thermodynamics - How do we know that heat is a differential form?

In thermodynamics, the first law can be written in differential form as

$$dU = \delta Q - \delta W$$ Here, $dU$ is the differential $1$-form of the internal energy but $\delta Q$ and $\delta W$ are inexact differentials, which is emphasized with the replacement of $d$ with $\delta$.

My question is why we regard heat (or work) as differential forms? Suppose our system can be described with state variables $(x_1,\ \cdots ,\ x_n)$. To my understanding, a general $1$-form is written as

$$df = f_1\ dx_1 + f_2\ dx_2 + \cdots + f_n\ dx_n$$ In particular, differential forms are linear functionals of our state variables. Is there any good reason to presuppose that $\delta Q$ is linear in the state variables?

In other words, if the infinitesimal heat transferred when moving from state $\mathbf{x}$ to $\mathbf{x} + d\mathbf{x}_1$ is $\delta Q_1$ and from $\mathbf{x}$ to $\mathbf{x} + d\mathbf{x}_2$ is $\delta Q_2$, is there any physical reason why the heat transferred from $\mathbf{x}$ to $\mathbf{x} + d\mathbf{x}_1 + d\mathbf{x}_2$ should be $\delta Q_1 + \delta Q_2$?

I apologize if the question is a bit unclear, I am still trying to formulate some of these ideas in my head.

P.S. I know that for quasi-static processes, we have $\delta Q = T\ dS$ (and $\delta W = p\ dV$) so have shown $\delta Q$ is a differential form in this case. I guess my question is about non-quasi-static processes in general.

Answer

We want to show that "infinitesimal" changes in heat along a given path in thermodynamic state space can be modeled via a differential 1-form conventionally called $\delta Q$.

The strategy.

1. 
   

   We introduce a certain kind mathematical object called a cochain.

   
   


2. 
   

   We argue that in thermodynamics, heat can naturally be modeled by a cochain.

   
   


3. 
   
   

   We note a mathematical theorem which says that to every sufficiently well-behaved cochain, there corresponds exactly one differential form, and in fact that the cochain is given by integration of that differential form.

   
   


4. 
   

   We argue that the differential form from step 3 is precisely what we usually call $\delta Q$ and has the interpretation of modeling "infinitesimal" changes in heat.

   
   

Some math.

In order to introduce cochains which we will argue should model heat, we need to introduce some other objects, namely singular cubes and chains. I know there is a lot of formalism in what follows, but bear with me because I think that understanding this stuff pays off in the end.

Cubes and chains.

Let the state space of the thermodynamic system be $\mathbb R^n$ for some positive integer $n$. A singular $k$-cube in $\mathbb R^n$ is a continuous function $c:[0,1]^k\to \mathbb R^n$. In particular, a singular 1-cube is simply a continuous curve segment in $\mathbb R^n$. Let $S_k$ be the set of all singular $k$-cubes in $\mathbb R^n$, and let $C_k$ denote the set of all functions $f:S_k\to\mathbb Z$ such that $f(c) = 0$ for all but finitely many $c\in S_k$. Each such function is called a $k$-chain.

The set of chains is a module.

It turns ut that the set of $k$-chains can be made into a vector space in the following simple way. For each $f,g\in C_k$, we define their sum $f+g$ as $(f+g)(c) = f(c) + g(c)$, and for each $a\in \mathbb Z$, we define the scalar multiple $af$ as $(af)(c) = af(c)$. I'll leave it to you to show that $f+g$ and $af$ are $k$-chains if $f$ and $g$ are. These operations turn the set $C_k$ into a module over the ring of integers $\mathbb Z$, the module of $k$-chains!

Ok so what the heck is the meaning of these chains? Well, if for each singular $k$-cube $c\in S_k$ we abuse notation a bit and let it also denote a corresponding $k$-chain $f$ defined by $f(c) = 1$ and $f(c') = 0$ for all $c'\neq c$, then one can show that every singular $k$-chain can be written as a finite linear combination of singular $k$-cubes:

$$\begin{align} a_1c_1 + a_2c_2 + \cdots + a_Nc_N \end{align}$$ For $k=1$, namely if we consider 1-chains, then it is relatively easy to visualize what these guys are. Recall that each singular 1-cube $c_i$ in the chain is just a curve segment. We can think of each scalar multiple $a_i$ of a given cube $c_i$ in the chain as an assignment of some number, a sort of signed magnitude, to that cube in the chain. We then think of adding the different cubes in the chain as gluing the different cubes (segments) of the chain together. We are left with an object that is just a piecewise-continuous curve in $\mathbb R^n$ such that each curve segment that makes up the curve is assigned a signed magnitude.

Drumroll please: introducing cochains!

Now here's where we get to the cool stuff. Recall that the set $C_k$ of all $k$-chains is a module. It follows that we can consider the set of all linear functionals $F:C_k\to \mathbb R$, namely the dual module of $C_k$. This dual module is often denoted $C^k$. Every element of $C^k$ is then called a $k$-cochain (the "co" here being reminiscent of "covector" which is usually used synonymously with the term "dual vector). In summary, cochains are linear functionals on the module of chains.

Heat as a $1$-cochain.

I'd now like to argue that heat can naturally be thought of as a $1$-cochain, namely a linear functional on $1$-chains? We do so in steps.

1. 
   

