Fluctuations and observables: volume fluctuations, critical dynamics

Fluctuations and observables: volume fluctuations, critical dynamics Vladimir Skokov (RBRC BNL) September 28, 2016 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 1 / 36

Outline Volume fluctuations: heuristic approach General properties of volume fluctuations Illustration based on Glauber MC Are ratios of susceptibilities volume independent? Answer with chiral model Role of vacuum term in chiral models Results with Functional Renormalization Group... critical dynamics VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 2 / 36

Signatures of phase transition and critical point I Dynamic: Divergent shear and bulk viscosities at CP. Model H universality class η ξ 19 1 and ζ ξ 2.8, where ξ is correlation length. Cavitation phenomena? Divergent heat and charge conductivities λ ξ 18/19. Possible observable: separation of entropy and charge clusters. Spinodal decomposition at 1-st phase transition J. Randrup 2011; V.S. and D. Voskresensky 2011 Clusters with lower entropy tend to move along the temperature gradient; Clusters with higher entropy tend to move opposite to the temperature gradient. T T - + + - + - + - - + entropy clusters: - δs <0 + δs >0 r Experimental results from C. Agosta et al J. Low Temp. Phys. 67, 237 (1987). J. Luettmer-Strathmann et al J. Chem. Phys 103, 7482 (1995). VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 3 / 36

Signatures of phase transition and critical point II µµ / 2 4 3 2 1 0-1 -2-3 -4 Static: O(4) crossover: universal sign structure of higher order cumulants, χ 6 Divergent cumulants of baryon number fluctuations at CP χ n ξ 3( nβγ 2 α 1) In vicinity of CP: universal sign structure of cumulants. Divergent baryon number susceptibility at off-equilibrium first order phase transition 0 0.2 0.4 0.6 0.8 1 1.2 1.4 n 3 q [fm ] T/T pc B. Friman, F. Karsch, K. Redlich, V. S. 2011 M. Stephanov 2011 C. Sasaki, K. Redlich, B. Friman 2007 0.02 χ B 6 0-0.02-0.04 1.0 0.8 0.6 0.4 µ q /T=0 PQM 0.5 1 1.5 Τ/T pc 0.2 0.0 0.5 1.0 1.5 2.0 µ/t pc 10 5 2 0 2 5 1 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 4 / 36

Fluctuations as a probe of phase diagram Listed predictions for an infinite static medium. In HIC: Finite lifetime (S. Mukherjee, R. Venugopalan, Y. Yin 2015/2016) Finite size and anisotropy (G. Almasi and V.S. work in progress) Conservation laws (A. Bzdak, V. Koch, V.S. 2011) Fluctuations not related to interesting physics (V.S., B. Friman, K. Redlich 2012) 1.0 0.8 R 4,2 R 6,2 0.6 0.4 0.2 0.0 0.2 0.4 0.0 0.2 0.4 0.6 0.8 1.0 p 1.0 0.5 0 N part 100 200 300 R 4,2 R 6,2 100 200 300 400 500 N ch p is a fraction of measured baryons µ = 0 Rn,m = χn/χm Modification due to finite life-time: S. Mukherjee, R. Venugopalan, Y. Yin 2015 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 5 / 36

Volume fluctuations: introduction 1 Consider fixed V, where net baryon number B described by P(B, V). n-th order moments of the net baryon number B n V = B n P(B, V) B= Reduced cumulants, corresponding to the net baryon number fluctuations per unit volume. The first four reduced cumulants are κ 1 (T, µ) = 1 V B V, κ 2 (T, µ) = 1 V (δb)2 V, κ 3 (T, µ) = 1 V (δb)3 V, κ 4 (T, µ) = 1 V [ (δb) 4 V 3 (δb) 2 V 2 ], where δb = B B and B = B V ; κ n = T 3 χ n. κ i are, to leading order, independent of volume V. Formal approach in V.S., B. Friman and K. Redlich, 1205.4756 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 6 / 36

