Self-Gravitating Lattice Gas

Self-Gravitating Lattice Gas Divana Boukari and Gerhard Müller University of Rhode Island Benaoumeur Bakhti and Philipp Maass Universität Osnabrück Michael Karbach Bergische Universität Wuppertal 11/9/2017 [selgra1 1/22]

Ideal Lattice Gas volume of cell: V c number of cells: N c number of particles: N vacant cell occupied cell density: ρ = N/N c D = 1 D = 2 D = 3 Equations of state (EOS): Lattice gas: FD gas: pv c k B T = ln `1 ρ p P T = f D/2+1 (z), p ρ (D+2)/D ρ. = v T v = f D/2(z) (isotherm) pv c kbt 3.0 2.5 2.0 1.5 1.0 0.5 p pt 5 0 0 1 2 3 v T v 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Ρ 11/9/2017 [selgra2 2/22]

Equilibrium Condition Heterogeneous environment: Presence of external potential U(r). Equilibrium condition has two parts: - equation of state (EOS): (local) thermal equilibrium from minimized free energy density, - equation of motion (EOM): mechanical equilibrium from balanced forces. Equilibrium establishes density profile ρ(r) and pressure profile p(r). Temperature T remains uniform. Potential U(r) here represented by long-range gravitational interactions. Mean-field approach proven exact. Ensemble inequivalence to be heeded. 11/9/2017 [selgra3 3/22]

Gravity in D = 1, 2, 3 Dimensions Interaction potential V ij and force F ij between occupied cells of mass m c at distance r ij : Gauss law: I V ij Gm 2 c 8 >< >: da g(r) = A D Gm in, r ij : D = 1, ln r ij : D = 2, r 1 ij : D = 3. A D = 2πD/2 Γ(D/2) = F ij = Gm2 c r D 1. ij 8 >< >: 2 : D = 1, 2π : D = 2, 4π : D = 3. Symmetric self-gravitating cluster with center of mass at r = 0. Gravitational field: g(r) = m in r D 1. Gravitational potential U(r) from g(r) = du dr. 11/9/2017 [selgra4 4/22]

Self-Gravitating Lattice Gas at Equilibrium Thermal and hydrostatic equilibrium: EOS: p(r)v c k B T = ln 1 ρ(r), EOM: dp dr = m c V c ρ(r)g(r), g(r) = Gm in r D 1, m in = m c V c Z r 0 dr `A D r D 1 ρ(r ). Scaled variables: ˆr. = r r s, ˆp. = p p s, ˆT. = k B T p s V c, Û. = U p s V c, rs D = NV cd, p s = A DG A D 2D Relations between profiles: m 2 c Vc 2 rs 2. EOS & EOM: ˆp(ˆr) = ˆT ln `1 Z dˆp ˆr ˆr ρ(ˆr), dˆr = 2Dρ(ˆr) dˆr «D 1 ρ(ˆr ), 0 ˆr «Û(ˆr) = ˆT 1 ρ(ˆr) ρ(0) ln. Reference value: Û(0) = 0. 1 ρ(0) ρ(ˆr) 11/9/2017 [selgra5 5/22]

Profiles at T = 0: Solid Cluster ρ(ˆr) = ˆp(ˆr) = ( ( 1 : 0 ˆr < 1, 0 : ˆr > 1, 1 ˆr 2 : 0 ˆr < 1, 0 : ˆr > 1, Û(ˆr) = ˆr2 : 0 ˆr 1, 8 >< 2ˆr 1 : D = 1, Û(ˆr) = 2 ln ˆr + 1 : D = 2, >: 3 2/ˆr : D = 3, 9 >= 3 2 1 Ρ p D 1 U D 2 D 3 >; ˆr 1. 0 0 1 2 3 Escape velocity finite in D > 2. Stable clusters of finite mass exist at all ˆT (D = 1), at ˆT < ˆTc (D = 2), at ˆT = 0 only (D = 3). 11/9/2017 [selgra6 6/22]

