# Analytical formulas, general properties and calculation of transport coefficients in the hadron gas: shear and bulk viscosities.

###### Abstract

Elaborated calculations of the shear and the bulk viscosities in the hadron gas, using the ultrarelativistic quantum molecular dynamics (UrQMD) model cross sections, are made. These cross sections are analyzed and improved. A special treatment of the resonances is implemented additionally. All this allows for better hydrodynamical description of the experimental data. The previously considered approximation of one constant cross section for all hadrons is justified. It’s found that the bulk viscosity of the hadron gas is much larger than the bulk viscosity of the pion gas while the shear viscosity is found to be less sensitive to the hadronic mass spectrum. The maximum of the bulk viscosity of the hadron gas is expected to be approximately in the temperature range with zero chemical potentials. This range covers the critical temperature values found from lattice calculations. We comment on some important aspects of calculations of the bulk viscosity, which were not taken into account or were not analyzed well previously. Doing this, a generalized Chapman-Enskog procedure, taking into account deviations from the chemical equilibrium, is outlined. Some general properties, features, the physical meaning of the bulk viscosity and some other comments on the deviations from the chemical equilibrium supplement this discussion. Analytical closed-form expressions for the transport coefficients and some related quantities within a quite large class of cross sections can be obtained. Some examples are explicitly considered. Comparisons with some previous calculations of the viscosities in the hadron gas and the pion gas are done.

###### pacs:

25.75.-q, 24.10.Pa, 47.45.Ab, 51.20.+d## I Introduction

The bulk and the shear viscosity coefficients are transport coefficients which enter in the hydrodynamic equations, and thus are important for studying of nonequilibrium evolution of any thermodynamic system.

There are two more additional reasons to study the shear viscosity. The first one is the experimentally observed
minimum of the ratio of the shear viscosity to the entropy density near the liquid-gas phase transition
for different substances, which may help in studying of the quantum chromodynamics phase diagram and finding of
the location of the critical point Csernai:2006zz (); lacey ()^{1}^{1}1Fireballs, created in heavy ion
collisions, have finite sizes and finite times of existence of their thermalized part. This puts important
restrictions on detection of the critical fluctuations of thermodynamic functions Stephanov:1999zu ().
Because of this it’s also important to consider nonequilibrium dissipative corrections and nonequilibrium
phenomenons like critical slow down/speed up.. Such a minimum was observed in theoretical results in several
models, see e. g. chakkap (); Dobado:2009ek (). For a counterexample see Chen:2010vf () and references
therein. The second reason is the calculation of the in strongly interacting systems, preferably real
ones, to compare physical inputs which provide small values of the . The conjectured lowest
bound^{2}^{2}2In Danielewicz:1984ww () the bound coming from the Heisenberg uncertainty principle was
obtained for the . However, it was obtained using a formula, which is justified in rarified gases with
short-range interactions. It’s well known already from the nonrelativistic kinetic theory that dense gases get
corrections over the particle number densities (see e. g. landau10 (), Section 18), corresponding to more
than binary collisions, and in very dense gasses this bound can be quite inaccurate. In liquids and other
substances the mechanism of appearance of the shear viscosity may be different (see Schafer:2009dj () for a
review). In particular, the shear viscosity of water can be very well described by a phenomenological formula
with an exponential dependence on the inverse temperature, see e. g. Sengers ().
adscftbound () was violated with different counterexamples. For some reasonable ones see
Buchel:2008vz (); Sinha:2009ev (). Also see the recent review Cremonini:2011iq (). The bulk viscosity, being
very sensitive to violation of the equation of state and being connected with fluctuations through the
fluctuation-dissipation theorem Callen:1951vq (), can have a maximum near a phase transition
Kharzeev:2007wb (); Karsch:2007jc (); chakkap (); Dobado:2012zf (). In chakkap () and Dobado:2012zf () sharp
maxima were observed in the bulk viscosity and the ratio in the linear -model for the
vacuum mass . Decreasing the vacuum mass the maximum eventually disappears. Any
maximum of the was not observed in the large-N limit of the linear -model in the
Dobado:2012zf (). Any maximum of the was not observed in the large-N limit of the -dimensional
Gross-Neveu model FernandezFraile:2010gu () (see also Sec. II for comments).

Whether one uses the Kubo^{3}^{3}3The Kubo formulas are
distinguished from the Greet-Kubo formulas e. g. in
Muronga:2003tb (); Kadanoff (). formula or the Boltzmann
equation one faces nearly the same integral equation for the
transport coefficients jeon (); jeonyaffe (); Arnold:2002zm (). The
preferable way to solve it is the variational (or Ritz) method.
Due to its complexity the relaxation time approximation is used
often in the framework of the Boltzmann equation. Though this
approximation is inaccurate, does not allow to control precision
of approximation and can potentially lead to large deviations. The
main difficulty in the variational method is in calculation of
collision integrals. To calculate any transport coefficient in the
lowest order approximation in a mixture with a very large number
of components (like in the hadron gas) one would need to
calculate roughly 12-dimensional integrals if only the
elastic collisions are considered. Fortunately, it’s possible to
simplify these integrals considerably and perform these
calculations in a reasonable time.

This paper contains calculations of the shear and the bulk viscosity coefficients for the hadron gas using the (corrected, see Sec. III) UrQMD cross sections. The calculations are done in the framework of the Boltzmann equation with the classical Maxwell-Boltzmann statistics, without medium effects and with the ideal gas equation of state. The Maxwell-Boltzmann statistics approximation allows one to obtain some relatively simple analytical closed-form expressions. Originally the calculations in the same approximations for the hadron gas but with one constant cross section for all hadrons were done in Moroz:2011vn (). The deviations in the worst cases are relatively small. In that paper some analytical formulas of the viscosities for 1-, 2- (explicitly) and N-component (up to solution of the matrix equation) gases with constant cross sections were obtained. Analogical formulas can be written down for quite a large class of non-constant cross sections, in particular, for the ones which appear in the chiral perturbation theory. The final expressions may become somewhat more cumbersome; anyway this is better than numerical integration at least in the speed of the computation. Explicit formulas for the viscosities with the elastic pion-pion isospin averaged cross section and somewhat more general one are obtained in the present paper. The results of the Moroz:2011vn () are partially reproduced in the present paper, improving the text and adding more detailed explanations. The presented calculations can be considered as quite precise ones at low temperatures where the elastic collisions dominate and the equation of state is close to the ideal gas equation of state. At higher temperatures the calculations with the total cross section are expected to give the qualitative description.

For comparison the calculations of the viscosities are performed
for the pion gas (throughout the paper the chemical potentials are
equal to zero if else is not stated). The results are relatively
close to the results in prakash (); davesne (). There the
calculations are made in the same approximations except for the
davesne (), where the Bose-Einstein statistics is used instead
of the Maxwell-Boltzmann one. The discrepancies from the used
classical statistics are not large at zero chemical potential and
become larger as the chemical potential grows (see Sec.
IV for the errors and comments). The comparison is
made with the results of the prakash (), see fig.
1. The discrepancies up to a factor of
for the bulk viscosity and up to a factor of for the shear
viscosity come most probably from somewhat different
elastic plus the quasielastic (through the intermediate
-resonance) cross section of the prakash ()^{4}^{4}4The
author could not reproduce this plotted total cross section by its
formula. In fact it was approximately 2.6 times larger. But the
plotted total cross section is quite close to the isospin averaged
(corrected) UrQMD total cross section. A notable
deviation is only at , when the UrQMD cross
section becomes times smaller. At
the UrQMD cross section is a little
larger instead. (the averaging over the scattering angle is
expected to give small errors; see also comments below the formula
(128)). The minima of the shear viscosities near
are attributed to the peaks from the
-resonances in the cross sections. It’s not
noticeable in the figure for the dash-dotted line. Nonzero values
of the bulk viscosities and theirs maxima are solely due to the
masses of the pions. The paper prakash () implements also the
isospin averaged current algebra elastic cross sections. These
cross sections can be reproduced in the lowest order in the chiral
perturbation theory Scherer:2002tk (). They obviously have
quite large deviations from the experimental data at high enough
energies and wrong asymptotic dependence, which can be
seen from the comparison of them with the isospin averaged elastic
plus the quasielastic experimental cross sections in the
prakash (). The elastic cross sections are rather
close to the constant Bass:1998ca (); Bleicher:1999xi ().

