Plasma Instabilities: Review

Plasma Instabilities: Review Guy Moore, McGill University And Arnold, Yaffe, Mrówczyński, Romatschke, Strickland, Rebhan, Lenaghan, Dumitru, Nara, Bödeker, Rummukainen, Berges, Venugopalan, Ipp, Schenke, Manuel usw Instabilities: general (Abelian) picture Requirements for instabilities to make sense Growth rate, dependence on anisotropy How big do they grow? What do they do? Universität Heidelberg, 12 Dez. 2011: Seite 1 aus 24

Suppose two streams of plasma collide: becomes What happens? Universität Heidelberg, 12 Dez. 2011: Seite 2 aus 24

Magnetic field growth! Consider the effects of a seed magnetic field ˆB ˆp = 0 and ˆk ˆp = 0 How do the particles deflect? Universität Heidelberg, 12 Dez. 2011: Seite 3 aus 24

Positive charges: No net ρ. Net current is induced as indicated. Universität Heidelberg, 12 Dez. 2011: Seite 4 aus 24

Negative charges: same-sign current contribution Induced B adds to seed B. Exponential Weibel instability Linearized analysis: B grows until bending angles become large. Universität Heidelberg, 12 Dez. 2011: Seite 5 aus 24

Note: particles in other directions are stabilizing Sum of J from two signs weakens seed magnetic field. Isotropy: effects from different directions cancel! Universität Heidelberg, 12 Dez. 2011: Seite 6 aus 24

What is B field growth rate? Force, velocity, deflection: F = eb ; v = tf p = ebt p ; v t y = 2 = eb 2p t2. concentration of charges: y/λ B or k y k the B-wavevector J en chg k y e 2 Bt 2 k d 3 p (2π) 3 2p f(p) Define the combination e 2 d 3 p f(p) (2π) 3 m2 Screening p mass squared Universität Heidelberg, 12 Dez. 2011: Seite 7 aus 24

From last slide: J (kb)m 2 t 2. Current must compete with field terms in Ampere s law: D B D t E = J For current to really matter, need J D B kb. This occurs when m 2 t 2 1 or t 1/m. Hence growth rate estimate: B B 0 e γt, γ m. Picture is self-consistent IF particles stay in same-sign B field for time scales tγ > 1. Universität Heidelberg, 12 Dez. 2011: Seite 8 aus 24

Assumptions I Built In in Proceeding Abelian fields? NO! Nonabelian works too! But: Classical field approximation: need α 1 I use e 2, g 2, α interchangeably Classical particle treatment: λ p,debroglie k 1 B or p m,k inst. Allows HL approx. Must be true in each direction! Former: concepts make no sense at strong coupling (I think) Latter: constraints on occupancies and level of anisotropy of excitations giving rise to instability Universität Heidelberg, 12 Dez. 2011: Seite 9 aus 24

Dependence on Occupancy and Anisotropy if typical momentum p Q, m 2 αq 2 f (f typical occupancy): Higher Occupancy: Larger m 2 (red), k inst (black) Universität Heidelberg, 12 Dez. 2011: Seite 10 aus 24

Dependence on Occupancy and Anisotropy For high anisotropy, m 2 goes down, k inst goes up! Smaller m 2 : less filled phase space, fewer part. Larger k inst : next! Universität Heidelberg, 12 Dez. 2011: Seite 11 aus 24

High anisotropy and larger k inst Modes unstable whenever particles stay in same-sign B for t > 1/m. Narrow momentum distrib: time can be longer! angle δ Dist. δ/m Distance 1/m Time scale t δ/k not 1/k. So k m/δ OK! Universität Heidelberg, 12 Dez. 2011: Seite 12 aus 24

High anisotropy m 2 e 2 d 3 p p f(p) δαq2 f (less phase-space) But k inst m/δ δ 1 2 larger. Treatment inconsistent if k inst m/δ > δq p z z = 1 / p z x = 1/p x Wave packet does not fit! Alternate analysis in this regime: k inst mq: more than δq, less than m/δ. Wave Packet Universität Heidelberg, 12 Dez. 2011: Seite 13 aus 24

