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This contribution presents an outline of a new mathematical formulation for Classical Non-Equilibrium Thermodynamics (CNET) based on a contact structure in differential geometry. First a non-equilibrium state space is introduced as the third key element besides the first and second law of thermodynamics. This state space provides the mathematical structure to generalize the Gibbs fundamental relation to non-equilibrium thermodynamics. A unique formulation for the second law of thermodynamics is postulated and it showed how the complying concept for non-equilibrium entropy is retrieved. The foundation of this formulation is a physical quantity, which is in non-equilibrium thermodynamics nowhere equal to zero. This is another perspective compared to the inequality, which is used in most other formulations in the literature. Based on this mathematical framework, it is proven that the thermodynamic potential is defined by the Gibbs free energy. The set of conjugated coordinates in the mathematical structure for the Gibbs fundamental relation will be identified for single component, closed systems. Only in the final section of this contribution will the equilibrium constraint be introduced and applied to obtain some familiar formulations for classical (equilibrium) thermodynamics.

The main objective of this paper is to derive a mathematical framework for macroscopic non-equilibrium thermodynamics, which is not based on some kind of an (implicit) equilibrium assumption. Instead it will be shown, that the equilibrium conditions are “contained” in the proposed framework by applying an additional constraint. There are two key components in this innovative derivation: 1) a generalization of Gibbs fundamental relation for non-equilibrium thermodynamics that is based on the dissipation of energy and 2) a unique formulation of the second law of thermodynamics, including a mathematical proper derivation of non-equilibrium entropy as a thermodynamic state function. This paper presents the mathematical framework of Classical Non-Equilibrium Thermodynamics (CNET) based on a harmonious integration of mathematics and physics.

In his monumental work “On the Equilibrium of Heterogeneous Substances” [

Gibbs postulates this relation for a mixture of

Based on the original work of Rudolf Clausius (1822-1888), the macroscopic entropy of systems in thermodynamic equilibrium is often defined as the quotient of heat flow and temperature. For non-equilibrium processes this definition is then simply extended to an inequality (e.g. Clausius inequality [

Section 2 starts with the introduction of a thermodynamic state and defines a corresponding state function. The mathematical machinery of contact geometry is used to postulate the so-called non-equilibrium state space

A powerful feature of phenomenological thermodynamics is its capability to make statements about the macroscopic state of a system without detailed knowledge of its microscopic details. This feature is based on the phenomenological observation, that a finite number of macroscopic properties is required to completely describe the thermodynamics of a system. Non-equilibrium thermodynamics will be described in a separate state space

A thermodynamic system consists of

Then the statistical average of some physical observable

This equation defines a macroscopic property of a system, which will be applied as a coordinate for the differential manifold

Non-equilibrium thermodynamics describes changes in a system, which depend on changes in its state

Definition 1 [Non-equilibrium state function]. The 0-form

In this definition symbol ^{1}.

The manifold

Postulate 1 [Gibbs state space]. Non-equilibrium thermodynamics is specified by a

The Gibbs state space

In the general context of this paper, it is very important to notice, that this postulate is neither the first law nor the second law of thermodynamics. The above postulate states that the contact structure

・ it is never equal to zero, so it is either everywhere positive or it is everywhere negative,

・ it is not completely integrable (Frobenius condition).

Both these properties play a key role in the formulation of the second law of thermodynamics in section 3. The physical interpretation of the quantity

According to Darboux’s theorem (see [

(summation over repeated indices). This equation can be identified as a generalization of the Gibbs fundamental relation, e.g. see Equation (1). Thus, the mathematical structure of the Gibbs fundamental relation is not postulated in this paper. Instead it is derived from a mathematical property of the contact structure

Definition 2 [Thermodynamic potential]. The canonical coordinate

Symbols

The distinction between exact and inexact 1-forms is an essential part of the mathematical formulation of the first and second laws of thermodynamics. The first law of thermodynamics states that for an arbitrary periodic process, the energy at the start and end of a complete period are identical. This statement complies with the definition of a non-equilibrium state function (e.g. see Definition 1). Both the amount of heat flow

Postulate 2 [First law of thermodynamics]. Summation of the inexact 1-forms for heat flow

The crucial feature in the above postulate is, that the sum of two inexact 1-forms

The internal energy

In nature, it is observed that, for closed systems, processes are such that the system eventually approximates an equilibrium state. Once a system has obtained such an equilibrium state, interactions with the surroundings of the system are required to disturb this equilibrium. Hence, processes are directed towards a mathematical extremum, which corresponds with the equilibrium state, and this is irreversible when there is no interference from outside the system. Observations of these phenomena are for the first time published by Sadi Carnot (1796-1832, see [^{2}

