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\author{Bernard Teissier}
\title{On Thom's interpretation of the Gibbs phase rule}
\maketitle

We give some details on Thom's interpretation of the Gibbs phase rule in the context of versal unfoldings of smooth functions with an algebraically isolated singularity. This hypothesis implies that we may assume, up to a diffeomorphism, that the function is a polynomial. We keep most notations of page 96 of the paper, and write a (mini-)versal $C^\infty$ unfolding as:
$$F_u=f+\sum_{i=1}^{\mu-1}u_ig_i,$$
where the functions $g_i$ are polynomials whose images, together with $1$, form a basis of the quotient algebra $${\mathbf R}\{x_1,\ldots,x_m\}/(f_{x_1},\ldots ,f_{x_m})$$ as a vector space over ${\mathbf R}$. We denote the number of internal variables by $m$ to avoid confusion with Thom's notation. The dimension $\nu$ of the parameter space of the versal unfolding is the Milnor number minus one, and not the Milnor number.\par
Thus we have a map germ ${\mathbf F}_u\colon ({\mathbf R}^m\times {\mathbf R}^{\mu-1},0)\longrightarrow ({\mathbf R}\times{\mathbf R}^{\mu-1},0)$ sending $(x,u)$ to $(F_u(x),u)$. We denote by $u_0$ a coordinate on ${\mathbf R}$.\par
The \textit{discriminant} $D_{{\mathbf F}_u}$ of the map ${\mathbf F}_u$ is the image of its critical locus $C_{{\mathbf F}_u}$. It is a semialgebraic hypersurface in ${\mathbf R}\times{\mathbf R}^{\mu-1}$. It is known (see \cite[\S 5]{Te2}) that $C_{F_u}$ is non singular and the map $C_{{\mathbf F}_u}\to D_{{\mathbf F}_u}$ is finite.\par
 The singular locus of $D_{{\mathbf F}_u}$ is if codimension one and its image $\Delta_{{\mathbf F}_u}$ in ${\mathbf R}^{\mu-1}$ by the second projection ${\mathbf R}\times{\mathbf R}^{\mu-1}\to {\mathbf R}^{\mu-1}$ is a semialgebraic hypersurface containing the bifurcation locus $\Sigma$ and the conflict strata. Indeed, a point $u$ is in ${\mathbf R}^{\mu-1}\setminus \Delta_{{\mathbf F}_u}$ if and only if the function $F_u$ is an excellent Morse function in a suitable neighborhood of $0$ in ${\mathbf R}^m$. In particular the Maxwell set, which corresponds to functions $F_u$ attaining at least twice their absolute minimum, is contained in $\Delta_{{\mathbf F}_u}$ because it is the image of a singular stratum of $D_{{\mathbf F}_u}$.\par\medskip
 
