Elementary Thermo-mechanical Systems and Higher Order Constraints

Abstract

In this paper we study a class of physical systems that combine a finite number of mechanical and thermodynamic observables. We call them elementary thermo-mechanical systems (ETMS). We introduce these systems by means of simple examples, and we obtain their (time) evolution equations by using, essentially, the Newton’s laws and the First Law of Thermodynamics only. We show that such equations are similar to those defining certain constrained mechanical systems. From that, and addressing the main aim of the paper, we give a general definition of ETMS, in a variational formalism, as a particular subclass of the Lagrangian higher order constrained systems.

Appendix A: Background on Thermodynamics

Below, we introduce the basic notation and terminology on thermodynamics that we shall use along all of the paper, and recall some fundamental concepts on the subject (see [7, 16, 24, 28]).

$*$ A thermodynamic system is typically defined by $2d+1$ observables, which we shall denote $x_{1},\ldots ,x_{d-1},T,y_{1},\ldots ,y_{d-1},S$ and U. The $x_{i}$’s and T (resp. $y_{i}$’s, S and U) are called intensive (resp. extensive) variables. The extensive variables depend on the “size” of the system, while the intensive ones do not. U is the internal energy, T is the temperature and S the entropy. For example, consider a mixture of $r\in {\mathbb {N}}$ different chemical components. In this case, we have that:

• $d=r+2$;

• for $i=1,\ldots ,r$, each variable $y_{i}:=N_{i}$ (resp. $-x_{i}:=\mu _{i}$) represents the number of moles (resp. the chemical potential) of a given type;

• $x_{r+1}=:P$ is the pressure and $y_{r+1}=:V$ the volume.

Remark I

Sometimes, systems can be seen as composed by “simpler” ones, i.e. those defined by a smaller number of variables. We say in this case that such a system is a composite system. In the last example, each chemical compound can be seen as a part of a composite system.

$*$ The possible values of the variables $x_{1},\ldots ,x_{d-1},T,y_{1},\ldots ,y_{d-1},S$ and U give rise to a manifold ${\mathcal {T}}\subset {\mathbb {R}}^{2d+1}$ (that we shall assume to be open), usually called the thermodynamical phase space (TPS): the set of states of the system. We can see these variables as coordinates for ${\mathcal {T}}$.

$*$ The manifold ${\mathcal {T}}$ is a contact manifold with contact form

$$\begin{aligned} \theta :=dU-T\ dS+{\displaystyle \sum \limits _{i=1}^{d-1}}x_{i}\,dy_{i}. \end{aligned}$$

(A.1)

Thus, $\left( x_{1},\ldots ,x_{d-1},-T,y_{1},\ldots ,y_{d-1},S,U\right)$ defines a global Darboux system for $\left( {\mathcal {T}},\theta \right)$, see [4, 26].

Remark II

For a composite system (see Remark I) formed out by two simpler ones, the TPS is a product manifold ${\mathcal {T}}={\mathcal {T}}_{1}\times {\mathcal {T}}_{2}$ with contact form

$$\begin{aligned} \theta :=dU_{1}-T_{1}\ dS_{1}+dU_{2}-T_{2}\ dS_{2}+{\displaystyle \sum \limits _{i=1}^{d_{1}-1}}x_{1,i}\,dy_{1,i}+{\displaystyle \sum \limits _{i=1}^{d_{2}-1}}x_{2,j}\,dy_{2,j}.\nonumber \\ \end{aligned}$$

(A.2)

Here $\left( x_{k,1},\ldots ,x_{k,d_{1}-1},-T_{k},y_{k,1},\ldots ,y_{k,d_{1}-1},S_{k},U_{k}\right)$, with $k=1,2$, are the global Darboux coordinates of ${\mathcal {T}}_{1}$ and ${\mathcal {T}}_{2}$.

$*$ By process we shall mean every curve $\Gamma :\left[ a,b\right] \rightarrow {\mathcal {T}}$. It represents a “continuum” of actions on the system that produce a “continuum” of changes on its states.

$*$ Among the states, a special role is played by a subset ${\mathcal {N}}\subset {\mathcal {T}}$, known as the space of equilibrium states, which is defined by the following two conditions on U. The first one says that, on the equilibrium states, U and the rest of the extensive variables $y_{i}$’s and S must be related by the formula

$$\begin{aligned} U=\Phi \left( y_{1},\ldots ,y_{d-1},S\right) , \end{aligned}$$

(A.3)

for some function $\Phi$ (typically homogeneous of degree one). Equation above is known as the Fundamental Equation of the system. The second condition says that, for any differentiable curve $\Gamma :\left[ a,b\right] \rightarrow {\mathcal {N}}\subset {\mathcal {T}}$, the variation $\Delta U:=U\left( \Gamma \left( b\right) \right) -U\left( \Gamma \left( a\right) \right)$ must satisfies

$$\begin{aligned} \Delta U={\mathbb {Q}}-W, \end{aligned}$$

(A.4)

where ${\mathbb {Q}}={\mathbb {Q}}\left( \Gamma \right)$ and $W=W\left( \Gamma \right)$ are the heat and the mechanical work, respectively, interchanged by the system and the environment along the process $\Gamma$. This is the First Law of Thermodynamics.

