4 particle descriptions. In conclusion  we have shown that a quantum- statistical model of nuclear matter  that includes the formation of clusters at densities below nuclear satura- tion  describes quite well the low-density symmetry en- ergy which was extracted from the analysis of heavy-ion collisions. Within such a theoretical approach the com- position and the thermodynamic quantities of nuclear matter can be modeled in a large region of densities  temperatures and asymmetries that are required  e.g.  in supernova simulations. Acknowledgement : This research was supported by the DFG cluster of excellence “Origin and Structure of the Universe”  by CompStar  a Research Networking Pro- gramme of the European Science Foundation  by US De- partment of Energy contract No. DE-AC02-06CH11357 (TK) and grant No. DE-FG03-93ER40773 (Texas A& M) and by Robert A. Welch Foundation grant No. A0330 (JBN). DB acknowledges support from the Polish Min- istry for Research and Higher Education  grant No. N N 202 2318 37 and from the Russian Fund for Basic Re- search  grant No. 08-02-01003-a. that the underlying RMF model for the quasiparticle de- scription with n$_{0}$ = 0 . 149 fm $^{$_{−}$3}$  E$_{sym}$ ( n$_{0}$ ) = 32 . 73 MeV gives a reasonable behavior at high density similar to the MDI  x =0 parametrization. We thus see that our approach successfully interpolates between the cluster- ing phenomena at low density and a realistic description around normal density. In the right panel of Fig. 2 we compare to the ex- perimental results  full (red) circles (Tab. I  col. 6) in an expanded low-density region. Besides the MDI parametrization we show the QS results [8] for T =1  4  and 8 MeV  which are in the range of the temperatures in the experiment. The QS results including cluster forma- tion agree well with the experimental data points  as seen in detail in Fig. 1. We conclude that medium-dependent cluster formation has to be considered in theoretical mod- els to obtain the low-density dependence of the symmetry energy that is observed in experiments. The temperatures and densities of columns 2 and 3 in Tab. I will be modified if medium effects on the light clusters are taken into account [24]. We have carried out a self-consistent determination of the temperatures T sc and densities n sc taking into account the medium- dependent quasiparticle energies as specified in Ref. [12] (columns 9 and 10 of Tab. I). Compared to the Albergo method results [20]  the temperatures T sc are about 10 % lower. Significantly higher values are obtained for the in- ferred densities n sc which are more sensitive to the inclu- sion of medium effects. We have also calculated the free and internal symmetry energies corresponding to these self-consistent values of T sc and n sc according to Ref. [8] (columns 11 and 12 of Tab. I). These results are also shown in the right panel of Fig. 2 as open (purple) cir- cles. The resultant internal symmetry energies are 15 to 20 % higher than the QS model values for T and n given in columns 2 and 3 in Tab. I. We have restricted our present work to that region of the phase diagram where heavier clusters with A > 4 are not relevant. The generalization of the given approach to account for clusters of arbitrary size would lead to an improvement in the low-density  low-temperature re- gion when nuclear statistical equilibrium is assumed. Al- ternatively  one can introduce the formation of heavier nuclei in the presence of a nucleon and cluster gas  cf. Refs. [25  26]. The simplest approach to model the formation of heavy clusters is to perform inhomogeneous mean-field calcu- lations in the Thomas-Fermi approximation assuming spherical Wigner-Seitz cells. In Fig. 2 (left panel) pre- liminary results for the zero-temperature symmetry en- ergy of such a calculation is shown by the long-dashed line using the same RMF parametrization as for the QS approach introduced above; for details see Ref. [27]. The symmetry energy in this model approaches a finite value at zero density in contrast to the behavior of the MDI parametrizations and conventional single-nucleon quasi- [1] J. M. Lattimer and M. Prakash  Phys. Rept. 442   109 (2007). [2] G. Watanabe  et al.   Phys. Rev. Lett. 103   121101 (2009). [3] P. Danielewicz  Nucl. Phys. A 727   233 (2003). [4] B. A. Li  et al.   Phys. Rept. 464   113 (2008). [5] C. Fuchs and H. H. Wolter  Eur. Phys. J. A 30   5 (2006). [6] T. Kl¨ ahn et al.   Phys. Rev. C 74   035802 (2006). [7] C. J. Horowitz and A. Schwenk  Nucl. Phys. A 776   55 (2006). [8] S. Typel  G. R¨ opke  T. Kl¨ ahn  D. Blaschke and H. H. Wolter  Phys. Rev. C 81   015803 (2010). [9] J. P. Bondorf  et al.   Phys. Rept. 257   133 (1995). [10] M. Schmidt  G. R¨ opke  and H. Schulz  Ann. Phys. (N.Y.) 202   57 (1990). [11] G. R¨ opke et al.   Nucl. Phys. A 379   536 (1982); 424   594 (1984). [12] G. R¨ opke  Phys. Rev. C 79   014002 (2009). [13] S. Typel  Phys. Rev. C 71   064301 (2005). [14] H. R. Jaqaman  Phys. Rev. C 38   1418 (1988). [15] H. Takemoto  et al.   Phys. Rev. C 69   035802 (2004). [16] V. Baran  et al.   Phys. Rept. 410   335 (2005). [17] M. B. Tsang et al.   Phys. Rev. Lett. 102   122701 (2009). [18] S. Wuenschel et al.   Phys. Rev. C 79   061602(R) (2009). [19] C. Sfienti  et al.  Phys. Rev. Lett. 102   152701 (2009) [20] S. Kowalski et al.   Phys. Rev. C 75   014601 (2007). [21] S. Albergo  et al.   Nuovo Cimento A 89   1 (1985) [22] M. B. Tsang et al.   Phys. Rev. Lett. 86   5023 (2001). [23] L. W. Chen et al.   Phys. Rev. Lett. 94   032701 (2005); Phys. Rev. C 76   054316 (2007). [24] S. Shlomo et al.   Phys. Rev. C 79   034604 (2009). [25] J. M. Lattimer  et al.   Nucl. Phys. A 535   331 (1991). [26] H. Shen et al.   Nucl. Phys. A 637   435 (1998). [27] S. Typel et al.  in preparation.