Differential Equations for Profiles Analysis of EOS & EOM at ˆT > 0: density: ρ + D 1 ˆr ρ 1 2ρ ρ(1 ρ) (ρ ) 2 + 2Ḓ T ρ2 (1 ρ) = 0, ρ (0) = 0, pressure: ˆp + D 1 ˆr ˆp 1 ρ ˆTρ (ˆp ) 2 + 2D(ρ) 2 = 0, ˆp (0) = 0, potential: Û + D 1 ˆr with ρ(ˆr) = 1 e ˆp(ˆr)/ ˆT ; ρ(ˆr) = Additional boundary conditions: density: D Z pressure: 2D(D 1) 0 Û 2Dρ = 0, Û(0) = Û (0) = 0, dˆr ˆr D 1 ρ(ˆr) = 1, Z 0 dˆr ˆr 2D 3ˆp(ˆr) = 1, in D = 1 only: ˆp(0) = 1, ρ(0) = 1 e 1/ ˆT. 1 e (Û(ˆr) ˆµ)/ ˆT + 1, ˆµ =. µ = ˆT ln p s V c «1 ρ(0). ρ(0) 11/9/2017 [selgra7 7/22]

Ideal Classical Gas Limit (1) ICG assumption: low density ρ(ˆr) 1 realized throughout the cluster. ρ ODE for density profile: ρ + D 1 ˆr with boundary conditions as in ILG. ρ ρ ρ ρ «2 + 2D ˆT ρ = 0, Analytic solutions for finite clusters: D = 1: ρ(ˆr) ICG = 1ˆT sech 2 ˆrˆT solution normalized at all ˆT, «, exponential decay: ρ(ˆr) e 2ˆr/ ˆT. D = 2: ρ(ˆr) ICG = 4cˆT " «# ˆr 2 2 1 + 2c, ˆT solution normalizable only for ˆTc = 1 2, Ρ 2.0 1.5 1.0 0.5 d T 0, 0.1, 0.5, 1.0, 2.0 0.0 0.0 0.5 1.0 1.5 2.0 power-law decay: ρ(ˆr) ˆr 4. 11/9/2017 [selgra8 8/22]

Ideal Classical Gas Limit (2) ICG in D = 3 confined to sphere of radius ˆR. = R/r s. Rescaled variables: r. = ˆr/ ˆR, ρ. = ˆR 3 ρ, T. = ˆR ˆT. ODE: ρ ρ + 2 r ρ ρ «2 ρ + 6 ρ T Z 1 ρ = 0, ρ (0) = 0, 3 0 d r r 2 ρ( r) = 1. Normalizable solutions exist for T TC = 0.794422... Vega and Sanchez (2006) predict spinodal point: TC = 0.79442, transition point: TT = 0.8215, based on computer simulations. Ρ 10.0 5.0 2.0 1.0 0.5 T T c T 0.8, 0.9, 1.0, 1.1 0.0 0.2 0.4 0.6 0.8 1.0 r 11/9/2017 [selgra22 9/22]

ILG Profiles in D = 1 p 1.0 0.8 0.6 a 1.0 0.8 0.6 b T 0, 0.1, 0.5, 1.0, 2.0 Ρ 0.4 0.2 T 0, 0.1, 0.5, 1.0, 2.0 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 Analytic ICG profile: ρ(ˆr) ICG = 1ˆT sech 2 ˆrˆT Exact ILG asymptotics: «Ρ 0.0 0.0 0.5 1.0 1.5 2.0 1.00 0.50 0.20 0.10 0.05 T 1.0, 2.0, 3.0 c 2ˆr/ ˆT ρ(ˆr) as e : ˆr 1 0.02 0.01 0 1 2 3 4 5 6 11/9/2017 [selgra9 10/22]

