# Thermo-dynamical limit function's seminar

Speaker: Yoshio Sano (National Institute of Informatics) Thermo-dynamical limit function's seminar Mon, Jun 20, 2011, 15:30 - 18:00 Balcony A So far, we introduced the topologically completed Hopf algebra $A[[Conf]]$, called the configuration algebra and the space $\mathcal{L}_A$ of all primitive elements in the Hopf algebra as a completion of the space generated by logarithmic growth functions $M(T)$ (free energies) for $T\in Conf_0$. Today, we introduced another topological basis $\{\varphi(S)\}_{S\in Conf_0}$ of $\mathcal{L}_A$, where the transformation matrix between the two basis is given by some combinatorially defined coefficients, called kabi-coefficients. The advantage of the new basis is that any element in $\mathcal{L}_A$ is uniquely developped as an infinite linear combination of them. We are now interested in the set $\Omega$ of all possible limits of sequences of free energy $M(T_n)/\#(T_n)$ where $T_n$ is suitably growing sequence of configurations in $Conf$ with $n\to \infty$ (where limit process is defined by the classical topology in $\mathcal{L}_R$ with $R=$ the real number field). Actually, the set $\Omega$ is a compact convex set in $\mathcal{L}_R$. Thus, we are interested in the extremal boundary points of the limit set. We'll give a formula where the limit function is developed in the new basis $\{\varphi(S)\}_{S\in Conf_0}$ whose coefficients are expressed by the proportions of some special value of certain generating functions. This gives a clue to calculate the limit functions in the next stage.