Essence of of critical phenomena

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Abbas K. Rizi

See https://complexity.sitpor.org/2020/04/10/cp/

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Essence of
Critical Phenomena:
Phase Transitions & RG
1
Abbas K. Rizi
CX Group, CS Dept., Aalto University
abbas.sitpor.org

Emergence of giant component in an ER Network
Networks, 2nd edition. Mark Newman, Oxford University Press (2018). © Mark Newman.
DOI: 10.1093/oso/9780198805090.001.0001
2

Phase Transitions
© cosmosmagazine.com
3

Phase Transitions
© cosmosmagazine.com
4

Phase Transitions
© cosmosmagazine.com
© Wikipedia
Control Parameter
OrderParameter
order disorder
5

Phase Transitions
© cosmosmagazine.com
© Wikipedia
Critical Point
OrderParameter
order disorder
6

Phase Transitions
© cosmosmagazine.com
© WikipediaOrder Parameters:
● Magnetization: <σ>
● Density of Particles: ρ
7

Ising Model & Phase Transition
Configurations of the Ising model on a two-dimensional square
lattice considered as a magnet (left) and as a lattice gas (right).
8

Network (Lattice)
Ising Model
j
k
l
n
i
Jij = 1
Jmj = 0
12

bit-player.org
18
#Spontaneous_Symmetry_Breaking

Hyperscaling Relations
Rushbrooke
Widom
Fisher
Josephson
22

Simon DeDeo, Ph.D. in Astrophysics
✘ Assistant Professor, Laboratory for Social
Minds, Carnegie Mellon University
✘ External Faculty at the Santa Fe Institute
✘ http://bit.ly/SFIrenorm
Introduction to Renormalization
26

coarse-graining
Lossycompression
Theory
28

coarse-graining
Lossycompression
Theory 1
Theory 2
29

coarse-graining
Lossycompression
Theory 1
Theory 2
RGFlow
30

Real Space Renormalization Group
(RSRG)
32

σ1
Renormalizing the Ising Model
σ4 σa σ2
σ3
decimation
33

σ1
σ4 σa σ2
σ3
#trace_out
Renormalizing the Ising Model
35

41
Inducing Quartets and Commutation Failure

Changing the network structure
Network J Network J’
47

49
#synergy
#emergence
Taking out of the Model Class

σ1
σ4 σa σ2
σ3
#trace_out
Coarse-graining: a single site
50

#trace_out
Coarse-graining: Whole Lattice
σ1
σ4 σa σ2
σ3
σa σa
σaσa
51

52
σ1
σ4 σa σ2
σ3
σa σa
σaσa
σ1
σ4 σ2
σ3

54
σ
σ σa σ
σ
σa σa
σaσa
σa
σa σa
σaσa
σ
σ σ
σ
σ
σa
σ
σ σa σ σaσ σσa
2s2

60
Changing the Square Structure
σa
2s2
s2
s4

61
1st NN: 2S2
2nd NN: S2
Emergent Term: S4
Hyper-Graph

65
Saving The Square Structure
1st NN
"Approx. RG Group"
J Isomorphism J’
J is #isomorphic to J ‘

66
Saving The Square Structure
1st NN
"Approx. RG Group"

67
RG RG RG RG
Saving The Square Structure
1st NN
"Approx. RG Group"

70
σ1
σa σ2
σ3
2s2
s2
σ4
A Better Transformation!

71
σ1
σa σ2
σ3
2s2
s2σ4
A Better Transformation!

72
σ1
σa σ2
σ3
3s2
σ4
A Better Transformation!

89
β0.0
∞
0.441
Crystal Statistics. I. A Two-Dimensional Model with an
Order-Disorder Transition
Phys. Rev. 65, 117 – Published 1 February 1944
Lars Onsager
Exact Solution for 2D Ising Model

92
https://www.youtube.com/watch?v=MxRddFrEnPc

Thanks!
Abbas K. Rizi
ABBAS.SITPOR.ORG
✘ bit.ly/SFIrenorm
95

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Essence of of critical phenomena

1. Essence of Critical Phenomena: Phase Transitions & RG 1 Abbas K. Rizi CX Group, CS Dept., Aalto University abbas.sitpor.org

2. Emergence of giant component in an ER Network Networks, 2nd edition. Mark Newman, Oxford University Press (2018). © Mark Newman. DOI: 10.1093/oso/9780198805090.001.0001 2

8. Ising Model & Phase Transition Configurations of the Ising model on a two-dimensional square lattice considered as a magnet (left) and as a lattice gas (right). 8

