Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions Rainer Verch ITP, Universit¨ at Leipzig Cortona, June 7, 2018 joint work w/ Thomas-Paul Hack arXiv:1806.00504 Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
Two pictures say more than thousand words ... ρ P ( t , x 1 ) /ρ β 1 , ρ = T 00 j P ( t , x 1 ) / j N , j = T 01 free m > 0 Klein-Gordon in d = 1 + 3, limit state is stationary and homogeneous but not KMS Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
Other models - conformal hydrodynamics in d = 1 + 2 [Bhaseen et al., Nature Phys. 11 (2015) 5] Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
Other models - CFTs in d = 1 + 1 [Bernard & Doyon 2012-2014, Hollands & Longo 2016 (pic.)] Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
Plan of the talk ... and other models / initial states, axiomatic analysis in quant. stat. mech. by [Ruelle 2000] , different NESS-setup in 1+3 QFT [Drago, Faldino, Pinamonti 2017] [Doyon et. al 2014] have studied the free Klein-Gordon field w/ m ≥ 0 in d = 1 + n , “semi-box” Fock space picture, sharp contact surface we analyse the case of a “smooth contact” to have better regularity of the initial state (Hadamard state), the limit NESS is the same as the one of [Doyon et. al 2014] with better regularity of the initial state, we can also construct and analyse interacting case (at first order) the interacting NESS does not appear to be closer to equilibrium than the free NESS, but both are stable w.r.t. spatially localised perturbations Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
NESS for the free Klein-Gordon field Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
Basic idea basic idea: construct an initial state ω G by gluing together initial data of Wightman correl. fct. of ω β 1 , ω β 2 , β 1 = ( k B T 1 ) − 1 � = β 2 = ( k B T 2 ) − 1 , evolve initial data in time (technical) problem: positivity (unitarity) of correl. fct. for observables in the contact region and cross-correl. ∆ + , G ( f , f ) . ! = ω P ( φ ( f ) φ ( f )) ≥ 0 f any test function � φ ( f ) . = dx φ ( x ) f ( x ) consider ω i , i = 1 , 2 , 3 with ∆ + , i ≥ ∆ + , 3 , i = 1 , 2, µ i initial data for ∆ + , i , 1 = χ 1 ( x 1 ) + χ 2 ( x 1 ) smooth part. of 1 for x 1 -axis ⇒ initial data for ∆ + , G ≥ 0 constructed by µ G . = ( χ 1 ⊗ χ 1 ) µ 1 + ( χ 2 ⊗ χ 2 ) µ 2 + 2( χ 1 ⊗ χ 2 + χ 2 ⊗ χ 1 ) µ 3 ≥ µ 3 Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
Thermal domination we choose ∂ x 1 χ 1 = − ∂ x 1 χ 2 of compact support = “contact region”, ∂ x 1 χ 1 determines contact profile, ω 3 is “state in contact region and for cross-correlations” correl. fct. of thermal (KMS) state ∆ + ,β ( x 1 , x 2 ) = ∆ + , ∞ ( x 1 , x 2 ) � d p 1 cos ( ω p ( t 1 − t 2 ) − p · ( x 1 − x 2 )) � | p | 2 + m 2 + ω p = (2 π ) d − 1 ω p exp( βω p ) − 1 ∆ + ,β 1 ≥ ∆ + ,β 2 for β 1 ≤ β 2 ⇒ ω 3 can be chosen e.g. as mixture of ω β with β ≥ max( β 1 , β 2 ), for practical computations vacuum is simplest choice physically we expect that contact state can also be “hotter” than left/right reservoirs Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
Generalisation of gluing procedure consider a smooth partition of unity 1 = ψ + (1 − ψ ) of the time-axis such that ˙ ψ has compact support and ψ vanishes for large negative times K . g . ∆ . = [ K , ψ ] = ¨ ψ + 2 ˙ = � + m 2 , ψ∂ t , = ∆ R − ∆ A � τφ . = ∆ g φ = dy ∆( x , y )[ K , ψ ( t y )] φ ( y ) is well-defined for any smooth function φ and the identity on solutions of the KGE, g φ is “thickened initial data” for φ localised in supp ˙ ψ for ω i , i = 1 , 2 , 3 with ∆ + , i ≥ ∆ + , 3 , i = 1 , 2, σ i . = τχ i = ∆ χ i g ∆ + , G . = ( σ 1 ⊗ σ 1 )∆ + , 1 +( σ 2 ⊗ σ 2 )∆ + , 2 +( σ 1 ⊗ σ 2 + σ 2 ⊗ σ 1 )∆ + , 3 ≥ ∆ + , 3 defines quasifree Hadamard state Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
The limit NESS consider large time limit of ∆ + , G ∆ + , N ( t 1 , x 1 , t 2 , x 2 ) . = lim τ →∞ ∆ + , G ( t 1 + τ, x 1 , t 2 + τ, x 2 ) for any (admissible) contact state ω 3 , any contact/switch-on profiles χ i , ψ we find ∆ + , N ( x 1 , x 2 ) = ∆ + , ∞ ( x 1 , x 2 ) � d p 1 cos ( ω p ( t 1 − t 2 ) − p · ( x 1 − x 2 )) + (2 π ) d − 1 ω p exp( β ( p 1 ) ω p ) − 1 β ( p 1 ) = β 1 Θ( p 1 ) + β 2 Θ( − p 1 ) convergence is O ( τ − 1 ), limit exists for any m ≥ 0 and d = 1 + n (for m = 0 , d = 1 + { 1 , 2 } limit of derivatives exist) ∆ + , N defines quasifree Hadamard state ω N Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