In several papers the bulk viscosity was calculated for the pion gas, using the chiral perturbation theory (or the unitarized chiral perturbation theory) and some other approaches, with quite large discrepancies between the quantitative results. In fernnicola () the calculations were done by the Kubo formula in a rough approximation. There the number-changing processes were neglected too, and the non-vanishing value of the bulk viscosity is obtained due to a trace anomaly and the pions’ masses. At small temperatures, where the effects of the trace anomaly are small, the magnitude of the bulk viscosity is large in compare to the results of this paper and the prakash (); davesne (). For example, at () it’s larger approximately in 39 (8) times than the bulk viscosity in this paper. The maximal values differ in approximately 24 times. In lumoore () the calculations are done in the framework of the Boltzmann equation and have a divergent dependence of the for because of remained weak number-changing processes (at the bulk viscosity is nearly 57 times larger than the bulk viscosity calculated in this paper). This dependence should change at low enough temperatures, or higher ones for the pion gas, see Sec. II for explanations. Joining the results of the calculations at low and high temperatures, the function may turn out to be not continuous at the middle temperatures (which is not a physical effect, see Sec. II), and the smooth function is to be obtained through some interpolation. In dobado () the bulk viscosity was calculated in the framework of the Boltzmann equation with the ideal gas equation of state and only the elastic collisions taken into account. The Inverse Amplitude Method was used to get the scattering amplitudes of the pions. The quantitative results are close to the results in this paper (discrepancies up to a factor of ). In chenwang () the calculations are done in the framework of the Boltzmann equation for the massless pions. There the bulk viscosity increases rapidly so that the ratio increases with the temperature.

Calculations of the shear viscosity in the hadron gas with a large
number of components were done in Gorenstein:2007mw (), using
some approximate phenomenological formula, and in toneev (),
using the relaxation time approximation. These results are in good
agreement with the calculations of this paper. Hence, as long as
the ratio calculated in the toneev () for the free
massive pion gas is times larger (in the temperature range
with the deviations growing as the
temperature decreases) than the one calculated in this paper, one
can suspect that the difference comes from the bulk viscosity
because of the used relaxation time approximation^{5}^{5}5In the
relaxation time approximation the bulk viscosity source term is
treated somewhat differently: the becomes proportional to
the integral of the squared source term (times some functions of
momentum) and not to the square of the integrated source term
(times some functions of momentum). Note that in the
fernnicola () the used formula has this relaxation time
approximation form. Also there the source term is the one of a
system with the inelastic processes. These facts could help to
understand the enlarged values of the bulk viscosity. Not small
quantitative discrepancies can be noticed between the calculations
of the chakkap () and the Dobado:2012zf (). and likely
not conserved particle numbers at low temperatures, provided that
the SHMC model’s cross sections, used in the toneev (), don’t
have large deviations from the UrQMD cross sections or the
experimental data, which seems to be the case. Also note that the
results in the toneev () for the free particles and the SHMC
model don’t differ very much. These facts may explain why the
of the hadron gas in the toneev () is
times larger (in the temperature range ) than
the calculated in this paper. At the low temperature
and the vanishing chemical potentials it is 11.3
times more (at the same temperature the factor is 8.2 for the case
of the pion gas). In nhngr () the calculation of the bulk
viscosity is done for the hadron gas (with an excluded-volume
equation of state) with the masses less than using some
special formula, obtained though some ansatz
Kharzeev:2007wb (). Its quantitative accuracy has not been
clarified. The ratio in the nhngr () deviates from the
of this paper up to a factor of 1.8 in the temperature
range and is different on at
.

Also the shear viscosity has been calculated using the Kubo formula (or the Green-Kubo formula) in a gas of mesons and their resonances Muronga:2003tb (). There the UrQMD simulations are performed to calculate the energy-momentum tensor, used in the calculations by the Kubo formula. The in the Muronga:2003tb () is times smaller then the for the hadron gas in this paper. At it is times smaller. In Muroya:2004pu () similar calculations, using the URASiMA event generator, are done for the shear viscosity with close results.

The structure of the paper is the following. A misleading viewpoint on the bulk viscosity, connected with the inelastic processes, is commented on in Sec. II together with some properties, features and physical meaning of the bulk viscosity. In that section some questions concerning the deviations from the chemical equilibrium are addressed too. Sec. III contains some comments on the constant cross sections, which are used in approximating calculations, and some other general comments on cross sections. Also it contains a description of the UrQMD cross sections, which are used in the main calculations, together with their analysis, corrections and the consequences of the corrections for the freeze-out temperatures. The applicability of the used through the paper approximations is discussed in Sec. IV. The system of the Boltzmann equations, its solution and formal expressions of the transport coefficients can be found in Sec. V. The numerical calculations for the hadron gas are presented in Sec. VI. In Sec. VII.1 analytical results for the single-component gas are presented. In particular, an analytical expression for the first order single-component shear viscosity coefficient with constant cross section, found before in anderson (), is corrected while the bulk viscosity coefficient remains the same. The nonequilibrium distribution function in the same approximation is written down. Also the viscosities with some non-constant cross sections are written down. Some analytical results for the binary mixture with constant cross sections are considered in Sec. VII.2. Integrals of source terms needed for the calculation of the transport coefficients can be found in Appendix A. The general entropy density formula can be found in Appendix B. It is used in the numerical calculations for the hadron gas. Transformations of collision brackets, being the 12-dimensional integrals which enter in the viscosities, and some analytical formulas for them can be found in Appendix C. The closed-form expressions for collision rates, mean free paths and mean free times are included in Appendix D.

## Ii Some features and properties of the bulk viscosity

First, it should be reminded that the transport coefficients are defined as coefficients next to their gradients in the formal expansion of the energy-momentum tensor and the charge density flows over the gradients of the thermodynamic functions or the flow velocity (see e. g. landau6 (), Section 136). The Kubo formulas are not definitions of the transport coefficients, as one might think. They may introduce some assumptions. In particular, the Kubo formulas in the form as in the jeon () have zero frequency and zero momentum limits, which neglect finite size and finite time effects. Zero momentum limit implies the thermodynamical limit. This limit is needed to avoid possible nonphysical contributions from inappropriate choice of a current and an ensemble Kadanoff (). The Kubo formulas in the form as in Muroya:2004pu (); Kubo () suppose thermal equilibrium in the initial moment of time . So that any infinite space-time scale cannot be connected with the transport coefficients by their definitions.

