Warming stripes

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Warming_stripes

An early (2018) warming stripes graphic published by their originator, climatologist Ed Hawkins. The progression from blue (cooler) to red (warmer) stripes portrays annual increases of global average temperature since 1850 (left side of graphic) until the date of the graphic (right side).

Warming stripes (sometimes referred to as climate stripes, climate timelines or stripe graphics) are data visualization graphics that use a series of coloured stripes chronologically ordered to visually portray long-term temperature trends. Warming stripes reflect a "minimalist" style, conceived to use colour alone to avoid technical distractions to intuitively convey global warming trends to non-scientists.

The initial concept of visualizing historical temperature data has been extended to involve animation, to visualize sea level rise and predictive climate data, and to visually juxtapose temperature trends with other data such as atmospheric CO₂ concentration, global glacier retreat, precipitation, progression of ocean depths, aviation emission's percentage contribution to global warming, biodiversity loss, soil moisture deviations, and fine particulate matter concentrations. In less technical contexts, the graphics have been embraced by climate activists, used as cover images of books and magazines, used in fashion design, projected onto natural landmarks, and used on athletic team uniforms, music festival stages, and public infrastructure.

Background, publication and content

Conventional graphic versus warming stripes graphic

This conventional graphic includes date ranges, explanatory legends, and technical terminology that warming stripes avoid.

 

This composite of a conventional line chart superimposed on a warming stripe graphic illustrates year-by-year correlation of data points and coloured stripes.

"I wanted to communicate temperature changes in a way that was simple and intuitive, removing all the distractions of standard climate graphics so that the long-term trends and variations in temperature are crystal clear. Our visual system will do the interpretation of the stripes without us even thinking about it."

— Ed Hawkins, May 2018

 

Hawkins chose colours from a broader palette originally designed for distinguishing areas in maps.

 

A colour field abstract artwork

In May 2016, to make visualizing climate change easier for the general public, University of Reading climate scientist Ed Hawkins created an animated spiral graphic of global temperature change as a function of time, a representation said to have gone viral. Jason Samenow wrote in The Washington Post that the spiral graph was "the most compelling global warming visualization ever made",^[27] before it was featured in the opening ceremony of the 2016 Summer Olympics.

Separately, by 10 June 2017, Ellie Highwood, also a climate scientist at the University of Reading, had completed a crocheted "global warming blanket" that was inspired by "temperature blankets" representing temperature trends in respective localities. Hawkins provided Highwood with a more user friendly colour scale to avoid the muted colour differences present in Highwood's blanket. Independently, in November 2015, University of Georgia estuarine scientist Joan Sheldon made a "globally warm scarf" having 400 blue, red and purple rows, but could not contact Hawkins until 2022. Both Highwood and Sheldon credit as their original inspirations, "sky blankets" and "sky scarves" which are based on daily sky colours.

On 22 May 2018, Hawkins published graphics constituting a chronologically ordered series of blue and red vertical stripes that he called warming stripes. Hawkins, a lead author for the IPCC 6th Assessment Report, received the Royal Society's 2018 Kavli Medal, in part "for actively communicating climate science and its various implications with broad audiences".

As described in a BBC article, in the month the big meteorological agencies release their annual climate assessments, Hawkins experimented with different ways of rendering the global data and "chanced upon the coloured stripes idea". When he tried out a banner at the Hay Festival, according to the article, Hawkins "knew he'd struck a chord". The National Centre for Atmospheric Science (UK), with which Hawkins is affiliated, states that the stripes "paint a picture of our changing climate in a compelling way. Hawkins swapped out numerical data points for colours which we intuitively react to".

Others have called Hawkins' warming stripes "climate stripes" or "climate timelines".

Warming stripe graphics are reminiscent of colour field painting, a style prominent in the mid 20th century, which strips out all distractions and uses only colour to convey meaning. Colour field pioneer artist Barnett Newman said he was "creating images whose reality is self-evident", an ethos that Hawkins is said to have applied to the problem of climate change.

Collaborating with Berkeley Earth scientist Robert Rohde, on 17 June 2019 Hawkins published for public use, a large set of warming stripes on ShowYourStripes.info. Individualized warming stripe graphics were published for the globe, for most countries, as well as for certain smaller regions such as states in the US or parts of the UK, since different parts of the world are warming more quickly than others.

Hawkins et al. updated the graphs of warming stripes on 2 April 2025. These stripes show temperature trends for both oceans and atmosphere, as ocean trends show that 90% of the extra heat is stored in global oceans. This report explains that "Delayed action in reducing carbon emissions leads to temperature records being significantly exceeded over the next century (with warming stripes moving into much darker shades of red), so that 2024 – the warmest year to date – would eventually be viewed as a cold year."

Data sources and data visualization

Effect of geographic selection: Warming stripes for the Northern and Southern Hemispheres show how different, but same-size, regions compare. Greater recent temperature anomalies in the North display as stripes that are off the red scale.

 

Effect of geographic size: Warming stripes for the Globe and for the Caribbean Islands region show that larger year-to-year variations, for geographical and statistical reasons, are to be expected for smaller regions (bottom graphic).

 

Effect of each colour's temperature range: one dataset, but with different temperature range per colour (colour scales shown on left side). In the top graphic (with 0.10 °C per colour), recent temperatures exceed the red scale; the bottom graphic (0.15 °C per colour) avoids this clipping.

 

Effect of reference period (baseline): One dataset, with averages over three "reference periods" (horizontal purple bars) determining blue/red boundaries. The earliest, lowest-temp baseline (top) causes recent temperatures to exceed the red scale; later baselines avoid this clipping.

 

Effect of choosing a baseline independent of any time period's average value: This stripe graphic of global average sea level change has a baseline that is less than all data values, producing a graphic having shades of only a single colour.

Warming stripe graphics are defined with various parameters, including:

• source of dataset (meteorological organization)
• geographical scope of measurement (global, country, state, etc.)
• time period (year range, for horizontal "axis")
• temperature range (range of anomaly (deviation) about a reference or baseline temperature)
• colour palette (usually, shades of blue and red),
• colour scale (assignment of colours to represent respective ranges of temperature anomaly),
• temperature boundaries (temperature above which a stripe is red and below which is blue, usually determined by an average annual temperature over a "reference period" or "baseline" of usually 30 years).