Volume fluctuations: introduction 2 V dependence of moments follows B V = κ 1 V, B 2 V = κ 2 V + κ 2 1 V2, B 3 V = κ 3 V + 3κ 2 κ 1 V 2 + κ 3 1 V3, B 4 V = κ 4 V + ( 4κ 3 κ 1 + 3κ 2 2) V 2 + 6κ 2 κ 2 1 V3 + κ 4 1 V4. The coefficients are from the Bell polynomials. Now allow for fluctuations of the volume P(V), corresponding moments V n = V n P(V)dV and reduced cumulants of volume fluctuations, v n. In presence of volume fluctuations the moments of the net baryon number are given by B n = dv P(V) B n P(B, V) = dv P(V) B n V Thus cumulants B= c 1 = κ 1, c 2 = κ 2 + κ 2 1 v 2, c 3 = κ 3 + 3κ 2 κ 1 v 2 + κ 3 1 v 3, c 4 = κ 4 + (4κ 3 κ 1 + 3κ 2 2 )v 2 + 6κ 2 κ 2 1 v 3 + κ 4 1 v 4 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 7 / 36

Volume fluctuations: general properties c 1 = κ 1, c 2 = κ 2 + κ 2 1 v 2, c 3 = κ 3 + 3κ 2 κ 1 v 2 + κ 3 1 v 3, c 4 = κ 4 + (4κ 3 κ 1 + 3κ 2 2 )v 2 + 6κ 2 κ 2 1 v 3 + κ 4 1 v 4 v 2 0 c 2 κ 2 c 3 receives contribution from ν 3. For very central events (defined by N ch ) ν 3 is negative: volume (or S ) has an upper bound V max, while systems with lower volume may produce large N ch owing to fluctuations. c 3 < κ 3 if ν 3 < 3ν 2 κ 2 /κ 3 1. At zero µ only c n with n 4 are modified, i.e. c 4 = κ 4 + 3κ 2 2 ν 2. Thus c 4 κ 4. At nonzero µ, c 4 : competing contributions with opposite signs VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 8 / 36

Volume fluctuations: illustrations for ν 2 3.0 Assumption: V N part Glauber + Negative binomial; Au-Au s = 200 (δnpart) 2 / Npart 2.5 2.0 1.5 1.0 0.5 0.0 Fixed N ch Centrality class averaged 100 200 300 400 500 600 700 800 N ch Suppression of fluctuations at very large and very small (not shown) N ch Centrality binning increases volume fluctuations and introduces non-monotonicity V.S., B. Friman and K. Redlich, 1205.4756 Points: centrality classes from right to left 0-5%, 5-10%, 10-20%, 20-30%,... VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 9 / 36

Volume fluctuations: illustrations for ν 3 (δnpart) 3 / Npart 10 5 0 5 10 15 20 Fixed N ch Centrality class averaged 100 200 300 400 500 600 700 800 N ch P (Npart) Negative skewness for central collisions; negative and large skewness for 0-5% centrality bin. Skewness changes sign at about N ch 500 0.05 0.04 0.03 0.02 0.01 N ch = 400 N ch = 650 N ch = 800 0.00 150 200 250 300 350 400 N part Points: centrality classes from right to left 0-5%, 5-10%, 10-20%, 20-30%,... B. Friman, K. Redlich, V.S., work in progress VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 10 / 36

Volume fluctuations: illustrations for ν 4 (δnpart) 4 / Npart 3 (δnpart) 2 2 / Npart 600 400 200 0 200 400 600 Fixed N ch Centrality class averaged 100 200 300 400 500 600 700 800 N ch Changes sign Centrality binning effect is strong Points: centrality classes from right to left 0-5%, 5-10%, 10-20%, 20-30%,... B. Friman, K. Redlich, V.S., work in progress VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 11 / 36

Volume fluctuations: energy dependence Kurtosis: opposite sign contributions may conspire to non-monotonic energy dependence. More studies are needed. B. Friman, K. Redlich, V.S., work in progress VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 12 / 36

Volume fluctuations: are κ n volume independent? Central assumption: δb n are volume independent. This is true for large volumes if surface effects can be neglected. Volume dependence of chiral condensate at T = 0 σ/fπ 1.35 1.30 1.25 1.20 1.15 1.10 1.05 1.00 0.95 PBC 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 L, fm Main ingredients: Functional Renormalization Group + Quark-meson model Equilibrium calculations in a box L 3 for physical pion mass. Expectation: cumulants are more sensitive Gabor Almasi, V.S. to appear soon VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 13 / 36