Weak anisotropy What if almost-isotropic, with a few (O(ǫ)) extras? = Isotropic part has no influence on what k s unstable! Repeat treatment using ǫm 2 in place of m 2. k inst ǫ 1 2m. But γ ǫ 3 2m. Universität Heidelberg, 12 Dez. 2011: Seite 14 aus 24

What limits B-field growth? Many things can do the job: Large angle change: y < 1/k or B < δmp/g Robust Nielsen-Olesen instability: B < k 2 /g m 2 /(δ 2 g) Robust Nonabelian Nonlinear Interactions: B < k k z /g m 2 /(δg) initial conditions? Whichever works at smallest B-value is relevant. Universität Heidelberg, 12 Dez. 2011: Seite 15 aus 24

Large angle change Current stops building when particles bend too much. y > 1/k p > δp, B > δmp/g Universität Heidelberg, 12 Dez. 2011: Seite 16 aus 24

Nielsen-Olesen Instability Uniform B field: circular orbits, quantized p : ( to B that is) p 2 = (1 2n)eB. Energies of allowed excitations FOR SPIN-1: E 2 = p 2 p 2 z 2e s B = p 2 z (2n 1 ± 2)eB One mode has negative E 2 ; exp growth, γ N-O gb Uniform B approx is OK if p 2 increments 2eB > k 2 inst. So k inst < eb to avoid instability Universität Heidelberg, 12 Dez. 2011: Seite 17 aus 24

Nonabelian Nonlinearities Suppose modes grow with many colors and k s. One color acts to rotate J s due to another color: No Blue Current No Blue Current Universität Heidelberg, 12 Dez. 2011: Seite 18 aus 24

Requirement this happens: A in covariant deriv, A m/g or B m 2 /(gδ). Gauge-invariant version: 1/k 1/m Color randomization when Wilson loop shown has O(1) phase. Universität Heidelberg, 12 Dez. 2011: Seite 19 aus 24

What do plasma instabilities do? Main thing: angle-change. θ 1 for self-consistency (we saw) Many small independent kicks : describe with ˆq ˆq dp2 dt ( p)2 t coh = F 2 t coh αb 2 t coh Now B m 2 /gδ, t coh 1/m ˆq m3 δ 2 Thermal-like, O(1) anisotropic: g 3 T 3 (elastic: g 4 T 3 ) Enhanced by 1/δ 2 when large anisotropy. Universität Heidelberg, 12 Dez. 2011: Seite 20 aus 24

Where are plasma instabilities important? Assume ONE typical momentum scale, eg, Q s Anisotropy δ = α d Occupancy f = α c Occupancy-anisotropy plane. f α c Θ(p Q s )Θ( p z α d Q s ) Universität Heidelberg, 12 Dez. 2011: Seite 21 aus 24

slope 9/23 Where are plasma instabilities important? Consider feedback of instab. on hard modes, radiation, merging: Kukela Moore I Anisotropy d=ln( δ)/ln( α) Plasma Instabilities slope 1 Soft Particle Bath Forms c=1/3 d=1/7 d=1/3 d=ln( ε )/ln( α) Scattering c=1 Occupancy c=ln(f)/ln( α) Nielsen Olesen Instabilities Universität Heidelberg, 12 Dez. 2011: Seite 22 aus 24

Application: Heavy Ions Kurkela Moore II Anisotropy d=ln( δ)/ln( α) slope 9/23 Plasma Instabilities slope 1 Attractor 1 Initial Condition Soft Particle Bath Forms c=1/3 d=1/7 d=1/3 Attractor 2 d=ln( ε )/ln( α) Scattering c=1 Occupancy c=ln(f)/ln( α) 8: Nielsen Olesen Instabilities Thermal bath dominates: t α 12 5 s equilibration: t α 5 2 s Q 1 s. Q 1 s Universität Heidelberg, 12 Dez. 2011: Seite 23 aus 24

Conclusions Plasma instabilities generic in anisotropic, α 1 plasmas Especially important in situations of high anisotropy More phenomena can limit growth in a nonabelian than in an abelian context Should play a pivotal role in the equilibration process in heavy ion collisions (in the toy case of α s 1) To make quantitative predictions we need to understand the weak anisotropy case. Universität Heidelberg, 12 Dez. 2011: Seite 24 aus 24