Hawkins' original graphics use the eight most saturated blues and reds from the ColorBrewer 9-class single hue palettes, which optimize colour palettes for maps and are noted for their colourblind-friendliness. Hawkins said the specific colour choice was an aesthetic decision ("I think they look just right"), also selecting baseline periods to ensure equally dark shades of blue and red for aesthetic balance. Hawkins chose the 1971-2000 average as a boundary between reds and blues because the average global temperature in that reference period represented the mid-point in the warming to date.

A Republik analysis said that "this graphic explains everything in the blink of an eye", attributing its effect mainly to the chosen colors, which "have a magical effect on our brain, (letting) us recognize connections before we have even actively thought about them". The analysis concluded that colors other than blue and red "don't convey the same urgency as (Hawkins') original graphic, in which the colors were used in the classic way: blue=cold, red=warm."

ShowYourStripes.info cites dataset sources Berkeley Earth, NOAA, UK Met Office, MeteoSwiss, DWD (Germany), specifically explaining that the data for most countries comes from the Berkeley Earth temperature dataset, except that for the US, UK, Switzerland & Germany the data comes from respective national meteorological agencies.

For each country-level #ShowYourStripes graphic (Hawkins, June 2019), the average temperature in the 1971–2000 reference period is set as the boundary between blue (cooler) and red (warmer) colours, the colour scale varying +/- 2.6 standard deviations of the annual average temperatures between 1901 and 2000. Hawkins noted in 2019 that the graphic for the Arctic "broke the colour scale" since it is warming more than twice as fast as the global average, and reported that the 2023 global average was so extreme that a new, darker shade of red was required.

For statistical and geographic reasons, it is expected that graphics for small areas will show more year-to-year variation than those for large regions. Year-to-year changes reflected in graphics for localities result from weather variability, whereas global warming over centuries reflects climate change.

The NOAA website warns that the graphics "shouldn't be used to compare the rate of change at one location to another", explaining that "the highest and lowest values on the colour scale may be different at different locations". Further, a certain colour in one graphic will not necessarily correspond to the same temperature in other graphics. A climate change denier generated a warming stripes graphic that misleadingly affixed Northern Hemisphere readings over one period to global readings over another period, and omitted readings for the most recent thirteen years, with some of the data being 29-year-smoothed—to give the false impression that recent warming is routine. Calling the graphic "imposter warming stripes", meteorologist Jeff Berardelli described it in January 2020 as "a mishmash of data riddled with gaps and inconsistencies" with an apparent objective to confuse the public.

Applications and influence

Comparing multiple datasets

 

A conventional line graph comparing several highly correlated temperature datasets

 

A "stacked" warming stripes graphic comparing essentially the same highly correlated temperature datasets as the line graph

 

A "stacked" warming stripe graphic compares temperature datasets for various layers of Earth's atmosphere and oceans

 

Logo, US House Select Committee on the Climate Crisis (formed 2019)

 

Warming stripes at the 2019 United Nations Climate Change Conference (COP25).

 

Demonstrators dressed as warming stripes during an Extinction Rebellion protest in Berlin, Germany (2019).

 

Warming stripes on a bus in Reading, Berkshire, U.K.

 

Warming stripes on the Saxons' Bridge in Leipzig, Germany

After Hawkins' first publication of warming stripe graphics in May 2018, broadcast meteorologists in multiple countries began to show stripe-decorated neckties, necklaces, pins and coffee mugs on-air, reflecting a growing acceptance of climate science among meteorologists and a willingness to communicate it to audiences. In 2019, the United States House Select Committee on the Climate Crisis used warming stripes in its committee logo, showing horizontally oriented stripes behind a silhouette of the United States Capitol, and three US Senators wore warming stripe lapel pins at the 2020 State of the Union Address.

On 17 June 2019, Hawkins initiated a social media campaign with hashtag #ShowYourStripes that encourages people to download their regions' graphics from ShowYourStripes.info, and to post them. The campaign was backed by U.N. Climate Change, the World Meteorological Organization and the Intergovernmental Panel on Climate Change. Called "a new symbol for the climate emergency" by French magazine L'EDN, the graphics have been embraced by climate activists, used as cover images of books and magazines, used in fashion design, projected onto natural landmarks, and used on athletic team uniforms, music festival stages, and public infrastructure. More specifically, warming stripes have been applied to knit-it-yourself scarves, a vase, neckties, cufflinks, bath towels, vehicles, and a music festival stage, as well as on the side of Freiburg, Germany, streetcars, as municipal murals in Córdoba, Spain, Anchorage, Alaska, and Jersey, on face masks during the COVID-19 pandemic, in an action logo of the German soccer club 1. FSV Mainz 05, on the side of the Climate Change Observatory in Valencia, on the side of a power station turbine house in Reading, Berkshire, on tech-themed shirts, on designer dresses, on the uniforms of Reading Football Club, on Leipzig's Sachsen Bridge, on a biomethane-powered bus, as a stage backdrop at the 2022 Glastonbury Festival, on the racer uniforms and socks and webpage banner of the Climate Classic bicycle race, on the World Bank's Climate Explainer Series, projected onto the White Cliffs of Dover, on an Envision Racing electric race car, and on numerous bridges and towers noted by Climate Central. Remarking that "infiltrating popular culture is a means of triggering a change of attitude that will lead to mass action", Hawkins surmised that making the graphics available for free has made them used more widely. Hawkins further said that any merchandise-related profits are donated to charity.

Through a campaign led by nonprofit Climate Central using hashtag #MetsUnite, more than 100 TV meteorologists—the scientists most laymen interact with more than any other—featured warming stripes and used the graphics to focus audience attention during broadcasts on summer solstices beginning in 2018 with the "Stripes for the Solstice" effort.

On 24 June 2019, Hawkins tweeted that nearly a million stripe graphics had been downloaded by visitors from more than 180 countries in the course of their first week.

In 2018, the German Weather Service's meteorological training journal Promet showed a warming stripes graphic on the cover of the issue titled "Climate Communication". By September 2019, the Met Office, the UK's national weather service, was using both a climate spiral and a warming stripe graphic on its "What is climate change?" webpage. Concurrently, the cover of the 21–27 September 2019 issue of The Economist, dedicated to "The climate issue," showed a warming stripe graphic, as did the cover of The Guardian on the morning of the 20 September 2019 climate strikes. The environmental initiative Scientists for Future (2019) included warming stripes in its logo. The Science Information Service (Germany) noted in December 2019 that warming stripes were a "frequently used motif" in demonstrations by the School strike for the climate and Scientists for Future, and were also on the roof of the German Maritime Museum in Bremerhaven. Also in December 2019, Voilà Information Design said that warming stripes "have replaced the polar bear on a melting iceberg as the icon of the climate crisis".