As is already stated before, the heat flow

Type of 1-form | Condition | Example | ||
---|---|---|---|---|

1) | exact | |||

2) | inexact | |||

3) | contact | |||

⁞ | ⁞ | ⁞ | ⁞ | |

(2n + 1) | contact |

in Postulate 2. In classical thermodynamics, the above definition is then often extended to non-equilibrium states by replacing the equal sign with an unequal sign or by adding some other term^{3}. In such an approach, it is not clear that entropy

In these relations

One of the less famous pioneers in thermodynamics is the Greek mathematician Constantin Carathéodory (1800-1900). His work [^{4}, although it is one of the first attempts for a rigorous formalization of thermodynamics. In the second axiom in his 1909 paper [

Axiom II: In jeder beliebigen Umgebung eines willkürlich vorgeschriebenen Anfangszustandes gibt es Zustände, die durch adiabatische Zustandsänderungen nicht beliebig approximiert werden können^{5}.

Above axiom is in the literature also known as Carathéodory’s inaccessibility condition or as Carathéodory’s principle [^{6}. So the heat transfer is not necessarily always equal to zero, which is in the textbook of Bamberg and Sternberg [

Based on the Frobenius integration theorem the second relation is derived from Axiom II and therefore also known as Carathéodory’s theorem [

So modern interpretations of mathematical formulations for the definition of entropy in classical thermodynamics show, that the Frobenius integration theorem plays a crucial role. But opposite to the before mentioned formulations, in this paper Frobenius integration theorem is not applied to the inexact 1-form

Postulate 3 [Second law of thermodynamics]. The difference between the inexact 1-form

Similar to the approach of Carathéodory, the concept of entropy is not defined in the above formulation for the second law of thermodynamics. Based on the integrability condition in Postulate 3 it can then be shown (see [

The exterior derivative of entropy

An integral part of the definition of a non-equilibrium state function is, that the initial value is identical to the final value at the end of a complete period in a periodic process. This means that the initial and final non-equilibrium state

Both the absolute temperature

In this section the thermodynamic potential

This equation does not add new information to the presented mathematical framework of non-equilibrium thermodynamics. It only specifies the set of coordinates that determines the work flow^{7}, where blackboard fonts are used to denote extensive quantities. The indices are

By comparing this equation with Equation (6), one could argue that the thermodynamic potential and the set of conjugated coordinates are identified. But this identification would not be a sound mathematical proof due to the presence of the contact 1-form

Opposite to most textbooks on thermodynamics, in this paper the definition of the Gibbs free energy

Theorem 1 [Gibbs free energy]. For a single component, closed system the thermodynamic potential

The corresponding pairs of conjugated coordinates

Proof. The proof starts with the combination of the generalized Gibbs fundamental relation and the second law of thermodynamics by substituting Equation (13) into Equation (6). Then replace the 1-form

Notice that due to these steps the Gibbs contact 1-form

The next step in this proof is based on a unique property of thermodynamic properties: thermodynamic properties are either extensive or intensive quantities. Extensive quantities are proportional to the size (or extend) of the thermodynamic system and therefore scalable, while intensive quantities are independent of the size of the system. The total volume

Consider the case that all conjugate coordinates denoted by ^{8}. Substitute every relation in Equation (20) to the corresponding terms in Equation (19) and apply the product rule of differentiation. Rearrange the resulting equation by collecting all terms with either

The total volume

and

Equation (23)_{2} is obtained by multiplying Equation (23)_{1} with the total volume

The final step in this proof is to identify the set of conjugated coordinates _{2} with

This is again a Pfaffian equation for a thermodynamic state function, either for the Gibbs energy

There are

Again, for an arbitrary thermodynamic process the above equation can only hold when all terms between brackets are equal to zero. The set of extensive conjugated coordinates

Equation (17), which is in this paper derived in the above proof, is in the literature also known as Euler’s relation (e.g. see [

Equation (16) can be derived by subtracting the above equation from Equation (27), where

Both Equations (16) and (27) are implementations of Equation (6) for single component, closed systems, which is therefore a generic formulation of the third key element of thermodynamics (besides both laws of thermodynamics). Both equations confirm that Gibbs free energy

So far, this paper has been focused on the general case of non-equilibrium states and arbitrary irreversible processes. For an isolated system (then there are no interactions with the surroundings), it is in nature observed, that the macro- scopic state of the system does not change any more if one waits sufficiently long. The corresponding thermodynamic state is called the equilibrium state of the system for which the change of a state function

