Finite size effect of hadronic matter on its transport coefficients
We have theoretically investigated the finite system size effect of hadronic matter on its transport coefficients like shear viscosity, bulk viscosity, and electrical conductivity. We have used a Hadron Resonance Gas (HRG) model to calculate the thermodynamical quantities like entropy density, speed of sound and also the above transport coefficients. All these quantities are found to be sensitive to finite system size effects of hadronic matter. The effect of finite system size is found to be more when the system is at low temperatures and gets reduced at high temperatures.
The research on estimation of transport coefficients for the medium, produced in high energy heavy ion collision experiments, received a considerable attention in the scientific community, when RHIC experiments announced that they have got a QCD medium, having a very small value of shear viscosity to entropy density () ratio Romatschke:2007mq (); Luzum:2008cw (); Roy:2012jb (). In high energy nuclear physics, the topic becomes more exciting when we notice that the traditional QCD theory Arnold () predicts 10-20 times larger value of than the experimental value or the lower bound KSS (). As an alternative treatments of QCD theory, different effective QCD models Purnendu (); Redlich_NPA (); Marty (); G_CAPSS (); Weise2 (); Kinkar (); G_IFT (); Deb (); Tawfik () and hadronic models Itakura (); Dobado (); Nicola (); Weise1 (); SSS (); Ghosh_piN (); Gorenstein (); HM (); Kadam:2015xsa (); Hostler () have attempted to estimate this in recent times. Some attempts are also made by using some transport simulations Bass (); Muronga (); Plumari (); Pal (), where Kubo-type correlators are generated for estimating shear viscosity. Some estimations are also done from Lattice QCD calculations Meyer_eta (); LQCD_eta2 (). Currently, one has a reasonable estimate of the values of shear viscosity for QCD matter. Other transport coefficients like bulk viscosity, thermal conductivity and electrical conductivity are also important to estimate from the same dynamical framework for the strongly interacting medium. Some of the work in the literature for the microscopic calculations of bulk viscosity can be found in the Refs. Paech1 (); Gavin (); Arnold_bulk (); Prakash (); Tuchin (); Tuchin2 (); Nicola (); Marty (); De-Fu (); Redlich_NPA (); Redlich_PRC (); G_IFT (); Deb (); Tawfik (); Purnendu (); Vinod (); Santosh (); Sarkar (); HM (); Kadam:2015xsa (); Kadam:2015fza (); Sarwar:2015irq (); Hostler (); Nicola_PRL (); SG_NISER (); Meyer_zeta (); Dobado_zeta1 (); Dobado_zeta2 (); Saha:2015lla (), and those for electrical conductivity can be found in the Refs. LQCD_Ding (); LQCD_Arts_2007 (); LQCD_Buividovich (); LQCD_Burnier (); LQCD_Gupta (); LQCD_Barndt (); LQCD_Amato (); Cassing (); Puglisi (); Greif (); Marty (); PKS (); Finazzo (); Lee (); Nicola_PRD (); Greif2 (); G_IFT2 (); Ghosh:2016yvt ().
The matter created due to the energy deposition of the colliding nuclei has a finite volume. The volume of the system depends on the size of the colliding nuclei, the center of mass energy and centrality of the collision. Different effects of finiteness of the system size have been discussed in the literature Ferdinand:1969zz (); Fisher:1972zza (); Luscher:1985dn (); Elze:1986db (); Gasser:1987ah (); Spieles:1997ab (); Gopie:1998qn (); Kiriyama:2002xy (); Fischer:2005nf (); Abreu:2006pt (); Shao:2006gz (); Yasui:2006qc (); Bazavov:2007ny (); Palhares:2009tf (); Luecker:2009bs (); Braun:2010vd (); Fraga:2011hi (); Braun:2011iz (); Abreu:2011rj (); Abreu:2011zzc (); Ebert:2011tt (); Khanna:2012zz (); Bhattacharyya:2012rp (); Bhattacharyya:2014uxa (); Bhattacharyya:2015zka (); Bhattacharyya:2015kda (); Magdy:2015eda (); Redlich:2016vvb (); Xu:2016skm (); Bhattacharyya:2015pra (). The present article is aimed to observe a finite system size effect of the medium on the estimation of transport coefficients. Specifically we discuss the relative change in the values of the transport coefficients because of transformation from infinite matter to finite matter. Since the absolute values of transport coefficients are not our main aim, so the present article is not going to discuss the comparison of estimates for the transport coefficients using the current HRG model with the corresponding estimates, done in references, mentioned above.
In a classical view we may not expect different values of transport coefficient for a fluid having different system size. However, for fluid where the system size is small enough for quantum effect to play a role, one may expect a possibility of finite system size effect. We have tried to explore this fact for the tiny QCD matter, produced in the high energy heavy ion collision experiment. We realize the lower bound of shear viscosity to entropy density ratio () arises due to lowest possible quantum fluctuation of fluid, which can never be ignored even in the infinite coupling limit. However, in classical view, one can easily think about in this infinite coupling limit. Since we know that the of RHIC matter is surprisingly close to this lower bound Song:2010mg (); Schenke:2010rr (), therefore, we may associate this matter with the lowest possible quantum fluctuation and we may also consider other possible quantum effect coming from the finite system size effect. A simplest possible idea of finite lower momentum cut-off has been adopted to incorporate the finite system size picture of the medium and we have applied it in different transport coefficients calculations.
