Identity

The following statements are equivalent:

  • Identity is one.

  • Identity is unique.

  • Identity is one, which is all together.

  • Identity is uniform, that is, Identity smoothly positions the various relative forms together to identify them as one.

  • Various forms can be identified together to give one Identity.

  • Expressing completeness, we say that a unique Identity is conserved.

To present these statements as one, we use the concept of a cycle. The symbol usually used to denote Identity is \(0\). I like to interpret this symbol to be a cycle: \(\circ\). When a cycle (\(\circ\), circle) is unique, there is no difference between any of its points which shows uniformity. Naming is Identification. One may Relatively Name the points in a cycle to obtain various relative forms (forming Phases in a cycle). The cycle itself is uniform without any Relative Names given. In other words, ‘Cycle’ is the unique name given to identify various forms which make up the cycle. This shows how the concept of a cycle expresses all of the above equivalent statements. Thus, Identity is Cycle. Cycle is Identity.

  • (Example). Water cycle is an example of Identity as a cycle. Water is the Identity. The Water Cycle can be relatively named at points to give a distribution of various forms. For example, Condensation and De-condensation are relative names given to a points. Correspondingly, Solid Liquid and Gas are Relative identities. In the Physical Water Cycle, the Identity is Amount of Water. When there is a unique cycle of Water (no interaction with anything else), the amount of water in the cycle remains conserved. The forms are named as water droplets, water bodies and water vapour. These forms cycle from water droplets to water bodies to water vapour in one way of flow. The name given to identify two forms in this flow is de-condensation (evaporation). The other half is Condensation which forms

Summary of what we have seen:

  • Identity as a cycle.

  • Naming of points in a cycle as identification of two phases.

  • Naming of Phases in a cycle as identifying two points.

Points and Phases are complementary to each other. A pair of points is identified by a Phase and a pair of Phases is identified by a point. Naming is Identifying. Identification can be done at relative levels.

Let Identity of Thermodynamics be Energy

Cycle is Identity. In Thermodynamics, let it be named (identified) as Energy. In Thermodynamics, we see how to smoothly translate from one form of Energy to another form of Energy. I repeat: various forms of Energy may be Identified to a unique Identity named Energy. Energy is allowed to translate smoothly from one form to another form in a cycle. The cycle which identifies all forms together is the Energy Identity.

Naming Relative Identities of Energy forms a Scale

The various forms of Energy are formed by giving names to them as Relative Identities. Every form (Relative Identity) has its complementary form which pairs together to give Energy as their Identity. To classify, we may name Energy as the Absolute Identity of Thermodynamics. Compared to Energy (Absolute Identity), the various forms of Energy are named as Relative Identities. Within Relative Identities, there are further classifications (relative forms of relative Identities). A Scale is naming (classification) of Relative Identities. So we have obtained a scale of Identities. Each classification name (level) in a scale is called a degree (level). At each Degree, Relative Identities cycle together (identify) to form that Degree as their Identity. This can be imagined as a family tree. Thus a scale of Relative Identities is a family of Identities. First Relative Identities connect to form the Absolute Identity. Second Relative Identites connect to form First Relative Identities. Third Relative Identities connect to form Second Relative Identities and so on. Any usual scale is a prototype example of a Scale. In Distance Scales, Distance is the Absolute Identity. Various forms of Distance are Relative Identities. For example, centimeter, meter and kilometer are Relative Identities. These classifications form degrees (levels) as centimeter is lower in degree than meter which is lower in degree than kilometer. Various ways to name (classify) distance gives rise to different scales. Each degree has its own scale. For example, centimeter scale exists at its own degree and so does the meter scale and the kilometer scale. Some examples are:

  • Let’s see the specific example of a metre scale. Distance is the Identity. That is, the total Length (distance) of the scale forms the Identity. Each marking on a metre scale is a relative Identity. On either side of a marking are two Relative Distances which together form the total Length (Identity) of the scale. For example, if the total legth of the scale is \(1\) metre and there are \(10\) markings, we have \((0,10), (1,9), (2,8), (3,7), (4,6), (5,5), (6,4), (7,3), (8,2), (9,1), (10, 0)\) as the relative Identities corresponding to the markings \(0,1,2,3,4,5,6,7,8,9,10\).

  • A family tree or Lineage is an example of a scale. At any given point, except the Highest and the Lowest, there are higher and lower points in the form of branches. A complementary pair (Parents) are joined together to form a child as the Identity. Further, the child may pair up and form another relative child.

  • In Thermodynamics, complementary pairs are named Intensive and Extensive. That is, Intensive and Extensive are Relative Identities which pair up to form Energy as the Identity.

For simplicity, at some point in the scale let us consider two forms of Identity. Name the two forms \(A\) and \(B\). Each of \(A\) and \(B\) are relative Identities when compared to Energy (\(E\)). Equilibrium is the name given when two relative Identities are Identified to a unique one. The name equilibrium corresponds to the mathematical concept of equals to (\(=\)) or balance. Thus, Unique Identification is Equilibrium. This can be simplified to: see a pair of complementary relatively lower Identites which identified together at the Boundary between them complete a cycle. This Cycle is the relatively Higher Identity.