Chemical potentials and condensates we obtain analogous results for non-vanishing chemical potentials (complex φ ) ... and for condensates (quasifree states with non-vanishing 1-point function) Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
NESS in local thermal equilibrium? [Buchholz, Ojima, Roos 2001] have introduced a concept of local thermal equilibrium (LTE) states for describing near-equilibrium situations, essentially these states have to satisfy ω LTE ( A ( x )) = ω β ( A ( x )) | β = β ( x ) for sufficiently many local thermal observables A ( x ) → Relation to “local KMS condition” [Gransee, Pinamonti, Verch 2017] may be generalised to mixtures of ω β | β = β ( x ) in different rest frames ω N is not (mixed) LTE except for m = 0 , d = 1 + 1 (where it is KMS in a different rest frame), otherwise it is too anisotropic this suggests a possible generalisation of LTE to “LNESS”, allowing for mixtures of thermal currents Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
NESS is mode-wise KMS form of 2PF suggests: ω N is a state in which “left/right-movers” are separately in equilibrium at different temperatures β 1 , β 2 (compare with Unruh state in Schwarzschild) rigorously: define time-translation α β 1 ,β 2 , shifts “left/right-moving t Fourier modes” by β 1 t , β 2 t , well-def. in the sense of exp. val. in states with tempered-distribution-type correl. fcts. → ω N is a KMS state w.r.t. α β 1 ,β 2 at inverse temp. β = 1. t Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
Reasons for non-equilibrium asymptotics expectation in the literature: generic state will evolve to generalised Gibbs ensemble (GGE) [Rigol et al. 2007. ...] , where entropy is maximised for all conserved quantities I i of system, formally ρ = 1 � Z exp( − λ i I i ) if only H (and N ) conserved, proper thermalisation formal density matrix for limit NESS ω N is [Doyon et. al 2014] ρ N = 1 H N = β 1 H L + β 2 H R = β 1 + β 2 H + β 1 − β 2 Z exp( − H N ) ( P 1 + Q ) 2 2 � � H = R d − 1 d x T 00 ( x ) P 1 = R d − 1 d x T 01 ( x ) Q non-local Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
Improving thermalisation expectation [Doyon et. al 2014] : in a non-linear QFT only conserved quantities are H and P i , initial state should thermalise, at least in a different rest frame. before looking at non-linear QFT, we analysed linear inhomogeneous models K = � + m 2 + U ( x 1 ) K φ = 0 two toy-models: U = δ , “phase-shift at x 1 = 0” → no (apparent) improved thermalisation in initial rest-frame Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
NESS for the interacting Klein-Gordon field Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
Finally ... a proper interaction consider d = 1 + 3 and Klein-Gordon field with homogeneous linear part and interaction (Lagrangean term) � dx f ( x ) φ 4 ( x ) V = V ( f ) = f ( x ) is a coupling function, we would like to consider adiabatic limit f → 1 however direct definition of adiabatic limit is already problematic for equilibrium states at T > 0, presumably because interacting field does not behave like a free field at large times [Buchholz & Bros 2002] solution given by [Fredenhagen & Lindner 2014] in perturbative algebraic QFT (pAQFT) based on earlier work by [Hollands & Wald 2003] Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
pAQFT for pedestrians in pAQFT [Brunetti, D¨ utsch, Fredenhagen, Hollands, Wald, . . . ] one defines interacting observables in the algebra of the free theory A 0 which is “the algebra of normal ordered (Wick) polynomials” elements of A 0 are functionals of smooth field configurations � ∞ � A = A ( φ ) = a 0 + dx 1 . . . dx n φ ( x 1 ) . . . φ ( x n ) f ( x 1 , . . . , x n ) n =1 corresponding to ∞ � � : A ( φ ):= a 0 1 + dx 1 . . . dx n : φ ( x 1 ) . . . φ ( x n ): f n ( x 1 , . . . , x n ) n =1 f n symm. distributions with prescribed singularity (wave front set), e.g. f n = f ( x 1 ) δ n ( x 1 , . . . , x n ), but generic f n is not localised on diagonal Rainer Verch Non-equilibrium steady states for the Klein-Gordon field in 1+3 dimensions
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