The article is organized as follows. Next section has covered the brief formalism part of transport coefficients and finite system size picture of HRG model. Then we have analyzed our numerical output in the result section and at last, we summarize our work.
According to Green-Kubo relation Zubarev (); Kubo (), different transport coefficients like shear viscosity , bulk viscosity and electrical conductivity are related with their respective thermal fluctuations or thermal correlation functions - and and , where stands for thermal average. The operators and can be found from energy-momentum tensor as
where is speed of sound in the medium and the index ; . Whereas, the operator is basically electromagnetic current of medium constituents. The explicit spectral representations of the transport coefficients in momentum space () are given below Nicola ():
In this work, we calculate the above transport coefficients within the hadron gas resonance (HRG) model, so we have to add the contributions of all mesons () and baryons () for getting total transport coefficients of hadronic matter. The mathematical structure of transport coefficients, obtained from the one-loop diagram in quasi-particle Kubo approach and relaxation time approximation (RTA) in kinetic theory approach, are exactly same, therefore, we start with the standard expressions of Nicola (); Weise1 (); G_IJMPA (), Nicola (); G_IFT () and Nicola_PRD (); Nicola (); Ghosh:2016yvt ():
where and are degeneracy factors for baryons (fermions) and mesons (bosons) respectively. The ( stand for particle and anti-particle respectively) is Fermi-Dirac (FD) distribution function of having energy and is Bose-Einstein (BE) distribution function of with energy , where and denote the masses of B and M; stands for their momentum. During calculation of , we have to take care of isospin degeneracy factors of charged hadrons only. More explicitly, we can write the charge factors of baryons and mesons as and with , where and are their electrical charge number in units of electrons; e.g. for it is .
HRG : In the HRG model, the system consists of all the hadrons and resonances. Different versions of HRG model may be found in Refs. Hagedorn:1980kb (); Rischke:1991ke (); Cleymans:1992jz (); BraunMunzinger:1994xr (); Cleymans:1996cd (); Yen:1997rv (); BraunMunzinger:1999qy (); Cleymans:1999st (); BraunMunzinger:2001ip (); BraunMunzinger:2003zd (); Karsch:2003zq (); Tawfik:2004sw (); Becattini:2005xt (); Andronic:2005yp (); Andronic:2008gu (); Begun:2012rf (); Andronic:2012ut (); Tiwari:2011km (); Fu:2013gga (); Tawfik:2013eua (); Bhattacharyya:2013oya (); Bhattacharyya:2015zka (); Kadam:2015xsa (); Kadam:2015fza (); Albright:2014gva (); Albright:2015uua (); Begun:2016cva (); Adak:2016jtk (); Xu:2016skm (); Bhattacharyya:2015pra ().
Here the transport properties of hadronic matter with a finite volume is studied by using the simplest version namely the ideal or non-interacting HRG model.
where sum is over all the hadrons, refers to ideal i.e., non-interacting HRG. For individual hadron ,
where is the single particle energy, is the mass of the hadron, is the volume of the system, is the degeneracy factor for the hadron and is the temperature of the system. In the above expression is the chemical potential and are respectively the baryon number, strangeness and charge of the hadron, being corresponding chemical potentials. The upper and lower sign of corresponds to baryons and mesons respectively. In this work we have incorporated all the hadrons listed in the particle data book Olive:2016xmw () up to mass of 3 GeV.
We incorporate the finite size of the system by considering a lower momentum cut-off (say) where is the size of a cubic volume Bhattacharyya:2012rp (); Bhattacharyya:2014uxa (); Bhattacharyya:2015zka () of the system. With this cut-off the partition function of individual hadron becomes
In principle one should sum over discrete momentum values but for simplicity we integrate over continuous values of momentum.
From partition function we can calculate various thermodynamic quantities of interest. The partial pressure , the energy density can be calculated using the standard definitions,
With help of , , we can get entropy density (for ) as
From these thermodynamical quantities, one can find the square of speed of sound as
Iii Numerical results and discussion
The transport coefficients are calculated for two cases. In one case, they are calculated by considering finite system size effect and in other case, without considering such an effect. In former case, the limit of integration in Eqs. (3), (4), (LABEL:sigma_G) is to , while in latter case, the standard thermodynamical limit from to is taken. Dividing former by latter, we will get ratios of different transport coefficients, which may provide us an insight about finite system size effect on different transport coefficients. The ratios for , and are shown in Figs. 1, 4 and 7 and the related discussions are presented below.
