EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH - PowerPoint PPT Presentation

ITP HEIDELBERG FEBRUARY 11 2020 DAVIDE RINDORI UNIVERSITY OF FLORENCE AND INFN FLORENCE DIVISION EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH ACCELERATION

EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH ACCELERATION DAVIDE RINDORI � 2 INTRODUCTION ‣ What is the entropy current? ‣ It is a vector current � that makes the entropy � extensive: s μ S ρ ) = ∫ Σ � . d Σ n μ s μ S = − tr( ̂ ρ log ̂ ‣ Why is it interesting? ‣ It enters the local version of the second law of thermodynamics. ‣ It is a postulated ingredient of Israel’s relativistic hydrodynamics. ‣ It is responsible for the constitutive equations of the conserved currents. ‣ What is the problem with it? ‣ It is not the TEV of a current dependent on quantum fields, unlike charge currents. ‣ In Israel’s theory it is postulated but not derived . ‣ We put forward a method to derive it including quantum corrections. ‣ We perform a specific calculation at thermodynamic equilibrium with acceleration. [Israel 1976]

EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH ACCELERATION DAVIDE RINDORI � 3 MOTIVATIONS ▸ Relativistic hydrodynamics ▸ Astrophysics and cosmology: expectation value of energy-momentum tensor at thermodynamic equilibrium with quantum corrections ▸ Quark-Gluon Plasma as relativistic quantum fluid at local thermodynamic equilibrium with acceleration and vorticity ▸ Quantum Field Theory ▸ Relativistic quantum effects at low temperature due to acceleration (Unruh effect)

EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH ACCELERATION DAVIDE RINDORI � 4 OUTLINE 1. Relativistic quantum statistical mechanics 2. Global thermodynamic equilibrium with acceleration 3. Thermal expectation values and Unruh effect [F. Becattini Phys.Rev. D97 (2018) no.8, 085013] 4. Entropy current and extensivity 5. Entropy current at global equilibrium with acceleration 6. Entanglement entropy and Unruh effect [F. Becattini and D.R. Phys.Rev. D99 (2019) no.12, 125011] 7. Summary

̂ EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH ACCELERATION DAVIDE RINDORI � 5 RELATIVISTIC QUANTUM STATISTICAL MECHANICS ▸ Thermal QFT: calculate thermal expectation values (TEVs) of operators � . ⟨𝒫⟩ = tr( ̂ ρ 𝒫 ) ▸ Need covariant expression for � . ρ ▸ Maximum entropy principle. ▸ Foliate spacetime with family � of Σ ( τ ) spacelike hypersurfaces. ▸ Give energy-momentum and (possible) charge densities on � Σ ( τ ) � . n μ T μν , n μ j μ ▸

̂ EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH ACCELERATION DAVIDE RINDORI � 6 LOCAL THERMODYNAMIC EQUILIBRIUM ‣ Maximize � with constraints on � − tr( ̂ ρ LE log ̂ Σ ( τ ) ρ LE ) ‣ n μ ⟨ ̂ n μ ⟨ ̂ � . T μν ⟩ LE = n μ T μν , j μ ⟩ LE = n μ j μ ‣ Solution: Local Thermodynamic Equilibrium ( LTE ) operator exp [ − ∫ Σ ( τ ) j μ ) ] 1 d Σ n μ ( ̂ T μν β ν − ζ ̂ � . ρ LE = ‣ Z LE ‣ � four-temperature (timelike) such that: β μ u μ = β μ / ‣ � four-velocity ‣ � β 2 with � chemical potential ζ = μ / T μ ‣ � proper temperature β 2 T = 1/ [Zubarev et al. 1979, Van Weert 1982] [Becattini et al. 2015, Hayata et al. 2015]

̂ ̂ EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH ACCELERATION DAVIDE RINDORI � 7 GLOBAL THERMODYNAMIC EQUILIBRIUM Require � to be � -independent: ρ LE τ Global Thermodynamic Equilibrium ( GTE ) state Z exp [ − ∫ Σ j μ ) ] ρ = 1 d Σ n μ ( ̂ T μν β ν − ζ ̂ � � -independence τ � ⇕ � -independence Σ � ⇕ � ∇ μ ζ = 0, ∇ μ β ν + ∇ ν β μ = 0 � timelike Killing vector β μ

