Abstract

See Equilibrium in Thermodynamics to be a form of mass balance. The “distance” of how far one is from equilibrium is measured by the Thermodynamic potentials. Scalar Potentials are to be seen as a form of distance (measure) with the appropriate Identity to be the Equilibrium Phase boundary. In mathematics language, Thermodynamics is a symmetry, which is same measure of asymmetric forms of energy (mass).

Intro

First, we must identify all physical quantities to be materials formed by phase composition of duals (two opposite end components in a Phase Diagram). For example,

  • Space-time is formed by composition (mixture) of Space as one component and time as the other

  • Energy in one form is formed by composition (mixture) of Volume (\(V\)) and pressure (\(p\))

  • Energy in another form is by composition (mixture) of Force (\(F\)) and displacement (\(x\))

  • Speed is formed by composition of Space (\(X\)) and time (\(t\))

  • Each fundamental quantity is itself formed by composition of High and low amounts

  • Thermodynamic Potentials are formed by composing (with orientations \(-\) and \(+\)) the composition of Extensive and intensive quatities which give forms of energy.

There are different levels at which Equilibrium occurs. Consider them one by one:

  1. Equilibrium for single chemical component material : Intensive and extensive variables can be treated as duals to each other (opposite ends that represent the components of a Phase Diagram). Their product (for example \(P \cdot V\)) is a measure of mass of energy in the form of \(PV\) (% wt fraction along one Phase) . On the other hand there is \(T \cdot S\) which gives a measure of mass of energy in the form of \(TS\) (% wt fraction along the other Phase). Equilibrium means both forms of mass are balanced. \(PV-TS=0\). ‘Gibbs free energy’ \(G= PV-TS\) is what measures how far one is from this balance.

  2. When the Equilibrium is to be considered among two materials, we must “match” the Energy in the form of mass of one material with the Energy in the form of Mass of the other material which is the condition \(G_1 = G_2\) or \(G_1 - G_2 = 0\), which in the differential form is \(dG=0\).

  3. When Equilibrium is to be considered at the scale of all materials, assigning the value of Gibbs free potential energy is relative. A material at higher free energy will go towards equilibrium with that of a lower one, hence spontaneous reactions are given by the condition \(dG<0\).

Heat flow in materials from higher temperature to lower temperature can be seen as the prototype flow which connects two end components (dual phases) to bring them together to equilibrium. Similar Heat-type flow connects positive and negative charges: arrows of electric and magnetic fields show the flows. This flow in fluids also connects higher potential to lower potential. Einstein’s Gravity (Ricci type flow) connects points in space of higher curvature with those of lower curvature and so on. Gravitational Field is a Heat type flow.

Differentiation and Integration

The investigation into thinking led to the following observations:

  • Mental processes may be reduced to two fundamental processes- thought and abstraction.

  • Thought is division. Thought is representing observations in particular form, ie. Language. Thought captures observations by naming. Creating Boundaries is needed to name things. Specifying something needs Boundaries.

    • For example, saying something about “this” needs creation of a boundary which separates “this” from the rest, which is called “that”. In the case of Real Numbers, we can talk about positive numbers only if we create a boundary stating that everything else is negative. Boundaries are needed to differentiate between things.

    • Mental structures or abstractions such as triangles can be distinguished from squares by forming equivalence relations to define partitions among all Polygons which separate the triangles from the squares. Such partitions are boundaries differentiating all Polygons into various disjoint Subsets (which I will call Phases). In this sense, abstract thoughts (or structures) are creation of boundaries (Phase Boundaries).

  • One may restate the above point as: thought is “Dualistic”, since creating a boundary implies creation of two sides. Positive numbers exist if only if negative numbers exist. In looking at one side, one misses out on the other. Division (Thought) destroys Symmetry. The same “Thing” gets Differentiated into two distingushable objects. Thought (Naming) Differentiates the two.

    • For example, in the process of Evolution, at one point, there’s only a single body but when it gets separated by a boundary of physical division, we name one as ‘This’ and the other as ‘That’. This symmetry breaking process gives rise to ‘Dual Phases’. Parent and Child both come into existence simultaneously. Positive and Negative, Higher and Lower, Left and Right, Rigid and Fluid, are all examples of Dual Phases.