Molecules do not stop moving in such an equilibrium state; there are many fluctuations of the microstates, which are not visible on the macroscopic level. On a macroscopic level, any perturbation away of an equilibrium state requires some kind of exchange with the surrounding of the system. The above equation implements a mathematical extremum, which is not present in the proposed mathematical framework for non-equilibrium thermodynamics due to the mathematical properties of the Gibbs contact 1-form

The absolute temperature

Based on the above considerations it becomes clear, that an additional mathematical construct is required to embed equilibrium thermodynamics within the proposed mathematical framework. Consider therefore an arbitrary irreversible process

Postulate 4 [Equilibrium constraint]. There exists an equilibrium map

This postulate can also be found in the original work of Hermann [^{9}: A Legendre manifold

The distinction between the equilibrium Legendre state space

Next some results will be obtained for the case of equilibrium thermodynamics. An underline will be used in this paper to denote symbols that refer to an

equilibrium state on the Legendre submanifold

Then a similar procedure can be applied to pairs of conjugated coordinates in Equation (6), where the 1-form

Thus, application of the equilibrium map in Postulate 4 to the generalized Gibbs fundamental relation in Equation (6) results in

The last relation is called a Pfaffian equation, where the 1-form

The relations in Equation (35)_{3} follow from the combination of Equations (34)_{2} and (35)_{2}, so these relations are only valid for equilibrium thermodynamics. To further clarify this point, consider internal energy _{2} for the non-equili- brium case^{10} and combine the result with Equation (16) to obtain

Since the Gibbs contact 1-form _{3} do not hold for non-equilibrium thermodynamics.

Another example of interesting formulations in equilibrium thermodynamics are the so-called Maxwell relations (e.g. see [

The second relation shows the correct set of canonical coordinates to completely specify internal energy _{2} and substitute the result in the above Pfaffian equation to obtain

Mixed derivatives that originate from a closed 1-form are symmetric^{11}. So, differentiate the first relation with respect to volume

Based on the Poincaré lemma, it can be proven that every exact 1-form is also a closed 1-form (see [

The term between parentheses has to vanish since_{2} shows that the coefficients

This is one of the familiar four Maxwell relations [

So, the thermodynamic potential (viz. here

Classical Non-Equilibrium Thermodynamics (CNET) is the science of phenomenological changes in the state of systems. Especially these changes make the tools of differential geometry extremely suited for a rigorous mathematical formulation. The presented formulation for non-equilibrium thermodynamics consists of the following three base postulates:

・ there exists a non-equilibrium thermodynamic phase space (Gibbs state space),

・ energy is conserved (first law of thermodynamics),

・ the difference between heat flow and dissipation is completely integrable (second law of thermodynamics).

The thermodynamic phase space is built upon macroscopic properties of a system. Neither temporal nor spatial coordinates are considered in the presented mathematical formulation of non-equilibrium thermodynamics. The direction of processes is not implemented by an explicit inequality in the second law of thermodynamics. Instead an implicit reverse is applied: there is a physical quantity that is nowhere equal to zero, viz. the dissipation of energy. Furthermore, no explicit signs are assigned to nowhere vanishing physical quantities. Still this formulation shows clearly that there can only be a periodic process if and only if it is driven by the appropriate heat flow.

Together with the definition of a state function, other relations are derived based on the mathematical properties of this formulation. The mathematical structure for the Gibbs fundamental relation follows from the Gibbs state space. Through a clever combination with both laws of thermodynamics it can be proven that the thermodynamic potential is specified by the Gibbs free energy. Other coordinates in the formulation of the Gibbs fundamental relation are identified for a single component, closed system. Notice that there is neither in the postulates, nor in the derivation of the mathematical framework for non- equilibrium thermodynamics any assumption that restricts the arbitrary processes to certain classes (e.g. periodic, adiabatic, sufficiently slow, etc.).

The introduction of an equilibrium state is postponed until the final section, where it is introduced as an extremum (viz. no change of a state function). This is mathematically implemented by the pull-back of an equilibrium map with the condition that there is no dissipation of energy. Such an elaborate construction is required, because the dissipation of energy is in the general case of non- equilibrium thermodynamics by definition unequal to zero. For the case of an equilibrium state, the Gibbs fundamental relation reduces to a Pfaffian equation. It is shown that the Maxwell relations in thermodynamics can be derived only for the case of an equilibrium state.

Knobbe, E. and Roekaerts, D. (2017) Coherent Application of a Contact Structure to Formulate Classical Non-Equilibrium Thermodynamics. Modern Mechanical Engineering, 7, 8-26. https://doi.org/10.4236/mme.2017.71002