ILG Profiles for D = 2: Unconfined Space p 1.0 0.8 0.6 a T 0.1, 0.2, 0.3, 0.4 1.0 0.8 0.6 b T 0.1, 0.2, 0.3, 0.4 Ρ 0.4 0.4 0.2 0.2 0.0 0.0 0.5 1.0 1.5 2.0 Critical ILG profile: ˆT ˆTc = 1 2 " ρ(ˆr) ICG = 4cˆT «# ˆr 2 2 1 + 2c ˆT Exact ILG asymptotics: 0 ˆT ˆT c 2/ ˆT ρ(ˆr) as ˆr : ˆr 1 Ρ 0.0 0.0 0.5 1.0 1.5 2.0 1 0.01 10 4 10 6 10 8 T 0.1, 0.2, 0.3, 0.4 c 1.0 1.5 2.0 3.0 5.0 11/9/2017 [selgra10 11/22]

ILG Profiles for D = 2: Confined Space 0.1 a 0.01 b T 0.5 Ρ 10 5 T 0.45 Ρ 10 4 T 0.55 10 7 1 2 5 10 20 50 100 10 5 10 6 1 2 5 10 20 50 100 ILG in D = 2 confined to disk of radius ˆR = 20, 30,50,100. ˆT = 0.45: stable cluster, ˆT = 0.5: stability limit, ˆT = 0.55: unstable cluster. 11/9/2017 [selgra23 12/22]

ILG Profiles for D = 3: Confined Space ILG in D = 3 confined to sphere of radius ˆR. Ρ 1.000 0.500 0.100 0.050 0.010 0.005 T 0.21, 0.22, 0.23, 0.235, 0.24, 0.25 0.0 0.5 1.0 1.5 2.0 2.5 3.0 ˆR = 3 one normalizable solution at all ˆT, gradual cluster formation, no phase transition. Ρ 1.000 0.500 0.100 0.050 0.010 0.005 T 0.19, 0.195, 0.20, 0.205, 0.21 0 1 2 3 4 ˆR = 4 three normalizable solutions in ˆT interval, two mechanically stable solutions, free energy minimum switches, first-order phase transition. 11/9/2017 [selgra24 13/22]

ILG Profiles for D = 3: First-Order Transition T t 0.25 0.20 0.15 0.10 0.05 a 0.17 0.16 0.16 0.15 0.15 0.14 0.14 0 0.01 0.17 0.99 1. 0.00 0.0 0.2 0.4 0.6 0.8 1.0 Ρ 0 Guggenheim plot T t 0.25 0.2 0.15 0.1 0.05 b 0 0 0.1 0.2 0.3 1 R transition temperature ˆT t critical temperature: ˆTc 0.23, critical radius: ˆRc 3.1 11/9/2017 [selgra25 14/22]

Open ILG Clusters Closed system: finite mass, confined or unconfined space. ρ + D 1 ˆr ρ 1 2ρ ρ(1 ρ) (ρ ) 2 + 2Ḓ Z ˆR T ρ2 (1 ρ) = 0, ρ (0) = 0, D dˆr ˆr D 1 ρ(ˆr) = 1. 0 ˆr. = r r s, ˆp. = p p s, ˆT. = k B T p s V c, Û. = Um c p s V c, rs D = NV cd, p s = A DG A D 2D m 2 c Vc 2 rs 2. Open system: finite or infinite mass, unconfined space. ρ + D 1 r ρ 1 2ρ 0 ρ ρ(1 ρ 0 ρ) r. = ` ρ 2 + ρ 2 (1 ρ 0 ρ) = 0, ρ(0) = 1, ρ (0) = 0. s 2Dρ 0 ˆT ˆr, ρ. = ρ ρ 0. ICG limit ρ 0 0: ρ ρ + D 1 r ρ ρ «2 ρ + ρ = 0 ρ( r) ρ 8 >< >: e 2 r : D = 1, r 4 : D = 2, r 2 : D = 3. 11/9/2017 [selgra26 15/22]