The Boltzmann equations will be used in what follows. In the case
of the elastic collisions they can be derived from the Liouville
equation^{6}^{6}6The Boltzmann equations can also be derived for
the case of the inelastic collisions from some physical
considerations, see groot () (Chap. I, Sec. 2). in the
approximations ( is the effective
radius of two particle interactions between the particles of the
species and ) that is for rarified gases with short-range
interactions^{7}^{7}7This is the case of interest. Coulomb
interactions can be neglected in heavy ion collisions at all the
considered energies in this paper.. Also the linear integral
equations for the viscosities and other transport coefficients,
derivable from the Boltzmann equation, can be obtained (with some
corrections) from the perturbative calculations for quantum field
theories at finite temperature (including the inelastic processes)
using the Kubo formulas jeon (); Gagnon:2006hi (); Gagnon:2007qt (),
which justifies application of the Boltzmann equation when the
inelastic processes are present.

The bulk viscosity can reveal itself only when there is a nonzero
divergence of the flow velocity. This nonequilibrium perturbation
should not be confused with another possible independent
perturbation (as was done in several papers, some of which are
mentioned below; the roots of the misleading viewpoint, perhaps,
can be found in landau6 (), Section 81). Namely, this is the
homogeneous perturbation. It can be both the chemical and the
kinetic one^{8}^{8}8The inclusion of this kinetic perturbation is
similar to the inclusion of the chemical one so that it is omitted
for simplicity below. Usually this perturbation should fade first
because also the inelastic processes are responsible for the
relaxation of the momentum spectra. However, see comments for
-dimensional systems below.. Then it can be generalized and
made dependent on the coordinate. It just should not be
proportional to any gradient. Then the constraints of the local
conservation laws should be imposed on these perturbations. The
perturbations for the particle numbers should be such that don’t
violate conservation of all charges. Considering the case of
homogeneous chemical perturbation in a gas with fixed volume, one
concludes that the temperature should change with time, being some
energy per particle. So that energy conservation should be
obtained varying the temperature. Adding an infinitesimal
correction to the temperature one gets a perturbation of the form
. Such perturbations don’t contribute to all
collision integrals. To describe purely chemical perturbations
they have to be chosen in the form of the momentum-independent
terms (except for the terms), otherwise there
will be contributions from the elastic collision integrals. Such
perturbations can be considered as chemical potentials-like ones
(being small, one can expand the distribution functions over them
and get these momentum-independent terms) with the arguments for
maximization of the entropy. To find the evolution of these terms
they should be separated. Let’s write this in some formulas.
Multicomponent gas distribution functions with the leading
perturbations can be represented in the form (detailed definitions
can be found in Sec. V.1)

(1) |

where are the perturbation due to the gradients and
are the chemical perturbations^{9}^{9}9Note
that if the -th species have conserved particle numbers, then
the nonequilibrium chemical potential is nonphysical or redefining
the usual (thermodynamic) chemical potential.. Following steps of
Sec. V.1, one can get the following linearized
equations from the Boltzmann equations:

(2) |

where and is the sum of the linearized
collision integrals (divided on the ) of all the
processes and of the inelastic processes correspondingly. The 2-nd
order gradients and the squared 1-st order gradients are neglected
in the l. h. s. because they are of the next order^{10}^{10}10The
question of validity of this expansion over the gradients (which
coincides with the usual order counting in the formal expansion
over the gradients in the hydrodynamics) for some profiles is not
discussed in this paper. and should be cancelled in the next
iteration by the next corrections to the distribution functions.
Also the smallness of the is used. If the
spatial covariant gradients (at
the initial moment of time) are of the same order as the gradients
of the thermodynamic functions or the flow velocity, then the
terms in the l. h. s. should be
retained^{11}^{11}11It’s a reasonable assumption in the case when
the hydrodynamical description is applicable. For example, the
chemical perturbations can be a result of a fast previous
expansion (faster than the chemical equilibration). Then the
inhomogeneities of the chemical perturbations should correlate
with the inhomogeneities of the thermodynamic functions, the flow
velocity or it’s divergence.. The covariant temporal derivatives
are needed to describe the temporal evolution
of the . Then the equations (2)
can be split onto the separate equations for the and the

(3) |

(4) |

The equations (2) can be split within the framework of the perturbation theory over the gradients. Let’s consider also the condition in the (1). Then neglecting the in the (1) and repeating the steps of Sec. V.1, one can get the following linearized equations:

(5) |

The equations (5) are precise in the homogeneous case (the approximation is only from the linearization). The 1-st order gradients and the are neglected. Then using the (5) and the (2), one can get

(6) |

Solving the system of equations (5) in the local rest
frame, one gets the leading exponential fading dependencies on
time^{12}^{12}12If the expansion rate is much larger than the
collision rates of the inelastic processes (e. g. because of a
substantial decrease of the temperature), then the chemical
perturbations should enlarge instead. If the r. h. s. of the
(3) is smaller than the second term of the l. h.
s., then one can consider another approximation, when the -th
species particle numbers are conserved. Then the chemical
perturbation becomes an addition to the thermodynamic chemical
potential. (in a covariant form this should be an explicit
space-time dependence). Such dependencies were obtained in some
previous studies, see e. g. Matsui:1985eu (); Song:1996ik (). The
equations (6) are different from the ones obtained
from the common Chapman-Enskog procedure (see e. g. groot (),
Chap. V) because of the terms. The
contributions from the small chemical perturbations can be
neglected in the considered order in the transport coefficients
because they are multiplied on the 1-st order gradients. The
terms can be cancelled, introducing
terms proportional to the into
the terms. If the spatial distributions of the are such that
are of the 2-nd or a higher order, then the can be neglected. This assumption or approximation is
used in the calculations of this paper. In the linear response
theory one can also introduce independent small chemical
perturbations with the same conclusions for the 1-st order
transport coefficients and find evolution of the perturbations
with time.

Note that the deviation from the chemical equilibrium itself is not necessarily a source of the bulk viscosity, as is stated in Paech:2006st (). If the bulk viscosity is not equal to zero only because of the particles’ masses and they are tended to zero, the bulk viscosity source term and the bulk viscosity tend to zero even if there are inelastic processes (see the end of Sec. V.1). In the Paech:2006st () the independent chemical perturbations and the perturbations due to the gradients were just connected through the perturbations of particle numbers, and the bulk viscosity became proportional to the chemical relaxation time. Formally infinite chemical relaxation time doesn’t imply any divergencies in the chemical perturbations , but rather approximation of conserved particle numbers. Note that the dependence on the strength of the inelastic processes is different for the chemical perturbations and the perturbations due to the gradients. Increasing the strength of the inelastic processes the chemical relaxation time decreases. And the gradients’ relaxation time increases, because the transport coefficients, at least in rarified gases with short-range interactions, roughly speaking, are inversely proportional to the integrated cross sections (in an ideal liquid the gradients’ relaxation time is infinite). What happens with the bulk viscosity if the inelastic processes become weaker is discussed below.

