%0 Book Section %1 statphys23_0077 %A Fidaleo, F.F. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K algebras bose condensation einstein operator statphys23 topic-8 %T Bose Einstein Condensation on Nonhomogeneous graphs %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=77 %X We discuss the Bose--Einstein condensation (BEC for short) for the pure hopping model on some nonhomogeneous amenable graphs, and in particular for the comb graphs $G_d$ with backbone $Z^d$ and fingers $Z^d$--copies of $Z$. It is shown that the BEC phenomena are connected with the transience of the graphs (relative to the adjacence operator), and not only with the finiteness of the critical density, the last properties being not equivalent for nonhomogeneous graphs considered here. The situation differs to that relative to $Z^d$, and $Z^d$--periodic lattices (Matsui) where the transience and the finiteness of the critical density are known to be equivalent, and determined by the dimension $d$ of the lattices. Namely, the comb graphs considered here, exhibit hidden spectrum near zero, thus the critical density is finite. But, they are transient iff $d\geq3$. By considering periodic boundary condition along the backbone for the finite--volume theories, it is proven that whenever $dłeq2$ and the sequence of chemical potential $\mu_Łambda_n\uparrow0$, it is impossible to obtain any infinite--volume state describing BEC and having a reasonable local description (i.e. non locally normal states for which $\displaystylełim_Łambda_n\uparrow G_d ømega_Łambda_n\big(a^+(\delta_j)a(\delta_j)\big)=+ınfty$). If $d\geq3$, a careful choice of the sequence $\mu_Łambda_n\uparrow0$ should be done in order to obtain locally normal states describing BEC. For all these infinite volume states $ømega$, the density $$ \rho(ømega):=łim_Łambda_nG_d1|Łambda_n| \sum_jınŁambda_nømega\big(a^+(\delta_j)a(\delta_j)\big) $$ coincides with the critical density $\rho_c$. The situation behaves as follows. Below the critical density, the system describes (pure hopping) Bosons allocated on a $d+1$--dimensional system (i.e. the dimension described by the vertices of the comb $G_d$). Above the critical density, a considerable portion of particles condensates in momentum space as well as in configuration space, due to nonhomogeneity. Such a condensate is essentially localized along the backbone. The system undergoes a ``dimensional transition''. If $dłeq2$ we cannot exhibit any locally normal state describing BEC. If $d\geq3$, we can construct locally normal states $ømega$ describing BEC only if the sequence $\mu_Łambda_n\uparrow0$ is carefully chosen such that $\rho(ømega)=\rho_c$. Other relevant examples are finite additive perturbations of periodic graphs, where the shape of the wave function of the condensate (coinciding with the Perron--Frobenious eigenvector for the adjacence) describes a condensate essentially localized in a finite portion of the perturbed graph under consideration, and we have again no locally normal states describing BEC. Finally, we easily exhibit locally normal states $ømega$ describing the BEC for $N$, which is transient but $\rho_c=+ınfty$, thus $\rho(ømega)=+ınfty$, and for the comb graph with backbone $N$ and fingers $N$--copies of $Z^2$, where locally normal states can be constructed with any $\rho(ømega)$ including $\rho(ømega)\geq\rho_c$.

@incollection{statphys23_0077, abstract = {We discuss the Bose--Einstein condensation (BEC for short) for the pure hopping model on some nonhomogeneous amenable graphs, and in particular for the comb graphs $G_{d}$ with backbone ${\mathbb Z}^{d}$ and fingers ${\mathbb Z}^{d}$--copies of ${\mathbb Z}$. It is shown that the BEC phenomena are connected with the transience of the graphs (relative to the adjacence operator), and not only with the finiteness of the critical density, the last properties being not equivalent for nonhomogeneous graphs considered here. The situation differs to that relative to ${\mathbb Z}^{d}$, and ${\mathbb Z}^{d}$--periodic lattices (Matsui) where the transience and the finiteness of the critical density are known to be equivalent, and determined by the dimension $d$ of the lattices. Namely, the comb graphs considered here, exhibit hidden spectrum near zero, thus the critical density is finite. But, they are transient iff $d\geq3$. By considering periodic boundary condition along the backbone for the finite--volume theories, it is proven that whenever $d\leq2$ and the sequence of chemical potential $\mu_{\Lambda_{n}}\uparrow0$, it is impossible to obtain any infinite--volume state describing BEC and having a reasonable local description (i.e. non locally normal states for which ${\displaystyle\lim_{\Lambda_{n}\uparrow G_{d}} \omega_{\Lambda_{n}}\big(a^{+}(\delta_{j})a(\delta_{j})\big)=+\infty}$). If $d\geq3$, a careful choice of the sequence $\mu_{\Lambda_{n}}\uparrow0$ should be done in order to obtain locally normal states describing BEC. For all these infinite volume states $\omega$, the density $$ \rho(\omega):=\lim_{\Lambda_{n}\uparrow G_{d}}\frac{1}{|\Lambda_{n}|} \sum_{j\in\Lambda_{n}}\omega\big(a^{+}(\delta_{j})a(\delta_{j})\big) $$ coincides with the critical density $\rho_{c}$. The situation behaves as follows. Below the critical density, the system describes (pure hopping) Bosons allocated on a $d+1$--dimensional system (i.e. the dimension described by the vertices of the comb $G_{d}$). Above the critical density, a considerable portion of particles condensates in momentum space as well as in configuration space, due to nonhomogeneity. Such a condensate is essentially localized along the backbone. The system undergoes a ``dimensional transition''. If $d\leq2$ we cannot exhibit any locally normal state describing BEC. If $d\geq3$, we can construct locally normal states $\omega$ describing BEC only if the sequence $\mu_{\Lambda_{n}}\uparrow0$ is carefully chosen such that $\rho(\omega)=\rho_{c}$. Other relevant examples are finite additive perturbations of periodic graphs, where the shape of the wave function of the condensate (coinciding with the Perron--Frobenious eigenvector for the adjacence) describes a condensate essentially localized in a finite portion of the perturbed graph under consideration, and we have again no locally normal states describing BEC. Finally, we easily exhibit locally normal states $\omega$ describing the BEC for ${\mathbb N}$, which is transient but $\rho_{c}=+\infty$, thus $\rho(\omega)=+\infty$, and for the comb graph with backbone ${\mathbb N}$ and fingers ${\mathbb N}$--copies of ${\mathbb Z}^{2}$, where locally normal states can be constructed with any $\rho(\omega)$ including $\rho(\omega)\geq\rho_{c}$.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Fidaleo, F.F.}, biburl = {https://www.bibsonomy.org/bibtex/2f3d83ead5436c36fd3a4a8468f67a74c/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {68946e5f3ee3c1c7f986c27112d37932}, intrahash = {f3d83ead5436c36fd3a4a8468f67a74c}, keywords = {algebras bose condensation einstein operator statphys23 topic-8}, month = {9-13 July}, timestamp = {2007-06-20T10:16:11.000+0200}, title = {Bose Einstein Condensation on Nonhomogeneous graphs}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=77}, year = 2007 }