On 18 January 2020, a 20-metre-wide artistic light-show installation of warming stripes was opened at the Gendarmenmarkt in Berlin, with the Berlin-Brandenburg Academy of Sciences building being illuminated in the same way. The cover of the "Climate Issue" (fall 2020) of the Space Science and Engineering Center's Through the Atmosphere journal was a warming stripes graphic, and in June 2021 the WMO used warming stripes to "show climate change is here and now" in its statement that "2021 is a make-or-break year for climate action". The November 2021 UN Climate Change Conference (COP26) exhibited an immersive "climate canopy" sculpture consisting of hanging, blue and red color-coded, vertical lighted bars with fabric fringes.

On 27 September 2019, the Fachhochschule (University of Applied Science) Potsdam announced that warming stripes graphics had won in the science category of an international competition recognising innovative and understandable visualisations of climate change, the jury stating that the graphics make an "impact through their innovative, minimalist design".

Hawkins was appointed Member of the Order of the British Empire (MBE) in the 2020 New Year Honours "For services to Climate Science and to Science Communication".

In April 2022, textiles from haute couture fashion designer Lucy Tammam with warming stripes won the Best Customer Engagement Campaign title in the Sustainable Fashion 2022 awards by Drapers fashion magazine.

In October 2022, the front cover of Greta Thunberg's The Climate Book features warming stripes.

In June 2023, Pope Francis was presented with a warming stripes stole.

In May 2024, Hawkins received the Royal Geographical Society's Geographical Engagement Award for his work in developing warming stripes.

In 2025, warming stripes were included in the "Pirouette: Turning Points in Design" exhibition at New York's Museum of Modern Art, the exhibition highlighting design as an agent of change.

Extensions of warming stripes

A warming stripes colour scheme is applied to a conventional bar chart to visually emphasize changes in temperature. Taller bars are more intensely coloured.

 

Average global temperature are depicted with chronologically ordered, concentric coloured rings.

In 2018, University of Reading post-doctoral research assistant Emanuele Bevacqua juxtaposed vertical-stripe graphics for CO₂ concentration and for average global temperature (August), and "circular warming stripes" depicting average global temperature with concentric coloured rings (November).

Bifurcated graphic of two futures.

In March 2019, German engineer Alexander Radtke extended Hawkins' historical graphics to show predictions of future warming through the year 2200, a graphic that one commentator described as making the future "a lot more visceral". Radtke bifurcated the graphic to show diverging predictions for different degrees of human action in reducing greenhouse gas emissions.

On or before 30 May 2019, UK-based software engineer Kevin Pluck designed animated warming stripes that portray the unfolding of the temperature increase, allowing viewers to experience the change from an earlier stable climate to recent rapid warming.

Comprehensive "stack" of 196 warming stripes for respective countries grouped by continent.

This "stack", technically a heat map, organizes temperatures by month (horizontally) and year (vertically).

By June 2019, Hawkins vertically stacked hundreds of warming stripe graphics from corresponding world locations and grouped them by continent to form a comprehensive, composite graphic, "Temperature Changes Around the World (1901–2018)".

On 1 July 2019, Durham University geography research fellow Richard Selwyn Jones published a Global Glacier Change graphic, modeled after and credited as being inspired by Hawkins' #ShowYourStripes graphics, allowing global warming and global glacier retreat to be visually juxtaposed. Jones followed on 8 July 2019 with a stripe graphic portraying global sea level change using only shades of blue. Separately, NOAA displayed a graphic juxtaposing annual temperatures and precipitation, researchers from the Netherlands used stripe graphics to represent progression of ocean depths, and the Institute of Physics used applied the graphic to represent aviation emission's percentage contribution to global warming.

In 2023, University of Derby professor Miles Richardson created sequenced stripes to illustrate biodiversity loss, and the German Meteorological Service represented soil moisture deviations using sequenced green and brown stripes. In August 2024, the website airqualitystripes.info published shareable "air quality stripes" graphics for world cities, using blue, yellow, orange, red and black stripes to represent fine particulate matter (PM2.5) concentrations over time.

Critical response

Some warned that warming stripes of individual countries or states, taken out of context, could advance the idea that global temperatures are not rising, though research meteorologist J. Marshall Shepherd said that "geographic variations in the graphics offer an outstanding science communication opportunity". Meteorologist and #MetsUnite coordinator Jeff Berardelli said that "local stripe visuals help us tell a nuanced story—the climate is not changing uniformly everywhere".

Others say the charts should include axes or legends, though the website FAQ page explains the graphics were "specifically designed to be as simple as possible, and to start conversations... (to) fill a gap and enable communication with minimal scientific knowledge required to understand their meaning". J. Marshall Shepherd, former president of the American Meteorological Society, lauded Hawkins' approach, writing that "it is important not to miss the bigger picture. Science communication to the public has to be different" and commending Hawkins for his "innovative" approach and "outstanding science communication" effort.

In The Washington Post, Matthew Cappucci wrote that the "simple graphics ... leave a striking visual impression" and are "an easily accessible way to convey an alarming trend", adding that "warming tendencies are plain as day". Greenpeace spokesman Graham Thompson remarked that the graphics are "like a really well-designed logo while still being an accurate representation of very important data".

CBS News contributor Jeff Berardelli noted that the graphics "aren't based on future projections or model assumptions" in the context of stating that "science is not left or right. It's simply factual."

A September 2019 editorial in The Economist hypothesized that "to represent this span of human history (1850–2018) as a set of simple stripes may seem reductive"—noting those years "saw world wars, technological innovation, trade on an unprecedented scale and a staggering creation of wealth"—but concluded that "those complex histories and the simplifying stripes share a common cause," namely, fossil fuel combustion.

Informally, warming stripes have been said to resemble "tie-dyed bar codes" and a "work of art in a gallery". Warming stripes were included in the "Pirouette: Turning Points in Design" exhibition (2025) at New York's Museum of Modern Art, the exhibition highlighting design as an agent of change.

Zeroth law of thermodynamics

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Zeroth_law_of_thermodynamics

 

Thermodynamics
The classical Carnot heat engine

The zeroth law of thermodynamics is one of the four principal laws of thermodynamics. It provides an independent definition of temperature without reference to entropy, which is defined in the second law. The law was established by Ralph H. Fowler in the 1930s, long after the first, second, and third laws had been widely recognized.

The zeroth law states that if two thermodynamic systems are both in thermal equilibrium with a third system, then the two systems are in thermal equilibrium with each other.