   For each piecewise-continuous path $c$ (aka a $1$-chain) in thermodynamic state space, there is a certain amount of heat that is transferred to a system when it undergoes a quasistatic process along that path. Mathematically, then, it makes sense to model heat as a functional $Q:C_k\to\mathbb R$ that associates a real number to each path that physically represents how much heat is transferred to the system when it moves along the path.

   
   
   


2. 
   

   If $c_1+c_2$ is a $1$-chain with two segments, then the heat transferred to the system as it travels along this chain should be the sum of the heat transfers as it travels along $c_1$ and $c_2$ individually;

   $$\begin{align} Q[c_1+c_2] = Q[c_1] + Q[c_2] \end{align}$$ In other words, the heat functional $Q$ should be additive.
   
   


3. 
   

   If we reverse the orientation of a chain, which physically corresponds to traveling along a path in state space in the reverse direction, then the heat transferred to the system along this reversed path should have the opposite sign;

   $$\begin{align} Q[-c] = -Q[c] \end{align}$$
   
   


4. 
   

   If we combine steps 2 and 3, we find that $Q$ is a linear functional on chains; it is a cochain! To see why this is so, let a chain $a_1c_1 + a_2c_2$ be given. Since $a_1$ and $a_2$ are integers, we can rewrite this chain as

   $$\begin{align} a_1c_1 + a_2c_2 = \mathrm{sgn}(a_1) \underbrace{(c_1 + \cdots + c_1)}_{\text{$|a_1|$ terms}} +\mathrm{sgn}(a_2) \underbrace{(c_2 + \cdots + c_2)}_{\text{$|a_2|$ terms}} \end{align}$$ and we can therefore compute: $$\begin{align} Q[a_1c_1 + a_2c_2] &= Q[\mathrm{sgn}(a_1) (c_1 + \cdots + c_1) +\mathrm{sgn}(a_2) (c_2 + \cdots + c_2)] \\ &= \mathrm{sgn}(a_1)|a_1|Q[c_1] + \mathrm{sgn}(a_2)|a_2|Q[c_2] \\ &= a_1Q[c_1] + a_2 Q[c_2] \end{align}$$ In summary, by thinking about heat as a functional on paths, and by imposing physically reasonable constraints on that functional, we have argued that heat is a $1$-cochain.
   
   

From cochains to differential forms.

Now that we have argued that heat can be thought of as a $1$-cochain, let's show how this leads to modeling "infinitesimal" changes in heat with a differential $1$-form.

This is where things get really mathematically interesting. We first recall the definition of a differential $k$-form over a $k$-chain. If $\omega$ is a $k$-form, and $c = a_1c_1 + \cdots a_Nc_N$ is a $k$-chain, then we define the integral of $\omega$ over $c$ as follows:

$$\begin{align} \int_c\omega = a_1\int_{c_1} \omega + \cdots + a_N\int_{c_N}\omega. \end{align}$$ In other words, we integrate $\omega$ over each $k$-cube $c_i$ in the chain multiplied by the appropriate signed magnitude $a_i$ associated with that cube, and then we add up all of the results to get the integral over the chain as a whole. For example, if $k=1$ then we have an integral of a $1$-form over a $1$-chain which is usually just called a line integral.

Now notice that given any $k$-form $\omega$, there exists a corresponding cochain, which we'll call $F_\omega$, defined by

$$\begin{align} F_\omega[c] = \int_c\omega \end{align}$$ for any $k$-chain $c$. In other words, integration of a form over a chain can simply be thought of as applying a particular linear functional to that chain.

But here's the really cool thing. The construction we just exhibited shows that to every differential form, there corresponds a cochain $F_\omega$ given by integration of $\omega$. A natural question then arises: is there a mapping that goes the other way? Namely, if $F$ is a given $k$-cochain, is there a corresponding $k$-form $\omega_F$ such that $F$ can simply be written as integration over $\omega$? The answer is yes! (provided we make suitable technical assumptions). In fact, there is a mathematical theorem which basically says that

Given a sufficiently smooth $k$-cochain $F$, there is a unique differential form $\omega_F$ such that

$$\begin{align} F[c] = \int_c\omega_F \end{align}$$ for all suitably non-pathological chains $c$.

If we apply this result to the heat $1$-cochain $Q$, then we find that there exists a unique corresponding differential $1$-form $\omega_Q$ such that for any reasonable chain $c$, we have

$$\begin{align} Q[c] = \int_c \omega_Q \end{align}$$ This is precisely what we want. If we identify $\omega_Q$ as $\delta Q$, then we have shown that

The heat transferred to a system that moves along a given path ($1$-chain) in thermodynamic state space is given by the integral of a differential $1$-form $\delta Q$ along the path.

This is a precise formulation of the statement that $\delta Q$ is a one-form that represents "infinitesimal" heat transfers.

Note. This is a new, totally revamped version of the answer that actually answers the OP's question instead of just reformulating it mathematically. Most of the earlier comments pertain to older versions.

Acknowledgement.

I did not figure this all out on my own. In the original form of the answer, I reformulated the question in a mathematical form, and I essentially posted this mathematical question on math.SE:

https://math.stackexchange.com/questions/658214/when-can-a-functional-be-written-as-the-integral-of-a-1-form

That question was answered by user studiosus who found that the theorem on cochains and forms to which I refer was proven by Hassler Whitney roughly 60 years ago in his good Geometric Integration Theory. In attempting to understand the theorem, and especially the concept of cochains, I found the paper "Isomorphisms of Differential Forms and Cochains" written by Jenny Harrison to be very illuminating. In particular, her discussion of theorem on forms and chains to which I refer above is nice.