Making the inelastic processes weaker in compare to the elastic
ones, the bulk viscosity eventually gets a formal dominant
contribution from them because of the approximate zero mode(s)
jeon (), connected with possible conservation of particle
number(s)^{13}^{13}13If the particles involved into
the inelastic processes are massive, then the formal dominant
contribution is the exponential one over the temperature and grows
as the temperature decreases. If the particles are massless or
approximately massless, as in high-temperature QCD
Arnold:2006fz (), then a more complicated situation can occur,
and one may need to compare some differences of processes’ rates
(and not just equilibrium collision rates), arising in the
collision matrix ( in assignments of the
Arnold:2006fz ()). Under the same pair of used test-functions
(indexed by , in the ), and for the
same pair of particle species, smaller differences of processes’
rates can be neglected. Comparing among different pairs of
test-functions the smallest nonzero contributions dominate, or
rather as can be obtained directly from the inverted collision
matrix. . As long as it’s clear that the bulk viscosity is not
responsible for the chemical equilibration, it’s also clear that
there may be the approximation of conserved particle numbers if
the momentum spectrum, as well as the gradients, can relax by
means of only the elastic collisions (which is usually the case)
and the elastic processes make a dominant contribution to the
collision rates. The question is only at what concrete temperature
does this approximation sets in. Let’s make an illustrative
example of
what nonphysical contributions one can get from formally remained
weak inelastic processes. Consider infinitely weak inelastic
processes and the perturbation of the flow velocity such that the
energy-momentum tensor gets a sizable contribution from the bulk
viscosity term, not large in compare to the pressure (cf.
(20), (27)) to remain the perturbation theory
applicable. Then it’s obvious that this contribution is not
physical because it is created by the practically absent processes
and the infinitesimal perturbation of the flow velocity. Instead,
this system is practically described by the equilibrium
thermodynamic functions. This also answers positively the question
whether the thermodynamic chemical potential can be introduced for
approximately conserved particle number in principle. As far as
the author knows, the first correct comment (albeit somewhat
inaccurate) on this issue can be found in the jeonyaffe ().
However, note that in fact there is no divergent mean free paths,
corresponding to the inelastic processes (IMFP) in this case. They are
cut by the mean free paths, corresponding to the elastic processes
(and the overall collision rate have the dominant contribution
from the elastic collisions). So that it may be not necessary for
the chemical relaxation time to be much larger than any relevant
time scale (like the gradients relaxation time or the time of
existence of the thermal part of the system) to switch off the
inelastic processes. That’s why a criterion based on comparison of
collision rates of elastic and inelastic processes can be
considered to switch off the inelastic processes. Such a
comparison is done in the UrQMD studies of the hadron gas in
Bleicher:2002dm () (see Sec. IV for farther
discussions). According to Goity (), the chemical relaxation
time of the processes in the pion gas is much
larger than the thermal relaxation time. And e. g. at
the chemical relaxation time is equal to , which is larger
than the typical lifetime of the thermal part of the expanding
fireball (see e. g. Bleicher:2002dm ()). So that it’s the
inelastic processes which should be neglected
in the pion gas at or even higher temperatures, which
wasn’t done in the lumoore ().
To show importance of the gradients relaxation time, let’s consider the following possible case. Let’s consider
the only perturbation - propagating sound wave, perturbed in a point. It’s possible for the IMFP to be much
larger than the gradients relaxation size (on which the wave can be considered as damped) and be much smaller
than the system’s size at the same time. Then, the bulk viscosity cannot be defined by the IMFP in this case,
because it enters in the sound attenuation constant. Thus, the gradients relaxation size and time are cutting
parameters. Note that they exists even in infinite systems considered during infinite time interval.

The bulk viscosity source terms increases substantially if
particle numbers are not conserved (cf. (120),
(121); in mixtures these particle numbers should also be
not small). This reflects additional fluctuations from not
conserved particle numbers. Though the inelastic processes have to
be effective enough to consider the approximation of not conserved
particle numbers. Perhaps, the point at which the bulk viscosities
in the different approximations cross can provide a criterion for
switching on/off the inelastic processes. If this is not so, then
one would have to make some interpolation in the intermediate
region^{14}^{14}14Perhaps, the bulk viscosity calculated without
constant test-functions (except for zero modes of the inelastic
collision integrals, used to conserve charges) can provide a good
interpolation.. Note that e. g. in the calculations by the Kubo
formulas through the direct calculations of the energy-momentum
tensor as in the Muronga:2003tb () it’s not needed to use the
approximation of conserved or not conserved particle numbers
(which defines the number of independent thermodynamic chemical
potentials, through which the chemical potentials of all particles
are expressed, cf. (18)). There the energy-momentum
tensor should be a smooth function of time and the thermodynamic
functions as long as the inelastic processes fade smoothly. Then
the bulk viscosity should be a smooth function of the temperature
and particles’ chemical potentials regardless of the number of the
independent chemical potentials.

In the Arnold:2006fz () a bottleneck for the relaxation to
equilibrium characterized by the bulk viscosity due to the weakest
processes’ rates is assumed. Instead, there are rather dominant
contributions from some test-functions^{15}^{15}15Not a bottleneck
from some perturbations, because one actually doesn’t have a
choice in the form of the momentum dependence of the perturbations
corresponding to the transport coefficients. The kinetic
perturbation can be of different forms of the momentum
dependence. (as is commented in the footnote 13), which
should not be specially treated though, except for the ones which
are the approximate zero modes making a dominant contribution. A
similar dominance^{16}^{16}16Another similar dominance can exist
from particle species interacting weakly with all particles. is
present also in other transport coefficients, in particular, when
there is only one type of processes. Although in QCD at high
enough temperatures the equilibrium elastic
collisions rate is parametrically the largest one^{17}^{17}17The
estimate can be easily inferred from Arnold:2000dr ().,
, because of cancellations the momentum transfer
takes place with the rate , which
is parametrically smaller than the particle number change rate
from the effective ””
processes. This provides an example when the equilibrium collision
rates may differ substantially from the relevant collision rates.
The ”” processes provide small chemical
relaxation time in compare to the thermal relaxation time, which
justifies the approximation of not conserved particle numbers and
the enhancement of the bulk viscosity from the source terms at
least at small enough , whereas the contributions to the
bulk viscosity from the collision integrals of the
”” processes are suppressed at small enough
(the inelastic processes are not
suppressed, but they are of the order ). To avoid misunderstanding it may be mentioned that taking
the total collision rate of the ”” processes
as formally infinite by taking the corresponding matrix elements
as formally infinite ones, one gets zero bulk viscosity and zero
mean free paths as long as both the gluons and quarks take part in
these processes (see also footnote 24).

In the case of a -dimensional single-component gas the
elastic collisions cannot result in the relaxation of the momentum
spectra and, hence, cannot stimulate the system to evolute towards
equilibrium^{18}^{18}18There are forward scatterings and momentum
interchange. As long as the particles are not distinguishable the
momentum interchange from the elastic collisions is equivalent to
the forward scatterings or absence of the elastic collisions at
all.. As a result, the exponentially divergent bulk viscosity was
obtained in the paper FernandezFraile:2010gu (). Considering
again the example about the infinitely small perturbation of the
flow velocity and assuming also a finite size of the system, it’s
again obvious that the weak inelastic processes may make
nonphysical contributions (in this case the mean free path is
formally cut by the system’s size). If this is the case, then the
hydrodynamical description becomes inapplicable, and might use
simulations of particles’ collisions or the Boltzmann equations in
the approximation without collisions (on a time scale much smaller
than the chemical relaxation time). If the -dimensional
description is only an approximate one (that is with small angle
elastic scatterings in higher dimensions), the relaxation of the
momentum spectrum by the elastic collisions should be considered.
And if a -dimensional gas has at least two components with
different masses, then a nontrivial momentum exchange in the
elastic collisions is possible. This results in the possibility of
the relaxation of the momentum spectra by only the elastic
collisions Cubero ().

