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Exposure (photography)

In photography, exposure is the amount of light per unit area reaching a frame of photographic film or the surface of an electronic image sensor. It is determined by shutter speed, lens F-number, and scene luminance. Exposure is measured in units of lux-seconds (symbol lxs), and can be computed from exposure value (EV) and scene luminance in a specified region.

An "exposure" is a single shutter cycle. For example, a long exposure refers to a single, long shutter cycle to gather enough dim light, whereas a multiple exposure involves a series of shutter cycles, effectively layering a series of photographs in one image. The accumulated photometric exposure (Hv) is the same so long as the total exposure time is the same.

Radiant exposure
Radiant exposure of a surface, denoted He ("e" for "energetic", to avoid confusion with photometric quantities) and measured in, is given by
 * $$H_\mathrm{e} = E_\mathrm{e}t,$$

where
 * Ee is the irradiance of the surface, measured in ;
 * t is the exposure duration, measured in s.

Luminous exposure
Luminous exposure of a surface, denoted Hv ("v" for "visual", to avoid confusion with radiometric quantities) and measured in, is given by
 * $$H_\mathrm{v} = E_\mathrm{v}t,$$

where
 * Ev is the illuminance of the surface, measured in lx;
 * t is the exposure duration, measured in s.

If the measurement is adjusted to account only for light that reacts with the photo-sensitive surface, that is, weighted by the appropriate spectral sensitivity, the exposure is still measured in radiometric units (joules per square meter), rather than photometric units (weighted by the nominal sensitivity of the human eye). Only in this appropriately weighted case does the H measure the effective amount of light falling on the film, such that the characteristic curve will be correct independent of the spectrum of the light.

Many photographic materials are also sensitive to "invisible" light, which can be a nuisance (see UV filter and IR filter), or a benefit (see infrared photography and full-spectrum photography). The use of radiometric units is appropriate to characterize such sensitivity to invisible light.

In sensitometric data, such as characteristic curves, the log exposure is conventionally expressed as log10(H). Photographers more familiar with base-2 logarithmic scales (such as exposure values) can convert using log2(H) ≈ 3.32 log10(H).

Optimum exposure
"Correct" exposure may be defined as an exposure that achieves the effect the photographer intended.

A more technical approach recognises that a photographic film (or sensor) has a physically limited useful exposure range, sometimes called its dynamic range. If, for any part of the photograph, the actual exposure is outside this range, the film cannot record it accurately. In a very simple model, for example, out-of-range values would be recorded as "black" (underexposed) or "white" (overexposed) rather than the precisely graduated shades of colour and tone required to describe "detail". Therefore, the purpose of exposure adjustment (and/or lighting adjustment) is to control the physical amount of light from the subject that is allowed to fall on the film, so that 'significant' areas of shadow and highlight detail do not exceed the film's useful exposure range. This ensures that no 'significant' information is lost during capture.

The photographer may carefully overexpose or underexpose the photograph to eliminate "insignificant" or "unwanted" detail; to make, for example, a white altar cloth appear immaculately clean, or to emulate the heavy, pitiless shadows of film noir. However, it is technically much easier to discard recorded information during post processing than to try to 're-create' unrecorded information.

In a scene with strong or harsh lighting, the ratio between highlight and shadow luminance values may well be larger than the ratio between the film's maximum and minimum useful exposure values. In this case, adjusting the camera's exposure settings (which only applies changes to the whole image, not selectively to parts of the image) only allows the photographer to choose between underexposed shadows or overexposed highlights; it cannot bring both into the useful exposure range at the same time. Methods for dealing with this situation include: using what is called fill lighting to increase the illumination in shadow areas; using a graduated neutral-density filter, flag, scrim, or gobo to reduce the illumination falling upon areas deemed too bright; or varying the exposure between multiple, otherwise identical, photographs (exposure bracketing) and then combining them afterwards in an HDRI process.

