I feel that the process of using statistical ensembles to find properties of thermal system is not rigorous enough. There are some operations that need to be defined precisely. Also, it is not generalized enough. Currently, the only generally used statistical ensembles are the microcanonical ensemble, the canonical ensemble, and the grand canonical ensemble, but there are actually other possible ensembles that are potentially useful. Therefore, I feel it necessary to try to have a mathematical formulation.

## Mathematical tools and notations

Suppose $(\Omega,\sigma(\Omega),P)$ is a probability space. Suppose $W$ is an affine space. For some map $f:\Omega\to W$, we define the $P$-expectation of $f$ as $\mathrm E_P\!\left[f\right]\coloneqq\int_{x\in\Omega}\left(f(x)-e_0\right)\mathrm dP(x)+e_0,$ where $e_0\in W$ is arbitrary. Here the integral is Pettis integral. The expectation is defined if the Pettis integral is defined, and it is then well-defined in that it is independent of the $e_0$ we choose.

Suppose $X,Y$ are Polish spaces. Suppose $(Y,\sigma(Y),\mu),(X,\sigma(X),\nu)$ are measure spaces, where $\mu$ and $\nu$ are σ-finite Borel measures. Suppose $\pi:Y\to X$ is a measurable map so that $\forall A\in\sigma(X):\nu(A)=0\Rightarrow\mu\!\left(\pi^{-1}\!\left(A\right)\right)=0.$ Then, for each $x\in X$, there exists a Borel measure $\mu_x$ on the measurable subspace $\left(\pi^{-1}(x),\sigma\!\left(\pi^{-1}(x)\right)\right)$, such that for any integrable function $f$ on $Y$, $\int_{y\in Y}f\!\left(y\right)\mathrm d\mu(y) =\int_{x\in X}\mathrm d\nu(x)\int_{y\in\pi^{-1}(x)}f\!\left(y\right)\mathrm d\mu_x(y).$

Proof

Proof. Because $\mu$ is σ-finite, we have a countable covering of $Y$ by pairwise disjoint measurable sets of finite $\mu$-measure, denoted as $\left\{Y_i\right\}$. Each $Y_i$ inherits the σ-algebra from $Y$, and $\left(Y_i,\sigma\!\left(Y_i\right),\mu\right)$ is a measure space.

Define $\pi_i:Y_i\to X$ as the restriction of $\pi$ to $Y_i$, then $\pi_i$ is automatically a measurable map from $Y_i$ to $X$, and for any $x\in X$, $\pi^{-1}(x)=\bigcup_i\pi_i^{-1}(x),$ and the terms in the bigcup are pairwise disjoint.

Let $\nu_i$ be a measure on $X$ defined as $\nu_i(A)\coloneqq\mu\!\left(\pi_i^{-1}\!\left(A\right)\right).$ This is a measure because $\pi_i$ is a measurable map. According to the disintegration theorem, for each $x\in X$, there exists a Borel measure $\mu_{i,x}$ on $Y_i$ such that for $\nu$-almost all $x\in X$, $\mu_{i,x}$ is concentrated on $\pi_i^{-1}(x)$ (in other words, $\mu_{i,x}\!\left(Y\setminus\pi_i^{-1}(x)\right)=0$); and for any integrable function $f$ on $Y_i$, $\int_{y\in Y_i}f\!\left(y\right)\mathrm d\mu(y) =\int_{x\in X}\mathrm d\nu_i(x)\int_{y\in\pi_i^{-1}(x)}f\!\left(y\right)\mathrm d\mu_{i,x}(y).$ From the condition in the original proposition, we can easily prove that $\nu_i$ is absolutely continuous w.r.t. $\nu$. Therefore, we have their Radon–Nikodym derivative $\varphi_i(x)\coloneqq\frac{\mathrm d\nu_i(x)}{\mathrm d\nu(x)}.$

For each $x\in X$, define the measure $\mu_x$ on $\pi^{-1}(x)$ as $\mu_x(A)\coloneqq\sum_i\varphi_i\!\left(x\right)\mu_{i,x}\!\left(A\cap Y_i\right).$ This is a well-defined measure because the sets $A\cap Y_i$ are pairwise disjoint, and $\mu_{i,x}$ is well-defined measure on $Y_i$.

Then, for any integrable function $f$ on $Y$, \begin{align*} \int_{y\in Y}f\!\left(y\right)\mathrm d\mu(y) &=\sum_i\int_{y\in Y_i}f\!\left(y\right)\mathrm d\mu(y)\\ &=\sum_i\int_{x\in X}\mathrm d\nu_i(x)\int_{y\in\pi_i^{-1}(x)}f\!\left(y\right)\mathrm d\mu_{i,x}(y)\\ &=\sum_i\int_{x\in X}\varphi_i\!\left(x\right)\mathrm d\nu(x) \int_{y\in\pi_i^{-1}(x)}f\!\left(y\right)\mathrm d\mu_{i,x}(y)\\ &=\int_{x\in X}\mathrm d\nu(x)\sum_i\int_{y\in\pi_i^{-1}(x)}f\!\left(y\right)\mathrm d\mu_x(y)\\ &=\int_{x\in X}\mathrm d\nu(x)\int_{y\in\pi^{-1}(x)}f\!\left(y\right)\mathrm d\mu_x(y). \end{align*} $\square$

Here, the family of measures $\left\{\mu_x\right\}$ is called the disintegration of $\mu$ w.r.t. $\pi$ and $\nu$.

For two vector spaces $\vec W_1,\vec W_2$, we denote $\vec W_1\times\vec W_2$ as the direct sum of them. Also, rather than calling the new vector space their direct sum, I prefer to call it the product vector space of them (not to be confused with the tensor product) so that it is consistent with the notion of product affine spaces, product measure spaces, product topology, etc. Those product spaces are all notated by “$\times$” in this article.

Also, “$\vec W_1$” can be an abbreviation of $\vec W_1\times\left\{0_2\right\}$, where $0_2$ is the zero vector in $\vec W_2$.

Suppose $W$ is an affine space associated with the vector space $\vec W$. For any $A\subseteq W$ and $B\subseteq\vec W$, we denote $A+B$ as the Minkowski sum of $A$ and $B$, i.e., $A+B\coloneqq\left\{a+b\,\middle|\,a\in A,\,b\in B\right\}.$ This extends the definition of usual Minkowski sums for affine spaces.

By the way, because of the abbreviating “$\vec W_1$” meaning $\vec W_1\times\left\{0_2\right\}$ above, we can abuse the notation and write $\vec W_1+\vec W_2=\vec W_1\times\vec W_2,$ where “$+$” denotes the Minkowski sum. This is true for any two vector spaces $\vec W_1,\vec W_2$ that do not share a non-trivial vector subspace.

In general, it is not necessarily possible to decompose a topology as a product of two topologies. However, it is always possible for locally convex Hausdorff TVSs. We can always decompose the topology of a locally convex Hausdorff TVS as the product of the topologies on a pair of its complementary vector subspaces, one of which is finite-dimensional. This is true because every finite-dimensional subspace in such a space is topologically complemented. The complete statement is the following:

Let $\vec W$ be a locally convex Hausdorff TVS. For any finite-dimensional subspace $\vec W^\parallel$ of $\vec W$, there is a complement $\vec W^\perp$ of it such that the topology $\tau\!\left(\vec W\right)$ is the product topology of $\tau\!\left(\vec W^\parallel\right)$ and $\tau\!\left(\vec W^\perp\right)$.