Let’s summarize this section with formulation of the physical
meaning of the bulk viscosity. The bulk viscosity reflects
deviation of the value of the pressure from its local equilibrium
value (as can be seen from the (27)), appearing when the
system expands/compresses, because of the delay in the
equilibration. The bulk viscosity is not responsible for the
restoration of the chemical or the kinetic equilibria - it’s
responsible for the relaxation of the divergence of the flow
velocity. If there are inelastic processes, then the particle
numbers also get nonequilibrium contributions (cf. (14),
(37), (68)) such that the charge is conserved
locally (cf. (74))^{19}^{19}19One should keep in mind that
while studying the chemical perturbations
through the thermodynamic functions first the contributions from
the transport coefficients’ terms should be subtracted.. Though
these contributions together with the contribution to the pressure
may become nonphysical because of the approximate zero modes (if
such ones appear in the calculations). The magnitude of the bulk
viscosity changes from theory to theory. Under some quite general
assumptions a nonzero value of the bulk viscosity can be connected
with violation of the scale invariance due to a nonzero value of
the energy-momentum tensor Coleman:1970je (); Callan:1970ze ().
Of course, the beta function can contribute to the energy-momentum
tensor and the bulk viscosity too jeon ().

## Iii The hard core interaction model and the UrQMD cross sections

In a non-relativistic classical theory of particle interactions there is a widespread model, used in approximate calculations, called the hard core repulsion model or the model of hard spheres with some radius . For its applications to the high-energy nuclear collisions see Gorenstein:2007mw () and references therein. The differential scattering cross section for this model can be inferred from the problem of scattering of point particle on the spherical potential if and if landau1 (). In this model the differential cross section is equal to . To apply this result to the gas of hard spheres with the radius one can notice that the scattering of any two spheres can be considered as the scattering of the point particle on the sphere of the radius , so that one should take . The total cross section is obtained after integration over the angles of the which results in the . For collisions of hard spheres of different radiuses one should take or replace the on the :

(7) |

The relativistic generalization of this model is the constant (not dependent on the scattering energy and angle) differential cross sections model.

The hard spheres model is classical, and connection of its cross
sections to cross sections, calculated in any quantum theory, is
needed. For particles, having a spin, the differential cross
sections averaged over the initial spin states and summed over the
final ones will be used.^{20}^{20}20It’s assumed that particle
numbers of the same species but with different spin states are
equal. If this were not so then in approximation, in which the
spin interactions are neglected and probabilities to have certain
spin states are equal, the numbers of the particles with different
spin states would be approximately equal in the mean free time.
With equal particle numbers their distribution functions are equal
too. This allows one to use the summed over the final states cross
sections in the Boltzmann equations. If colliding particles are
identical and their differential cross section is integrated over
the momentums (or the spatial angle to get the total cross
section) then it should be multiplied on the factor to
cancel double counting of the momentum states. These factors are
exactly the factors next to the collision integrals
in the Boltzmann equations (30). The differential cross
sections times these factors will be called the classical
differential cross sections.

The UrQMD cross sections are used in the numerical calculations of
Sec. VI^{21}^{21}21Very high energy dependence of any
used UrQMD cross section is not important because of the
exponential suppression . The used cross sections
were cut on the and were continued by a
corresponding constant continuously at higher energies. At small
enough momentums there is another somewhat weaker suppression. The
momentum space density of each particle provides
suppression. This may (partially) suppress some deviations from
the experimental data of some UrQMD cross sections (like for the
pair) at . To estimate at
what temperatures some discrepancies in cross sections can appear
one can equate the to the sum of the averaged
one-particle energies (23) of the two colliding
particles.. These cross sections are described in
Bass:1998ca (); Bleicher:1999xi (). More details can be found in
the UrQMD program codes. Below there is some description mainly of
what is different or new.

The UrQMD cross sections are averaged over the initial spin states and summed over the final ones. As long as the UrQMD cross sections are total ones (integrated over the scattering angle), the factors are already absorbed into them (in what follows only such cross sections will be considered in this section tacitly). Dividing them on the , one gets the classical differential cross sections, averaged over the scattering angle.

The UrQMD codes (version 1.3) were modified to get accurately
tabulated (with a step of ) cross sections. Resonances’
masses and widths (they are tuned in their uncertainty regions to
describe the experimental data better), used in the UrQMD codes,
have somewhat different values than the ones in the
Bass:1998ca (). Influence of variation of these parameters was
studied in Gerhard:2012fj (). The UrQMD codes implement
somewhat different averaging of the c. m. momentums over the
resonances’ masses^{22}^{22}22Averaged powers of the momentums are
used, not powers of the averaged momentums. than in the papers
Bass:1998ca (); Bleicher:1999xi (). It was found that using the
resonance dominating cross sections from the papers
Bass:1998ca (); Bleicher:1999xi () some of these cross sections
could have a large rise at small c. m. momentums if constant
widths are used in the calculations of the averaged c. m.
momentums in the energy dependent widths. So that one should be
aware of this fact^{23}^{23}23It may be mentioned that one should be
also aware of possible differences in storing of the floating
point numbers in different programming languages or while using
different compilers.. The UrQMD codes have a low energy cut-off
at (and a similar one over the
c. m. momentum if triggered) for the resonance dominating cross
sections, and no large low energy rise was found there.

An important ingredient of the UrQMD model is the Additive Quark Model (AQM), which is used for unknown cross sections. Universality of hadrons, based on jet quenching arguments, is used to support this model. This model describes the experimentally known cross sections well at sufficiently high energies. Application of this model is better than elimination of the corresponding hadrons, which is the same as equating their all cross sections to zero and, hence, exclusion of their contributions from the thermodynamic functions (infinite mean paths, no thermalization).

At this point an interruption should be made to consider some important questions related to different types of the UrQMD cross sections. These different types are used due to several reasons and are the following: the elastic cross section(s) (ECS(s)), the elastic plus the quasielastic cross section(s) (EQCS(s)), the total cross section(s) (TCS(s)) and the previous two types with enhanced in some way resonances’ cross sections (index ”2” is appended in the abbreviations).

Of course, the system of the Boltzmann equations would have a solution with any of these cross sections. Usage of the ECSs is completely self-consistent as long as only the elastic collision integrals are used in the calculations of the viscosities. However, there are reasons to consider also the EQCSs. Exactly this type of cross sections, being averaged over the isospin, is implemented in prakash (). The quasielastic cross sections can be used as rightful contributions to the ECSs in the approximation that the 4-momentum of the intermediate resonance does not change (the effects of the exclusion of the resonances as independent particles are considered in Sec. VI). The mean free paths of the intermediate resonances without contributions of the decays, being not equal to zero, also introduce some errors, which are neglected. The EQCSs conserve particle numbers, which is consistent with the only elastic collision integrals, implemented in the calculations. There are also some additional arguments for the usage of these cross sections. From the phenomenological considerations one can take into account shortening of the mean free paths (or enlarging of the collision rates) due to creation of the resonances. In other words, there would have to be contributions from the inelastic collision integrals next to the elastic collision integrals, and they are taken into account approximately by the contributions from the quasielastic cross sections.