Overexposure and underexposure

A photograph may be described as overexposed when it has a loss of highlight detail, that is, when important bright parts of an image are "washed out" or effectively all white, known as "blown-out highlights" or "clipped whites". A photograph may be described as underexposed when it has a loss of shadow detail, that is, when important dark areas are "muddy" or indistinguishable from black, known as "blocked-up shadows" (or sometimes "crushed shadows", "crushed blacks", or "clipped blacks", especially in video). As the adjacent image shows, these terms are technical ones rather than artistic judgments; an overexposed or underexposed image may be "correct" in the sense that it provides the effect that the photographer intended. Intentionally over- or underexposing (relative to a standard or the camera's automatic exposure) is casually referred to as "exposing to the right" or "exposing to the left" respectively, as these shift the histogram of the image to the right or left.

Manual exposure
In manual mode, the photographer adjusts the lens aperture and/or shutter speed to achieve the desired exposure. Many photographers choose to control aperture and shutter independently because opening up the aperture increases exposure, but also decreases the depth of field, and a slower shutter increases exposure but also increases the opportunity for motion blur.

"Manual" exposure calculations may be based on some method of light metering with a working knowledge of exposure values, the APEX system and/or the Zone System.

Automatic exposure

A camera in automatic exposure or autoexposure (usually initialized as AE) mode automatically calculates and adjusts exposure settings to match (as closely as possible) the subject's mid-tone to the mid-tone of the photograph. For most cameras, this means using an on-board TTL exposure meter.

Aperture priority (commonly abbreviated as A, or Av for aperture value) mode gives the photographer manual control of the aperture, whilst the camera automatically adjusts the shutter speed to achieve the exposure specified by the TTL meter. Shutter priority (often abbreviated as S, or Tv for time value) mode gives manual shutter control, with automatic aperture compensation. In each case, the actual exposure level is still determined by the camera's exposure meter.

Exposure compensation
The purpose of an exposure meter is to estimate the subject's mid-tone luminance and indicate the camera exposure settings required to record this as a mid-tone. In order to do this it has to make a number of assumptions which, under certain circumstances, will be wrong. If the exposure setting indicated by an exposure meter is taken as the "reference" exposure, the photographer may wish to deliberately overexpose or underexpose in order to compensate for known or anticipated metering inaccuracies.

Cameras with any kind of internal exposure meter usually feature an exposure compensation setting which is intended to allow the photographer to simply offset the exposure level from the internal meter's estimate of appropriate exposure. Frequently calibrated in stops, also known as EV units, a "+1" exposure compensation setting indicates one stop more (twice as much) exposure and "–1" means one stop less (half as much) exposure.

Exposure compensation is particularly useful in combination with auto-exposure mode, as it allows the photographer to bias the exposure level without resorting to full manual exposure and losing the flexibility of auto exposure. On low-end video camcorders, exposure compensation may be the only manual exposure control available.

Exposure control
An appropriate exposure for a photograph is determined by the sensitivity of the medium used. For photographic film, sensitivity is referred to as film speed and is measured on a scale published by the International Organization for Standardization (ISO). Faster film, that is, film with a higher ISO rating, requires less exposure to make a readable image. Digital cameras usually have variable ISO settings that provide additional flexibility. Exposure is a combination of the length of time and the illuminance at the photosensitive material. Exposure time is controlled in a camera by shutter speed, and the illuminance depends on the lens aperture and the scene luminance. Slower shutter speeds (exposing the medium for a longer period of time), greater lens apertures (admitting more light), and higher-luminance scenes produce greater exposures.

An approximately correct exposure will be obtained on a sunny day using ISO 100 film, an aperture of and a shutter speed of 1/100 of a second. This is called the sunny 16 rule: at an aperture of on a sunny day, a suitable shutter speed will be one over the film speed (or closest equivalent).

A scene can be exposed in many ways, depending on the desired effect a photographer wishes to convey.

Reciprocity
An important principle of exposure is reciprocity. If one exposes the film or sensor for a longer period, a reciprocally smaller aperture is required to reduce the amount of light hitting the film to obtain the same exposure. For example, the photographer may prefer to make his sunny-16 shot at an aperture of (to obtain a shallow depth of field). As is 3 stops "faster" than, with each stop meaning double the amount of light, a new shutter speed of (1/125)/(2·2·2) = 1/1000 s is needed. Once the photographer has determined the exposure, aperture stops can be traded for halvings or doublings of speed, within limits.