  • Abstraction is the “reverse” of thinking. Abstraction is dissolving the boundaries that were created by thought. Abstraction relases captured observations from particular forms (Language). Abstraction forgets naming, which requires dissolution of boundaries that create the names. Why “reverse” was used will be clear later.

    • For example, even though the bodies may get separated by a boundary of physical division, but we share a common name for both the Parent and the Child: the abstraction which dissolves the two different names (boundary) gives the notion of a Family. This symmetry restoring process Integrate the Parent and Child into one simultaneously.

  • In other words, Abstraction and Thought are Duals to each other.

  • Knowledge is conserved. The “same amount of Knowledge” is represented in various forms of a Language. This statement resembling “Mass Balance” or any other “Conservation Law” will take some time to explain, just hold on.

I would like to focus only on these two fundamental operations involving mental structures- thought and abstraction as explained above. To avoid much debate away from the focus of this write up, we introduce two new terms to replace ‘thought’ and ‘abstraction’. Let us call the process of creating boundaries (thought) as ‘Differentiation’ and the process of dissolving them (abstraction) as ‘Integration’. Note that the capital D and I are used to avoid confusion with the terms from the corresponding operations used in the Language of Mathematics, which we will later see to be particular cases of the former.

  • Language Duality as a symmetry:
Differentiation Integration

Theory of States of Matter - An Elementary Language

The simplest theory to pick and demonstrate the ideas involved in Language is probably the Language of States of Matter.

  • Take a simple ( restrict to one dimension) system like Water which exists in different states of Matter. In this Language, Differentiation takes the form of Condensation and Integration takes the form of Evaporation. To keep consistency with the rest of the write up, we say that here Condensation plays the role of Differentiation and Evaporation plays the role of Integration. Starting from a Fluid form (water vapor or liquid water), Differentiation gives Solid form (ice) (hereafter I shall replace the word ‘Solid’ by ‘Condensed’). The Phase Boundary or the interface between the two is the Structure of this Language. Each form- Condensed or Fluid is a representation of Water in the Language of States of Matter.

  • Differentiation in this Language gives the notion of Rigidity. The Condensed Language is called relatively more Rigid. We see that Differentiation (adding structure) leads to greater Rigidity whereas Integration leads to lesser Rigidity. Lesser Rigidity means less structure and hence the boundaries of fluids (gases and liquids) are not well formed. Instead of writing more rigid and less rigid, I shall write more Rigid and more Fluid.

  • Differentiation destroys Symmetry, Integration creates Symmetry. This is because Differentiation is an additional boundary which differentiates between what should have been indistinuguishable.

  • The Differentiated forms are now restricted by the additional boundary compared to its Integral form. This is decrease in the Degrees of Freedom. Integrated forms have less boundary which allows more freedom. This is increase in the Degrees of Freedom.

  • At the Phase Boundary (which is the interface), both the Condensed and Fluid forms meet. There is “Equilibrium” at the Phase Boundary if there is “No Differentiation”. That is, the Language of the Condensed Form matches the Language of the Fluid Form. To reiterate: The Structure of Condensed Form must match the Structure of Fluid Form for “Equilibrium” at the Boundary. Since there is no way to Differentiate between the two Structures at Equilibrium on the Phase Boundary, there are no Boundaries of this Language (Structure) within the Phase Boundary at equilibrium. Thus, when restricted to a Phase Boundary at Equilibrium, within itslef there is No Differentiation of Phases in this Language. Simply stated, this means that one cannot distinguish between ice (Condensed Phase) and Water (Fluid Phase) at the interface where there is Equilibrium between the two phases. There maybe sub-Boundaries and sub-Phases of the given Phase

  • The act of differentiating once creates a boundary and further differentiating within the equilibrium boundary gives nothing (no further differentiation is possible). Since Differentiation in general, decreases the Degrees of Freedom, it must be the case that the final differentiation is not possible since the Degrees of Freedom have already all been reduced to nothing. That is, the Equilibrium Language at the Phase Boundary must be the most Rigid Structure before all its Degrees of Freedom vanish on Differentiating further and it becomes the Language with no Degree of Freedom (Absolutely Rigid Language).