Resonances are not just intermediate particles, and they can
collide with other particles. They make not negligibly small
contribution to the thermodynamic functions and the viscosities.
So that they are also included in the calculations as independent
particles with their parameters and corresponding
collision integrals. They would have to
have shortening of their mean free paths from their decays and
contributions from the inelastic collision
integrals too. These contributions may be taken into account from
the following collision rate considerations. A resonance’s decay
rate can be approximately replaced with just its total width.
Then, given a resonance, one would have to redistribute its width
(that is not changing the whole collision rate containing the
contribution of the decay rate) in such a way that the cross
section of the collision of this resonance with a resonance of the
same species gets an addition^{24}^{24}24This
enhancement leads to the shortening of the mean free paths of the
resonances of only this species, as needed. In the formal
limit of this infinitely large enhancement other collision
integrals can be neglected and the Boltzmann equation for this
species decouples. Then from the solution of the Boltzmann
equation for a single-component gas (see Sec. VII.1)
one concludes that the nonequilibrium perturbation to the
distribution function of this species vanishes in this limit. Note
that infinitely strong interactions also with particles of all
other particle species would result in zero transport
coefficients.. Using an approximate expression for the collision
rates (in the nonrelativistic approximation, applicable in this
case) from Appendix D, one easily finds the addition
(where
is the width) to the . Such
cross sections seem to be the most physically preferable ones
because they take into account more realistic mean free paths than
in the previous case while not violating the conservation of the
particle numbers too.

The TCSs are used to take into account even larger
shortening^{25}^{25}25Not the largest one. The effect of the
enhancements of the resonances’ TCSs is of for the bulk
viscosity and of for the shear viscosity so that TCS2s are
additionally considered. of the mean free paths than in the case
of the EQCSs. However, such cross sections introduce some
inconsistency, implying that the conservation of the particle
numbers is violated. As long as there may be contributions from
some partial cross sections to the UrQMD ECSs or the EQCSs which
were not taken into account (see below), the TCSs can be used as
the upper bounds for the ECSs and the EQCSs. However, it’s
expected that these bounds are excessively high. If so, the TCSs
(rather TCS2s) can be considered not only as the approximation
taking into account real mean free paths but also as some measure
of deviation from the approximation of only the elastic and the
quasielastic collisions with the following arguments. If the TCSs
were approximately equal to the ECSs or the EQCSs, or the numbers
of particles with large inelastic cross sections were small, then
one could expect small errors due to the negligibility of the
inelastic collisions.

Continuing the discussion of the details of the UrQMD cross
sections, it should be mentioned that the UrQMD TCSs are the most
reliable ones. The sum of the partial cross sections is not always
equal to the TCSs by their construction. If this is the case, then
some partial cross sections are rescaled depending on their
reliability^{26}^{26}26This information, including some other
information about the cross sections, is stored in the array
SigmaLn of the file blockres.f.

The magnitudes of the partial cross sections, implemented in the UrQMD codes, are used to determine, what a partial cross section to choose in a given collision, using a random number generator. Among these partial cross sections there are the ECSs. Exactly these ECSs are used in the present calculations. However, if a partial cross section with a string excitation is chosen in a given collision, there is a probability to end up with the elastic collision if the is too small. These contributions to the ECSs are not calculated and are not added to the ECSs. Also the string excitations can, possibly, end up with creation of a resonance. Contributions to the EQCSs from the string excitations are taken into account partially (see below).

The ECSs, if not known from the experiment, are taken in the form of some extrapolations, discussed below, or the AQM is used. The normalization on the corresponding TCSs can change the ECSs notably. The meson meson (MM) ECSs are equal to . The meson baryon (MB) ECSs are equal to the AQM rescaled experimental cross sections. But after the normalization they become equal to zero in the resonances dominated energy range (approximately below ). The anti-baryon baryon () ECSs are equal to the AQM rescaled experimental cross sections. Other ECSs are equal to the AQM ECSs.

Before discussing the quasielastic cross sections first let’s write for convenience the resonance dominated cross sections formula for a reaction . Correcting a typo and rewriting it in a somewhat different form than in Bass:1998ca (); Bleicher:1999xi (), one gets

(8) |

where is the partial energy-dependent width of the decay of the resonance into particles of types and without specification of their isospin projection, is the total energy-dependent width of the decay of the resonance , is the spin degeneracy factor, is the energy-dependent branching ratio. The squared Clebsch-Gordan coefficients allow to specify the branching ratio for the pair of the particles with concrete isospin projections. The squared Clebsch-Gordan coefficients should be normalized in such a way that they give unity after summation over all isospin projections in a given multiplet. This formula represents contributions from all possible resonances through which the reaction can take place. Now it’s easy to write down the cross sections for the quasielastic scatterings:

(9) |

One more multiplier takes into account the fact that a resonance decays only into the pair and represents the probability of this decay.

The TCS is not described by the formula (8) completely, and a partial cross section, attributed to the s-channel strings excitations, is added in the UrQMD model to fit the TCS to the experimental data. In the UrQMD model this s-channel strings cross section is added also to other strange meson nonstrange baryon TCSs when annihilation is possible due to the quark content. From comparison with the experimental data for the ECS Beringer () (actually it’s believed to be the EQCS because smaller peaks from the resonances are reproduced there) it was found that the half of the s-channel strings cross section is enough to describe well this experimental cross section. Then the half of the s-channel strings cross section is added to other strange meson nonstrange baryon EQCSs when annihilation is possible. These contributions from the strings excitations are the most low energetic ones. They are the only contributions from the strings excitations which are added. The next in the energy scale possible contributions to the EQCSs may be in the cross sections. In other pairs the string excitations appear approximately from .

There is an important omission, found in the UrQMD codes (present also in the last version 3.3). The function fcgk returns incorrect (two times smaller) values of the squared Clebsch-Gordan coefficients for the resonances dominated cross sections in some cases. The first case is for the pairs of unflavored mesons from the same multiplet with the isospin . For example, the function fcgk returns for the only possible isospin decomposition of the to the pair, because the states and are counted as different ones. As a result, the peak from the -resonance becomes two times smaller than e. g. in the -resonance isospin decomposition. The second less important case is for the pairs of unflavored mesons with the isospin and anti-nucleons. The third even less important case is for the pair and it’s charge conjugate.

Let’s make some comments on the errors what the above-mentioned omissions cause in some quantities at zero chemical potentials, which in turn demonstrate sensitivity to different changes in the cross sections. The errors in the viscosities with the ECSs are less than . The errors in the shear viscosity with the EQCSs (the TCSs) reach () at . Outside the temperature range the errors reach (). The errors in the bulk viscosity with the EQCSs (the TCSs) reach () at . Outside the temperature range the errors reach (). The errors in the total number of collisions per unit time per unit volume (using the TCSs and including the decay rates) reach (at ). Outside the temperature range the errors reach . In view of the errors for the total number of collisions the kinetic freeze-out temperatures found in the UrQMD studies Bleicher:2002dm () should decrease, becoming closer to the experimentally extracted ones (see Heinz:2007in () and references therein). The chemical freeze-out temperature may change in a less extent. This is because both the inelastic and the quasielastic processes’ cross sections (like of the quasielastic collision of the pair and of the reaction ) increase, so that the temperature at which the inelastic processes cease to be dominant may almost not change.

It’s observed that some of the UrQMD detailed balance cross sections (e. g. for the pair) are not symmetric under the particle interchange. This is because the function W3j, calculating the Wigner symbols, doesn’t return zero in some cases. Namely, the selection rule for the sum is not included. In principle, such omission could result in negative values of the essentially non-negative viscosities but, as long as only small fraction of cross sections is affected, this omission has caused only negligibly small errors in the viscosities. But e. g. the error in the TCS is approximately .