On chiral model Polyakov-loop extended Quark Meson model L = ψ [ iγ µ D µ g(σ + iγ 5 τ π) ] ψ + 1 2 ( µσ) 2 + 1 2 ( µπ) 2 U(σ, π) U(l, l ) Mean-field approximation: integrate out fermion fields & disregard fluctuations of mesonic field, Ω = Ω q U(σ, π) U(l, l ) Ω q = 1 V Tr log ( iγ µ D µ gσ ) = Ω v + Ω T = d 4 p 2N f N c (2π) 4 log(p2 + (gσ) 2 ) d 3 p 2TN f (2π) log ( 1 + e ) (E µ)/t+2πiq a/3 + antiq. 3 a Ω v is divergent; this is not a good reason to ignore Ω v V. S. and B. Friman et al, 1005.3166; R. Pisarski and V. S., 1604.00022 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 14 / 36

On chiral model: vacuum term To extract finite contribution: dimensional regularization Ω v = N ( ) cn f gσ 8π 2 g4 σ 4 log. Λ dr Similar contribution, but for bosons (change of overall sign), first analyzed by S. Coleman and E. Weinberg in PRD 7, 1888, 1973: Fluctuation induced first-order phase transition Gross-Neveu model, PRD 10, 3235, 1974: Dynamical symmetry breaking in asymptotically free field theories ; 1+1 dimensional NJL model VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 15 / 36

On chiral model: close to chiral transition For m q /T = gσ/t 1, the expansion of thermal part Ω T N c N f T 4 7π2 180 + 1 m 2 q 12 T + 1 m 4 [ q log 2 8π 2 T 4 ( mq πt ) + γ E 3 ] 4 Close to chiral phase transition, thermal part brings Ω T = + N ( cn f gσ ) 8π 2 g4 σ 4 log + πt Compare to vacuum part Ω v = N ( cn f gσ 8π 2 g4 σ 4 log 7ζ(3) m 6 q 192π 4 T + O 6 Exact cancellation of σ 4 log(σ) dependence close to transition! Λ dr ) m8 q T 8 V. S. and B. Friman et al, 1005.3166 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 16 / 36

On chiral model: is Ω v important? Lets neglect Ω v. Close to chiral transition, Ω T receives contribution σ 4 log σ < 0. W/o vacuum term, Landau theory Ω = 1 2 A(T T c)σ 2 + 1 4 Bσ4 ( 1 + C log(σ/σ 0 ) ) 1 0.5 Without vacuum term C.f. Coleman-Weinberg s fluctuation induced first-order phase transition: /f 4 0-0.5-1 Reduced T: t=-0.05 t=-0.01 t=0. t=0.01 t=0.05 First order phase transition, a la Coleman&Weinberg 0 0.2 0.4 0.6 0.8 1 /f Shift of transition temperature T PT = T c + BC 4A #2 Most important: first-order phase transition instead of expected second order. V. S. and B. Friman et al, 1005.3166 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 17 / 36

On chiral model: is Ω v important? Lets restore Ω v. With vacuum term, Landau theory Ω = 1 2 A(T T c)σ 2 + 1 4 Bσ4 /f 0.05 0.04 0.03 0.02 0.01 0-0.01 With vacuum term 0 0.1 0.2 0.3 0.4 /f t=-0.01 t=0. t=0.01 Ω v appears to be important in chiral limit; but for physical pion mass?! V. S. and B. Friman et al, 1005.3166 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 18 / 36

On chiral model: is Ω v important at m π? Crossover for both models Transition properties are however very different: Lee-Yang zeros in finite volume, type of singularities in complex chemical potential plane, etc Baryon number fluctuations, kurtosis, R 4,2 = χ 4 /χ 2 = κσ 2 : 9R 4,2 60 50 40 30 20 10 no vacuum term V.S. and B. Friman et al, 1005.3166 with vacuum term 0.2 0.21 0.22 0.23 0.24 [GeV] physical pion mass; zero chemical potential VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 19 / 36

On chiral model: is Ω v important at m π? Without vacuum term: With vacuum term: 250 15.9 8.1 5.5 3.8 250 10 5 3.3 2.5 200 3.6 200 T MeV 150 100 CEP T MeV 150 100 50 50 1.4 0 0 50 100 150 200 250 300 350 Μ MeV Focusing of isentropic due to remnant of first order phase transition in chiral limit 0 0 50 100 150 200 250 300 350 Μ MeV Expected behavior: isentropes are tangential to continuation of first order phase transition E. Nakano et al, 0907.1344 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 20 / 36

