# Entropy of the universe tends to a maximum A visual of Clausius' famous 1865 the "entropy of the universe tends to a maximum" statement, an overly condensed and truncated version of the second law of thermodynamics.

In statements, entropy of the universe tends to a maximum (LH:2) is the model which states that if the universe is divided into volumes, each volume expanded and contracted by heat, in heat cycles, i.e. Clausius cycles, that at the end of a series of such transformative cycles, the volumes will produce work, on the surroundings, and a "change" will accrue in the "working substance" (or working body), i.e. matter, of the volume, this change measured by what is called the sum of the "equivalence values" of "uncompensated transformations", symbol N, a quantity that will be at a maximum "numerical" value, i.e. a positive integers (whole numbers) 1, 2, 3, etc, at the end of the series of cyclical transformations.

In 1865, Rudolf Clausius, reduced the phrase "sum of the equivalence values of uncompensated transformations", symbol N, into the the term phrase "transformation content", then renamed this, using Greek word for transformation (τροπή), into the shortened term "entropy". Hence, in stead of saying the "value N", of all cyclical transformations, of all systems, tends to a maximum, he truncated all of this into the condensed phrase: the "entropy" of the "universe" tend to a maximum, wherein all the former meaning is lost.

In short, whenever work is produced by heat, in the cyclical passage of heat from a hot body, e.g. the sun, to a "cold body", e.g. snow, in a cyclical manner, an irreversible "change" occurs in the "working body", e.g. a field of grass, related to the work the molecules of the system do on each other, irreversibly. It is convoluted way of saying that "caloric" particles (Lavoisier, 1787) are conserved entities. It is sort of a thermodynamic way of saying that "you can never step in the same river twice." (Heraclitus, 485BC).

## Overview

In 1850 to 1865, Rudolf Clausius, in his Mechanical Theory of Heat, a collection of nine journal article publications and talks, upgraded the Lavoisier-Carnot model of the heat engine, wherein NO change occurs in the "working body" of a heat engine, e.g. body of "water" expanded and contracted in the steam engine, to one wherein change DOES occur in the "working body". The working body can be an volume of the universe, such as the field of grass shown adjacent, expanded and contracted by heat, thereby producing external work therefrom.

In this publication, Clausius developed a number of "characteristic functions" (rows: 11 to 14) to quantify and explain this "change", which occurs irreversibly in the working body, via concepts such as: equivalence values, positive transformations, negative transformations, uncompensated transformations, sum of the equivalence values of all uncompensated transformations, etc.,

In 1865, Clausius tried to condense all of these complex phrases into the catch phrase "entropy of the universe tends to a maximum", aka "entropy increase" as we now think of it, which has resulted in an immense amount of ongoing confusion in modern thermodynamics. In short, modern people, owing to lack of detailed explanation, lost by truncating about five different terms, phrases, and concepts into the phrase "entropy tends to a maximum", means "disorder tends to a maximum", which is not the case. In fact, Clausius, in all of his publications and lectures, never used the term "disorder" once!

### Maximum

To explain, if we key word search though the 1865 edition of Mechanical Theory of Heat for the term "maximum" we find the following meaning employed:

“A transmission of heat from a warm body to a cold one certainly takes place in those cases where work is produced by heat, and the condition fulfilled that the body in action is in the same state at the end of the operation as at the commencement. It is this maximum of work that must be compared with the transmission of the heat; and we hereby find that it may reasonably be assumed, with Carnot, that the work depends solely upon the quantity of heat transmitted, and upon the temperatures t and τ of both bodies A and B, but not upon the nature of the substance which transmits it. This maximum has the property, that, by its consumption, a quantity of heat may be carried from the cold body B to the warm one A equal to that which passed from A to B during its production.”
— Rudolf Clausius (1850), Mechanical Theory of Heat (§: “On the Moving Force of Heat and the Laws of Heat which may be Deduced Therefrom”, pgs. 43-44)

This "maximum" work obtainable, from the passage of heat from the cold body, through the working body, to the hot body, occurring such that a change in the working body occurs during the cycle, remains Clausius' focus in the next 15-years of publications, of his nine memoirs which comprise his Mechanical Theory of Heat textbook. The only other time he uses "maximum" is in respect to the maximum density of vapor. The term "maximum" used here, to clarify, has NOTHING to do with maximal disorder.