Also some fixes of the UrQMD cross sections are made. It’s found that the UrQMD ECS has large deviations from the experimental data Beringer () in the range (the UrQMD cross section reaches in the region ). To fit this cross section to the experimental data it is replaced with the AQM ECS in the range and is interpolated smoothly with the sine function in the range with the cross section being equal to at . This replacement is also applied to other MB ECSs, when annihilation is not possible due to the quark content.

The next fix is for the BB ECSs. It’s found that the UrQMD ECS has quite large deviations form the experimental data Beringer () too. To fit this cross section to the experimental data it is replaced with the AQM ECS in the range and interpolated smoothly with the sine function in the range with the cross section being equal to the AQM TCS at . This replacement is also applied to other BB ECSs.

Some other found lacks result in negligible errors in the viscosities. However, errors in the corresponding mean free paths and possible other quantities may be not negligible ones. Two of such lacks can be mentioned. The first one is the following. The and cross sections are fitted to the experimental data. And their charge conjugates are calculated using general formulas and so cause deviations up to for . The second lack is the following. In some not large energy regions with the resonance dominated cross sections are equal to zero for some small numbers of pairs because there is no resonances which could be created by this pair. These regions are replaced by a constant continuously.

Let’s also comment on the deviations from the fixes described in the last four preceding paragraphs. The altogether deviations in the viscosities and the total number of collisions with the TCSs are less than . The altogether deviations in the viscosities with the ECS or the EQCS are in the range . The largest contribution is from the MB cross sections’ fixes. At the deviations are less than (the temperatures above are not studied).

## Iv Conditions of applicability

Before proceeding forth first the applicability of the Boltzmann equation and of the calculations of the transport coefficients should be clarified.

Although the Boltzmann equations are valid for any perturbations
of the distribution functions they should be slowly varying
functions of the space-time coordinates to justify that they can
be considered as functions of macroscopic quantities like the
temperature, the chemical potentials or the flow velocity or, in
other words, that one can apply thermodynamics locally. Then one
can make the expansion over the independent gradients of the
thermodynamic functions and the flow velocity (the Chapman-Enskog
method), which vanish in equilibrium. Smallness of these
perturbations of the distribution functions in compare to their
leading parts ensures the validity of this expansion and that the
gradients are small^{27}^{27}27The magnitudes of thermodynamic
quantities can also be restricted by this condition or,
conversely, not restricted even if transport coefficients diverge.
See also Sec. VII.1 of this paper. The smallness of the
shear and the bulk viscosity gradients can also be checked by the
condition of smallness of the (27) in
compare to the (20). Of course, the next
corrections should be small too.. Because these perturbations are
inversely proportional to coupling constants one can say that they
are proportional to some product of particles’ mean free paths and
the gradients. So that, in other words, the mean free paths should
be much smaller than characteristic lengths, on which the
macroscopic quantities change considerably^{28}^{28}28 It’s clear
that the mean free paths should be smaller than the system’s size
too. .

As is discussed in Sec. II, the inelastic processes may need addition treatment in the calculations of the bulk viscosity. There is a need to specify reasonable conditions when the inelastic processes can be neglected. One could use the following reliable criterion, which takes into account both the particle number densities and the intensity of the interactions:

(10) |

where is the system’s volume, is the number of reactions
of particles of the -th species^{29}^{29}29Primed indexes run
over the particle species without regard to their spin states.
This assignment is clarified more in Sec. V.1. over
all channels per unit time per unit volume (analog of
(299)), is chosen to satisfy the inequality, and
is equal to the moment of time at which the divergence of
the flow velocity is relaxed (if this time can be estimated
reliably with remained inelastic processes) or to the moment of
time at which the system becomes practically not interacting
(after expansion) because of large cumulative mean free path in
compare to the system’s size. Though this criterion is likely to
be too strict, and at some higher temperatures the approximation
of conserved particle numbers should still work well. The main
alternative criterion is based on comparison of collision rates of
elastic and inelastic processes (as implemented in the
Bleicher:2002dm ()). Using this criterion and some other ones,
the chemical freeze-out line^{30}^{30}30This is an approximation. In
fact this should be a range in which particles of different
particle species have their own freeze-out points. in the
plane can be built for the hadron gas, see e. g.
Cleymans:2005xv (); Andronic:2005yp (). At zero chemical
potentials the chemical freeze-out temperature is approximately
equal to . The remaining question is how
good is the approximation of only the elastic collisions at . From the hydrodynamical description of the
elliptic flow at RHIC it’s found that near
the chemical freeze-out Dusling:2011fd (). The constant value
provides a good description of the elliptic flow both
at RHIC and LHC Bozek:2011ph (). It seems that the
approximation of conserved particle numbers is not implemented in
the bulk viscosity formula used in the nhngr (). The bulk
viscosity obtained from it is very close to the one of this paper.
These results support the choice of the approximation of only the
elastic collisions at and show that the
deviations are likely no more than in 2-3 times. Anyway, the
numerical calculations by the Kubo formula through simulations of
collisions are desirable along and around the chemical freeze-out
line for more accurate calculations (though the procedure of
collisions of particles introduce some errors itself
Bass:1998ca (), which should be kept in mind).

Errors due to the Maxwell-Boltzmann statistics, used instead of
the Bose-Einstein or the Fermi-Dirac ones, were found to be small
for the vanishing chemical potentials^{31}^{31}31It should be
mentioned that if the particles of the -th particle species are
bosons and if then there is a (local)
Bose-Einstein condensation for them, which should be treated in a
special way.. According to calculations for the pion gas in
davesne (), the bulk viscosity becomes larger at
and larger at for the vanishing
chemical potential. Although the relative deviations of the
thermodynamic quantities of the pion gas at the nonvanishing
chemical potential are not more than
^{32}^{32}32The relative deviations of the thermodynamic
quantities grow with the temperature for some fixed value of the
chemical potential and tend to some constant. the bulk viscosity
becomes up to times more. The shear viscosity becomes
less at and less at for the
vanishing chemical potential and less at and
less at for the . The
corrections to the bulk viscosity of the fermion gas, according to
calculations of the bulk viscosity source term, not presented in
this paper, are of the opposite sign and approximately of the same
magnitude. So that for the hadron gas the error due to the used
classical statistics can be even smaller than for the pion gas.

The numerical calculations in Sec. VI of the viscosities with the total cross sections justify the choice of one constant cross section for all hadrons. It’s approximately equal to , corresponding to the effective radius (as given by the (7)), which is used in the estimations below.

The condition of applicability of the ideal gas equation of state is controlled by the dimensionless parameter which appears in the first correction from the binary collisions in the virial expansion and should be small. Here is the so called excluded volume parameter and is the mean volume per particle. One finds at , at and at for the vanishing chemical potentials. Along the chemical freeze-out line (its parametrization can be found in Gorenstein:2007mw ()) the grows from to with the temperature. From comparison with lattice calculations Borsanyi:2010cj () one can find that the corrections to the ideal gas equation of state are small at . One could suspect that even small corrections to the thermodynamic quantities can result in large corrections for the bulk viscosity, though this seems to be not the case. The errors in the bulk viscosity from the scale-violating contributions of the hadrons’ masses are less than the errors from the contributions to the trace of the energy-momentum tensor (for more details see Sec. VI).