̂ ̂ ̂ EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH ACCELERATION DAVIDE RINDORI � 8 GTE IN MINKOWSKI SPACETIME In Minkowski spacetime: Hence the GTE density operator: Z exp [ − b μ ̂ � β μ = b μ + ϖ μν x ν Q ] ρ = 1 P μ + 1 J μν + ζ ̂ 2 ϖ μν ̂ � ‣ � constant b μ ϖ μν = − 1 ‣ � ‣ � � ( ̂ P μ , ̂ generators of Poincaré group 2( ∂ μ β ν − ∂ ν β μ ) J μν ) ‣ constant thermal vorticity Different choices of � correspond to different GTEs. Set � for simplicity. ( b μ , ϖ μν ) ζ = 0 ‣ Homogeneous GTE: b μ = 1 � (1,0,0,0), ϖ μν = 0 T 0 Z exp [ − T 0 ] β μ = 1 ρ = 1 H � . (1,0,0,0), T 0

̂ ̂ ̂ ̂ ̂ ̂ EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH ACCELERATION DAVIDE RINDORI � 9 GTE IN MINKOWSKI SPACETIME ‣ GTE with rotation : b μ = 1 ϖ μν = ω T 0 ( g 1 μ g 2 ν − g 1 ν g 2 μ ) � (1,0,0,0), T 0 Z exp [ − J z ] β μ = 1 ρ = 1 H + ω � (1, ω × x ), T 0 T 0 T 0 ‣ GTE with acceleration : b μ = 1 ϖ μν = a T 0 ( g 0 ν g 3 μ − g 3 ν g 0 μ ) � (1,0,0,0), T 0 Z exp [ − K z ] T 0 ( a + z ,0,0, t ) , β μ = a 1 ρ = 1 H + a � T 0 T 0

EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH ACCELERATION DAVIDE RINDORI � 10 GTE WITH ACCELERATION IN MINKOWSKI SPACETIME ‣ Shift � : z ′ � = z + 1/ a T 0 ( a + z ,0,0, t ) = a β μ = a 1 � . ( z ′ � ,0,0, t ) T 0 z ′ � 2 − t 2 ‣ Flow lines are hyperbolae with constant � : β μ 1 1 T 0 � u μ = ( z ′ � ,0,0, t ), β 2 = T = β 2 = z ′ � 2 − t 2 z ′ � 2 − t 2 a ‣ Proper four-acceleration 1 A μ = u ν ∂ ν u μ = � z ′ � 2 − t 2 ( t ,0,0, z ′ � ) ‣ constant magnitude � along flow lines, hence A 2 the name “ GTE with acceleration ”. ‣ � is bifurcated Killing horizon : | z ′ � | = t ‣ Decompose � with ϖ μν = α μ u ν − α ν u μ � timelike and future-oriented only β μ α 2 = A 2 T 2 = − a 2 A μ α μ = ℏ � , hence � constant. in Right Rindler Wedge ( RRW ) . T 2 ck B T 0

̂ ̂ ̂ ̂ ̂ ̂ EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH ACCELERATION DAVIDE RINDORI � 11 GTE WITH ACCELERATION IN MINKOWSKI SPACETIME Z exp [ − ∫ Σ T μν β ν ] ρ = 1 d Σ n μ ̂ Recall : � ‣ ‣ is � -independent. Σ ‣ β μ = 0 Note : � at � . z ′ � = 0 ‣ Consequence : For any � through � Σ z ′ � = 0 � ρ R ⊗ ̂ [ ̂ ρ R , ̂ ρ = ρ L , ρ L ] = 0 ‣ with � involving DOFs only in RRW/LRW: ρ R/L exp [ − ∫ z ′ � >0 T μν β ν ] ρ R = 1 d Σ n μ ̂ � , Z R exp [ − ∫ z ′ � <0 T μν β ν ] ‣ Consequence : If � RRW, then ρ L = 1 x ∈ d Σ n μ ̂ � . Z L ⟨ ̂ ρ ̂ ρ R ̂ � . 𝒫 ( x ) ⟩ = tr( ̂ 𝒫 ( x )) = tr R ( ̂ 𝒫 ( x ))