Two systems are said to be in thermal equilibrium if they are linked by a wall permeable only to heat, and they do not change over time.

Another formulation by James Clerk Maxwell is "All heat is of the same kind". Another statement of the law is "All diathermal walls are equivalent".

The zeroth law is important for the mathematical formulation of thermodynamics. It makes the relation of thermal equilibrium between systems an equivalence relation, which can represent equality of some quantity associated with each system. A quantity that is the same for two systems, if they can be placed in thermal equilibrium with each other, is a scale of temperature. The zeroth law is needed for the definition of such scales, and justifies the use of practical thermometers.

Equivalence relation

A thermodynamic system is by definition in its own state of internal thermodynamic equilibrium, that is to say, there is no change in its observable state (i.e. macrostate) over time and no flows occur in it. One precise statement of the zeroth law is that the relation of thermal equilibrium is an equivalence relation on pairs of thermodynamic systems. In other words, the set of all systems each in its own state of internal thermodynamic equilibrium may be divided into subsets in which every system belongs to one and only one subset, and is in thermal equilibrium with every other member of that subset, and is not in thermal equilibrium with a member of any other subset. This means that a unique "tag" can be assigned to every system, and if the "tags" of two systems are the same, they are in thermal equilibrium with each other, and if different, they are not. This property is used to justify the use of empirical temperature as a tagging system. Empirical temperature provides further relations of thermally equilibrated systems, such as order and continuity with regard to "hotness" or "coldness", but these are not implied by the standard statement of the zeroth law.

If it is defined that a thermodynamic system is in thermal equilibrium with itself (i.e., thermal equilibrium is reflexive), then the zeroth law may be stated as follows:

If a body C, be in thermal equilibrium with two other bodies, A and B, then A and B are in thermal equilibrium with one another.

This statement asserts that thermal equilibrium is a left-Euclidean relation between thermodynamic systems. If we also define that every thermodynamic system is in thermal equilibrium with itself, then thermal equilibrium is also a reflexive relation. Binary relations that are both reflexive and Euclidean are equivalence relations. Thus, again implicitly assuming reflexivity, the zeroth law is therefore often expressed as a right-Euclidean statement:

If two systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with each other.

One consequence of an equivalence relationship is that the equilibrium relationship is symmetric: If A is in thermal equilibrium with B, then B is in thermal equilibrium with A. Thus, the two systems are in thermal equilibrium with each other, or they are in mutual equilibrium. Another consequence of equivalence is that thermal equilibrium is described as a transitive relation:

If A is in thermal equilibrium with B and if B is in thermal equilibrium with C, then A is in thermal equilibrium with C.

A reflexive, transitive relation does not guarantee an equivalence relationship. For the above statement to be true, both reflexivity and symmetry must be implicitly assumed.

It is the Euclidean relationships which apply directly to thermometry. An ideal thermometer is a thermometer which does not measurably change the state of the system it is measuring. Assuming that the unchanging reading of an ideal thermometer is a valid tagging system for the equivalence classes of a set of equilibrated thermodynamic systems, then the systems are in thermal equilibrium, if a thermometer gives the same reading for each system. If the system are thermally connected, no subsequent change in the state of either one can occur. If the readings are different, then thermally connecting the two systems causes a change in the states of both systems. The zeroth law provides no information regarding this final reading.

Foundation of temperature

Nowadays, there are two nearly separate concepts of temperature, the thermodynamic concept, and that of the kinetic theory of gases and other materials.

The zeroth law belongs to the thermodynamic concept, but this is no longer the primary international definition of temperature. The current primary international definition of temperature is in terms of the kinetic energy of freely moving microscopic particles such as molecules, related to temperature through the Boltzmann constant ${\displaystyle k_{\mathrm{B}}}$. The present article is about the thermodynamic concept, not about the kinetic theory concept.

The zeroth law establishes thermal equilibrium as an equivalence relationship. An equivalence relationship on a set (such as the set of all systems each in its own state of internal thermodynamic equilibrium) divides that set into a collection of distinct subsets ("disjoint subsets") where any member of the set is a member of one and only one such subset. In the case of the zeroth law, these subsets consist of systems which are in mutual equilibrium. This partitioning allows any member of the subset to be uniquely "tagged" with a label identifying the subset to which it belongs. Although the labeling may be quite arbitrary, temperature is just such a labeling process which uses the real number system for tagging. The zeroth law justifies the use of suitable thermodynamic systems as thermometers to provide such a labeling, which yield any number of possible empirical temperature scales, and justifies the use of the second law of thermodynamics to provide an absolute, or thermodynamic temperature scale. Such temperature scales bring additional continuity and ordering (i.e., "hot" and "cold") properties to the concept of temperature.

In the space of thermodynamic parameters, zones of constant temperature form a surface, that provides a natural order of nearby surfaces. One may therefore construct a global temperature function that provides a continuous ordering of states. The dimensionality of a surface of constant temperature is one less than the number of thermodynamic parameters, thus, for an ideal gas described with three thermodynamic parameters P, V and N, it is a two-dimensional surface.

For example, if two systems of ideal gases are in joint thermodynamic equilibrium across an immovable diathermal wall, then ⁠P₁V₁/N₁⁠ = ⁠P₂V₂/N₂⁠ where P_i is the pressure in the ith system, V_i is the volume, and N_i is the amount (in moles, or simply the number of atoms) of gas.

The surface ⁠PV/N⁠ = constant defines surfaces of equal thermodynamic temperature, and one may label defining T so that ⁠PV/N⁠ = RT, where R is some constant. These systems can now be used as a thermometer to calibrate other systems. Such systems are known as "ideal gas thermometers".

In a sense, focused on the zeroth law, there is only one kind of diathermal wall or one kind of heat, as expressed by Maxwell's dictum that "All heat is of the same kind". But in another sense, heat is transferred in different ranks, as expressed by Arnold Sommerfeld's dictum "Thermodynamics investigates the conditions that govern the transformation of heat into work. It teaches us to recognize temperature as the measure of the work-value of heat. Heat of higher temperature is richer, is capable of doing more work. Work may be regarded as heat of an infinitely high temperature, as unconditionally available heat." This is why temperature is the particular variable indicated by the zeroth law's statement of equivalence.

Dependence on the existence of walls permeable only to heat

In Constantin Carathéodory's (1909) theory, it is postulated that there exist walls "permeable only to heat", though heat is not explicitly defined in that paper. This postulate is a physical postulate of existence. It does not say that there is only one kind of heat. This paper of Carathéodory states as proviso 4 of its account of such walls: "Whenever each of the systems S₁ and S₂ is made to reach equilibrium with a third system S₃ under identical conditions, systems S₁ and S₂ are in mutual equilibrium".