On chiral model: is Ω v important at m π? C. Herold, M. Nahrgang and Co, e.g. 1601.04839 T (MeV) 160 150 140 130 120 110 100 90 4 3.5 3 2.5 2 1.5 1 80 50 100 150 200 250 µ (MeV) Unfortunately these calculations do not include vacuum term... 10 9 8 7 6 5 4 3 2 1 0 e/e 0 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 21 / 36

On chiral model: is Ω v important at m π? T/T pc 1.1 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.0 0.5 1.0 1.5 2.0 µ/t pc 10 5 2 0 2 5 10 Is the PQM model from J. Steinheimer, J. Randrup, V. Koch, PRC 89, 034901, 2014? V. S., QM 2012 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 22 / 36

On chiral model: missing vacuum term in literature Vacuum term was also overlooked in numerous papers... O. Scavenius, A. Mocsy, I. N. Mishustin and D. H. Rischke, PRC 64, 045202 (2001) B.-J. Schaefer and J. Wambach, PRD 75, 085015 (2007) E. S. Bowman and J. I. Kapusta, PRC 79, 015202 (2009) U. S. Gupta and V. K. Tiwari, arxiv:0911.2464 D. Nickel, PRD 80, 074025 (2009) T. Kahara and K. Tuominen, PRD 80, 114022 (2009) J. I. Kapusta and E. S. Bowman, NPA 830, 721C (2009)... VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 23 / 36

Volume fluctuations: are κ n volume independent? Central assumption: δb n are volume independent. This is definitely true for large volumes if surface effects can be neglected. Volume dependence of chiral condensate at T = 0 σ/fπ 1.35 1.30 1.25 1.20 1.15 1.10 1.05 1.00 0.95 PBC 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 L, fm Main ingredients: Functional Renormalization Group + Quark-meson model Equilibrium calculations in a box L 3 for physical pion mass. Expectation: cumulants are more sensitive Gabor Almasi, V.S. to appear soon VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 24 / 36

Functional renormalization group The general flow equation for the scale-dependent effective action Γ k. Γ k : loosely speaking modes with momenta greater than k are integrated out, k is IR cut-off. k Γ k [Φ, ψ] = 1 2 Tr { k R kb ( Γ (2,0) k [Φ, ψ] + R kb ) 1 } Tr The flow equation for the PQM model in infinite volume k Ω(k, ρ 1 { 2 [σ2 + π 2 ]) = k4 3 12π 2 E π [ ] 1 1 + 2n B (E σ ; T) 4N f N c E σ E q [ ] 1 + 2n B (E π ; T) { k R kf ( Γ (0,2) k [Φ, ψ] + R kf ) 1 } [ 1 N(l, l ; T, µ q ) N(l, l ; T, µ q ) n B (E; T) is the boson distribution functions N(l, l ; T, µ q ) are fermion distribution function modified owing to coupling to gluons E σ and E π are the functions of k, Ω/ ρ and ρ 2 Ω/ ρ 2 E q = k 2 + 2gρ FRG defines Ω(k, ρ; T, µ Q, µ B ). Physically relevant quantity is the thermodynamical potential Ω(T, µ Q, µ B ) Ω(k 0, ρ ρ 0 ; T, µ Q, µ B ), where ρ 0 is the minimum of Ω. V.S. et al 1004.2665 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 25 / 36 + ]}

Ω(k, σ), a.u. σ, MeV Functional renormalization group: evolution Scale-dependent effective potential: 21.79 21.785 minimum at k 0 FRG in infinite volume close to 1st order phase transition k 0 Minimum of scale-dependent effective potential: 100 80 fermion dominated/mean-field 21.78 60 21.775 21.77 21.765 transient minimum 40 20 fermions decouple/ soft mesonic excitations 50 60 70 80 90 100 σ, MeV Soft mesonic excitations drive evolution at very small k. 0 200 400 k, MeV 600 800 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 26 / 36