One more important requirement, which one needs to justify the Boltzmann equation approach, is that the mean free time should be much larger than (the is the characteristic single-particle energy) danielewicz () or the de Broglie wavelength should be much smaller than the mean free path Arnold:2002zm () to distinguish independent acts of collisions and for particles to have well-defined on-shell energy and momentum. This condition gets badly satisfied for high temperatures or densities. The mean free path of the particle species is given by the formula (315) or the formula (310) if the inelastic processes can be neglected. The wavelength can be written as , where the averaged modulus of the momentum of the -th species is

(11) |

where , is the modified Bessel function of the second kind. As it follows from the (11) the largest wavelength is for the lightest particles, the -mesons. The elastic collision mean free paths are close to each other for all particle species. Hence, the smallest value of the ratio is for the -mesons. Its value is close to the value of the and is exponentially suppressed for small temperatures too. At the temperature () and the vanishing chemical potentials this ratio is equal to 0.18 (0.7). Along the chemical freeze-out line it grows from to with the temperature.

To go beyond these conditions one can use the Kubo (or Green-Kubo)
formulas, for instance. In the jeon () the Kubo formulas were
used to perform perturbative calculations of the viscosities in
the leading order. Basing on this result, an example of effective
weakly coupled kinetic theory of quasiparticle excitations with
thermal masses and thermal scattering amplitudes was presented in
the jeonyaffe (). There the function (appearing because
of the temperature dependence of the mass) takes into account the
next in the coupling constant correction to the energy-momentum
tensor and the equation of state^{33}^{33}33In the hadron gas it’s
believed that the vacuum masses are large in compare to their
thermal corrections for the most of the hadrons at temperatures
or even higher ones. Then expanding over the
thermal correction in the matrix elements, one would get even
smaller corrections than the ones to the equation of state in
coupling constants (because of coupling constants next to the
matrix elements) in a perturbation theory, e. g. chiral
perturbation theory.. For further developments see
Arnold:2002zm (); Gagnon:2006hi (); Gagnon:2007qt (). For some other
approaches see
Blaizot:1992gn (); Calzetta:1986cq (); Calzetta:1999ps () and
Arnold:1997gh () with references therein.

## V Details of calculations

### v.1 The Boltzmann equation and its solution

The calculations in this paper go close to the ones in
groot () though with some differences and generalizations.
Let’s start from some definitions. Multi-indices will be
used to denote particle species with certain spin states. Indexes
will be used to denote particle species without
regard to their spin states (and run from 1 to the number of the
particle species ) and to denote conserved quantum
numbers^{34}^{34}34In systems with only the elastic collisions each
particle species have their own ”conserved quantum number”, equal
to 1.. Quantifiers with respect to the indexes are
omitted in the text where they may be needed which won’t result in
a confusion. Because nothing depends on spin variables one has for
every sum over the multi-indexes

(12) |

where is the spin degeneracy factor. The following assignments will be used:

(13) |

where denotes values of conserved quantum numbers of the -th kind of the -th particle species. Everywhere the particle number densities are summed, the spin degeneracy factor appears and then gets absorbed into the or the by the definition. All other quantities with primed and unprimed indexes don’t differ, except for rates, the mean free times and the mean free paths defined in Appendix D, the commented below, the coefficients , and, of course, quantities, whose free indexes set the indexes of the particle number densities . Also the assignment will be used for compactness somewhere.

The particle number flows are^{35}^{35}35The metric
signature is used throughout the paper.

(14) |

where the assignment is introduced. The energy-momentum tensor is

(15) |

The local equilibrium distribution functions are

(16) |

where is the chemical potential of the -th particle
species, is the temperature and is the relativistic
flow 4-velocity such that (with a frequently
used consequence ). The local equilibrium
is considered as perturbations of independent thermodynamic
variables and the flow velocity over a global equilibrium such
that they can depend on the space-time coordinate .
Additional chemical perturbations could also be considered, but
they don’t enter in the first order transport coefficients if they
are small, as is discussed in Sec. II. The chemical
equilibrium implies that the particle number densities are equal
to their global equilibrium values. The global equilibrium is
called the time-independent stationary state with the maximal
entropy^{36}^{36}36The kinetic equilibrium implies that the momentum
distributions are the same as in the global equilibrium. Thus, a
state of a system with both the pointwise (for the whole system)
kinetic and the pointwise chemical equilibria is the global
equilibrium.. The global equilibrium of an isolated system can be
found by variation of the total nonequilibrium entropy functional
landau5 () over the distribution functions with condition of
the total energy and the total net charges conservation:

(17) |

where are the Lagrange coefficients. Equating the first variation to zero, one easily gets the function (16) with , and

(18) |

where are the independent chemical potentials coupled to the conserved net charges.

With , substituted in the (14) and the (15), one gets the leading contribution in the gradients expansion of the particle number flow and the energy-momentum tensor:

(19) |

(20) |

where the projector

(21) |

is introduced. The is the ideal gas particle number density,

(22) |

the is the ideal gas energy density,

(23) |

and the is the ideal gas pressure,

(24) |

Also the following assignments are used:

(25) |

Above is the enthalpy per particle, is the energy per particle and , are the enthalpy and the energy per particle of the -th particle species correspondingly, which are well defined in the ideal gas.

In the relativistic hydrodynamics the flow velocity needs somewhat extended definition. The most convenient condition which can be applied to the is the Landau-Lifshitz condition landau6 () (Section 136). This condition states that in the local rest frame (where the flow velocity is zero though its gradient can have a nonzero value) each imaginary infinitesimal cell of fluid should have zero momentum, and its energy density and the charge density should be related to other thermodynamic quantities through the equilibrium thermodynamic relations (without a contribution of nonequilibrium dissipations). Its covariant mathematical formulation is

(26) |

The next to leading correction over the gradients expansion to the
can be written as an expansion over the 1-st order
Lorentz covariant gradients, which are rotationally and space
inversion invariant and satisfy the Landau-Lifshitz
condition^{37}^{37}37 Also this form of respects the
second law of thermodynamics landau6 () (Section 136).
(26):

(27) |

where for any tensor the symmetrized traceless tensor assignment is introduced:

(28) |

The equation (27) is the definition of the shear and the bulk viscosity coefficients. The term in the (27) can be considered as a nonequilibrium contribution to the pressure which enters in the (20).

By means of the projector (21) one can split the space-time derivative as

(29) |

where , . In the local rest frame (where ) the becomes the time derivative and the becomes the spacial derivative. Then the Boltzmann equations can be written in the form

(30) |

where represents the inelastic or
number-changing collision integrals (it is omitted in calculations
in this paper if the opposite is not stated explicitly) and
is the elastic collision
integral. The collision integral has the form of
the sum of positive gain terms and negative loss terms. Its
explicit form is^{38}^{38}38The factor cancels double
counting in integration over momentums of identical particles. The
factor comes from the relativistic normalization of the
scattering amplitudes. (cf. jeon (); Arnold:2002zm ())

(31) | |||||

where if and denote the same particle species without regard to the spin states and otherwise, is the square of the dimensionless elastic scattering amplitude averaged over the initial spin states and summed over the final ones. Index designates that and are different variables. Introducing as

(32) |

one can rewrite the collision integral (31) in the form as in groot () (Chap. I, Sec. 2)

(33) |

The is related to the elastic differential cross section as groot () (Chap. I, Sec. 2)

(34) |

where is the usual Mandelstam variable. The has properties (due to time reversibility and a freedom of relabelling of order numbers of particles taking part in reaction). And e.g. in the general case. The elastic collision integrals have important properties which one can easily prove groot () (Chap. II, Sec. 1):

(35) |

(36) |

Also th