It is the function of this statement in the paper, not there labeled as the zeroth law, to provide not only for the existence of transfer of energy other than by work or transfer of matter, but further to provide that such transfer is unique in the sense that there is only one kind of such wall, and one kind of such transfer. This is signaled in the postulate of this paper of Carathéodory that precisely one non-deformation variable is needed to complete the specification of a thermodynamic state, beyond the necessary deformation variables, which are not restricted in number. It is therefore not exactly clear what Carathéodory means when in the introduction of this paper he writes

It is possible to develop the whole theory without assuming the existence of heat, that is of a quantity that is of a different nature from the normal mechanical quantities.

It is the opinion of Elliott H. Lieb and Jakob Yngvason (1999) that the derivation from statistical mechanics of the law of entropy increase is a goal that has so far eluded the deepest thinkers. Thus the idea remains open to consideration that the existence of heat and temperature are needed as coherent primitive concepts for thermodynamics, as expressed, for example, by Maxwell and Max Planck. On the other hand, Planck (1926) clarified how the second law can be stated without reference to heat or temperature, by referring to the irreversible and universal nature of friction in natural thermodynamic processes.

History

Writing long before the term "zeroth law" was coined, in 1871 Maxwell discussed at some length ideas which he summarized by the words "All heat is of the same kind". Modern theorists sometimes express this idea by postulating the existence of a unique one-dimensional hotness manifold, into which every proper temperature scale has a monotonic mapping. This may be expressed by the statement that there is only one kind of temperature, regardless of the variety of scales in which it is expressed. Another modern expression of this idea is that "All diathermal walls are equivalent". This might also be expressed by saying that there is precisely one kind of non-mechanical, non-matter-transferring contact equilibrium between thermodynamic systems.

According to Sommerfeld, Ralph H. Fowler coined the term zeroth law of thermodynamics while discussing the 1935 text by Meghnad Saha and B.N. Srivastava.

They write on page 1 that "every physical quantity must be measurable in numerical terms". They presume that temperature is a physical quantity and then deduce the statement "If a body A is in temperature equilibrium with two bodies B and C, then B and C themselves are in temperature equilibrium with each other". Then they italicize a self-standing paragraph, as if to state their basic postulate:

Any of the physical properties of A which change with the application of heat may be observed and utilised for the measurement of temperature.

They do not themselves here use the phrase "zeroth law of thermodynamics". There are very many statements of these same physical ideas in the physics literature long before this text, in very similar language. What was new here was just the label zeroth law of thermodynamics.

Fowler & Guggenheim (1936/1965) wrote of the zeroth law as follows:

... we introduce the postulate: If two assemblies are each in thermal equilibrium with a third assembly, they are in thermal equilibrium with each other.

They then proposed that

... it may be shown to follow that the condition for thermal equilibrium between several assemblies is the equality of a certain single-valued function of the thermodynamic states of the assemblies, which may be called the temperature t, any one of the assemblies being used as a "thermometer" reading the temperature t on a suitable scale. This postulate of the "Existence of temperature" could with advantage be known as the zeroth law of thermodynamics.

The first sentence of this present article is a version of this statement. It is not explicitly evident in the existence statement of Fowler and Edward A. Guggenheim that temperature refers to a unique attribute of a state of a system, such as is expressed in the idea of the hotness manifold. Also their statement refers explicitly to statistical mechanical assemblies, not explicitly to macroscopic thermodynamically defined systems.

Inclusive fitness

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Inclusive_fitness  

 

Inclusive fitness is a conceptual framework in evolutionary biology first defined by W. D. Hamilton in 1964. It is primarily used to aid the understanding of how social traits are expected to evolve in structured populations. It involves partitioning an individual's expected fitness returns into two distinct components: direct fitness returns - the component of a focal individual’s fitness that is independent of who it interacts with socially; indirect fitness returns - the component that is dependent on who it interacts with socially. The direct component of an individual's fitness is often called its personal fitness, while an individual’s direct and indirect fitness components taken together are often called its inclusive fitness..

Under an inclusive fitness framework direct fitness returns are realised through the offspring a focal individual produces independent of who it interacts with, while indirect fitness returns are realised by adding up all the effects our focal individual has on the (number of) offspring produced by those it interacts with weighted by the relatedness of our focal individual to those it interacts with. This can be visualised in a sexually reproducing system (assuming identity by descent) by saying that an individual's own child, who carries one half of that individual's genes, represents one offspring equivalent. A sibling's child, who will carry one-quarter of the individual's genes, will then represent 1/2 offspring equivalent (and so on - see coefficient of relationship for further examples).

Neighbour-modulated fitness is the conceptual inverse of inclusive fitness. Where inclusive fitness calculates an individual’s indirect fitness component by summing the fitness that focal individual receives through modifying the productivities of those it interacts with (its neighbours), neighbour-modulated fitness instead calculates it by summing the effects an individual’s neighbours have on that focal individual’s productivity. When taken over an entire population, these two frameworks give functionally equivalent results. Hamilton’s rule is a particularly important result in the fields of evolutionary ecology and behavioral ecology that follows naturally from the partitioning of fitness into direct and indirect components, as given by inclusive and neighbour-modulated fitness. It enables us to see how the average trait value of a population is expected to evolve under the assumption of small mutational steps.

Kin selection is a well known case whereby inclusive fitness effects can influence the evolution of social behaviours. Kin selection relies on positive relatedness (driven by identity by descent) to enable individuals who positively influence the fitness of those they interact with at a cost to their own personal fitness, to outcompete individuals employing more selfish strategies. It is thought to be one of the primary mechanisms underlying the evolution of altruistic behaviour, alongside the less prevalent reciprocity (see also reciprocal altruism), and to be of particular importance in enabling the evolution of eusociality among other forms of group living. Inclusive fitness has also been used to explain the existence of spiteful behaviour, where individuals negatively influence the fitness of those they interact with at a cost to their own personal fitness.

Inclusive fitness and neighbour-modulated fitness are both frameworks that leverage the individual as the unit of selection. It is from this that the gene-centered view of evolution emerged: a perspective that has facilitated much of the work done into the evolution of conflict (examples include parent-offspring conflict, interlocus sexual conflict, and intragenomic conflict).