Functional renormalization group in finite volume The general flow equation for the effective action k Γ k [Φ, ψ] = 1 { ( ) } 1 { 2 Tr ( ) k R kb Γ (2,0) k [Φ, ψ] + R kb Tr k R kf Γ (0,2) 1 } k [Φ, ψ] + R kf The flow equation involve summation over modes n 1,n 2,n 3. All principal components were considered in 2010, but it was not until 2016 when we were able to solve the technical issues and the flow equation based on Chebyshev collocation pseudo spectral lattice. With parallel realization of code, it takes seconds to compute thermodynamics to very high precision at a given T and µ. This become possible due to support of Gabor Almasi by HGS-HIRE and NTG BNL. Gabor Almasi, V. S. to appear soon VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 27 / 36

Results: PQM + FRG in finite model µ = 0 Anisotropy coefficient A = L /L χ4/χ2 1.5 1.4 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0.6 µ/t = 0.0; A = 1 L =1.5 L =2.25 L =2.5 L =3 L =5 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 T/T pc χ4/χ2 1.3 1.2 1.1 1.0 0.9 0.8 0.7 µ/t = 0.0; A = 2 L =1.5 L =2.25 L =2.5 L =3 L =5 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 T/T pc Gabor Almasi, V. S. to appear soon VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 28 / 36

Results: PQM + FRG in finite model µ 0 Anisotropy coefficient A = L /L χ4/χ2 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 µ/t = 1.0; A = 1 L =1.5 L =2.25 L =2.5 L =3 L =5 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 T/T pc χ4/χ2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 µ/t = 1.0; A = 2 L =1.5 L =2.25 L =2.5 L =3 L =5 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 T/T pc Approximation of volume independence of κ n breaks down at about L = 3 4 fm: this makes analysis of volume fluctuations significantly more complicated and tractable only in a fully dynamical model. Gabor Almasi, V. S. to appear soon VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 29 / 36

Conclusions Any sourse of fluctuations may strongly affect higher order cumulants Volume fluctuaions: some properties are model independent, i.e. negative ν 3. In general, v n are non-monotonic functions of centrality and energy Ratios of cumulants depend on volume for L 3 fm Chiral model: the vacuum term should not be omitted. VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 30 / 36

Discussion Session VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 31 / 36

From hydrodynamics to model H Navier-Stokes equation, continuity equation, and equation for heat transport: ( mn [ t u i + (u )u i ] = i P + k [η k u i + i u k 2 ) ] d δ ikdivu + ζδ ik divu t n + div(nu) = 0 [ ] ( s T t + div(su) = div(κ T) + η k u i + i u k 2 2 ikdivu) d δ + ζ(divu) 2. Slowest mode only VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 32 / 36

From hydrodynamics to model H Navier-Stokes equation, continuity equation, and equation for heat transport: mn [ ] ( t u i + (u )u i = i P + k [η k u i + i u k 2 ) ] d δ ikdivu + ζδ ik divu t n + div(nu) = 0 [ ] ( s T t + div(su) = div(κ T) + η k u i + i u k 2 ) 2 d δ ikdivu + ζ(divu) 2 Slowest mode only Neglect u 2. VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 32 / 36

From hydrodynamics to model H Navier-Stokes equation, continuity equation, and equation for heat transport: mn t u i = i P + k η ku i + i u k 2 d δ ikdivu + ζδ ikdivu t n + div(nu) = 0 [ ] s T t + div(su) = div(κ T) Slowest mode only Neglect u 2. Neglect longitudinal excitations k i u i, which are known to be fast (linearize and consider small deviations from background, see e.g. 1007.1538) VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 32 / 36

From hydrodynamics to model H Navier-Stokes equation, continuity equation, and equation for heat transport: mn t u i = i P + η 2 u i t n + div(nu) = 0 [ ] s T t + div(su) = div(κ T) Slowest mode only Neglect u 2. Neglect longitudinal excitations k i u i, which are known to be fast (linearize and consider small deviations from background, see e.g. 1007.1538) VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 32 / 36

From hydrodynamics to model H Navier-Stokes equation, continuity equation, and equation for heat transport: Slowest mode only Neglect u 2. mn t u i = i P + η 2 u i [ ] s nt t + u s = κ 2 T Neglect longitudinal excitations k i u i, which are known to be fast (linearize and consider small deviations from background, see e.g. 1007.1538) Specific entropy s = s/n. Neglect density dynamics; it is also related to sound mode VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 32 / 36

