PDF《BNL-NT-07/38, BI-TP 2007/20, CU-TP-》

1 V ?(T, V ) , ? = T

2 V ??(T, V )/T ?T , (2) which by construction both vanish at vanishing temperature. Using these relations one can express the di?erence between ? and 3p, i.e the thermal contribution to the trace of the energy-momentum tensor Θ?? (T ), in terms of a derivative of the pressure with respect to temperature, i.e. Θ?? (T ) T

4 ≡ ? ? 3p T

4 = T ? ?T (p/T

4 ) , (3) In fact, it is Θ?? (T ) which is the basic thermodynamic quantity conveniently calculated on the lattice. All other bulk thermodynamic observables, e.g. p/T

4 , ?/T

4 as well as the entropy density, s/T

3 ≡ (? + p)/T

4 , can be deduced

3 from this using the above thermodynamic relations. In particular, we obtain the pressure from Θ?? (T ) through integration of Eq. 3, p(T ) T

4 ? p(T0) T

4 0 = T T0 dT ′

1 T ′5 Θ?? (T ′ ) . (4) Usually, the temperature for the lower integration limit, T0, is chosen to be a temperature su?ciently deep in the hadronic phase of QCD where the pressure p(T0), receives contributions only from massive hadronic states and is already exponentially small. We will discuss this in more detail in Section V. Eq.

4 then directly gives the pressure at temperature T . Using p/T

4 determined from Eq.

4 and combining it with Eq. 3, we obtain ?/T

4 as well as s/T

3 . This makes it evident, that there is indeed only one independent bulk thermodynamic observable calculated in the thermodynamic (large volume) limit. All other observables are derived through standard thermodynamic relations so that thermodynamic consistency of all bulk thermodynamic observables is insured by construction! We stress that the normalization introduced here for the grand canonical potential, Eq. 1, forces the pressure and energy density to vanish at T = 0. As a consequence of this normalization, any non-perturbative structure of the QCD vacuum, e.g. quark and gluon condensates, that contribute to the trace anomaly Θ?? (0), and would lead to a non-vanishing vacuum pressure and/or energy density, eventually will show up as non-perturbative contributions to the high temperature part of these thermodynamic observables. This is similar to the normalization used, e.g. in the bag model and the hadron resonance gas, but di?ers from the normalization used e.g. in resummed perturbative calculation at high temperature [15, 16] or phenomenological (quasi-particle) models for the high temperature phase of QCD [17]. This should be kept in mind when comparing results for the EoS with perturbative and model calculations. We also note that ambiguities in normalizing pressure and energy density at zero temperature drop out in a calculation of the entropy density which thus is the preferred observable for such compa........