9. Ising Model 9

10. Ising Model 10

11. σi : +1, -1 i Ising Model 11

12. Network (Lattice) Ising Model j k l n i Jij = 1 Jmj = 0 12

17. Partition Function Ising Model 17

18. bit-player.org 18 #Spontaneous_Symmetry_Breaking

19. github.com/VictorSeven/IsingModel 19

20. Scalings & Universalities TTc m 20

21. Scalings & Universalities TTc m 21

22. Hyperscaling Relations Rushbrooke Widom Fisher Josephson 22

23. 23 https://youtu.be/Kd4UvhUsBAU

24. 24

25. Renormalization 25

26. Simon DeDeo, Ph.D. in Astrophysics ✘ Assistant Professor, Laboratory for Social Minds, Carnegie Mellon University ✘ External Faculty at the Santa Fe Institute ✘ http://bit.ly/SFIrenorm Introduction to Renormalization 26

27. Theory Renormalization 27

28. coarse-graining Lossycompression Theory 28

29. coarse-graining Lossycompression Theory 1 Theory 2 29

30. coarse-graining Lossycompression Theory 1 Theory 2 RGFlow 30

31. 31

32. Real Space Renormalization Group (RSRG) 32

33. σ1 Renormalizing the Ising Model σ4 σa σ2 σ3 decimation 33

34. 34

35. σ1 σ4 σa σ2 σ3 #trace_out Renormalizing the Ising Model 35

36. 36

37. 37

38. 38

39. coarse-graining 39

40. 40

41. 41 Inducing Quartets and Commutation Failure

42. 42

43. 43

44. σ1 σ4 σ2 σ3 44

45. σ1 σ4 σ2 σ3 45

46. σ1 σ4 σ2 σ3 46

47. Changing the network structure Network J Network J’ 47

48. RG Renormalizing the Ising Model 48

49. 49 #synergy #emergence Taking out of the Model Class

50. σ1 σ4 σa σ2 σ3 #trace_out Coarse-graining: a single site 50

51. #trace_out Coarse-graining: Whole Lattice σ1 σ4 σa σ2 σ3 σa σa σaσa 51

52. 52 σ1 σ4 σa σ2 σ3 σa σa σaσa σ1 σ4 σ2 σ3

53. 53 σ1 σ4 σa σ2 σ3 σa σa σaσa S2+S2

54. 54 σ σ σa σ σ σa σa σaσa σa σa σa σaσa σ σ σ σ σ σa σ σ σa σ σaσ σσa 2s2

55. 55 45°

56. 56 Square Lattice! 1st NN 2s2

57. 57 Square Lattice! 1st NN

58. 58 σ1 σa σ2 σ3 2s2 s2 1st NN 2nd NN σ4

59. 59 1st NN 2nd NN

60. 60 Changing the Square Structure σa 2s2 s2 s4

61. 61 1st NN: 2S2 2nd NN: S2 Emergent Term: S4 Hyper-Graph

62. 62

63. 63

64. 64

65. 65 Saving The Square Structure 1st NN "Approx. RG Group" J Isomorphism J’ J is #isomorphic to J ‘

66. 66 Saving The Square Structure 1st NN "Approx. RG Group"

67. 67 RG RG RG RG Saving The Square Structure 1st NN "Approx. RG Group"

68. 68 Bad Decimation Transformation!

69. 69 A Better Transformation!

70. 70 σ1 σa σ2 σ3 2s2 s2 σ4 A Better Transformation!

71. 71 σ1 σa σ2 σ3 2s2 s2σ4 A Better Transformation!

72. 72 σ1 σa σ2 σ3 3s2 σ4 A Better Transformation!

73. 73 A Better Transformation!

74. 74

75. 75 β f(β)

76. 76 β f(β)

77. 77 β f(β)

78. 78 β f(β)

79. 79 β f(β)

80. 80 β f(β)

81. 81 β f(β) Low T

82. 82 β f(β)

83. 83 β f(β) ∞

84. 84 β f(β) ∞

85. 85 β f(β) Self-similarity! 0.508

86. 86 β f(β) ∞ β0.0 ∞ 0.508

87. 87

88. 88 β f(β) ∞ β0.0 ∞ 0.441

89. 89 β0.0 ∞ 0.441 Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition Phys. Rev. 65, 117 – Published 1 February 1944 Lars Onsager Exact Solution for 2D Ising Model

90. 90 bit-player.org

91. 91 β 0.0 ∞ 0.441 bit-player.org

92. 92 https://www.youtube.com/watch?v=MxRddFrEnPc

93. 93 kineticallyconstrained.com

94. 94 https://youtu.be/fi-g2ET97W8

95. Thanks! Abbas K. Rizi ABBAS.SITPOR.ORG ✘ bit.ly/SFIrenorm 95

Essence of of critical phenomena

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