This decomposition is also valid for affine spaces. If an affine space $W$ is associated with a locally convex Hausdorff TVS $\vec W$, then for any finite-dimensional vector subspace $\vec W^\parallel$ of $\vec W$, we can topologically decompose $W$ into $W^\perp+\vec W^\parallel$.

Because the product topology of subspace topologies is the same as the subspace topology of the product topology, we can also decompose $E^\perp+\vec W^\parallel$ as the product topological space of $E^\perp$ and $\vec W^\parallel$ if $E^\perp\subseteq W^\perp$.

Such decompositions are useful because they allow us to disintegrate Borel measures. If we already have a σ-finite Borel measure on $E^\perp+\vec W^\parallel$ and we can define a σ-finite Borel measure on $\vec W^\parallel$, then we can define a measure on $E^\perp$ by the disintegrating, and we guarantees that the disintegration is also σ-finite and Borel.

When I want to use multi-index notations, I will use “$\bullet$” to denote the indices. For example, $\Sigma\alpha_\bullet\coloneqq\sum_\bullet\alpha_\bullet.$ $\alpha_\bullet\beta_\bullet\coloneqq\sum_\bullet\alpha_\bullet\beta_\bullet.$ $\alpha_\bullet^{\beta_\bullet}\coloneqq\prod_\bullet\alpha_\bullet^{\beta_\bullet}.$ $\alpha_\bullet!\coloneqq\prod_\bullet\alpha_\bullet!.$

## Extensive quantities and macrostates

First, I need to point out that the most central state function of a thermal system is not its energy, but its entropy. The energy is regarded as the central state function in thermodynamics, which can be seen from the fundamental equation of thermodynamics $\mathrm dU=-p\,\mathrm dV+T\,\mathrm dS+\mu\,\mathrm dN.$ We also always do the Legendre transformations on the potential function $U$ to get other potential functions instead of doing the transformation on other extensive quantities. All such practices make us think that $S$ is just some quantity that is similar to $V$ and $N$, and mathematically we can just regard it as an extensive quantity whose changing is a way of doing work.

However, this is not the case. The entropy $S$ is different from $U,V,N$ in the following sense:

• The entropy is a derived quantity due to a mathematical construction from the second law of thermodynamics, while $U,V,N$ are observable quantities that have solid physical meanings before we introduce anything about thermodynamics.
• The entropy may change in an isolated system, while $U,V,N$ do not.
• We may have an intuitive understanding of how different systems in contact may exchange $U,V,N$ with each other, but $S$ cannot be “exchanged” in such a sense.
• In statistical mechanics, $U,V,N$ restrict what microstates are possible for a thermal system, but $S$ serves as a totally different role: it represents something about the probability distribution over all the possible microstates.

Therefore, I would rather rewrite the fundamental equation of thermodynamics as $\mathrm dS=\frac1T\,\mathrm dU+\frac pT\,\mathrm dV-\frac\mu T\,\mathrm dN.$$\p{1}$ Equation 1 embodies how different quantities serve different roles more clearly, but it becomes vague in its own physical meaning. Does it mean different ways of changing the entropy in quasi-static processes? Both mathematically and physically, yes, but it is not a useful interpretation. Because what we are doing is mathematical formulation of physical theories, we do not need to try to assign physical meanings to anything we construct. This new equation is purely mathematical, and the only way we use it is to relate intensive variables to derivatives of $S$ w.r.t. extensive quantities.

From now on, I will call quantities like $U,V,N$ the extensive quantities, not including $S$. However, this is not a good statement as part of our mathematical formulation. Considering that there is a good notion of how different systems may exchange values of extensive quantities and that we can scale a system by multiplying the extensive quantities by a factor, we require that the extensive quantities must support at least linear operations… do we?

Well, actually we will see that if we require a space to be a vector space, things would be a little bit complex because sometimes we need to construct a new space of extensive quantities out of the affine subspace of an existing one, which is not a vector space by nature. If we require the space to be a vector space, we need to translate that affine subspace to make it pass through the zero element of the vector space, which is possible but does not give any insight about the physics except adding complication to our construction. Therefore, I will not require the space of extensive quantities to be a vector space, but be an affine space.

You may ask, OK then, but how do we “add” or “scale” extensive quantities if they live on an affine space? First, regarding the addition operation, we will use an abstraction for such operations so that the actual implementation about how we combine the summands is hidden under this abstraction. We will see that this abstraction is useful because it also applies to other senarios or useful operations that does not necessarily involve any meaningful addition. Regarding the scaling operation, I would argue that now we do not need them. I have generalized the notion of extensive quantities so that now the notion “extensive quantities” includes some quantities that are not really extensive quantities in any traditional sense. They are no longer meant to be scaled because they simply cannot. Actually, rather than calling them extensive quantities, I would like to call them a macrostate, with the only difference from the general notion macrostate being that it has an affine structure so that I can take the ensemble average of it to get its macroscopic value. I would stick to the term “extensive quantities” because they are actual extensive quantities in all my examples and because it is a good way to understand its physical meaning with this name, but you need to keep in mind that what I actually refer to is a macrostate.

There is another difficulty. If we look closely, Equation 1 actually does not make much sense in that $N$ is quantized (and also $U$ if we are doing quantum). If we are doing real numbers, we can always translate a quantized quantity to something that is not allowed, which means that we cannot have the full set of operations on the allowed values of the extensive quantities. Therefore, we need to specify a subset on the affine space to represent the allowed values of the extensive quantities.

We also see that Equation 1 is a relation between differentials. Do we need to require that we have differential structure on the space of extensive quantities? Not yet, because it actually is somehow difficult. The same difficulty about the quantized quantities applies. The clever way is to just avoid using the differentials. (Mathematicians are always skeptical about differentiating something while physicists just assume everything is differentiable…) It may seem surprising, but actually differentials are evitable in our mathematical formulation if you do not require intensive variables to be well-defined inside the system itself (actually, they are indeed not well-defined except when you have a system in thermal equilibrium and take the thermaldynamic limit).

If we have to use differentials, we can use the Gateaux derivative. It is general enough to be defined on any locally convex TVS, and it is intuitive when it is linear and continuous.

Although differential structure is not necessary, there is an inevitable structure on the space of extensive quantities. Remember that in canonical and grand canonical ensembles, we allow $U$ or $N$ to fluctuate, so we should be able to describe such fluctuations on our space of extensive quantities. To do this, I think it is safe to assume that we can have some topology on the allowed subset to make it a Polish space, just like how probabilists often assume about the probability space they are working on.

A final point. Here is a difference in how physicists and mathematicians describe probability distributions: physicists would use a probability density function while mathematicians would use a probability measure. Mathematically, to have a probability density function, we need to have an underlying measure on our space for a notion of “volume” on the space, and then we can define the probability density function as the Radon–Nikodym derivative of the probability measure w.r.t. the underlying volume measure. Also, for the Radon–Nikodym derivative to exist, the probability measure must be absolutely continuous w.r.t. the volume measure, which means that we have to sacrifice all the probability distributions that are not absolutely continuous to take the probability density function approach. Then, it seems that if we use the probability density function approach, we are introducing an excess measure structure on the space of extensive quantities and losing some possibilities and generalizabilities, but it would turn out that the extra structure is useful. Therefore, I will use the probability density function approach.

Here is our final definition of the space of extensive quantities:

Definition. A space of extensive quantities is a tuple $(W,E,\lambda)$, where

• $W$ is an affine space associated with a reflexive vector space $\vec W$ over $\mathbb R$, and it is equipped with topology $\tau(W)$ that is naturally constructed from the topology $\tau\!\left(\vec W\right)$ on $\vec W$;
• $E\subseteq V$ is a topological subspace of $W$, and its topology $\tau(E)$ makes $E$ a Polish space; and
• $\lambda:\sigma(E)\to[0,+\infty]$ is a non-trivial σ-finite Borel measure, where $\sigma(E)\supseteq\mathfrak B(E)$ is a σ-algebra on $E$ that contains the Borel σ-algebra on $E$.

