Transport Coefficients in Classical Summary Relativistic Field - PowerPoint PPT Presentation

Transport Coefficients in Classical Relativistic Field Theories Marietta M. Homor Introduction Energy and Temperature Mass Classical viscosity Transport Coefficients in Classical Summary Relativistic Field Theories Marietta M. Homor October 9, 2014

Viscosity of hadronic matter Transport Coefficients in Classical Relativistic Field Theories Marietta M. Homor Introduction Energy and Temperature Mass Classical viscosity Summary Figure: Heavy-ion collisions ◮ observation: nearly ideal liquid ◮ relevant quantity: η/ s (damping of hydrodynamic waves) Figure: η/ s [1]

Motivation Transport Coefficients in Classical Relativistic Field Theories Marietta M. Homor ◮ small transport coefficient ⇔ strongly interacting system (strength: g ) Introduction η Energy and 1 s ∼ ◮ g 4 log g Temperature ◮ non-perturbative Mass Classical viscosity ◮ Monte Carlo simulation: less sensitive for ω → 0 , Summary large systematic errors ◮ Boltzmann equation: only if 2-2 collision is enough ◮ objective: transport coefficient in classical field theory ◮ scheme: „leap-frog” algorithm (dynamics) → correlators (direct count) → transport coefficient from Green-Kubo - formula ◮ test: classical Φ 4 theory 2 ( ∇ Φ) 2 + m 2 2 Φ 2 + λ H = 1 2 Π 2 + 1 24 Φ 4 (1)

Simulation Transport Coefficients in Classical Relativistic Field Theories Marietta M. Homor Introduction Energy and ◮ canonical equations, periodic boundary Temperature conditions, leap-frog algorithm Mass t 0 + δ t Classical viscosity ◮ initial conditions: � � � � Π , Φ( t 0 ) 2 Summary ◮ Canonical eq. of ˙ Φ (1st part of time step): Initial condition → Φ( t 0 + δ t ) ◮ Canonical eq. of ˙ Π (2nd part of time step): t 0 + δ t t 0 + 3 δ t � � �� � � Φ( t 0 + δ t ) , Π → Π 2 2 ◮ energy controlled simulation ◮ input parameters: N 3 lattice size, a = 1 (grid), λ (interaction), m 2 Lagrangian-mass

Energy and Temperature Transport Coefficients in Classical Relativistic Field Theories Marietta M. Homor ◮ time independent total energy: Introduction Energy and i + m 2 Temperature 1 i + 1 i + λ � 2 Π 2 2 ( ∇ Φ) 2 2 Φ 2 24 Φ 4 E = (2) Mass i i ∈ U Classical viscosity Summary ◮ in canonical ensemble: �| Π k | 2 � = 1 � � d Π k d Φ k | Π k | 2 e − β H (3) Z k k ◮ Fourier transformation: x ∈ U exp − 2 π i ( kx ) / N f x ∈ U U = � f k ∈ ¯ ◮ Temperature using numerical schemes: �| Π k | 2 � = 2 N 3 T (4)

Temperature Transport Coefficients in Classical Relativistic Field Theories Marietta M. Homor ◮ Temperature in this simulation : given Π lattice → �| Π k | 2 � counted directly Introduction ◮ equilibrium: equipartition and time independece Energy and Temperature Mass Typical data Time average Classical viscosity Temperature ( 2 N 3 T ) Summary Mode ( ¯ k ) k 2 = � 3 Figure: Equipartition ( ¯ i = 1 sin 2 � 2 π ke i � ) N

Energy and Temperature Transport Coefficients in Classical Relativistic Field Theories Marietta M. Homor Data ax + b Introduction cx + d Energy and Temperature Mass Total Energy ( E ) Classical viscosity Summary a = 0 , 63 , b = 1 , 00 c = 0 , 71 , d = − 9392 , 47 intersect. = 125987 Temperature ( 2 N 3 T ) Figure: Temperature dependence of total energy ( N 3 = V = 50 3 ), λ = 5 , m 2 = − 0 , 5

Definitions Transport Coefficients in Classical Relativistic Field Theories Marietta M. Homor 1. based on the oscillation of Φ( x ) Introduction ◮ neglectable △ Φ and ∇ Φ Energy and ◮ weak interaction → Gauss-approximation: Temperature � Φ 4 � ≈ 3 � Φ 2 �� Φ 2 � Mass m 2 + λ ◮ ¨ � 2 Φ 2 � Φ + Φ = 0 Classical viscosity ◮ M = 2 π Summary T 2. G ( t , x ) := � Φ( t + t ′ , x )Φ( t , y ) � correlator ◮ Fourier → � t max 1 d t ′ Φ( t + t ′ , k )Φ( t ′ , − k ) G ( t , k ) = t max − t min t min (5) ◮ non-interacting, oscillating solution: | Φ k | 2 � � G ( t , k ) = cos ω k t , (6) k = k 2 + m 2 where ω 2

Mass definition Transport Coefficients in Classical Relativistic Field Theories ◮ non-interacting theory: Marietta M. Homor � Φ( t ′ , k )Φ( t ′ + t , − k ) � = �| Φ k | 2 � cos ω k t Introduction ◮ Interacting system: Energy and Temperature � Φ( 0 , | k | = 0 )Φ( t , 0 ) � = a exp ( − t /τ ) cos ( ω t + φ ) Mass Classical viscosity � Φ( 0 , | k | = 0 )Φ( t , 0 ) � Summary Time ( t ) Figure: Effective mass determined by correlator