Let us first come to the shear viscosity case, shown in Fig. 1, where the ratios are plotted versus . Here, denotes the shear viscosity of hadronic matter, having a system volume with radius , while stands for shear viscosity of infinite hadronic matter. In other words, the calculations of and are the shear viscosity calculations with and without considering finite system size effect, respectively. We have taken six different sizes of fm, which are motivated from the values of extracted by a HRG model fits to particle yields in high energy heavy ion collision experiment STAR_17 (). Fig. 1 shows that the ratio decreases when values of decreases. It also decreases with decrease in of the system. The increasing value of corresponds to the system is approaching towards the thermodynamic limit, so the ratio also approaches towards unity. The case of small values of and high values of is also associated with tending to thermodynamic limit, so the ratio will again approach towards unity.
Fig. 2 shows variation of the ratio of shear viscosity () to entropy density () with the temperature for fixed value of the relaxation time fm. The is calculated for different sizes of the system. Similar to , volume dependence is observed significantly in this ratio especially at low temperature. Interestingly, the increases when we decrease the . There is opposite dependency between and . The dependence of entropy density , shown in Fig. 3, will help us to understand the difference. In y-axis of Fig. 3, the entropy density is normalized by its Stefan-Boltzmann (SB) limit () and a dimensionless quantity is presented. For 3 flavor quarks, the SB limit of entropy density is . We notice that is decreasing as decreases and in , rate of decreasing for becomes dominant over that for . Therefore, ultimately increases as decreases. LQCD data of from the WB group Borsanyi:2013bia () (triangles) and Hot QCD group Bazavov:2014pvz () (circles) are in well agreement with our estimations from the HRG model for the temperature range studied. The straight horizontal dotted line in Fig. 2 denotes the lower bound of , known as KSS bound KSS () . So, our investigation suggests that finite system size effect of hadronic matter leads to a shift in away from the KSS bound. The numerical strength of will be proportionally controlled by the the relaxation time , which is chosen as 1 fm for this Fig. 2. However, we only focus on the changes of with , instead of its absolute value.
The Fig. 4 is same as Fig. 1 but for bulk viscosity of hadronic matter. It is observed that the ratio is more than unity at low temperature for very small system sizes. If we look at Eq. (3) for and Eq. (4) for , one can identify that the expression of contain an additional term
which vanishes in the limits of and . At high temperature QCD, this limits hold and QCD matter reaches to a scale independent or conformal symmetric situation, which can alternatively be realized from the vanishing values of for QCD matter at high temperature. In this context, the bulk viscosity calculation in HRG model is trying to measure indirectly the breaking of this conformal symmetric nature of QCD matter in the temperature range, where it is non-perturbative. The present investigation has tried to capture more delicate issue - finite system size effect on this breaking of conformal symmetry by studying the dependence of . The reason may be well understood from the Fig. 5, which shows temperature dependence of square of speed of sound () for different values of . We notice that is suppressed at low temperature due to finite system size effect. The conformal symmetry breaking term is enhanced due to this suppression in . Therefore, the ratio becomes more than unity in the low temperature region. LQCD data of from the WB group Borsanyi:2013bia () (triangles) and Hot QCD group Bazavov:2014pvz () (circles) are included in Fig. 5. Our estimations of in HRG model are in reasonable agreement with LQCD calculations in the temperature range studied.
Fig. 6 shows the vs for different values of . For an infinite system increases with increase of temperature, but for a small system size, the initially decreases with increase of temperature and then at higher temperature it increases slowly. The overall conclusion from the studies related to both and is that these ratios may be enhanced if we incorporate the finite system size effect of the hadronic matter.
Figure 7 shows the results for the electrical conductivity of hadronic matter. The presentation follows the same pattern as we have taken in earlier Figs. 1 and 4. With respect to other transport coefficients like and , the electrical conductivity is quite sensitive to the system size effect even at a higher temperature. For example, at MeV, suppression in , and for respectively fm are , and with respect to case. The temperature dependence of for different system sizes calculated at fm is shown in the Fig. 8. This dimensionless ratio for electrical conductivity is found to have the same qualitative dependence as is for the .
Iv Summary and Perspectives
In summary, we have tried to investigate the finite system size effect on different transport coefficients of hadronic matter. We have used an ideal HRG model to describe the thermodynamic behavior of hadronic medium constituents. We have adopted the simplest possible idea of a finite lower momentum cut-off to study the finite system size effect on the transport coefficients of the medium. Using standard expressions of transport coefficients like shear viscosity, bulk viscosity and electrical conductivity, we have calculated them as a function of temperature for various system sizes. We observe a significant finite system size effect on these transport coefficients at low temperatures. However, this effect reduces as we go to higher temperatures. The values of shear viscosity and electrical conductivity decrease due to finite system size effect but the values of bulk viscosity at low temperature increases. We find that the speed of sound gets smaller in the hadronic medium because of the finite system size effect at low temperatures. This suppression causes the enhancement of conformal breaking term as well as of bulk viscosity values for the medium. The entropy density also gets suppressed due to finite system size effect. The competing dependence of and with on system size gets reflected in the dimensionless ratios and . Both ratios are ultimately enhanced due to finite system size effect.
Acknowledgment: SG is financially supported from University Grant Commission (UGC) Dr. D. S. Kothari Post Doctoral Fellowship (India) under grant No. F.4-2/2006 (BSR)/PH/15-16/0060. SS and BM are supported by DAE and DST, which is greatly acknowledged.
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