Overview

The British evolutionary biologist W. D. Hamilton showed mathematically that, because other members of a population may share one's genes, a gene can also increase its evolutionary success by indirectly promoting the reproduction and survival of other individuals who also carry that gene. This is variously called "kin theory", "kin selection theory" or "inclusive fitness theory". The most obvious category of such individuals is close genetic relatives, and where these are concerned, the application of inclusive fitness theory is often more straightforwardly treated via the narrower kin selection theory. Hamilton's theory, alongside reciprocal altruism, is considered one of the two primary mechanisms for the evolution of social behaviors in natural species and a major contribution to the field of sociobiology, which holds that some behaviors can be dictated by genes, and therefore can be passed to future generations and may be selected for as the organism evolves.

Belding's ground squirrel provides an example; it gives an alarm call to warn its local group of the presence of a predator. By emitting the alarm, it gives its own location away, putting itself in more danger. In the process, however, the squirrel may protect its relatives within the local group (along with the rest of the group). Therefore, if the effect of the trait influencing the alarm call typically protects the other squirrels in the immediate area, it will lead to the passing on of more copies of the alarm call trait in the next generation than the squirrel could leave by reproducing on its own. In such a case natural selection will increase the trait that influences giving the alarm call, provided that a sufficient fraction of the shared genes include the gene(s) predisposing to the alarm call.

Synalpheus regalis, a eusocial shrimp, is an organism whose social traits meet the inclusive fitness criterion. The larger defenders protect the young juveniles in the colony from outsiders. By ensuring the young's survival, the genes will continue to be passed on to future generations.

Inclusive fitness is more generalized than strict kin selection, which requires that the shared genes are identical by descent. Inclusive fitness is not limited to cases where "kin" ('close genetic relatives') are involved.

Hamilton's rule

Hamilton's rule is most easily derived in the framework of neighbour-modulated fitness, where the fitness of a focal individual is considered to be modulated by the actions of its neighbours. This is the inverse of inclusive fitness where we consider how a focal individual modulates the fitness of its neighbours. However, taken over the entire population, these two approaches are equivalent to each other so long as fitness remains linear in trait value. A simple derivation of Hamilton's rule can be gained via the Price equation as follows. If an infinite population is assumed, such that any non-selective effects can be ignored, the Price equation can be written as:

${\displaystyle \overline{w}\mathrm{\Delta }\overline{z}=\mathrm{cov}(w_i,z_i)}$

Where ${\displaystyle z}$ represents trait value and ${\displaystyle w}$ represents fitness, either taken for an individual ${\displaystyle i}$ or averaged over the entire population. If fitness is linear in trait value, the fitness for an individual ${\displaystyle i}$ can be written as:

${\displaystyle w_i=\alpha +(-c)z_i+b{z}_n}$

Where ${\displaystyle \alpha }$ is the component of an individual's fitness which is independent of trait value, ${\displaystyle -c}$ parameterizes the effect of individual ${\displaystyle i}$'s phenotype on its own fitness (written negative, by convention, to represent a fitness cost), ${\displaystyle z_n}$ is the average trait value of individual ${\displaystyle i}$'s neighbours, and ${\displaystyle b}$ parameterizes the effect of individual ${\displaystyle i}$'s neighbours on its fitness (written positive, by convention, to represent a fitness benefit). Substituting into the Price equation then gives:

${\displaystyle \overline{w}\mathrm{\Delta }\overline{z}=\mathrm{cov}(\alpha -c{z}_i+b{z}_n,z_i)=\mathrm{cov}(\alpha ,z_i)-c\mathrm{cov}(z_i,z_i)+b\mathrm{cov}(z_n,z_i)}$

Since ${\displaystyle \alpha }$ by definition does not covary with ${\displaystyle z_i}$, this rearranges to:

${\displaystyle \mathrm{\Delta }\overline{z}=\frac{\mathrm{cov}(z_i,z_i)}{\overline{w}}\left(b\frac{\mathrm{cov}(z_n,z_i)}{\mathrm{cov}(z_i,z_i)}-c\right)}$

Since ${\displaystyle \frac{\mathrm{cov}(z_i,z_i)}{\overline{w}}=\frac{\mathrm{var}(z_i)}{\overline{w}}}$ this term must, by definition, be greater than 0. This is because variances can never be negative, and negative mean fitness is undefined (if mean fitness is 0 the population has crashed, similarly 0 variance would imply a monomorphic population, in both cases a change in mean trait value is impossible). It can then be said that that mean trait value will increase (${\displaystyle \mathrm{\Delta }\overline{z}>0}$) when:

${\displaystyle b\frac{\mathrm{cov}(z_n,z_i)}{\mathrm{cov}(z_i,z_i)}-c>0}$

or

${\displaystyle rb>c}$

Giving Hamilton's rule, where relatedness (${\displaystyle r}$) is a regression coefficient of the form ${\displaystyle \frac{\mathrm{cov}(z_n,z_i)}{\mathrm{cov}(z_i,z_i)}}$, or ${\displaystyle \frac{\mathrm{cov}(z_n,z_i)}{\mathrm{var}(z_i)}}$. Relatedness here can vary between a value of 1 (only interacting with individuals of the same trait value) and -1 (only interacting with individuals of a [most] different trait value), and will be 0 when all individuals in the population interact with equal likelihood.

Fitness in practice, however, does not tend to be linear in trait value -this would imply an increase to an infinitely large trait value being just as valuable to fitness as a similar increase to a very small trait value. Consequently, to apply Hamilton's rule to biological systems the conditions under which fitness can be approximated to being linear in trait value must first be found. There are two main methods used to approximate fitness as being linear in trait value; performing a partial regression with respect to both the focal individual's trait value and its neighbours average trait value, or taking a first order Taylor series approximation of fitness with respect to trait value. Performing a partial regression requires minimal assumptions, but only provides a statistical relationship as opposed to a mechanistic one, and cannot be extrapolated beyond the dataset that it was generated from. Linearizing via a Taylor series approximation, however, provides a powerful mechanistic relationship (see also causal model), but requires the assumption that evolution proceeds in sufficiently small mutational steps that the difference in trait value between an individual and its neighbours is close to 0 (in accordance with Fisher's geometric model): although in practice this approximation can often still retain predictive power under larger mutational steps.

As a first order approximation (linear in trait value), Hamilton's rule can only inform about how the mean trait value in a population is expected to change (directional selection). It contains no information about how the variance in trait value is expected to change (disruptive selection). As such it cannot be considered sufficient to determine evolutionary stability, even when Hamilton's rule predicts no change in trait value. This is because disruptive selection terms, and subsequent conditions for evolutionary branching, must instead be obtained from second order approximations (quadratic in trait value) of fitness.