Here, I also added a requirement of σ-finiteness. This is necessary when constructing product measures. At first I also wanted to require that $\lambda$ has some translational invariance, but I then realized that it is not necessary, so I removed it from the definition (but we will see that we need them as a property of baths).

Example. Here is an example of a space of extensive quantities. \begin{align*} W&\coloneqq\mathbb R^3,\\ E&\coloneqq(0,+\infty)\times(0,+\infty)\times\mathbb Z^+,\\ \lambda(A)&\coloneqq\sum_{N\in\mathbb Z^+}\operatorname{area}(A\cap(0,+\infty)\times(0,+\infty)\times\{N\}). \end{align*} Physically we may think of this as the extensive quantities of the system of ideal gas. The three dimensions of $W$ are energy, volume, and number of particles.

Example. Here is another example of a space of extensive quantities. \begin{align*} W&\coloneqq\mathbb R^2,\\ E&\coloneqq\{(3N/2+n,N)\,|\,N\in\mathbb Z^+,n\in\mathbb N\},\\ \lambda(A)&\coloneqq\operatorname{card}A. \end{align*} Physically we may think of this as the extensive quantities of the system of Einstein solid with $\hbar\omega=1$. The two dimensions of $W$ are energy and number of particles.

## Thermal systems and the number of microstates

Remember I said above that, in statistical mechanics, $U,V,N$ restrict what microstates are possible for a thermal system. We can translate this as such: for each possible values of extensive quantities, denoted as $e\in E$, here is a set of possible microstates, denoted as $M_e$ (you can then see why we excluded the entropy from the extensive quantities: otherwise we cannot do such a classification of microstates).

Now the problem is what structures we should add to $M_e$ for each $e\in E$. Recall that in statistical mechanics, we study probability distribution over all possible microstates. Therefore, we need to be able to have a probability measure on $M_e$. In other words, $M_e$ should be a measurable space. As said before, we can either use a probability measure directly, or use a volume measure together with a probability density function. This time, we seem to have no choice but the probability density function approach because there is a natural notion of volume on $M_e$: the number of microstates.

Wait! There is a problem. Recall that in microcanonical ensemble, we allow the energy to fluctuate. The number of microstates at exactly a certain energy is actually zero in most cases, so we are actually considering those microstates with some certain small range of energy. In other words, we are considering the microstate density: the number of microstates inside unit range of energy. Similarly, we should define a measure on $M_e$ to represent the microstate density, which is the number of microstates inside unit volume of extensive quantities, where the “volume” is measured by the measure $\lambda$ in the space of the extensive quantities.

This makes our formulation a little bit different from the microcanonical ensemble: our formulation would allow all extensive quantities to fluctuate while the microcanonical ensemble would only allow the energy to fluctuate. This is inevitable because we are treating extensive quantities like energy, volume, and number of particles as the same kind of quantity. It is not preferable to separate a subspace out from our affine space $W$ to say “these are the quantities that may fluctuate, and those are not.” Therefore, we need to justify why we may allow all extensive quantities to fluctuate. The justification is: mathematically, we are actually not allowing any extensive quantities to fluctuate. There is no actual fluctuation, and we are directly considering the microstate density without involving any change in the extensive quantities. In other words, using the language of microcanonical ensemble, we are considering the area of the surface of the energy shell instead of the volume of the energy shell with a small thickness.

Another important point is that we must make sure that specifying all the extensive quantities should be enough to restrict the system to finite number of microstates. In other words, the total microstate density should be finite for any possible $e\in E$. Also, there should be at least some possible microstates in $M_e$, so the total microstate density should not be zero.

We may then sum up the above discussion to give $M_e$ enough structure to make it the set of microstates of a thermal system with the given extensive quantities $e$. Then, the disjoint union of all of them (the family of measure spaces) is the thermal system.

Definition. A thermal system is a pair $\left(\mathcal E,\mathcal M\right)$, where

• $\mathcal E\coloneqq\left(W,E,\lambda\right)$ is a space of extensive quantities;
• $\mathcal M\coloneqq\bigsqcup_{e\in E}M_e$ is a family of measure spaces; and
• For each $e\in E$, $M_e$ is a measure space equipped with a measure $\mu_e$ such that $\mu_e\!\left(M_e\right)$ is finite and nonzero.

From now on, I will use a pair $(e,m)\in\mathcal M$ to specify a single microstate, where $e\in E$ and $m\in M_e$.

Example. For the thermal system of a solid consisting of spin-$\frac12$ particles, where each particle has two possible states with energy $0$ and $1$, we can construct \begin{align*} W&\coloneqq\mathbb R^2,\\ E&\coloneqq\left\{\left(U,N\right)\in\mathbb N\times\mathbb Z^+\,\middle|\,U\le N\right\},\\ \lambda(A)&\coloneqq\operatorname{card}A,\\ M_{U,N}&\coloneqq\left\{n\in\left\{0,1\right\}^N\,\middle|\,\sum_in_i=U\right\},\\ \mu_{U,N}(A)&\coloneqq\operatorname{card}A. \end{align*} This should be the simplest example of a thermal system.

Example. We may complete the example of the system of ideal gas. Suppose we are considering the system of ideal atomic gas inside a cubic box. The construction of the space of extensive quantities is the same as before. Denote possible values of extensive quantities in coordinates $e=(U,V,N)$. Now the measure spaces $M_e$ may be constructed as such: \begin{align*} M_{U,V,N}&\coloneqq\left\{\left(\ldots\right)\in \left(\left[0,\sqrt[3]V\right]^3\times\mathbb R^3\right)^N \,\middle|\,\text{lexicographic order, }\sum_i\frac{\left|\mathbf p_i\right|^2}{2m}=U\right\},\\ \mu_{U,V,N}(A)&\coloneqq\frac{H^{6N-1}(A)}{h^{3N}}. \end{align*} The “lexicographic order” here means that only those configurations where particle indices coincides with the lexicographic order are included in $M_e$. This is because the particles are indistinguishable, and the order of particles is irrelevant. The lexicographic order restriction is the same as using the quotient of the $N$-fold Cartesian product by permutation actions, but then defining $\mu_e$ would be difficult. Alternatively, we may still make them ordered, but divide the result by $N!$ in the definition of $\mu_e$, but this way is less clear in its physical meaning.

Here $H^d$ is the $d$ dimensional Hausdorff measure. To understand, the expression $H^{6N-1}(A)$ is just the $(6N-1)$-dimensional “volume” of $A$.

Since we have microstate density, why do not we have the true number of microstates? We can define a measure on $\mathcal M$ to represent the number of microstates.

Definition. The measure of number of microstates is a measure $\mu:\sigma(\mathcal M)\to\left[0,+\infty\right]$, where $\sigma(\mathcal M)\coloneqq\left\{\bigsqcup_{e\in A}B_e\,\middle|\,A\in\sigma(E),\,B_e\in\sigma(M_e)\right\},$ and the measure is defined by $\mu(A)\coloneqq\iint\limits_{(e,m)\in A}\mathrm d\mu_e(m)\,\mathrm d\lambda(e).$ The uniqueness of $\mu$ is guaranteed by the σ-finiteness of $\lambda$ and $\mu_e$. The expression $\mu(A)$ is called the number of microstates in $A$.