Cycle \(\circ\) Energy is Conserved

Cycle is Identity. Identity is Conserved. This is the same as “Identity is unique”. Identity Cycle is unique. Energy is the Identity is the Unique Cycle in Thermodynamics. Stated the other way round, a conserved quantity is an Identity (at appropriate scale).

\(\circ\) Connection is Smooth Translation (Flow)

A cycle is called “not conservative” or “non-conservative” if it is not the Unique Cycle (Identity) but one part of it which is separated from the rest. A separate part is formed if only if there is a division of the Unique Identity Cycle into two or more parts by boundaries. Let us consider a Unique Boundary which divides the cycle into two parts. These two parts are a Complementary pair that are connected by a boundary to all together form the Identity Cycle. The two parts are relative Identities in themselves. Hence one may consider them as cycles individually that are connected through the Boundary as their Identity. That is to say that if a cycle is not unique, it is completed (complemented) by connecting it to its “complementary pair” and form a Unique Identity Cycle. The process of Connection is smooth translation to continuously extend a form to its complete Cycle. This can be done if the boundary which bounds one from being complete with it’s other part is Identified. This is the same as one part identifying the other. It is the same as meeting of two parts. All processes essentially do the same as Identifying a Unique Cycle.

Cyclic \(\circ\) Connection of System and Surrounding by Boundary

A system in thermodynamics is the part of Space Identity bounded by a Unique Cyclic Boundary (think of a loop). That is, the Universal Space as a Unique Cyclic Identity is divided into system and its complementary pair (the surrounding which is the Space other than the System), by a boundary which separates the two. Thus, System, boundary and Surrounding together form the Unique Space Identity. The System, and Surrounding are Relative Identities, connected together by the Boundary which becomes their meeting or their Identity.

Reservoir is Relative Identity

Identity Cycle can be separated into System and Reservoir. Reservoir of Heat or other relative Identities of Energy are relative Identities which remain relatively unaffected by changes in the system connected to it. This is a statement of scales (measure). Reservoir is assigned a relatively much larger Measure than the system which has a relatively smaller measure so that the changes in Reservoir are negligible at the scale of the system where measures are being recorded.

ASymmetry is Balance at Relatively less Scale and Imbalance at Relatively more Scale

Each Relative Identity is itself a cycle. Hence it is relatively conserved . Whenever there are two complementary pairs which form a Cycle, they are distinguishable because of their Relative Scales (Measure). If they had the same scale (measure), they would be in perfect balance with each other and hence in Equilibrium (by definition, equilibrium is balance or symmetry (same measure)). Thus, Asymmetry is inherent to distinguish between two complementary pairs which together form a cycle. For uniformity in naming, we name the relative less measure Identity as Differential form and the relative more measure Identity as Integral form. These have been given a variety of names in literature. Some of the names are:

  • Mathematics: Less and More, or, \(-\) and \(+\)

  • Natural Science: Condensed and Fluid

  • Calculus: Differential Form and Integral Form

  • Symmetry: Image and Object, or, Reflection and Object

  • Directions: Left and Right

  • English : Differentiated and Identified

  • Scale: Imbalance and Balance

  • Thermodynamics:

    • Intensive and Extensive

    • Non-Equilibrium and Equilibrium

  • Groups : Inverse \(a^{-1}\) and Element \(a\).

Some examples of Complementary Pairs in an Identity Cycle are:

  • Odd and Even are complementary pairs. Odd is relatively less in measure and Even is relatively more in measure.

  • \(-\) and \(+\) are complementary pairs which Identify to \(0\) (Cycle). \(-\) is relatively less in measure and \(+\) is relatively more in measure. These may also identify to \(S^1\) (Cycle).

  • Complex Numbers \(\bar{z}\) and \(z\) are complementary pairs which Identify to \(\| z \|^2\) (Identity called distance or Modulus). \(\bar{z}\) is relatively less in measure and \(z\) is relatively more in measure.

Revisiting the usual Story of Thermodynamics (BTS, Behind the Scenes)

Zeroth Law

This names a Cyclic Identity called Temperature of a System to be a relative Identity of Energy (The other relative part which completes the Identity of Energy cycle is called Entropy, it is the complementary pair of Temperature which will be introduced later). Further states that two systems are Identified with each other (to be at the same Temperature) if they are individually Identified to a common unique system (Identity). This is a statement of associative composition of the Temperature Identity. Usually, the Identification of Temperature of two systems is named Thermal Equilibrium. In Mathematical Language, we may see Thermal Equilibrium to be a Cyclic Identity with the three systems in equilibrium forming a cyclic group of degree three. This is isomorphic to the cube roots of Identity \(\{1,\omega,\omega^2\}\). Thus, the Zeroth Law says that whenever \(\omega\) and \(\omega^2\) are Identified to \(1\), we way complete the cycle and Identify \(\omega\) with \(\omega^2\). Written algebraically, if the two systems are named \(B\) and \(C\) which are in thermal equilibrium with a common system \(A\) then this is expressed as:

\[B-A-C \iff B=A=C\]

First Law of Thermodynamics

This names Energy to be a Cyclic Identity of a System. The two complementary forms of Energy are given relative Identities (names) Work and Heat. Thus, Energy Identity of a system may be

\[E = W + Q\]

Second Law of Thermodynamics