Identifications between different phase spaces

Question

I've discovered Adam's lecture notes on statistical mechanics after posting my first question about Minlo's discussion on continuous Gibbs measures. Adam's lecture notes are really good, but there is some little difference between his and Minlo's book that I'd like to clarify.

On page 27, Adams introduces two sets, as follows (with $\Lambda \subset \mathbb{R}^{d})$: $$\Gamma_{\Lambda, N} :=\{\omega \subset \Lambda \times \mathbb{R}^{d}, \omega =\{(q,p_{q}), q \in \hat{\omega}\}, |\hat{\omega}|=N\} \quad (1)$$ $$ \Gamma_{\Lambda} := \{\omega \subset \Lambda \times \mathbb{R}^{d}, \omega = \{(q,p_{q}), q \in \hat{\omega}\}, |\hat{\omega}|<+\infty\} \quad (2) $$ In addition, for each $\Delta \subset \Lambda$ Borel-measurable, define a counting variable $N_{\Delta}$ to be the function $N_{\Delta}:\Gamma_{\Lambda} \to \mathbb{R}$ by $N_{\Delta}(\omega) = |\omega \cap \Delta|$. The $\sigma$-algebra on $\Gamma_{\Lambda}$ generated by the family of counting variables $\{N_{\Delta}\}$ is denoted by $\mathcal{B}_{\Lambda}^{\infty}$.

Then, the notes procceeds by introducing the following definition.

Definition: Let $\Lambda \subset \mathbb{R}^{d}$, $\beta > 0$ and $\mu \in \mathbb{R}$. Define the phase space $\Gamma_{\Lambda}:=\bigcup_{N=0}^{\infty}\Gamma_{\Lambda, N}$, where $\Gamma_{\Lambda, N} := (\Lambda\times \mathbb{R}^{d})^{2N}$ is the phase space of exactly $N$ particles and equip it with $\mathcal{B}_{\Lambda}^{\infty}$. The probability mesure $\gamma_{\Lambda, \beta}$ on $(\Gamma_{\Lambda},\mathcal{B}_{\Lambda}^{\infty})$ such that the restrictions $\gamma_{\Lambda, \beta}|_{\Gamma_{\Lambda, N}}$ have densities: $$\rho_{\Lambda,\beta}^{(N)}(x) = Z_{\Lambda}(\beta, \mu)^{-1}e^{-\beta H_{\Lambda}^{(N)}(x)-\mu N}$$ where $H_{N}^{(N)}$ is the Hamiltonian for $N$ particles in $\Lambda$ is called grand canonical ensemble in $\Lambda$.

Well, Adams seems to be using two different notions of phase space at the same time. In the above definition, $\Gamma_{\Lambda} = \bigcup_{N=0}^{\infty}\Gamma_{\Lambda,N}$ but the $\sigma$-algebra $\mathcal{B}_{\Lambda}^{\infty}$ only makes sense in $\Gamma_{\Lambda}$ given by (2). Thus, it seems to me that he's using some identification between $\Gamma_{\Lambda,N}$ as given by (1) and $(\Lambda \times \mathbb{R}^{d})^{2N}$. But what is this idetification? Note that nothing prevents us to take an element in $(\Lambda \times \mathbb{R}^{d})^{2N}$ which has equal entries and this would led to a single point in $\Gamma_{\Lambda,N}$ as given by (1). What m I getting wrong here?

Answer 1

$\newcommand\La{\Lambda} \newcommand\Ga{\Gamma} \newcommand\om{\omega} \newcommand\R{\mathbb R}$ To me, all this looks quite terrible. In formula (4.17) on page 28 of the linked lecture notes (corresponding to your formula (1)), Adams "defines" the "phase space for exactly $N$ particles in box $\La\subset\R^d$" as $$\Ga_{\La,N} :=\{\om\subset\La\times \R^d\colon \om=\{(q,p_q), q\in\hat\om\}, |\hat\om|=N\},$$ "where $\hat\om$, the set of positions occupied by the particles, is a locally finite subset of $\La$, and $p_q$ is the momentum of the particle at positions $q$." Here, unfortunately, Adams mixes a mathematical definition with its informal, physical interpretation. In particular, $p_q$ is only "defined" as a "momentum". Also, "locally finite" does not seem to be defined anywhere in the notes, and why one needs here "locally" is at best unclear, because we have $|\hat\om|=N$, which already implies that $\hat\om$ is just finite (assuming that the number $N$ of "particles" is a natural number). You can see how many problems just this one little "definition" has.

With this only physically "defined" $p_q$, my best guess is that $\Ga_{\La,N}$ was meant to be the set of all functions of the form $\om\colon\hat\om\to\R^d$, where $\hat\om$ is any subset $\La$ with $|\hat\om|=N$. Any such function is (or, if you prefer, can be identified with) a set of the form $\{(q_1,p_1),\dots,(q_N,p_N)\}$, where $q_1,\dots,q_N$ are pairwise distinct points in $\La$ and $p_1,\dots,p_N$ are arbitrary points in $\R^d$. So, $\Ga_{\La,N}$ is the image of the subset of the set $(\La\times\R^d)^N$ consisting of all $N$-tuples $((q_1,p_1),\dots,(q_N,p_N))\in(\La\times\R^d)^N$ with pairwise distinct $q_1,\dots,q_N$ under the map that maps the $N$-tuples $((q_1,p_1),\dots,(q_N,p_N))$ to the corresponding sets $\{(q_1,p_1),\dots,(q_N,p_N)\}$. So, $\Ga_{\La,N}$ may be viewed as a set of dimension $2Nd$.

On the other other hand, in Definition 4.6 on page 28 of the same lecture notes, $\Ga_{\La,N}$ is defined as the plain product set $(\La\times\R^d)^{2N}$ of dimension $2d\times2N=4Nd$, which is twice the dimension of $\Ga_{\La,N}$ according to the previous definition. So, there seems to be no way to reconcile these two different definitions of $\Ga_{\La,N}$, and then to reconcile the corresponding two different definitions of $\Ga_\La$. And indeed, the definition of the $\sigma$-algebra $\mathcal B_\La^\infty$ is applicable only to the first definition of $\Ga_\La$.

---

By the way, I strongly suggest that you avoid using commas in place of colons in definitions of sets; e.g., avoid writing $A:=\{x\in X,x>0,x<1,x^2>1/2\}$ instead of $A:=\{x\in X\colon x>0,x<1,x^2>1/2\}$. The colon here stands for "such as" and hence plays a role quite different from that of the commas (which stand for "and"), and that should be reflected in the notation.