## States and the entropy

Here is a central idea in statistical ensembles: a state is a probability distribution on the microstates of a thermal system. It is among the ideas upon which the whole theory of statistical ensembles is built. I will take this idea, too.

As said before, I have taken the probability density approach of defining a probability distribution. Therefore, a state is just a probability density function.

Definition. A state of a thermal system $(\mathcal E,\mathcal M)$ is a function $p:\mathcal M\to\left[0,+\infty\right]$ such that $(\mathcal M,\sigma(\mathcal M),P)$ is a probability space, where $P:\sigma(\mathcal M)\to\left[0,1\right]$ is defined by $P(A)\coloneqq\int_Ap\,\mathrm d\mu.$$\p{2}$ Two states are the same if they are equal $\mu$-almost everywhere.

A probability space is just a measure space with a normalized measure, and here the physical meaning of $p$ is the probability density on $\mathcal M$, and $P(A)$ is the probability of finding a microstate in $A$.

Note that a state is not necessarily an equilibrium state (thermal state). We will introduce the concept of equilibrium states later.

Now we may introduce the concept of entropy.

I need to clarify that the entropy that we are talking about here is just the entropy in statistical mechanics. The reason I add this clarification is that we may also formally define an entropy in the language of measure theory, which is defined for any probability space and does not depend on any so-called probability density function or a “volume” measure (which is the number of microstates in our case). The definition of this entropy is (if anyone is interested) $S^{\mathrm{info}}\coloneqq\sup_\Pi\sum_{A\in\Pi}-P(A)\ln P(A),$ where $P$ is the probability measure on the probability space, and the supremum is taken over all $P$-almost partition $\Pi$ of the probability space ($\Pi$ is a subset of the σ-algebra so that $P(\bigcup_{A\in\Pi}A)=1$ and $P(A\cap B)=0$ for $A,B\in\Pi$). This definition looks intuitive and nice, and not surprisingly it is… not consistent with the entropy in statistical mechanics. The discrepancy happens when we are doing classical statistical mechanics because the entropy defined above will diverge to infinity for those “continuous” probability distributions. A quick check is that the entropy of the uniform distribution over $[0,1]$ is $+\infty$.

Definition. The entropy of a state $p$ is defined by $S[p]\coloneqq\int_\mathcal M-p\ln p\,\mathrm d\mu.$

Different from extensive quantities, the entropy is a functional of $p$. The entropy here is consistent with the entropy in thermodynamics or statistical mechanics.

This definition of entropy is called the Gibbs entropy formula. It agrees with the entropy defined in thermodynamics, but we are unable to show that at this stage because we have not defined temperature or heat yet.

Note that the base of the logarithm is not important, and it is just a matter of unit system. In SI units, the base would be $\exp k_\mathrm B^{-1}$, where $k_\mathrm B$ is the Boltzmann constant.

Physically, the extensive quantities may be measured macroscopically. The actual values that we get when we measure them are postulated to be the ensemble average. Therefore, for a given state $p$, we can define the measured values of extensive quantities by taking the $P$-expectation of the extensive quantities.

Definition. For a thermal system $(\mathcal E,\mathcal M)$ and a state $p$ of it, the measured value of extensive quantities of the state $p$ is the $P$-expectation of the $E$-valued random variable $(e,m)\mapsto e$. Explicitly, the definition is $\varepsilon[p]\coloneqq\mathrm E_P\!\left[\left(e,m\right)\mapsto e\right],$ where the probability measure $P$ on $\mathcal M$ is defined in Equation 2.

In the definition, it involves taking the $P$-expectation of a $W$-valued function. This involves doing a Pettis integral, which I claim to exist. It exists because the map $(e,m)\mapsto e-e_0$ must be weakly $P$-measurable, and such a function must be Pettis-integrable on a reflexive space.

Note that $\varepsilon[p]\in W$, and it is not necessarily in $E$.

The usage of the measured value of extensive quantities is that we can use it to get the fundamental equation of a thermal system, which describes the relationship between the extensive quantities and the entropy at any equilibrium state. Suppose that we postulate a family of states $p_t^\circ$ of the thermal system (or its slices, which will be introduced below), labeled by different $t$’s, and call them the possible equilibrium states. Then, we can have the following two equations: $\begin{cases} S^\circ=S\!\left[p_t^\circ\right],\\ \varepsilon^\circ=\varepsilon\!\left[p_t^\circ\right]. \end{cases}$$\p{3}$ By cancelling out the $t$ in the two equations (which may be impossible but assumed to be possible), we can get the fundamental equation in this form: $S^\circ=S^\circ\!\left(\varepsilon^\circ\right).$$\p{4}$ Then, here we get the function $S^\circ:E^\circ\to\mathbb R$, where $E^\circ$ is a subset of $W$ consisting of all possible measured values of extensive quantities among equilibrium states. If we can possibly define some differential structure on $E^\circ$ so that we can possibly take the differential of $S^\circ$ and write something sensible like $\mathrm dS^\circ=i\!\left(\varepsilon^\circ\right)(\mathrm d\varepsilon^\circ),$ where $i^\circ\!\left(\varepsilon^\circ\right)\in\vec W'$ is a continuous linear functional, then we can define $i^\circ\!\left(\varepsilon^\circ\right)$ to be the intensive quantities at $\varepsilon^\circ$. A proper comparison with differential geometry is that we may analogly call $i^\circ$ be a covector field on $E^\circ$ defined as the differential of the scalar field $S^\circ$.

However, as I have said before, I did not postulate there to be any differential structure on $E^\circ$, so the intensive quantities should not be generally defined in this way.

## Slicing

A good notion about thermal systems is that we can get new thermal systems from existing ones (although they are physically essentially the same system, they have different mathematical structure and contain different amount of information about them). There are two ways of constructing new thermal systems from existing ones:

• By fixing some extensive quantities. I call this way slicing.
• By allowing some extensive quantities to change freely. I call this way contracting.

I chose the words “slicing” and “contracting”. They are not present in actual physics textbooks, but I found the notion of them necesesary.

Slicing fixes extensive quantities. How we do it is to pick out a subset of $E$ and make it our new accessible values of extensive quantities. I find a special way of picking out such a subset is especially useful: picking it from an affine subspace of $W$. In this way, we can use a smaller affine space as the underlying space of our new thermal system. Then we see why I chose the word “slicing”: we are slicing the original affine space into parallel pieces, and picking one piece as our new affine space, and picking the corresponding accessible values of extensive quantities and possible microstates within that piece to form our new thermal system.

Definition. A slicing of a space of extensive quantities $\left(W,E,\lambda\right)$ is a pair $\left(W^\parallel,\lambda^\parallel\right)$, where

• $W^\parallel\subseteq W$ is an affine subspace of $W$;
• $E^\parallel\coloneqq E\cap W^\parallel$ is non-empty, and it is Polish as a topological subspace of $E$; and
• $\lambda^\parallel:\sigma\!\left(E^\parallel\right)\to\left[0,+\infty\right)$ is a non-trivial σ-finite Borel measure on $E^\parallel$, where $\sigma\!\left(E^\parallel\right)\subseteq\mathfrak B\!\left(E^\parallel\right)$ is a σ-algebra on $E^\parallel$ that contains the Borel σ-algebra on $E^\parallel$.

This constructs a new space of extensive quantities $\left(W^\parallel,E^\parallel,\lambda^\parallel\right)$, called a slice of the original space of extensive quantities $\left(W,E,\lambda\right)$.

