltonian Evolution Pfaff dimension 2n+1 = 5 T< 0 T = 0 T = T1 T = T3 T = T2 Note how dissipative irreversible process (friction) reverses the Angular Momentum!

61. Cartan’s Magic formula combines continuous topological evolution and thermodynamics Physical Systems of Pfaff dimension 4 generate a unique continuous evolutionary process which is thermodynamically irreversible. Without Topological Evolution, there is no Arrow of Time and no Thermodynamic Irreversibility. Summary

62. Recall the Axioms of Topological Thermodynamics 1. Topological Structure of Physical Systems is encoded by an Action differential 1-form A. 2. Physical Processes can be defined in terms of contravariant Vector/Spinor direction fields, V. The Action 1-form, A, is usually defined as Action per unit Source The unit source is mole number, charge, molecules, stars galaxies, nucleons, baryon mass.....

63. Recall the Axioms of Topological Thermodynamics 1. Topological Structure of Physical Systems is encoded by an Action differential 1-form A. 2. Physical Processes can be defined in terms of contravariant Vector/Spinor direction fields, V. CLASSIC EXAMPLES OF Different Action 1-forms Lagrange Action A = L(q k,V k,t)dt (isolated-equilibrium only) Cartan-Hilbert Action A = L(q k,V k,t)dt + p k (dq k -V k dt) Euler-Fluid Action A = p k dq k - H(p k,q k,t)dt Maxwell Action A = A k (x,y,z,t)dx k -  (x,y,z,t)dt.

64. Maximal Pfaff Topological Dimension and induced manifold structure of the different CLASSIC Action 1-forms Lagrange Action PTD(A) = 2 (Isolated) Cartan-Hilbert Action PTD(A) = 2n+2 (Symplectic) Euler-Fluid Action PTD(A) = 2n+1 (Contact) Maxwell Action PTD(A) = 4 (Symplectic)

65. Special Direction Fields Relative to a 1-form, A Associated V A i(V A )A = 0 Extremal V E i(V E )dA = 0 Characteristic V C i(V C )A = 0 and i(V C )dA = 0 Note that V C preserves the PTD(A) and the Cartan Topology: L(V C )A = 0 and L(V C )dA = 0 Recall Process can be defined in terms of contravariant Vector/Spinor direction fields, V.

66. Associated V A are locally Adiabatic as i(V A )A = 0  i(V A )Q=0 Extremal V E produce zero Thermodyamic Work and Hamiltonian mechanics i(V E )dA = W = 0 Characteristic V C lead to Hamilton Jacobi theory. Hamiltonian and adiabatic Recall Process can be defined in terms of contravariant Vector/Spinor direction fields, V.

67. Note that the “flow of Work” is always transverse to the Process i(V)W = i(V)i(V)dA = 0. But the “flow of Heat” is not transverse, i(V)Q ≠ 0, unless the process is Adiabatic, and i(V)Q = 0 This demonstrates the topological difference between Work, W, and Heat, Q.

68. Based on the Pfaff Topological Dimension of W Thermodynamic Reversible Processes imply that the Heat 1-form, Q is integrable. Q^dQ = 0 Extremal:PTD(W) = 0, W = 0, Hamiltonian Bernoulli-Casimir: PTD(W) = 1, W exact Hamiltonian Helmholtz: PTD(W) = 1, W closed Conservation of Vorticity Each of these flows are thermodynamically reversible, as dW = 0 = dQ, implies Q ^ dQ = 0.

69. Construct the 3-form A^F = A^dA Topological Torsion vector T 4 A ^ dA = i(T 4 )(dx^dy^dz^dt) T 4 is a 4 component Torsion Direction Field As L (T) A =  A, Q^dQ =  2 A^dA  0. Then  2  Entropy Production Rate  (1/2 divergence of T 4 ) 2 “Entropy” production Rate Bulk Viscosity