Remark

When ${\mathbb {Q}}=0$ along a process, one says that such a process is adiabatic.

At a differential level, Eq. (A.4) translates to

(A.5)

Here, and are 1-forms (non-necessarily closed) on ${\mathcal {T}}$ such that, given a process $\Gamma$,

Idem for . For instance, for a mixture of chemical components (see above), and are given by

at least for some processes. Accordingly,

$$\begin{aligned} dU=T\ dS-P\ dV-{\displaystyle \sum \limits _{i=1}^{r}}x_{i}\,dy_{i}. \end{aligned}$$

In general, we must have

$$\begin{aligned} dU=T\ dS-{\displaystyle \sum \limits _{i=1}^{d-1}}x_{i}\,dy_{i}. \end{aligned}$$

(A.6)

Combining Eqs. (A.3) and (A.6), it follows that the subset ${\mathcal {N}}$ is defined by the equations

$$\begin{aligned} x_{i}= & {} -\frac{\partial \Phi }{\partial y_{i}}\left( y_{1},\ldots ,y_{d-1},S\right) ,\nonumber \\ T= & {} \frac{\partial \Phi }{\partial S}\left( y_{1},\ldots ,y_{d-1},S\right) \quad \text {and }U=\Phi \left( y_{1},\ldots ,y_{d-1},S\right) , \end{aligned}$$

(A.7)

known as state equations. This means that ${\mathcal {N}}$ is a Legendre submanifold of $\left( {\mathcal {T}},\theta \right)$ (see [4, 26]).

Remark III

As explained in [16], the function $\Phi$ or, equivalently, the internal energy U, is defined by the allowed mechanical work on the system. In other words, U is completely determined if we know the work done $W\left( \Gamma \right)$ along any process $\Gamma$. (This information, in fact, not only determines U, but also ${\mathbb {Q}}$).

$*$ For instance, the fundamental equation of the so-called ideal gas, with only one chemical component, is given by 

$$\begin{aligned} \Phi (N,S,V)=N\,u_{0}\left( \frac{N\,v_{0}\ e^{\frac{S-N\,s_{0}}{N\,R}}}{V}\right) ^{\frac{1}{\alpha }}, \end{aligned}$$

(A.8)

where $\alpha$ is a dimensionless constant, R is the universal constant of ideal gases, and $s_{0}$, $v_{0}$ and $u_{0}$ are constants with units of entropy, volume and energy per mole, respectively (see [7] for more details). Thus, the related state equations read

$$\begin{aligned} \mu =\frac{U}{N}\,\left( 1+\frac{1}{\alpha }\,\left( 1-\frac{S}{NR}\right) \right) ,\quad T=\frac{U}{NR\alpha },\quad P=\frac{U}{V\,\alpha }, \end{aligned}$$

(A.9)

with

$$\begin{aligned} U=N\,u_{0}\left( \frac{N\,v_{0}\ e^{\frac{S-N\,s_{0}}{N\,R}}}{V}\right) ^{\frac{1}{\alpha }}. \end{aligned}$$

(A.10)

Let us mention that the last two equations in (A.9) and the Eq. (A.10) are usually written as

$$\begin{aligned} U=\alpha NRT,\quad PV=NRT\quad \text {and }S=N\,s_{0}+NR\ \mathrm {ln}\left( \frac{T^{\alpha }V}{t_{0}^{\alpha }Nv_{0}}\right) , \end{aligned}$$

(A.11)

where $t_{0}:=u_{0}/R\alpha$.

$*$ The differentiable curves along the equilibrium states $\Gamma :\left[ a,b\right] \rightarrow {\mathcal {N}}$ are usually called quasi-static processes (and we shall take this convention). Note that, along such curves, the state equations (A.7) are satisfied for every $t\in \left[ a,b\right]$ (by definition of ${\mathcal {N}}$).

Remark IV

In practice, in order to have a quasi-static process $\Gamma$, the velocity of $\Gamma$ must be small (w.r.t. certain characteristic lengths and times related to the microscopic properties of the system). That is to say, the action that defines the process must produce changes in the states at a very slow rate. This justifies the name “quasi-static.” However, in this paper, when we say that a process is quasi-static we will not be assuming that the rate of change of states is necessarily slow. We will only assume that the state equations are satisfied for all time along such a process.

$*$ Not every process $\Gamma :\left[ a,b\right] \rightarrow {\mathcal {N}}$ is allowed. According to the Second Law of Thermodynamics, a process $\Gamma$ must satisfy

(A.12)

where

(A.13)

In infinitesimal terms

(A.14)

For adiabatic processes, since , we must have $\Delta S\ge 0$.

$*$ A process $\Gamma :\left[ a,b\right] \rightarrow {\mathcal {N}}$ is say to be reversible if there exists another process $\Gamma ^{-}:\left[ a,b\right] \rightarrow {\mathcal {N}}$ such that $\Gamma ^{-}\left( t\right) =\Gamma \left( a+b-t\right)$. Otherwise, $\Gamma$ is say to be irreversible. Then, a process is reversible if and only if the equation

(A.15)

holds.