User:Marc Goossens/Gurtin Axiomatics of Thermodynamics

Introduction
Consider a body B, interacting with its exterior Bext.

Thermomechanics sets forth the general scheme for describing such an interaction classically (i.e. not according quantum mechanics) as described below. The backdrop is assumed to be classical (Newtonian / Galilean) spacetime, with Euclidean 3-space as "space-slices".

The formulation of the "thermodynamical laws" governing such interactions adopted below, is due to Gurtin and Williams; though at first sight unfamiliar and sophisticated, it has the advantage of great conceptual clarity: it brings out the essential dynamical relationships between energy, heat and entropy of a body with finite (i.e. not "infinitesimal") volume.

At the same time, one can recover the more traditional statements of both laws, such as the Clausius-Duhem form of the second law through successive steps of specialization.

The presentation given here is merely schematic.

General setting; "body"
In the remainder of this section, only a single body B is considered, which is taken to be a sufficiently regular subset of 3-dimensional Euclidean space. The exterior of this body is simply its complement in this 3-space.

Even in this simple setting, in order to arrive at a concise measure theoretic formulation of all concepts and statements introduced below, the notion of a (mathematically suitable and physically plausible) "material universe" of subbodies of B would need to be made precise, and the laws below stated tho hold "for all subbodies". This aspect is dropped here for brevity. Its full treatment is due to Walter Noll.

Internal energy and entropy of a body; flux of energy and entropy
The goal is to describe dynamics of bodies undergoing processes, hence time is featured explicitly: at each time t, the body is assigned two real-valued, positive functions $$E_t(\mbox {B})$$, its internal energy, and $$ S_t(\mbox {B}) $$, its entropy. For now, these are merely names.

Likewise, the interaction is characterised by two real-valued, functions as follows: $$\mathcal

{E}_t(\mbox {B},\mbox {B}^\mbox {ext})$$, representing the energy flux from the exterior into B, and $$ \mathcal {S}_t(\mbox {B},\mbox {B}^\mbox {ext}) $$, the entropy flux associated with the interaction (or process).

As the notation suggests, $$E_t $$ and $$S_t $$ are to be regarded functions taking values on "bodies" B, being sufficiently regular subsets of Euclidean space. The fluxes are defined on pairs of disjoint bodies.

In order to derive any further theorems or indeed their tradional formulation from the laws as given below, all functions and their time derivative, as well as the fluxes mentioned in this subsection must be assumed (= postulated!) to be measures which are bounded with respect to Euclidean volume of the sets they are defined on. (This excludes surface, line or point densities from our scope.)

From this volume boundedness, it follows that both $$E$$ and $$\dot {E}$$ may be represented by a volume integral of density functions $$e (\bar {x}, t)$$ and $$\dot {e} (\bar {x}, t)$$, such that for a subbody $$\mathcal {P}$$:


 * $$E_t(\mathcal {P}) = \int _{\mathcal {P}} e (\bar {x},t) dV(x)$$, and $$\dot {E}_t(\mathcal {P}) = \int _{\mathcal {P}} \dot {e} (\bar {x},t)  dV(x)$$

with $$\dot {e} (\bar {x},t) = \partial _t e (\bar {x},t) $$. For a given t, the density function $$e$$ is the Radon-Nikodym derivative of $$E$$.

The integral represenation will be an element later on in recovering the more familiar form of the thermodynamical laws, which are first formulated in terms of the measures themselves, to bring out the underlying concepts more clearly.

First law
According to thermomechanics, all thermomechanical processes comply to two principal postulates. The first one extends the picture of classical continuum mechanics by also allowing non-mechanical transfer of energy into (out of) a body. This is expressed in the form of a balance law,


 * $$\dot{E} = \mathcal {E}, \mbox {with} ~ \mathcal {E} := \mathcal {Q} + \mathcal {W} $$ (1), known as the first law.

In (1), $$\mathcal {W}=\mathcal {W}_t(\mbox {B},\mbox {B}^\mbox{ext}) $$ stands for mechanical energy flux or working from Bext into B, as given by continuum mechanics, whereas $$\mathcal {Q} = \mathcal {Q}_t(\mbox {B},\mbox {B}^\mbox{ext}) $$ is a notion specific to thermomechanics, representing "non-mechanical energy flux", commonly termed heat flux.

Thus, the first law states that the rate of change in internal energy $$\dot{E} = \frac {dE}{dt}$$ precisely equals the total influx of energy (mechanical + heat)

Second law
The second postulate goes on to impose a constraint on this heat flux, by linking it to another quantity, the entropy, in the form of an inequality,


 * $$\dot{S} \ge \mathcal {S} $$ (2a), together with a coupling condition claiming that


 * if $$\mathcal {Q} = 0 $$, then also $$\mathcal {S} = 0 $$ (2b), both statements combined forming the second law.

With (2a), the second law stipulates to begin with that all materials constituting bodies will be such that for all processes, the rate of increase in entropy $$\dot{S} = \frac {dS}{dt}$$ of the body is never less than, but may well exceed the influx of entropy. Expressed by (2b), it goes on to say that a non-zero flux of entropy is always accompanied by a non-zero flux of heat.

As (2a) suggests, we may look at the entropy production, defined as $$ S^+ := \dot{S} - \mathcal {S}

$$, the excess of entropy increase in the body over entropy influx, which, still according to the second law, never decreases: $$ S^+ \ge 0 $$.

Requirement (2b) effectively couples heat and entropy, by ensuring the existence of a function relating both: temperature.

Recovering the traditional formulation
In order to regain the familiar form of both laws, specializing assumptions need to be introduced. Focusing on the thermodynamical side of things, the purely mechanical term $$\mathcal {W} $$ will be dropped in the remainder of this section.

The first law in integral and differential form
TODO: obtain first law in integral form / mention differential form.

Another reasonable assumption, adding further physical relevance to the laws as given above, is that the heat flux into a body is suitably bounded by the volume of the body and the area of the contact surface with the exterior part considered. This implies that the heat flux admits a unique decomposition $$\mathcal {Q} = \mathcal {Q}^{(Rad)} + \mathcal {Q}^{(Cond)}$$ into a radiative and a conductive part. The former is then shown to behave as a measure bounded with respect to volume, the latter as one bounded with respect to the area of the contact surface.

As always, the integral form of any physical law is more univerally valid than its differential "equivalent".

The former is more tolerant to discontinuities.

The second law in integral and differential form
TODO: obtain 2nd law in integral form / mention differential form

$$\int_{\mathcal {A}} \dot {s} (\bar x,t)\, dV(x) \ge \int_{\partial \mathcal {A}} \frac {\bar q(\bar x,t) \cdot \bar n} {\varphi (\bar x,t)} \, dA(x) + \int_{\mathcal {A}} \frac {r(\bar x,t)} {\vartheta (\bar x,t)} \, dV(x)$$

Clausius-Duhem form of the second law
TODO: reduce 2nd law to C-D form

Processes: principle of thermodynamic determinism
TODO

Thermokinetic process for a body fixes E(t), Q(t) and S(t).

functionals

e, q, s, r, ...

Gurtin & Williams p 112 (simple heat conductor without memory)

assumptions: no internal radiation + conductive temp independent of S at each point

given rad temp (x, t) + its grad g

then q(x,t) = q (rTemp, g)

Criticism of the Gurtin-Williams formalism
Mathematically sound + brings physical insight.

Not an axiomatic basis for thermodynamics, as "laws" do not express evident first principles or primitive notions do not immediately correspond to "known concepts" from pretheories.