### N value

Correctly, building on Carnot's model that one can calculate the "maximum" efficiency of a heat engine, solely by knowledge of the temperatures of the hot body and cold body, Clausius deduces his "N" value (see: characteristic functions, rows: 11 to 12), which is the measure of the so-called "equivalence change" in the transformation of the working body. This value, is equal to zero in a ideal "reversible" scenario, i.e. that conceived by Lavoisier and Carnot, but will have a positive value in real "reversible" scenarios, of work cycles, actually seen in nature; this is formulated as follows: | Reversible cycle | Irreversible cycle

Hence, as all real processes are "irreversible", the value of N will always increase.

### Tends to maximum

Thus, what Clausius means when he says "tends to a maximum", translates as "value of N tends to a maximum", which, scaled up to the entire universe, divided into volumes, is what he means when, at the end of his Mechanical Theory of Heat, when he says that the entropy of the universe tends to a maximum:

“For the present I will confine myself to the statement of one result. If for the entire universe we conceive the same magnitude to be determined, consistently and with due regard to all circumstances, which for a single body I have called entropy, and if at the same time we introduce the other and simpler conception of energy, we may express in the following manner the fundamental laws of the universe which correspond to the two fundamental theorems of the mechanical theory of heat.
1. The energy of the universe is constant.
2. The entropy of the universe tends to a maximum.
— Rudolf Clausius (1865), Mechanical Theory of Heat (pg. 365)

Here, to repeat again, this does not translated as the "disorder of the universe tends to a maximum", but rather, as he says in the previous 1850 quote, that this "maximum" has a property related to "consumption", of heat transforming into work, in the working substance or body, which tends to a maximum. The numerical value of this "maximum" is Clausius' N value (characteristic function rows: 11-14).

### Transformation equivalents

But, to understand this so-called maximum "N value" correctly, one has to digest Clausius model of "transformation equivalents" and uncompensated transformations, as defined by rows 10-14 above, which is not simple; as Maxwell famously stated:

“By the introduction of the expression ‘without compensation’ (verses ‘of itself’), combined with a full interpretation of this phrase, the statement of the second principle: ‘that heat cannot without compensation pass from a colder to a warmer body’, becomes complete and exact; but in order to understand it we must have a previous knowledge of the theory of transformation-equivalents, or in other words ‘entropy’, and it is to be feared that we shall have to be taught thermodynamics for several generations before we can expect beginners to receive as axiomatic the theory of entropy.”
James Maxwell (1878), “Tait’s Thermodynamics” 

Basically, "entropy tending to a maximum", in the universe, means that "change", in nature, occurs, which is quantified by N > 0, meaning that when heat produces work in the universe, the bodies or systems that produce this work "change" irreversibly, in respect to the heat NOT being conserved, but rather transformed, irreversibly in respect to the non-reversible work the molecules of the system do on each other.

## Other

### Planck model

See main: Planck entropy

In 1900 to 1910s, Max Planck, in order to solve the problem of how the brightness of light from a light bulb is function temperature, developed a so-called "hypothesis of elementary disorder", wherein he argued that atomic systems are disordered elementally, that they are governed by the "law of accidents", and that a dice throwing model explains what "entropy increases" means. Planck's model of entropy, however, was confused in his conceptual understanding of which of "body" of the heat engine is where the entropy increase occurs, Planck thinking that this occurs in the two heat reservoirs, i.e. the hot body and cold body. Nevertheless, his model, became very popular with physicists, and many people believe his model to be correct, and hence that entropy = disorder, and that entropy increase means "disorder increase". This scientific folklore began to become prevalent in the 1925s and thereafter.