Definition. A slice of a thermal system $\left(\mathcal E,\mathcal M\right)$ defined by the slicing $\left(W^\parallel,\lambda^\parallel\right)$ of $\mathcal E$ is a new thermal system $\left(\mathcal E^\parallel,\mathcal M^\parallel\right)$ constructed as such:

• $\mathcal E^\parallel\coloneqq\left(W^\parallel,E^\parallel,\lambda^\parallel\right)$ is the slice of $\mathcal E$ corrsponding to the given slicing; and
• $\mathcal M^\parallel\coloneqq\bigsqcup_{e\in E^\parallel}M_e$.

The idea behind slicing is to make some extensive quantities become extrinsic parameters and not part of the system itself. It would physically mean fixing some extensive quantities. However, here is a problem: if we fix some extensive quantities, the dimension (“dimension” as in “dimensional analysis”) of the volume element in the space of extensive quantities would be changed. In other words, the dimension of $\lambda$ does not agree with $\lambda^\parallel$. This is physically not desirable because we want to keep the number of microstates dimensionless so that its logarithm does not depend on the units we use. However, this is not a problem because here is an argument: in any physical construction of a thermal system, it is fine to have dimensionful number of microstates, the cost is that the model must not be valid under low temperature; in mathematical construction, dimension is never a thing, so we do not even need to worry about it. In low temperature, we must use quantum statistical mechanics, where all quantities are quantized so that the number of microstates is literally the number of microstates, which must be dimensionless. In high temperature, we do not need the third law of thermodynamics, which is the only law that restricts how we should choose the zero (ground level) of the entropy, and in this case we may freely change our units because it only affects the entropy by an additive constant.

Example. In the example of a system of ideal gas, we may slice the space of extensive quantities to the slice $V=1$ to fix the volume.

## Isolations and the microcanonical ensemble

Here is a special type of slicing. Because a single point is an (zero-dimensional) affine subspace, it may form a slicing. Such a slicing fixes all of the extensive quantities. We may call it an isolating.

A thermal system with a zero-dimensional space of extensive quantities is called an isolated system. The physical meaning of such a system is that it is isolated from the outside so that it cannot exchange any extensive quantities with the outside. We may construct an isolated system out of an existing thermal system by the process of isolating.

Definition. An isolating (at $e^\circ$) of a space of extensive quantities $\left(W,E,\lambda\right)$ is a slicing $\left(W^\parallel,\lambda^\parallel\right)$ of it, constructed as \begin{align*} W^\parallel&\coloneqq\left\{e^\circ\right\},\\ \lambda^\parallel(A)&\coloneqq\begin{cases}1,&A=\left\{e^\circ\right\},\\0,&A=\varnothing,\end{cases} \end{align*} where $e^\circ\in E$.

Definition. An isolated system is a thermal system whose underlying affine space of its space of extensive quantities is a single-element set.

Definition. An isolation (at $e^\circ$) of a thermal system $\left(\mathcal E,\mathcal M\right)$ is the slice of it corresponding to the isolating at $e^\circ$ of $\mathcal E$. An isolation is an isolated system.

Here is an obvious property of isolated systems: the measured value of extensive quantities of any state of an isolated system is $e^\circ$, the only possible value of the extensive quantities.

After introducing isolated systems, we can now introduce the equal a priori probability postulate. Although we may alternatively use other set of axioms to develop the theory of statistical ensembles, using the equal a priori probability postulate is a simple and traditional way to do it. Most importantly, this is a way that does not require us to define concepts like the temperature beforehand, which is a good thing for a mathematical formulation because it would require less mathematical structures or objects that are hard to well define at this stage.

Axiom (the equal a priori probability postulate). The equilibrium state of an isolated system is the uniform distribution.

Actually, instead of saying that this is an axiom, we may say that formally this is a definition of equilibrium states. However, I still prefer to call it an axiom because it only defines the equilibrium state of isolated systems rather than any thermal systems.

The equilibrium state of an isolated system $\left(\mathcal E,\mathcal M\right)$ may be written mathematically as $p^\circ\!\left(\cdot\right)\coloneqq\frac1{\mu\!\left(\mathcal M\right)}$ (the circle in the superscript denotes equilibrium state). After writing this out, we have successfully derived the microcanonical ensemble. We can then calculate the entropy of the state, which is $S^\circ\coloneqq S\!\left[p^\circ\right]=\ln\mu(\mathcal M).$$\p{5}$ Mentioning the entropy, a notable feature about the equilibrium state of an isolated system is that it is the state of the system that has the maximum entropy, and any state different from it has a lower entropy.

Theorem. For an isolated system, for any state $p$ of it, $S[p]\le S^\circ,$ where $S^\circ$ is the entropy of the equilibrium state of it. The equality holds iff $p$ is the same state as the equilibrium state.

Proof

Proof. Define a probability measure $P^\circ$ on $\mathcal M$ by $P^\circ(A)\coloneqq\frac{\mu(A)}{\mu(\mathcal M)},$ then $\left(\mathcal M,\sigma\!\left(\mathcal M\right),P^\circ\right)$ is a probability space. Any state $p$, as a function on $\mathcal M$, can be regarded as a random variable in the probability space $\left(\mathcal M,\sigma\!\left(\mathcal M\right),P^\circ\right)$.

Define the real function $\varphi(x)\coloneqq\begin{cases} x\ln x,&x\in\left(0,+\infty\right),\\ 0,&x=0. \end{cases}$ It is a convex function, so according to the probabilistic form of Jensen’s inequality, $\varphi\!\left(\mathrm E_{P^\circ}\!\left[p\right]\right) \le\mathrm E_{P^\circ}\!\left[\varphi\circ p\right].$ In other words, $\frac1{\mu(\mathcal M)}\ln\frac1{\mu(\mathcal M)} \le\int_{m\in\mathcal M}p\!\left(m\right)\ln p\!\left(m\right) \,\frac{\mathrm d\mu\!\left(m\right)}{\mu(\mathcal M)}.$ Then, it follows immediately that $S[p]\le S^\circ$. The equality holds iff $\varphi$ is linear on a convex set $A\subseteq\left[0,+\infty\right)$ such that the value of the random variable $p$ is $P^\circ$-almost surely in $A$. However, because $\varphi$ non-linear on any set with more than two points, the only possibility is that the value of $p$ is $P^\circ$-almost surely a constant, which means that the probability distribution defined by the probability density function $p$ is equal to the uniform distribution $\mu$-almost everywhere. Therefore, the equality holds iff $p$ is the same state as the equilibrium state. $\square$

This theorem is the well-known relation between the entropy and the equilibrium state: for an isolated system, the equilibrium state is the state with the maximum entropy.

By Equation 5, we can now derive the relationship between the entropy and the extensive quantities at equilibrium states by the process of isolating. Define a family of states $\left\{p^\circ_e\right\}_{e\in E}$, where each state $p^\circ_e$ is the equilibrium state of the system isolated at $e$. Then, we have the fundamental equation $S^\circ(e)=\ln\Omega(e),$$\p{6}$ where $\Omega(e)\coloneqq\mu_e\!\left(M_e\right)$ is called the counting function (I invented the phrase), which is the microscopic characteristic function of microcanonical ensembles. This defines a function $S^\circ:E\to\mathbb R$, which may be used to give a fundamental equation in the form of Equation 4, and it is the macroscopic characteristic function of microcanonical ensembles.

We will encounter microscopic or macroscopic characteristic functions for other ensembles later.

Example. In the example of a system of a tank of ideal atomic gas, we have the fundamental equation $S^\circ=\ln\!\left(\frac1{h^{3N}N!}V^NS_{3N-1}\!\left(\sqrt{2mU}\right)\right),$ where $S_n(r)$ is the surface area of an $n$-sphere with radius $r$, which is proportional to $r^n$. Taking its derivative w.r.t. $U,V,N$ and taking the thermodynamic limit will recover familiar results.