Gardner et al. (2007) suggest that Hamilton's rule can be applied to multi-locus models, but that it should be done at the point of interpreting theory, rather than the starting point of enquiry. They suggest that one should "use standard population genetics, game theory, or other methodologies to derive a condition for when the social trait of interest is favoured by selection and then use Hamilton's rule as an aid for conceptualizing this result". It is now becoming increasingly popular to use adaptive dynamics approaches to gain selection conditions which are directly interpretable with respect to Hamilton's rule.

Altruism

See also: Human inclusive fitness

The concept serves to explain how natural selection can perpetuate altruism. If there is an "altruism gene" (or complex of genes) that influences an organism's behaviour to be helpful and protective of relatives and their offspring, this behaviour also increases the proportion of the altruism gene in the population, because relatives are likely to share genes with the altruist due to common descent. In formal terms, if such a complex of genes arises, Hamilton's rule (rbc) specifies the selective criteria (in terms of cost, benefit and relatedness) for such a trait to increase in frequency in the population. Hamilton noted that inclusive fitness theory does not by itself predict that a species will necessarily evolve such altruistic behaviours, since an opportunity or context for interaction between individuals is a more primary and necessary requirement in order for any social interaction to occur in the first place. As Hamilton put it, "Altruistic or selfish acts are only possible when a suitable social object is available. In this sense behaviours are conditional from the start." In other words, while inclusive fitness theory specifies a set of necessary criteria for the evolution of altruistic traits, it does not specify a sufficient condition for their evolution in any given species. More primary necessary criteria include the existence of gene complexes for altruistic traits in gene pool, as mentioned above, and especially that "a suitable social object is available", as Hamilton noted. The American evolutionary biologist Paul W. Sherman gives a fuller discussion of Hamilton's latter point:

To understand any species' pattern of nepotism, two questions about individuals' behavior must be considered: (1) what is reproductively ideal?, and (2) what is socially possible? With his formulation of "inclusive fitness," Hamilton suggested a mathematical way of answering (1). Here I suggest that the answer to (2) depends on demography, particularly its spatial component, dispersal, and its temporal component, mortality. Only when ecological circumstances affecting demography consistently make it socially possible will nepotism be elaborated according to what is reproductively ideal. For example, if dispersing is advantageous and if it usually separates relatives permanently, as in many birds, on the rare occasions when nestmates or other kin live in proximity, they will not preferentially cooperate. Similarly, nepotism will not be elaborated among relatives that have infrequently coexisted in a population's or a species' evolutionary history. If an animal's life history characteristicsusually preclude the existence of certain relatives, that is if kin are usually unavailable, the rare coexistence of such kin will not occasion preferential treatment. For example, if reproductives generally die soon after zygotes are formed, as in many temperate zone insects, the unusual individual that survives to interact with its offspring is not expected to behave parentally.

The occurrence of sibling cannibalism in several species underlines the point that inclusive fitness theory should not be understood to simply predict that genetically related individuals will inevitably recognize and engage in positive social behaviours towards genetic relatives. Only in species that have the appropriate traits in their gene pool, and in which individuals typically interacted with genetic relatives in the natural conditions of their evolutionary history, will social behaviour potentially be elaborated, and consideration of the evolutionarily typical demographic composition of grouping contexts of that species is thus a first step in understanding how selection pressures upon inclusive fitness have shaped the forms of its social behaviour. Richard Dawkins gives a simplified illustration:

If families [genetic relatives] happen to go around in groups, this fact provides a useful rule of thumb for kin selection: 'care for any individual you often see'."

Evidence from a variety of species including primates and other social mammals suggests that contextual cues (such as familiarity) are often significant proximate mechanisms mediating the expression of altruistic behaviour, regardless of whether the participants are always in fact genetic relatives or not. This is nevertheless evolutionarily stable since selection pressure acts on typical conditions, not on the rare occasions where actual genetic relatedness differs from that normally encountered. Inclusive fitness theory thus does not imply that organisms evolve to direct altruism towards genetic relatives. Many popular treatments do however promote this interpretation, as illustrated in a review:

[M]any misunderstandings persist. In many cases, they result from conflating "coefficient of relatedness" and "proportion of shared genes," which is a short step from the intuitively appealing—but incorrect—interpretation that "animals tend to be altruistic toward those with whom they share a lot of genes." These misunderstandings don't just crop up occasionally; they are repeated in many writings, including undergraduate psychology textbooks—most of them in the field of social psychology, within sections describing evolutionary approaches to altruism. (Park 2007, p860)

Such misunderstandings of inclusive fitness' implications for the study of altruism, even amongst professional biologists utilizing the theory, are widespread, prompting prominent theorists to regularly attempt to highlight and clarify the mistakes. An example of attempted clarification is West et al. (2010):

In his original papers on inclusive fitness theory, Hamilton pointed out a sufficiently high relatedness to favour altruistic behaviours could accrue in two ways—kin discrimination or limited dispersal. There is a huge theoretical literature on the possible role of limited dispersal, as well as experimental evolution tests of these models. However, despite this, it is still sometimes claimed that kin selection requires kin discrimination. Furthermore, a large number of authors appear to have implicitly or explicitly assumed that kin discrimination is the only mechanism by which altruistic behaviours can be directed towards relatives... [T]here is a huge industry of papers reinventing limited dispersal as an explanation for cooperation. The mistakes in these areas seem to stem from the incorrect assumption that kin selection or indirect fitness benefits require kin discrimination (misconception 5), despite the fact that Hamilton pointed out the potential role of limited dispersal in his earliest papers on inclusive fitness theory.

Green-beard effect

See also: Kin recognition

As well as interactions in reliable contexts of genetic relatedness, altruists may also have some way to recognize altruistic behaviour in unrelated individuals and be inclined to support them. As Dawkins points out in The Selfish Gene and The Extended Phenotype, this must be distinguished from the green-beard effect.

The green-beard effect is the act of a gene (or several closely linked genes), that:

1. Produces a phenotype.
2. Allows recognition of that phenotype in others.
3. Causes the individual to preferentially treat other individuals with the same gene.