Open ILG Cluster in D = 1 Use inverse function r( ρ) for analytic solution: r + 1 2ρ ρ(1 ρ) r 1 ρ 2 (1 ρ)( r ) 3 = 0, ρ = ρ 0 ρ, r(ρ 0 ) = 0, r (ρ 0 ) = ρ 0 r( ρ) = r ρ0 2 Z 1 ρ( r) as e ν 1(ρ 0 ) r s ν 1 (ρ 0 ) = ρ d ρ q : 0 ρ 1 ρ (1 ρ 0 ρ ) ln 1 ρ 0 ρ 1 ρ 0 2 ρ 0 ln(1 ρ 0 ) ρ( r) ICG = sech 2 r 2 «finite mass, exponential decay law Ρ 1 0.1 0.01 10 4 10 5 10 6 Ρ 0 0, 0.9, 0.99, 0.999 2 4 6 8 10 11/9/2017 [selgra27 16/22]

Open ILG Cluster in D = 2 Numerical solution: ρ( r) as r ν 2(ρ 0 ), ρ( r) ICG = 1 `1 + r 2 /8 2 finite mass, algebraic decay law ν 2 (ρ 0 ) ρ 0 1 2 ln(1 ρ 0 ) Ρ 0.1 10 5 10 7 1 Ρ 0 0, 0.9, 0.5 0.75 1. 5 0.99, 0.999 10 9 0.1 1 10 100 1000 ν2 20 15 10 4 3 2 ν 2 / 2ln(1-ρ 0 ) 0 0.0 0.2 0.4 0.6 0.8 1.0 ρ 0 11/9/2017 [selgra28 17/22]

Open ILG Cluster in D = 3 Numerical solution: ρ( r) as 2 r 2 infinite mass, algebraic decay law ICG: Bonnor-Ebert sphere gaseous core gaseous halo ILG: emerging horizon condensed matter core gaseous shell gaseous halo Ρ 1 0.01 10 4 10 6 Ρ 0 0, 0.9, 0.99, 0.999 10 8 0.1 1 10 100 1000 10 4 11/9/2017 [selgra29 18/22]

ICG Asymptotics in D 3 ρ ρ + D 1 r ρ ρ «2 ρ + ρ = 0 ρ( r) ρ 8 >< >: e 2 r : D = 1, r 4 : D = 2, r 2 : D = 3. ρ( r) a r α ; α = 2, a = 2(D 2). infinite mass, algebraic decay law 0.100 (a) 1 0.500 (b) ρ ρ 0.100 0.050 10-5 10-7 D =3 0.1 1 10 100 1000 10 4 r 0.010 0.005 D =3,4,5 1 2 5 10 20 50 r 11/9/2017 [selgra30 19/22]

ICG Asymptotics in 2 D 3 infinite mass, crossover between algebraic decay laws 0.100 (a) 0.100 (b) ρ ρ 10-5 10-5 10 - D =2.5 10-7 D =2.3 0.1 1 10 100 1000 10 4 r 0.1 1 10 100 1000 10 4 r 0.100 (c) 0.100 ( ) ρ ρ 10-5 10-10 -7 D =2.1 10 - D =2.05 0.1 1 10 100 1000 10 4 r 0.1 1 10 100 1000 10 4 r 11/9/2017 [selgra31 20/22]

ICG Asymptotics in 1 D 2 ρ ρ + D 1 r ρ ρ «2 ρ + ρ = 0 ρ( r) ρ 8 >< >: e 2 r : D = 1, r 4 : D = 2, r 2 : D = 3. ρ( r) exp b r β ; β = 2 D, lim D 1 b(d) = 2, finite mass, stretched exponential decay law lim β(d)b(d) = 4. D 2 ln ρ 0-5 -10-15 = -20 0 6 8 10 12 2-D r 7 6 5 4 3 2 1 0 1.0 1.2 1.4 1.6 1.8 2.0 D b(d) (2-D) b(d) 11/9/2017 [selgra32 21/22]

Outlook Rotating clusters: Competition between gravitational and centrifugal forces. Multivalued functional relation between L and ω. instabilities of density profiles: - binary configurations in D = 1, - ring configurations in D = 2, - effects of hysteresis. Quantum gas clusters: ODE for fugacity profile z(r). Comparison between FD and BE gases, CS generalization. FD Pauli principle vs ILG hardcore repulsion. Relativistic effects. BE condensation vs gravitational collapse. FD cluster drifting into region of magnetic field. 11/9/2017 [selgra33 22/22]