## Contracting

I have previously mentioned that the other way of deriving a new system out of an existing one is called contracting. Now we should introduce this concept because it is very useful later when we need to define the contact between subsystems of a composite system (whose definition will be given later).

The idea behind contracting is also to reduce the dimension of the space of extensive quantities. However, rather than making some of the extensive quantities extrinsic parameters, it makes them “intrinsic” within the space of microstates. A vivid analogy is this: imagine a thermal system as many boxes of microstates with each box labeled by specific values of extensive quantities, then we partition those boxes to classify them, and put all the boxes in each partition into one larger box. The new set of larger boxes are labeled by a specific values of fewer extensive quantities, and it is the so-called contraction of the origional set of boxes.

I call it contracting because it is like contracting the affine space of extensive quantities into a flat sheet of its subspace. The way we do this should be described by a projection. A projection in affine space maps the whole space into one of its affine subspace, and the preimage of each point in the subspace is another affine subspace of the original space. The preimages forms a family of parallel affine subspaces labeled by their image under the projection. The family of affine subspaces may be used to define a family of slices of the space of extensive quantities or the thermal system, which are useful when defining the contraction of the space of extensive quantities or the system.

Definition. A contracting of a space of extensive quantities $\left(W,E,\lambda\right)$ is given by a tuple $\left(\pi,\lambda^\perp\right)$, where

• $\pi:W\to W^\perp$ is a projection map from $W$ to an affine subspace $W^\perp$ of $W$;
• $E^\perp\coloneqq\pi(E)$, the image of $E$ under $\pi$, is equipped with the minimal topology $\tau\!\left(E^\perp\right)$ so that $\pi$ is continuous, and the topology makes $E^\perp$ Polish;
• $\lambda^\perp:\sigma\!\left(E^\perp\right)\to\left[0,+\infty\right]$ is a non-trivial σ-finite Borel measure on $E^\perp$, where $\sigma\!\left(E^\perp\right)\supseteq\mathfrak B\!\left(E^\perp\right)$ is a σ-algebra of $E^\perp$ that contains the Borel σ-algebra of $E^\perp$; and
• For any $A\in\sigma\!\left(E^\perp\right)$, $\lambda^{\perp}(A)=0$ iff $\lambda\!\left(\pi^{-1}(A)\right)=0$.

This contracting defines a new space of extensive quantities $\left(W^\perp,E^\perp,\lambda^\perp\right)$, called a contraction of the original space of extensive quantities $\left(W,E,\lambda\right)$.

Definition. The contractive slicings of a space of extensive quantities $\left(W,E,\lambda\right)$ defined by a contracting $\left(\pi,\lambda^\perp\right)$ of it is a family of slicings $\bigsqcup_{e\in W^\perp}\left(W^\parallel_e,\lambda^\parallel_e\right)$, where

• $W^\parallel_e\coloneqq\pi^{-1}(e)$ is the preimage of $\left\{e\right\}$ under $\pi$, an affine subspace of $W$; and
• $\lambda_e^\parallel:\sigma\!\left(E_e^\parallel\right)\to\left[0,+\infty\right]$ is a Borel measure; the family of measures is the disintegration of $\lambda$ w.r.t. $\pi$ and $\lambda^\perp$.

Definition. A contraction of a thermal system $\left(\mathcal E,\mathcal M\right)$ defined by the contracting $\left(\pi,\lambda^\perp\right)$ of $\mathcal E$ is a new thermal system $\left(\mathcal E^\perp,\mathcal M^\perp\right)$ constructed as such:

• $\mathcal E^\perp\coloneqq\left(W^\perp,E^\perp,\lambda^\perp\right)$ is the contraction of $\mathcal E$ corresponding to the given contracting;
• $\mathcal M^\perp\coloneqq\bigsqcup_{e\in E^\perp}M_e^\perp$, where for each $e\in E^\perp$, $M_e^\perp\coloneqq\mathcal M_e^\parallel$; the family of systems $\left(\mathcal E_e^\parallel,\mathcal M_e^\parallel\right)$ (labeled by $e\in E^\perp$) are slices of $\left(\mathcal E,\mathcal M\right)$ corresponding to the contractive slicings of $\mathcal E$ defined by the contracting $\left(\pi,\lambda^\perp\right)$; the measure equipped on $\mathcal M_e^\parallel$ is the measure of number of microstates of $\left(\mathcal E_e^\parallel,\mathcal M_e^\parallel\right)$.

In some cases, the total number of microstates in $\mathcal M^\parallel_e$ is not finite for some $e$, then the contraction is not defined in this case.

Example. For the thermal system of a solid consisting of spin-$\frac12$ particles, define a constracting $\left(\pi,\lambda^\perp\right)$ by \begin{align*} \pi\!\left(U,N\right)&\coloneqq N,\\ \lambda^\perp\!\left(A\right)&\coloneqq\operatorname{card}A. \end{align*} Then the corresponding contraction of the thermal system may be written as a thermal system $\left(\left(W,E,\lambda\right),\bigsqcup_{e\in E}M_e\right)$, where \begin{align*} W&\coloneqq\mathbb R,\\ E&\coloneqq\mathbb Z^+,\\ \lambda\!\left(A\right)&\coloneqq\operatorname{card}A,\\ M_N&\coloneqq\left\{0,1\right\}^N,\\ \mu_N\!\left(A\right)&\coloneqq\operatorname{card}A. \end{align*}

Different from a slice of a system, a contraction of a system does not have the problem about the dimension (“dimension” as in “dimensional analysis”) of the measure on the space of extensive quantities. Although the dimension of $\lambda^\perp$ is different from $\lambda$, the dimension of $\mu^\perp_e$ (the measure on $M^\perp_e$) is also different from $\mu$, and they change together in such a way that the resultant $\mu^\perp$ (the measure of number of microstates on $\mathcal M^\perp$) has the same dimension as $\mu$.

This fact actually hints us that a contraction of a thermal system is essentially the same as the original thermal system in such a sense that the microstates in the two systems are naturally one-to-one connected. Indeed, the natural bijection from $\mathcal M$ to $\mathcal M^\perp$ is given by $\left(e,m\right)\mapsto\left(\pi(e),\left(e,m\right)\right)$. It is obvious that for any measurable function $f$ on $\mathcal M^\perp$ we have $\int_{\left(e,m\right)\in\mathcal M}f\!\left(\pi(e),(e,m)\right)\mathrm d\mu(e,m) =\int_{\left(e,m\right)\in\mathcal M^\perp}f\!\left(e,m\right)\mathrm d\mu^\perp(e,m).$ Using this map, we can pull back any function $f^\perp$ on $\mathcal M^\perp$ to become a function on $\mathcal M$ by $f\!\left(e,m\right)\coloneqq f^\perp\!\left(\pi(e),\left(e,m\right)\right)$ and the other way around. I want to call $f$ the contractional pullback of $f^\perp$ under $\pi$ and call $f^\perp$ the contractional pushforward of $f$ under $\pi$. Specially, we may pull back any state $p^\perp$ of a contraction to become a state $p$ on the original thermal system. We will see that pullbacks of states are rather useful.

