[PPT] - Transport Coefficients in Classical Summary Relativistic Field PowerPoint Presentation

SLIDE 1

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

Transport Coefficients in Classical Relativistic Field Theories

Marietta M. Homor October 9, 2014

SLIDE 2

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

Viscosity of hadronic matter

Figure: Heavy-ion collisions Figure: η/s [1]

◮ observation: nearly ideal

liquid

◮ relevant quantity: η/s

(damping of hydrodynamic waves)

SLIDE 3

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

Motivation

◮ small transport coefficient ⇔ strongly interacting

system (strength: g)

◮ η s ∼ 1 g4 log g ◮ non-perturbative ◮ Monte Carlo simulation: less sensitive for ω → 0,

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 Φ4 theory

H = 1 2Π2 + 1 2(∇Φ)2 + m2 2 Φ2 + λ 24Φ4 (1)

SLIDE 4

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

Simulation

◮ canonical equations, periodic boundary

conditions, leap-frog algorithm

◮ initial conditions:

Π
t0 + δt

2

, Φ(t0)
◮ Canonical eq. of ˙

Φ (1st part of time step): Initial condition → Φ(t0 + δt)

◮ Canonical eq. of ˙

Π (2nd part of time step):

Φ(t0 + δt), Π
t0 + δt

2

→ Π
t0 + 3δt

2

◮ energy controlled simulation

◮ input parameters: N3 lattice size, a = 1 (grid), λ

(interaction), m2 Lagrangian-mass

SLIDE 5

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

Energy and Temperature

◮ time independent total energy:

E =

i∈U

1 2Π2

i + 1

2(∇Φ)2

i + m2

2 Φ2

i + λ

24Φ4

i

(2)

◮ in canonical ensemble:

|Πk|2 = 1 Zk

k

dΠkdΦk|Πk|2e−βH (3)

◮ Fourier transformation:

fk∈¯

U = x∈U exp−2πi(kx)/N fx∈U ◮ Temperature using numerical schemes:

|Πk|2 = 2N3T (4)

SLIDE 6

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

Temperature

◮ Temperature in this simulation: given Π lattice

→ |Πk|2 counted directly

◮ equilibrium: equipartition and time independece

Temperature (2N3T) Mode (¯ k)

Typical data Time average

Figure: Equipartition (¯ k2 = 3

i=1 sin2

2π kei

N

)

SLIDE 7

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

Energy and Temperature

Total Energy (E) Temperature (2N3T)

Data

ax + b cx + d

a = 0,63, b = 1,00 c = 0,71, d = −9392,47

intersect. = 125987

Figure: Temperature dependence of total energy (N3 = V = 503), λ = 5, m2 = −0,5

SLIDE 8

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

Definitions

1. based on the oscillation of Φ(x)

◮ neglectable △Φ and ∇Φ ◮ weak interaction → Gauss-approximation:

Φ4 ≈ 3Φ2Φ2

◮ ¨

Φ +

m2 + λ

2 Φ2

Φ = 0

◮ M = 2π

T

2. G(t, x) := Φ(t + t′, x)Φ(t, y) correlator

◮ Fourier →

G(t, k) = 1 tmax − tmin tmax

tmin

dt′Φ(t + t′, k)Φ(t′, −k) (5)

◮ non-interacting, oscillating solution:

G(t, k) =

|Φk|2

cos ωkt, (6) where ω2

k = k2 + m2

SLIDE 9

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

Mass definition

◮ non-interacting theory:

Φ(t′, k)Φ(t′ + t, −k) = |Φk|2 cos ωkt

◮ Interacting system:

Φ(0, |k| = 0)Φ(t, 0) = a exp(−t/τ) cos(ωt + φ)

Time (t) Φ(0, |k| = 0)Φ(t, 0)

Figure: Effective mass determined by correlator

SLIDE 10

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

Temperature dependency

Mass controlled simulation

◮ initial conditions:

Π
t0 + δt

2

, Φ(t0)
◮ time developement then calculation of E,T,M

◮ if M = Mwanted system change ◮ additive and multiplicative noise → exponential

decay of total energy → new system → recalculation

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Temperature (2T)

Effective mass (M)

(a) various volumes

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 100000 200000 300000 400000 500000 600000

Effective mass (M)

Temperature (2N3T)