 The map ${\mathbf F}_u$ is a stable map-germ and as such can be Thom-stratified: there exist finite partitions of the source and target into semialgebraic non singular strata such that:
 \begin{itemize}
 \item The restriction of ${\mathbf F}_u$ to a stratum of the source is a submersion onto a stratum of the target;
 \item The Thom isotopy theorem is valid: the topological type of the germ of ${\mathbf F}_u$ at a point of a stratum of the source depends only on the stratum.
 \end{itemize}\par\medskip\noindent
 It is not clear in which cases there exists such a stratification which is minimal in some sense, since already minimal Whitney stratifications do not exist in general for semialgebraic sets (see \cite{BT}). However, the statement we are interested in concerns the largest possible dimension of the stratum of the origin in such a stratification.\par\noindent
Let us denote by $T_0$ the stratum in  ${\mathbf R}\times{\mathbf R}^{\mu-1}$ containing the origin.\par\noindent The discriminant hypersurface $D_{{\mathbf F}_u}$ has a very special geometry (see \cite[\S 5]{Te2}), in particular all limit directions of its tangent hyperplanes at smooth points tending to the origin are the hyperplane $u_0=0$. This implies that the images in ${\mathbf R}^{\mu-1}$ by the second projection ${\mathbf R}\times{\mathbf R}^{\mu-1}\to {\mathbf R}^{\mu-1}$ of the strata in ${\mathbf R}\times{\mathbf R}^{\mu-1}$ form a stratification and the map from $T_0$ to its image $T'_0\subset {\mathbf R}^{\mu-1}$ is an isomorphism. Let $S_0\subset{\mathbf R}^m\times {\mathbf R}^{\mu-1}$ be the stratum of the origin at the source; it is contained in the critical locus and so is a finite submersion onto its image $T'_0$ so that the maps $S_0\to T_0\to T'_0$ are algebraic diffeomorphisms. Thus, as a point $u$ moves in $T'_0$, the corresponding functions $F_u$ have a unique critical point at which their germs are all topologically equivalent. From the viewpoint of catastrophe theory the parameters corresponding to coordinates on $T'_0$ are not significant and the number $p_f$ of significant parameters is the minimum codimension of the stratum $T'_0\subset{\mathbf R}^{\mu-1}$ in a Thom stratification of ${\mathbf F}_u$.\par
On the other hand, the maximum number of stable attractors (for the gradient vector field), that is, the maximum number of non degenerate minima of a function $F_u$ which can appear represents the number of phases which can coexist in the sense that the system can make a transition from one to an other.\par\noindent Let us denote by $\phi_f$ the maximum number of minima which a function $F_u$ in a versal unfolding of $f$ can present (in a suitable neighborhood of $0$). They have to be non degenerate. \par\noindent A rather coarse form of the classical Gibbs phase rule states that the number $\phi$ of coexisting phases in a thermodynamical system is at most equal to the number of parameters of the system plus one: $\phi\leq p+1$. In our model, this translates as :
$$\phi_f\leq p_f+1={\rm codim}_{{\mathbf R}^{\mu-1}}T'_0+1={\rm codim}_{{\mathbf R}\times{\mathbf R}^{\mu-1}}T_0.$$
See also \cite{Th}. Some ideas for a proof can be found in \cite{Te1}, based on the real avatar of the product decomposition theorem for the discriminant of complex miniversal unfoldings of \cite[\S 5]{Te2}. Let $\delta_{\mathbf R}({\mathbf F}_u)$ denote the maximum number of singular points that can be found in a fiber of a function $F_u$, in a suitable neighborhood of $0$ in ${\mathbf R}^m$ and for $u$ in a small neighborhood of $0$ in ${\mathbf R}^{\mu-1}$. By the product decomposition theorem this corresponds to points in $D_{{\mathbf F}_u}$ where this hypersurface is locally the union of $\delta_{\mathbf R}({\mathbf F}_u)$ non singular hypersurfaces in general position. The locus of such points is of codimension  $\delta_{\mathbf R}({\mathbf F}_u)$ in ${\mathbf R}\times{\mathbf R}^{\mu-1}$ and since the topology of the situation is constant along $T_0$ and this configuration of non singular hypersurfaces is topologically  invariant on $D_{{\mathbf F}_u}$ (again a consequence of the product decomposition theorem), this locus must contain $T_0$ in its closure, so that $$\delta_{\mathbf R}({\mathbf F}_u)\leq {\rm codim}_{{\mathbf R}\times{\mathbf R}^{\mu-1}}T_0.$$
The next step is to prove that $\phi_f\leq \delta_{\mathbf R}({\mathbf F}_u)$. This should follow (the argument is not complete) from a Smale-type argument (\cite{GWPL}) asserting that since the $\phi_f$ minima of one of the $F_u$ have the same index they can be moved to the same level by a perturbation of that function small enough to be represented in ${\mathbf F}_u$.

\begin{thebibliography}{6}
\bibitem{BT} H. Brodersen and D. Trotman, Whitney b) regularity is weaker than Kuo's ratio test for real algebraic stratifications, \textit{Math. Scand.}, {\bf 45}, 1979, 27-34.

\bibitem{CG} J. Cerf et A. Gramain. Ñ Le th\'eor\`eme du h-cobordisme (Smale).
Cours profess\'e au printemps 1966 \`a la Facult\'e des Sciences dÕOrsay.
Secr\'etariat Math\'ematique de lÕ\'Ecole Normale Sup\'erieure, 1968.
Available at
https://www.maths.ed.ac.uk/~v1ranick/surgery/
cerf-gramain.pdf

\bibitem{GWPL} C.G. Gibson, K. Wirthm\"uller, A. du Plessis and E.J.N. Looijenga.
Ñ Topological stability of smooth mappings, Springer Lecture Notes in Mathematics 552 (1976).

\bibitem{Te1} B. Teissier,  Sur la version catastrophiste de la r\`egle des phases de Gibbs, \textit{
Rencontre de Carg\`ese sur les singularit\'es et leurs applications, Institut d'Etudes scientifiques de Carg\`ese, 1-6 Septembre 1975}.
Available at https://webusers.imj-prg.fr/~bernard.teissier/documents/Gibbs.pdf

\bibitem{Te2} B. Teissier, The hunting of invariants in the geometry of discriminants, in \textit{Real and complex singularities 
(Proc. Ninth Nordic Summer School NAVF Sympos. Math., Oslo, 1976)}, pp. 
565--678. Sijthoff and Noordhoff, Alphen aan den Rijn, 1977.
Available at https://webusers.imj-prg.fr/~bernard.teissier/documents/Thehunting.pdf

\bibitem{Th} R. Thom, Langage et Catastrophe, \'el\'ements pour une s\'emantique topologique, in \textit{Dynamical systems}, Academic Press, 1973.
\end{thebibliography}



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