The true characteristic of most photographic emulsions is not actually linear (see sensitometry), but it is close enough over the exposure range of about 1 second to 1/1000 of a second. Outside of this range, it becomes necessary to increase the exposure from the calculated value to account for this characteristic of the emulsion. This characteristic is known as reciprocity failure. The film manufacturer's data sheets should be consulted to arrive at the correction required, as different emulsions have different characteristics.

Digital camera image sensors can also be subject to a form of reciprocity failure.

Determining exposure

The Zone System is another method of determining exposure and development combinations to achieve a greater tonality range over conventional methods by varying the contrast of the film to fit the print contrast capability. Digital cameras can achieve similar results (high dynamic range) by combining several different exposures (varying shutter or diaphragm) made in quick succession.

Today, most cameras automatically determine the correct exposure at the time of taking a photograph by using a built-in light meter, or multiple point meters interpreted by a built-in computer, see metering mode.

Negative and print film tends to bias for exposing for the shadow areas (film dislikes being starved of light), with digital favouring exposure for highlights. See latitude below.

Latitude
Latitude is the degree by which one can over, or under expose an image, and still recover an acceptable level of quality from an exposure. Typically negative film has a better ability to record a range of brightness than slide/transparency film or digital. Digital should be considered to be the reverse of print film, with a good latitude in the shadow range, and a narrow one in the highlight area; in contrast to film's large highlight latitude, and narrow shadow latitude. Slide/Transparency film has a narrow latitude in both highlight and shadow areas, requiring greater exposure accuracy.

Negative film's latitude increases somewhat with high ISO material, in contrast digital tends to narrow on latitude with high ISO settings.

Highlights
Areas of a photo where information is lost due to extreme brightness are described as having "blown-out highlights" or "flared highlights".

In digital images this information loss is often irreversible, though small problems can be made less noticeable using photo manipulation software. Recording to RAW format can correct this problem to some degree, as can using a digital camera with a better sensor.

Film can often have areas of extreme overexposure but still record detail in those areas. This information is usually somewhat recoverable when printing or transferring to digital.

A loss of highlights in a photograph is usually undesirable, but in some cases can be considered to "enhance" appeal. Examples include black and white photography and portraits with an out-of-focus background.

Blacks
Areas of a photo where information is lost due to extreme darkness are described as "crushed blacks". Digital capture tends to be more tolerant of underexposure, allowing better recovery of shadow detail, than same-ISO negative print film.

Crushed blacks cause loss of detail, but can be used for artistic effect.

title: Exposure (photography)


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River Dane

The River Dane is a tributary of the River Weaver that originates in the Peak District area of England. The name of the river (earlier Daven) is probably from the Old Welsh dafn, meaning a "drop or trickle", implying a slow-moving river.

Course
It forms the border between first Cheshire and Derbyshire on the west and east, and then between Cheshire and Staffordshire where it then flows north-west through Cheshire before meeting the Weaver in Northwich.

The river rises close to the source of the River Goyt just to the south west of Buxton, at Dane Head on Axe Edge Moor. Flowing southwest, it forms county borders for around 10 mi before flowing west through Congleton and past Holmes Chapel. The point on the river where the three counties meet, at Panniers' Pool Bridge, is called Three Shire Heads (sometimes Three Shires Head). Passing just north of Middlewich, it merges first with the River Croco near the site of the old Roman fort in Harbutt's Field, and then with the River Wheelock near the aqueduct carrying the Trent and Mersey Canal, and runs the remaining 5 mi north to Northwich where it flows into the River Weaver.

The River Dane is the longest, cleanest and thought to be the fastest flowing river through Cheshire. The route of the Dane is followed as closely as possible by the Dane Valley Way, a 48 mile walking route from Buxton to Northwich.

Although the main river is part of the Mersey catchment and flows into the Irish Sea, a portion of the water can be diverted via canal feeders into Rudyard Lake and subsequently the Caldon Canal. This water discharges eventually into the River Trent and ultimately into the North Sea, having crossed the English watershed.