Obviously, the family of affine subspaces $\left\{W^\parallel_e\right\}_{e\in W^\perp}$ are parallel to each other. Therefore, their associated vector subspaces are the same vector subspace $\vec W^\parallel$ of $\vec W$, which is a complement of the vector subspace $\vec W^\perp$, the vector space that $W^\perp$ is associated with. We can write $\vec W=\vec W^\perp+\vec W^\parallel,\quad W=W^\perp+\vec W^\parallel.$ Each point in $W$ can be written in the form of $e+s$, where $e\in W^\perp$ and $s\in\vec W^\parallel$. Furthermore, for any $e\in W^\perp$, the map $s\mapsto e+s$ is a bijection from $\vec W^\parallel$ to $W^\parallel_e$. This bijection can then push forward linear operations from $\vec W^\parallel$ to $W^\parallel_e$. For example, we can define the action of some continuous linear functional $i\in\vec W^{\parallel\prime}$ on a point $e'\in W^\parallel_e$ as $i\!\left(e'\right)\coloneqq i\!\left(e'-\pi\!\left(e'\right)\right),$$\p{7}$ where $\pi\!\left(e'\right)$ is just $e$.

However, we need to remember that there is no generally physically meaningful linear structure on $W^\parallel_e$. The linear structure that we have constructed is just for convenience in notations.

An interesting fact about slicing, isolating, and contracting is that: an isolation of a contraction is a contraction of a slice.

Suppose we have a thermal system $\left(\mathcal E,\mathcal M\right)$, and by a contracting $\left(\pi,\lambda^\perp\right)$ we derive its contraction $\left(\mathcal E^\perp,\mathcal M^\perp\right)$.

Now, consider one of its contractive slices $\left(\mathcal E^\parallel_{e^\circ},\mathcal M^\parallel_{e^\circ}\right)$, where $e^\circ\in E^\perp$. Then, we contract this slice by the contracting $\left(\pi,\lambda^{\perp\prime}\right)$, where $\pi$ is the same $\pi$ as used above but whose domain is restricted to $W^\parallel_{e^\circ}$, and $\lambda^{\perp\prime}$ is the counting measure. Because the whole $W^\parallel_{e^\circ}$ is mapped to $e^\circ$ under $\pi$, the contraction becomes an isolated system whose only possible value of extensive quantities is $e^\circ$. Its spaces of microstates consist of only one measure space, which is $\mathcal M^\parallel_{e^\circ}$.

On the other hand, consider isolating $\left(\mathcal E^\perp,\mathcal M^\perp\right)$ at $e^\circ$. Its isolation at $e^\circ$ is an isolated system whose only possible value of extensive quantities is $e^\circ$. Its spaces of microstates consist of only one measure space, which is $M^\perp_{e^\circ}$, which is the same as $\mathcal M^\parallel_{e^\circ}$.

Therefore, an isolation of a contraction is a contraction of a slice.

This fact is useful because it enables us to find the equilibrium state of a slice. Using microcanonical ensemble, we can already find the equilibrium state of any isolated system, so we can find the equilibrium state of an isolation of a contraction. Then, it is the equilibrium state of a contraction of a slice. Then, by the contractional pullback, it is the equilibrium state of a slice.

## Thermal contact

Composite systems are systems that are composed of other systems. This is a useful concept because it allows us to treat multiple systems as a whole. The motivation of develop this concept is that we should use it to derive the canonical ensemble and the grand canonical ensemble. In those ensembles, the system is not isolated but in contact with a bath. To consider them as a whole system, we need to define composite systems.

The simplest case of a composite system is where the subsystems are independent of each other. Physically, this means that the subsystems do not have any thermodynamic contact between each other. I would like to call the simplest case a product thermal system just as how mathematicians name their product spaces constructed out of existing spaces.

Definition. The product space of extensive quantities of two spaces of extensive quantities $\left(W^{(1)},E^{(1)},\lambda^{(1)}\right)$ and $\left(W^{(2)},E^{(2)},\lambda^{(2)}\right)$ is a space of extensive quantities $\left(W,E,\lambda\right)$ constructed as such:

• $W\coloneqq W^{(1)}\times W^{(2)}$ is the product affine space of $W^{(1)}$ and $W^{(2)}$;
• $E\coloneqq E^{(1)}\times E^{(2)}$ is the product topological space as well as the product measure space of $E^{(1)}$ and $E^{(2)}$; and
• $\lambda$ is the product measure of $\lambda^{(1)}$ and $\lambda^{(2)}$, whose uniqueness is guaranteed by the σ-finiteness of $\lambda^{(1)}$ and $\lambda^{(2)}$.

Definition. The product thermal system of two thermal systems $\left(\mathcal E^{(1)},\mathcal M^{(1)}\right)$ and $\left(\mathcal E^{(2)},\mathcal M^{(2)}\right)$ is a thermal system $\left(\mathcal E,\mathcal M\right)$ constructed as such:

• $\mathcal E\coloneqq\left(W,E,\lambda\right)$ is the product space of extensive quantities of $\mathcal E^{(1)}$ and $\mathcal E^{(2)}$; and
• $\mathcal M\coloneqq\bigsqcup_{(e_1,e_2)\in E}M_{e_1,e_2}$, where $M_{e_1,e_2}\coloneqq M^{(1)}_{e_1}\times M^{(2)}_{e_2}$ is the product measure space of $M^{(1)}_{e_1}$ and $M^{(2)}_{e_2}$, equipped with measure $\mu_{e_1,e_2}$, the product measure of $\mu^{(1)}_{e_1}$ and $\mu^{(2)}_{e_2}$.

By this definition, $\mathcal M$ is naturally identified with $\mathcal M^{(1)}\times\mathcal M^{(2)}$, and the measure of number of microstates $\mu$ on $\mathcal M$ is in this sense the same as the product measure of $\mu^{(1)}$ and $\mu^{(2)}$ (the measures of number of microstates on $\mathcal M^{(1)}$ and $\mathcal M^{(2)}$). We can project elements in $\mathcal M$ back into $\mathcal M^{(1)}$ and $\mathcal M^{(2)}$ by the map $(e_1,e_2,m_1,m_2)\mapsto(e_1,m_1)$ and the map $(e_1,e_2,m_1,m_2)\mapsto(e_2,m_2)$.

This hints us that a probability distribution on $\mathcal M$ (which may be given by a state $p$ of $(\mathcal E,\mathcal M)$) can be viewed as a joint probability distribution of the two random variables on $\mathcal M$: $(e_1,e_2,m_1,m_2)\mapsto(e_1,m_1)$ and $(e_1,e_2,m_1,m_2)\mapsto(e_2,m_2)$. As we all know, a joint distribution encodes conditional distributions and marginal distributions. Therefore, given any state of a product thermal system, we can define its conditional states and marginal states of the subsystems. Conditional states are not very useful because they are not physically observed states of subsystems. The physically observed states of subsystems are marginal states, so marginal states are of special interest.

Definition. Given a state $p$ of the product thermal system $(\mathcal E,\mathcal M)$ of $\left(\mathcal E^{(1)},\mathcal M^{(1)}\right)$ and $\left(\mathcal E^{(2)},\mathcal M^{(2)}\right)$, its marginal state of the subsystem $\left(\mathcal E^{(1)},\mathcal M^{(1)}\right)$ is a state $p^{(1)}$ of the system $\left(\mathcal E^{(1)},\mathcal M^{(1)}\right)$ defined by $p^{(1)}\!\left(e_1,m_1\right)\coloneqq\int_{\left(e_2,m_2\right)\in\mathcal M^{(2)}} p\!\left(e_1,e_2,m_1,m_2\right)\mathrm d\mu^{(2)}\!\left(e_2,m_2\right).$

Physically, if a product thermal system is in equilibrium, then each of its subsystems is in equilibrium as well. Therefore, if $p^\circ$ is an equilibrium state of the product thermal system, then the marginal states of $p^\circ$ are equilibrium states of the subsystems.