  • To capture The above Paragraph using symbols, let \(=\) represent the words ‘is the same in Structure as’ let \(d\) represent differentiation particular to the Language whose Phases are being distinguished by it, let the symbol \(0\) denote the Language with no Degrees of Freedom with respect to this Language, that is the absolutely Rigid Language, let the symbol \(E\) stand as a variable for all the various possible Phases within this Language, and let the symbol \(E*\) denote the Equilibrium Phase at the Phase Boundary of the Language. In symbols the above paragraph says:

    • \(- + = 0\) or \(-dd+ =0\)

    • \((d E)_{E*} = 0\).

    • The first equation above, holds as a property of the Language of Differentiation. That is, be it any Language (Structure), one cannot create a Boundary of that Language within a Boundary already made in that Language. This is clear since, any sub-Boundary within itself within two Phases on either side, any other boundary which also Differentiates the two must coincide with the same. The second equation above holds at every Degree of a Language independently.

  • (As a sidenote) The above paragraph is an example of a language differentiation process which takes English Language and reduces its degrees of freedom by compressing it to rigid symbols. That is, a statement of some length (one dimension) was reduced to a Rigid symbol of zero dimension, which is a loss of degree of freedom. This differential form of English Language is called Language of Mathematics (this will be seen later).

  • To clarify: this doesn’t mean that there is no structure in equilibrium at the phase boundary, there is an Equilibrium Phase (\(E*\)) at the Boundary which is a structure (compilation of boundaries), just that with respect to this Language (Structure), there is no further differentiation (boundary) possible at equilibrium. Or, there is no way to create another boundary of this Language within the Equilibrium Phase at the Boundary. To state in other words, differentiation (the particular form of Differentiation within this Language) of the Structures in this Language within the Phase Boundary is not possible, which is equivalent to saying that the Difference between two Phases of this Language doesn’t exist at the Equilibrium Phase boundary.

  • Let’s Start with some given Phase (SubLanguage) of a Language. We may obtain another Phase in this Language by Differentiation, that is increasing boundaries (decreasing degrees of freedom) or Integration (increasing degrees of freedom). The more the degree of Differentiation, meaning more boundaries created in a structure, the lesser there are degrees of freedom in a Language. This is going away from Equilibrium. The more the degree of Differentiation, meaning more boundaries created in a structure, the lesser there are degrees of freedom in a Language. This is going away from Equilibrium. This is why the degree of Differentiation in one Phase must Match the degree of Differentiation in its Dual Phase. This takes the form of Maxwell’s Equations or Stokes Theorem in Calculus.

  • Corresponding to every Phase (or degrees of freedom) there are as many ways to partition the Energy. This is what is known as Equipartitioning of Energy. The Measure of Energy given to each Phase across the Boundary must be equal however they take different forms. This is the behaviour that we want to see in all Languages. These are the common features (universal)- Structure as symmetry od Duality or Language as a form of Symmetry of Duality, that we will look for.

  • Thus we have seen Condensed-Fluid Duality:

Condensed Fluid

Intensive-Extensive Duality

  • I shall reinterpret The Language of Thermodynamics in terms of Language Duality introduced above. The Language of Thermodynamics Differentiates Physical Variables into Intensive and Extensive Variables. In this Language, Intensive Variable and Extensive Variable are Dual to each other.

  • Intensive Variable plays the role of Differential Form (Condensed) whereas Extensive Variable plays the role of Integral Form (Fluid).

    • For example, Pressure is an Intensive Variable (Condensed Form). Volume is its corresponding Dual Extensive Variable (Fluid form). Volume when Restricted by a Boundary (decrease in Degrees of Freedom) gives Pressure. Or, Pressure is obtained by Differentiation of Volume. Volume is obtained by Integration of Pressure, to give it additional Degrees of Freedom.
    Pressure Volume
    • For example, Temperature is an Intensive Variable (Condensed Form). Entropy is its corresponding Dual Extensive Variable (Fluid form). Entropy when Restricted by a Boundary (decrease in Degrees of Freedom) gives Temperature. Or, Temperature is obtained by Differentiation of Entropy. Entropy is obtained by Integration of Temperature, to give it additional Degrees of Freedom.
    Temperature Entropy
    • Force-Extension

  • The “Amount of Mass” in Thermodynamics is the Potential Energy. Thermodynamics says that Energy is conserved no matter what forms it is divided into.

### Summary

Thermodynamics is identified as Mass Balance. Calculus is identified as Thermodynamics.