The green-beard effect was originally a thought experiment by Hamilton in his publications on inclusive fitness in 1964, although it hadn't yet been observed. As of today, it has been observed in few species. Its rarity is probably due to its susceptibility to 'cheating' whereby individuals can gain the trait that confers the advantage, without the altruistic behaviour. This normally would occur via the crossing over of chromosomes which happens frequently, often rendering the green-beard effect a transient state. However, Wang et al. has shown in one of the species where the effect is common (fire ants), recombination cannot occur due to a large genetic transversion, essentially forming a supergene. This, along with homozygote inviability at the green-beard loci allows for the extended maintenance of the green-beard effect.

Equally, cheaters may not be able to invade the green-beard population if the mechanism for preferential treatment and the phenotype are intrinsically linked. In budding yeast (Saccharomyces cerevisiae), the dominant allele FLO1 is responsible for flocculation (self-adherence between cells) which helps protect them against harmful substances such as ethanol. While 'cheater' yeast cells occasionally find their way into the biofilm-like substance that is formed from FLO1 expressing yeast, they cannot invade as the FLO1 expressing yeast will not bind to them in return, and thus the phenotype is intrinsically linked to the preference.

Parent–offspring conflict and optimization

Early writings on inclusive fitness theory (including Hamilton 1964) used K in place of B/C. Thus Hamilton's rule was expressed as

${\displaystyle K>1/r}$ is the necessary and sufficient condition for selection for altruism.

Where B is the gain to the beneficiary, C is the cost to the actor and r is the number of its own offspring equivalents the actor expects in one of the offspring of the beneficiary. r is either called the coefficient of relatedness or coefficient of relationship, depending on how it is computed. The method of computing has changed over time, as has the terminology. It is not clear whether or not changes in the terminology followed changes in computation.

Robert Trivers (1974) defined "parent-offspring conflict" as any case where

${\displaystyle 1<K<2}$

i.e., K is between 1 and 2. The benefit is greater than the cost but is less than twice the cost. In this case, the parent would wish the offspring to behave as if r is 1 between siblings, although it is actually presumed to be 1/2 or closely approximated by 1/2. In other words, a parent would wish its offspring to give up ten offspring in order to raise 11 nieces and nephews. The offspring, when not manipulated by the parent, would require at least 21 nieces and nephews to justify the sacrifice of 10 of its own offspring.

The parent is trying to maximize its number of grandchildren, while the offspring is trying to maximize the number of its own offspring equivalents (via offspring and nieces and nephews) it produces. If the parent cannot manipulate the offspring and therefore loses in the conflict, the grandparents with the fewest grandchildren seem to be selected for. In other words, if the parent has no influence on the offspring's behaviour, grandparents with fewer grandchildren increase in frequency in the population.

By extension, parents with the fewest offspring will also increase in frequency. This seems to go against Ronald Fisher's "Fundamental Theorem of Natural Selection" which states that the change in fitness over the course of a generation equals the variance in fitness at the beginning of the generation. Variance is defined as the square of a quantity—standard deviation —and as a square must always be positive (or zero). That would imply that e fitness could never decrease as time passes. This goes along with the intuitive idea that lower fitness cannot be selected for. During parent-offspring conflict, the number of stranger equivalents reared per offspring equivalents reared is going down. Consideration of this phenomenon caused Orlove (1979) and Grafen (2006) to say that nothing is being maximized.

According to Trivers, if Sigmund Freud had tried to explain intra-family conflict after Hamilton instead of before him, he would have attributed the motivation for the conflict and for the castration complex to resource allocation issues rather than to sexual jealousy.

Incidentally, when k=1 or k=2, the average number of offspring per parent stays constant as time goes by. When k<1 or k>2 then the average number of offspring per parent increases as time goes by.

The term "gene" can refer to a locus (location) on an organism's DNA—a section that codes for a particular trait. Alternative versions of the code at that location are called "alleles." If there are two alleles at a locus, one of which codes for altruism and the other for selfishness, an individual who has one of each is said to be a heterozygote at that locus. If the heterozygote uses half of its resources raising its own offspring and the other half helping its siblings raise theirs, that condition is called codominance. If there is codominance the "2" in the above argument is exactly 2. If by contrast, the altruism allele is more dominant, then the 2 in the above would be replaced by a number smaller than 2. If the selfishness allele is the more dominant, something greater than 2 would replace the 2.

Opposing view

A 2010 paper by Martin Nowak, Corina Tarnita, and E. O. Wilson suggested that standard natural selection theory is superior to inclusive fitness theory, stating that the interactions between cost and benefit cannot be explained only in terms of relatedness. This, Nowak said, makes Hamilton's rule at worst superfluous and at best ad hoc. Gardner in turn was critical of the paper, describing it as "a really terrible article", and along with other co-authors has written a reply, submitted to Nature. The disagreement stems from a long history of confusion over what Hamilton's rule represents. Hamilton's rule gives the direction of mean phenotypic change (directional selection) so long as fitness is linear in phenotype, and the utility of Hamilton's rule is simply a reflection of when it is suitable to consider fitness as being linear in phenotype. The primary (and strictest) case is when evolution proceeds in very small mutational steps. Under such circumstances Hamilton's rule then emerges as the result of taking a first order Taylor series approximation of fitness with regards to phenotype. This assumption of small mutational steps (otherwise known as δ-weak selection) is often made on the basis of Fisher's geometric model and underpins much of modern evolutionary theory.

In work prior to Nowak et al. (2010), various authors derived different versions of a formula for ${\displaystyle r}$, all designed to preserve Hamilton's rule. Orlove noted that if a formula for ${\displaystyle r}$ is defined so as to ensure that Hamilton's rule is preserved, then the approach is by definition ad hoc. However, he published an unrelated derivation of the same formula for ${\displaystyle r}$ – a derivation designed to preserve two statements about the rate of selection – which on its own was similarly ad hoc. Orlove argued that the existence of two unrelated derivations of the formula for ${\displaystyle r}$ reduces or eliminates the ad hoc nature of the formula, and of inclusive fitness theory as well. The derivations were demonstrated to be unrelated by corresponding parts of the two identical formulae for ${\displaystyle r}$ being derived from the genotypes of different individuals. The parts that were derived from the genotypes of different individuals were terms to the right of the minus sign in the covariances in the two versions of the formula for ${\displaystyle r}$. By contrast, the terms left of the minus sign in both derivations come from the same source. In populations containing only two trait values, it has since been shown that ${\displaystyle r}$ is in fact Sewall Wright's coefficient of relationship.

Engles (1982) suggested that the c/b ratio be considered as a continuum of this behavioural trait rather than discontinuous in nature. From this approach fitness transactions can be better observed because there is more to what is happening to affect an individual's fitness than just losing and gaining.