From hydrodynamics to model H Navier-Stokes equation, continuity equation, and equation for heat transport: Slowest mode only Neglect u 2. mn t u i = i P + η 2 u i [ ] s nt t + u s = κ 2 T Neglect longitudinal excitations k i u i, which are known to be fast (linearize and consider small deviations from background, see e.g. 1007.1538) Specific entropy s = s/n. Neglect density dynamics; it is also related to sound mode Relate δt and δp VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 32 / 36

From hydrodynamics to model H Navier-Stokes equation, continuity equation, and equation for heat transport: Slowest mode only Neglect u 2. mn t u i = i P + η 2 u i + θi u [ ] s nt t + u s = κ 2 T + θ ψ Neglect longitudinal excitations k i u i, which are known to be fast (linearize and consider small deviations from background, see e.g. 1007.1538) Specific entropy s = s/n. Neglect density dynamics; it is also related to sound mode Relate δt and δp Add appropriate noise VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 32 / 36

From hydrodynamics to model H Navier-Stokes equation, continuity equation, and equation for heat transport: Slowest mode only Neglect u 2. mn t u i = i P + η 2 u i + θi u [ ] s nt t + u s = κ 2 T + θ ψ Neglect longitudinal excitations k i u i, which are known to be fast (linearize and consider small deviations from background, see e.g. 1007.1538) Specific entropy s = s/n. Neglect density dynamics; it is also related to sound mode Relate δt and δp Add appropriate noise Consider only small deviatons s = s c + δ s and expand EoS near s c VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 32 / 36

Careful analysis in Vasil ev The Field Theoretic Renormalization Group in Critical Behavior Theory and Stochastic Dynamics VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 33 / 36 From hydrodynamics to model H mn t u i = i P + η 2 u i + θi u [ ] s nt t + u s = κ 2 T + θ ψ From Hohenberg-Halperin, Rev. of Mod. Phys. 49, 1977: noise mode-mode coupling heat transport transverse projector noise Navier-Stokes surface tension EoS EoS

Volume fluctuations: formal approach Volume fluctuations is one of effects that may change cumulants. Higher order cumulants are most sensitive. Assumptions volume fluctuations are not critical (independent of baryon fluctuations) P(V, B) = P(V)P(B) other thermodynamic quantities do not fluctuate The last assumption can be justified at high energies (µ B 0). If system thermalizes, the finite temperature is independent of initial one. Cumulant generating functions (CGF) for both fluctuations CGF B (t) = ln CGF V (s) = ln B= 0 P(B) exp (B t) dv P(V) exp (Vs) The additivity of cumulants and thermodynamic principles imply that CGF B (t) = V ζ B (t), where ζ B is a volume-independent function. VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 34 / 36

Volume fluctuations Aim is to obtain cumulants with volume fluctuations. These cumulants are obtained from the cumulant generating function φ B (t) = ln dv P(V) P(B)e Bt But and consequently B P(B)e Bt = e VζB (t) B φ B (t) = ln dvp(v)e VζB (t) From comparison with definition of cumulant generating function of volume fluctuations: φ B (t) = CGF V ( ζ B (t) ) Corresponding reduced cumulants are given by Taylor expansion of about t = 0 c n = 1 d n V dt n φb (t) t=0 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 35 / 36

Volume fluctuations: cumulants Let κ n are the cumulants for baryon fluctuations and v n are for volume fluctuations (δx = X X ) κ 1 = 1 V B, κ 2 = 1 V (δb)2, κ 4 = 1 V [ (δb) 4 3 (δb) 2 2] v 1 = 1 V V V, v 2 = 1 V (δv)2, v 4 = 1 V [ (δv) 4 3 (δv) 2 2] Then wanted cumulants of baryon number fluctuations including volume fluctuations are given by c 1 = κ 1 c 2 = κ 2 + κ 2 1 v 2 c 3 = κ 3 + 3κ 2 κ 1 v 2 + κ 3 1 v 3 c 4 = κ 4 + (4κ 3 κ 1 + 3κ 2 2 )v 2 + 6κ 2 κ 2 1 v 3 + κ 4 1 v 4 VSkokov@bnl.gov Fluctuations RHIC/AGS Users meeting 36 / 36