Now, we need to consider how to describe the thermodynamic contact between subsystems. In the simplest case, where there is no thermodynamic contact between subsystems, the composite system is just the product thermal system of the subsystems, and the dimension of its space of extensive quantities is the sum of the that of the subsystems’. If there is some thermal contact between subsystems, then the dimension of the space of extensive quantities of the composite system will be less than that of the product thermal system. For example, if the subsystems are allowed to exchange energy, then two original extensive quantities (the energy of the first subsystem and that of the second subsystem) will be replaced by a single extensive quantity (the total energy of the composite system). Such a reduction in the dimension of the space of extensive quantities is the same as contracting that we defined above. Therefore, we can define a thermally composite system as a contraction of the product thermal system. Denote the projection map of the contracting as $\pi:W\to W^\perp:(e_1,e_2)\mapsto e$. (From now on in this section, composite systems refer to thermally composite system. I will introduce non-thermally composite systems later (in part 2), which describe non-thermal contacts between subsystems and are more complicated.)

Besides being the contraction of the product thermal system, there is an additional requirement. Given the extensive quantities of the composite system and those of one of the subsystems, we should be able to deduce those of the other subsystem. For example, if the subsystems are allowed to exchange energy, then the total energy of the composite system minus the energy of one of the subsystems should be the energy of the other subsystem, which is uniquely determined (if this is an allowed energy). Mathematically, thie means that for any $e_1\in W^{(1)}$ and $e_2\in W^{(2)}$, the two maps $\pi\!\left(e_1,\cdot\right)$ and $\pi\!\left(\cdot,e_2\right)$ are both injections.

Definition. A (thermally) composite thermal system of two thermal systems is the contraction of their product thermal system corresponding to a contracting $(\pi,\lambda^\perp)$, where $\pi:W\to W^\perp:(e_1,e_2)\mapsto e$ satisfies that for any $e_1\in W^{(1)}$ and $e_2\in W^{(2)}$, the two maps $\pi\!\left(e_1,\cdot\right)$ and $\pi\!\left(\cdot,e_2\right)$ are both injections.

We may define projection maps to get the extensive quantities of the subsystems from those of the composite system: $c^{(1)}:W\to W^{(1)}:(e_1,e_2)\mapsto e_1,\quad c^{(2)}:W\to W^{(2)}:(e_1,e_2)\mapsto e_2.$ Then, for each $e\in W^\perp$, the two spaces $W^{\parallel(1)}_e\coloneqq c^{(1)}\!\left(W_e^\parallel\right),\quad W^{\parallel(2)}_e\coloneqq c^{(2)}\!\left(W_e^\parallel\right)$ are respectively affine subspaces of $W^{(1)}$ and $W^{(2)}$, where $W_e^\parallel\coloneqq\pi^{-1}\!\left(e\right)$. The two affine subspaces are actually isomorphic to each other because of our additional requirement on the projection map $\pi$. Because $\pi\!\left(e_1,\cdot\right)$ is an injection, for any $e_1\in W^{\parallel(1)}_e$ there is a unique $e_2\in W^{\parallel(2)}_e$ such that $\pi\!\left(e_1,e_2\right)=e$, and vice versa. This gives a correspondence between the two affine subspaces. In other words, for each $e\in W^\perp$, there is a unique bijection $\rho_e:W^{\parallel(1)}_e\to W^{\parallel(2)}_e$ such that $\forall e_1\in W^{\parallel(1)}_e: \pi\!\left(e_1,e_2\right)=e\Leftrightarrow e_2=\rho_e\!\left(e_1\right).$$\p{8}$ The bijection $\rho_e$ is an affine isomorphism from $W^{\parallel(1)}_e$ to $W^{\parallel(2)}_e$.

What is more, $c^{(1)}$ is an affine isomorphism from $W^{\parallel}_e$ to $W^{\parallel(1)}_e$, and $c^{(2)}$ is an affine isomorphism from $W^{\parallel}_e$ to $W^{\parallel(2)}_e$. The three affine spaces $W^{\parallel}_e,W^{\parallel(1)}_e,W^{\parallel(2)}_e$ are then mutually isomorphic.

Example. Suppose we have two thermal systems, each of them have two extensive quantities called the energy and the number of particles. We write them as $\left(U_1,N_1\right)$ and $\left(U_2,N_2\right)$. They are in thermal contact so that they can exchange energy but not particles. Then, the extensive quantities of the composite system may be written as $\left(U/2,U/2,N_1,N_2\right)$, with $\pi:\left(U_1,U_2\right)\mapsto\left(U/2,U/2\right)$ defined as $\pi\!\left(U_1,U_2\right)\coloneqq\left(\frac{U_1+U_2}2,\frac{U_1+U_2}2\right).$ The isomorphism $\rho_{U/2,U/2,N_1,N_2}$ is then $\rho_{U/2,U/2,N_1,N_2}\!\left(U_1,N_1\right)=\left(U-U_1,N_2\right).$ The contracting is not unique. For example, $\left(U_1,U_2\right)\mapsto\left(3U/4,U/4\right)$ is another valid projection for constructing the composite thermal system, and it has exactly the same physical meaning as the one I constructed above.

The isomorphism from $W^{\parallel}_e$ can push forward the measure $\lambda^\parallel_e$ on $E^\parallel_e$ to a new measure $\lambda^{\parallel(1)}_e$ on $E^{\parallel(1)}_e$. Then, $\left(W^{\parallel(1)}_e,\lambda^{\parallel(1)}_e\right)$ is a slicing of $\left(W^{(1)},E^{(1)},\lambda^{(1)}\right)$, and we can get a slice $\left(\mathcal E^{\parallel(1)}_e,\mathcal M^{\parallel(1)}_e\right)$ of $\left(\mathcal E^{(1)},\mathcal M^{(1)}\right)$ out of this slicing. I would like to call this slice the compositing slice of $\left(\mathcal E^{(1)},\mathcal M^{(1)}\right)$ at $e$. Similarly, we define compositing slices of $\left(\mathcal E^{(2)},\mathcal M^{(2)}\right)$, denoted as $\left(\mathcal E^{\parallel(2)}_e,\mathcal M^{\parallel(2)}_e\right)$.

Similarly to how we can define marginal states of subsystems of a product thermal system, we can define marginal states of the compositing slices given a state of a contractive slice of the composite system. However, this time, there is a key difference: the subsystems (compositing slices) have isomorphic and completely dependent (deterministic) extensive quantities instead of having completely independent extensive quantities. Taken this into account, we can define marginal states of compositing slices as follows: $p^{\parallel(1)}\!\left(e_1,m_1\right) \coloneqq\int_{m_2\in M^{(2)}_{\rho_e(e_1)}}p^\parallel\!\left(e_1,\rho_e(e_1),m_1,m_2\right) \mathrm d\mu^{(2)}_{\rho_e(e_1)}\!\left(m_2\right),$$\p{9}$ where $p^{\parallel(1)}$ is a state of $\left(\mathcal E^{\parallel(1)}_e,\mathcal M^{\parallel(1)}_e\right)$, and $p^\parallel$ is a state of $\left(\mathcal E^{\parallel}_e,\mathcal M^{\parallel}_e\right)$ (a contractive slice of the composite system).

There is an additional property that $\rho_e$ has.

As we all know, an affine map is a linear map combined with a translation: $\rho_e\!\left(e_1\right)=\vec\rho\!\left(e_1-e_0\right)+\rho_e\!\left(e_0\right),$$\p{10}$ where $e_0$ is a fixed point in $W^{\parallel(1)}_e$, and $\vec\rho:\vec W^{\parallel(1)}_e\to \vec W^{\parallel(2)}_e$