Temperature dependency Transport Coefficients in Classical Relativistic Field Theories Mass controlled simulation Marietta M. Homor t 0 + δ t ◮ initial conditions: � � � � Π , Φ( t 0 ) Introduction 2 ◮ time developement then calculation of E , T , M Energy and Temperature ◮ if M � = M wanted system change Mass ◮ additive and multiplicative noise → exponential Classical viscosity decay of total energy → new system → Summary recalculation Effective mass ( M ) Effective mass ( M ) 1.1 1.1 1 1 0.9 0.9 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 0 100000 200000 300000 400000 500000 600000 Temperature ( 2 N 3 T ) Temperature ( 2 T ) (a) various volumes (b) various Lagrangian m Figure: Temperature dependence of the effective mass

Spontaneous symmetry breaking Transport Coefficients in Classical Relativistic Field Theories ◮ if m 2 < 0 → min. of potential energy at: Marietta M. Homor � 6 | m | 2 Introduction Φ 0 = ± λ Energy and ◮ 2nd order phase transition Temperature Effective Mass (M) Mass 1.2 Classical viscosity 1.1 Summary 1 0.9 0.8 0.7 0.6 0.5 0.4 various ( λ ) 0.3 0 100000 200000 300000 400000 Temperature ( 2 N 3 T ) Figure: Temperature dependence of the effective mass

Mass near phase-shift Transport Coefficients in Classical Relativistic Field Theories Marietta M. Homor data data � Φ( 0 , 0 )Φ( t , 0 ) � � Φ( 0 , 0 )Φ( t , 0 ) � fit fit Introduction Energy and Temperature Mass Classical viscosity Summary Time ( t ) Time ( t ) (a) m = 0 . 88 (b) m = 0 . 69 data data � Φ( 0 , 0 )Φ( t , 0 ) � � Φ( 0 , 0 )Φ( t , 0 ) � fit fit Time ( t ) Time ( t ) (c) m = 0 . 53 (d) m = 0 . 43 Figure: Different mass, same interaction strength λ = 5

Viscosity from Green-Kubo - formula Transport Coefficients in Classical Relativistic Field Theories Marietta M. Homor ◮ Spectral function for bosonic A and B operators: S BA ( t ) = � B ( t ) A ( 0 ) � . Introduction ◮ Fluctuation-dissipation theorem on finite Energy and Temperature temperature: Mass Classical viscosity � [ B , A ] � ( ω, k = 0 ) = ( 1 − e − βω ) S BA ( ω, k = 0 ) . (7) Summary ◮ if βω << 1 ( ω → 0 ): 1 lim ω � [ B , A ] � ( ω, k = 0 ) = β S BA ( ω = 0 , k = 0 ) . (8) ω → 0 ◮ Linear response theory (Kubo-formula for transport): � [ T 12 , T 12 ] � ( ω, k = 0 ) η = lim , (9) ω ω → 0 where T 12 = ∂ x Φ ∂ y Φ . ◮ Green-Kubo - formula: η = β � T 12 T 12 � ( k = 0 )

� T 12 T 12 � Transport Coefficients in Classical Relativistic Field Theories � T 12 T 12 � ( t , 0 ) Marietta M. Homor 12000 10000 Introduction 8000 Energy and Temperature 6000 Mass 4000 Classical viscosity 2000 Summary 0 0 5 10 15 20 25 30 35 40 ( t ) � T 12 T 12 � ( ω, 0 ) 180000 160000 140000 120000 100000 80000 60000 40000 20000 0 0 2 4 6 8 10 12 14 16 ( ω ) Figure: Result of simulation and Fourier-transform

Classical viscosity Transport Coefficients in Classical Relativistic Field Theories Classical viscosity Marietta M. Homor 0.7 0.6 η = β � T 12 T 12 � Introduction 0.5 Energy and Temperature 0.4 Mass 0.3 Classical viscosity 0.2 Summary 0.1 λ = 5 0 0 100000 200000 300000 400000 Temperature ( 2 N 3 T ) 3.5e-06 3e-06 2.5e-06 2e-06 η/ T 1.5e-06 1e-06 5e-07 λ = 5 0 0 100000 200000 300000 400000 Temperature ( 2 N 3 T ) Figure: Classical viscosity

Interpretation Transport Coefficients in Classical Relativistic Field Theories Marietta M. Homor Introduction Energy and Temperature ◮ validity range of classical approximation: p max ≪ T Mass Λ 3 ◮ role of cut-off: phase-volume: 6 π 2 , extreme case: Classical viscosity Summary Λ ≈ T ◮ 1st assumption: transport is dominated by classical fields T 3 ◮ ⇒ η = η class 6 π 2 ◮ 2nd assumption: key element in entropy is the entropy of free boson gas (QM)

η/ s Transport Coefficients in Classical Relativistic Field Theories Marietta M. Homor Introduction 0.6 Energy and Temperature λ = 5 1 / ( 4 π ) Mass 0.5 Classical viscosity Summary 0.4 η/ s 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 T / M

Summary Transport Coefficients in Classical Relativistic Field Theories Marietta M. Homor Introduction ◮ canonical equations + leap-frog algorithm Energy and Temperature ◮ given Π and Φ fields → E , T Mass ◮ mass by correlator, various system sizes and Classical viscosity Summary interaction strengths ◮ viscosity determinated by Green-Kubo formula ( η = β � T 12 T 12 � ( ω = 0 , k = 0 ) ) ◮ ⇒ practicable method for determining transport coefficients ◮ extra: spontaneous symmetry breaking, η/ s ◮ future plans: more complicated and physically relevant systems