(b) various Lagrangian m Figure: Temperature dependence of the effective mass

SLIDE 11

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

Spontaneous symmetry breaking

◮ if m2 < 0 → min. of potential energy at:

Φ0 = ±

6|m|2

λ ◮ 2nd order phase transition

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 100000 200000 300000 400000

Effective Mass (M) Temperature (2N3T) various (λ)

Figure: Temperature dependence of the effective mass

SLIDE 12

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

Mass near phase-shift

Φ(0, 0)Φ(t, 0) Time (t)

data fit

(a) m = 0.88

Φ(0, 0)Φ(t, 0) Time (t)

data fit

(b) m = 0.69

Φ(0, 0)Φ(t, 0) Time (t)

data fit

(c) m = 0.53

Φ(0, 0)Φ(t, 0) Time (t)

data fit

(d) m = 0.43 Figure: Different mass, same interaction strength λ = 5

SLIDE 13

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

Viscosity from Green-Kubo - formula

◮ Spectral function for bosonic A and B operators:

SBA(t) = B(t)A(0).

◮ Fluctuation-dissipation theorem on finite

temperature: [B, A](ω, k = 0) = (1 − e−βω)SBA(ω, k = 0). (7)

◮ if βω << 1 (ω → 0):

lim

ω→0

1 ω[B, A](ω, k = 0) = βSBA(ω = 0, k = 0). (8)

◮ Linear response theory (Kubo-formula for

transport): η = lim

ω→0

[T12, T12](ω, k = 0) ω , (9) where T12 = ∂xΦ∂yΦ.

◮ Green-Kubo - formula: η = βT12T12(k=0)

SLIDE 14

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

T12T12

2000 4000 6000 8000 10000 12000 5 10 15 20 25 30 35 40

(t)

T12T12(t, 0)

20000 40000 60000 80000 100000 120000 140000 160000 180000 2 4 6 8 10 12 14 16

(ω)

T12T12(ω, 0) Figure: Result of simulation and Fourier-transform

SLIDE 15

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

Classical viscosity

0.1 0.2 0.3 0.4 0.5 0.6 0.7 100000 200000 300000 400000

η = βT12T12

Temperature (2N3T) Classical viscosity λ = 5

5e-07 1e-06 1.5e-06 2e-06 2.5e-06 3e-06 3.5e-06 100000 200000 300000 400000

η/T

Temperature (2N3T) λ = 5

Figure: Classical viscosity

SLIDE 16

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

Interpretation

◮ validity range of classical approximation: pmax ≪ T ◮ role of cut-off: phase-volume: Λ3 6π2 , extreme case:

Λ ≈ T

◮ 1st assumption: transport is dominated by

classical fields

◮ ⇒ η = ηclass

T3 6π2

◮ 2nd assumption: key element in entropy is the

entropy of free boson gas (QM)

SLIDE 17

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

η/s

0.1 0.2 0.3 0.4 0.5 0.6 0.2 0.4 0.6 0.8 1 1.2

η/s T/M

λ = 5 1/(4π)

SLIDE 18

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

Summary

◮ canonical equations + leap-frog algorithm ◮ given Π and Φ fields → E, T ◮ mass by correlator, various system sizes and

interaction strengths

◮ viscosity determinated by Green-Kubo formula

(η = βT12T12(ω = 0, k = 0))

◮ ⇒ practicable method for determining transport

coefficients

◮ extra: spontaneous symmetry breaking, η/s ◮ future plans: more complicated and physically

relevant systems

SLIDE 19

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

Bibliography

Roy A. Lacey, N. N. Ajitanand, J. M. Alexander, P . Chung,

W. G. Holzmann, M. Issah, A. Taranenko, P

. Danielewicz, and Horst Stöcker. Has the qcd critical point been signaled by observations at the bnl relativistic heavy ion collider?

Phys. Rev. Lett., 98:092301, Mar 2007.

Edward Shuryak. Why does the quark gluon plasma at RHIC behave as a nearly ideal fluid? Prog.Part.Nucl.Phys., 53:273–303, 2004. Jakovác Antal. Véges h˝

mérséklet˝

u térelméletek jegyzet. El˝

adás jegyzet.

Pavel Kovtun, Dam T. Son, and Andrei O. Starinets. Holography and hydrodynamics: Diffusion on stretched horizons. JHEP, 0310:064, 2003.

SLIDE 20

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