̂ ̂ EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH ACCELERATION DAVIDE RINDORI � 12 THERMAL EXPECTATION VALUES IN THE RRW AND UNRUH EFFECT ‣ Free scalar field theory in the RRW: Klein-Gordon equation ( □ + m 2 ) ̂ � . ϕ = 0 ‣ Introduce (hyperbolic) Rindler coordinates : 2 a log ( 2 a log [ a 2 ( z ′ � 2 − t 2 ) ] , z ′ � − t ) , z ′ � + t τ = 1 ξ = 1 � . x T = ( x , y ) ‣ Solution: ϕ = ∫ + ∞ d ω ∫ ℝ 2 d 2 k T ( u ω , k T ̂ ω , k T ) � a R a R† ω , k T ̂ ω , k T + u * 0 ‣ with modes a ( 4 π 4 a sinh ( ) e − i ( ωτ − k T ⋅ x T ) m T e a ξ a ) K i ω 1 πω � u ω , k T = a ‣ orthonormalized with respect to Klein-Gordon inner product, � . m 2 T = k 2 T + m 2 ‣ � are creation and annihilation operators. a R† a R ω , k T , ̂ ω , k T [Crispino et al. 2008]

EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH ACCELERATION DAVIDE RINDORI � 13 THERMAL EXPECTATION VALUES IN THE RRW AND UNRUH EFFECT ‣ TEVs of physical interest can be calculated once the following are known 1 � a R† a R e ω / T 0 − 1 δ ( ω − ω ′ � ) δ 2 ( k T − k ′ � ⟨ ̂ ω , k T ̂ T ⟩ = T ) ω ′ � , k ′ � ω , k T ⟩ = ( e ω / T 0 − 1 + 1 ) δ ( ω − ω ′ � ) δ 2 ( k T − k ′ � 1 � a R a R† ⟨ ̂ T ̂ T ) ω ′ � , k ′ � � . a R a R a R† a R† ⟨ ̂ ω , k T ̂ T ⟩ = ⟨ ̂ ω , k T ̂ T ⟩ = 0 ω ′ � , k ′ � ω ′ � , k ′ � ‣ The � gives rise to divergences � needs renormalization. ⇒ +1 ‣ TEVs in Minkowski vacuum � : same TEVs as above with � . In particular | 0 M ⟩ T 0 = a /2 π 1 � . a R† a R e 2 πω / a − 1 δ ( ω − ω ′ � ) δ 2 ( k T − k ′ � ⟨ 0 M | ̂ ω , k T ̂ T | 0 M ⟩ = T ) ω ′ � , k ′ � This is the content of the Unruh effect , and � is the Unruh temperature . a /2 π

EXTENSIVITY AND ENTROPY CURRENT AT THERMODYNAMIC EQUILIBRIUM WITH ACCELERATION DAVIDE RINDORI � 14 THERMAL EXPECTATION VALUES IN THE RRW AND UNRUH EFFECT TEVs of operators quadratic in the field, once the ▸ � contribution is subtracted, vanish at | 0 M ⟩ � and become negative for � . T 0 = a /2 π T 0 < a /2 π For instance the energy density ρ Minkowski = ( ⟨ ̂ T μν | 0 M ⟩ ) u μ u ν T μν ⟩ − ⟨ 0 M | ̂ � turns out to be ▸ ρ Minkowski = ( 12 ) T 4 [ 1 − (2 π ) 4 ] 30 − α 2 π 2 α 4 � α 2 = − (2 π ) 2 ‣ where at � we have � . T 0 = a /2 π ‣ At � the proper temperature is T 0 = a /2 π A 2 T 2 = − a 2 − A 2 In the Minkowski vacuum � . ⇒ T = = T U T 2 2 π 0