Tributaries

 * Gad Brook (R)
 * Eldersbriar Brook (L)
 * Puddlinglake Brook (R)
 * River Wheelock (L)
 * Hoggins Brook (L)
 * Fowle Brook (L)
 * River Croco (L)
 * Sanderson's Brook (L)
 * Small Brook (L)
 * Swettenham Brook
 * Midge Brook (Ls)
 * Chapel Brook (R)
 * Dighill Brook (R)
 * Clonter Brook (Rs)
 * Loach Brook
 * Dairy Brook (L)
 * The Howty (L)
 * Dane in Shaw Brook (L)
 * Biddulph Brook
 * Timbers Brook (R)
 * Cow Brook (R)
 * Ravensclough Brook (L)
 * Shell Brook (R)
 * Clough Brook (R)
 * Highmoor Brook (R)
 * Rabb Brook (R)
 * Oaken Brook (R)
 * Cumberland Brook (L)
 * Yarnshaw Brook (L)
 * Correction Brook (L)
 * Tor Brook (R)
 * Black Brook
 * Flash Brook (L)
 * Far Brook (R)
 * Robins Brook (R)
 * Three Shires Brook (L)

History
In 1451 when the River Dane flooded in Congleton, it destroyed the town's corn mill, half of the timber-framed buildings and the wooden bridge over the river. To prevent it happening again, the river was diverted away from the town. Congleton's textile industry grew from the 14th century, with many water-powered mills built along the river by the 18th century.

A small stone bridge over the river near Holmes Chapel, Hermitage Bridge, was built in 1772 by a local ironmaster. Some years the river floods widely across the meadows here. Nearby a tall red brick railway viaduct, built in 1841, spans the broad Dane Valley between Holmes Chapel and Twemlow. It has 23 arches and is Grade II listed. It crosses the River Dane at Saltersford, where Cheshire salt traders once drove their horse-drawn carts through a ford along the toll road.

Reading downstream, the following crossings are all designated listed buildings or scheduled monuments: Dane Bridge, Hug Bridge, Lymford Bridge (early 19th century), Dane Aqueduct over the Macclesfield Canal (1830), Congleton Railway Viaduct (West Coast Main Line, 1849), Colley Mill Bridge, Havannah Bridge (early to mid-19th century, originally leading to a cigar factory), Hermitage Bridge (1772) and Shipbrook Bridge.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: River Dane


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River Manifold

The River Manifold is a river in Staffordshire, England. It is a tributary of the River Dove (which also flows through the Peak District, forming the boundary between Derbyshire and Staffordshire).

The Manifold rises at Flash Head just south of Buxton near Axe Edge, at the northern edge of the White Peak, known for its limestone beds. It continues for 12 mi before it joins the Dove. For part of its course, it runs underground (except when in spate), from Wetton Mill to Ilam. During this section it is joined by its major tributary, the River Hamps.

Villages on the river include Longnor, Hulme End and Ilam.

Its name may come from Anglo-Saxon manig-fald = "many folds", referring to its meanders.

Manifold Way
The Manifold Way is an 8 mi long-distance footpath and cycle track from Hulme End to Waterhouses, along the former route of the narrow-gauge (2' 6") Leek and Manifold Valley Light Railway which operated between 1904 and 1934. Opened in July 1937 after the LMS handed over the trackbed to Staffordshire County Council, it is tarmacked throughout.

The Manifold Valley Visitor Centre is housed in Hulme End Station, which also has a model of the railway.

Limestone crags and caves
The limestone cliffs that fringe the valley contain several rock-climbing areas, and named rock features, including Thor's Cave and Beeston Tor, which overlooks the confluence with the River Hamps.

Mining in the Manifold valley
The Manifold valley was famous for the mining of copper and lead, and the mines at Ecton were some of the richest in the country. The discovery of Stone Age implements in some of the caves imply that minerals were mined around the Manifold valley thousands of years ago. Nowadays there is little trace of the industry that made many people (mainly the Duke of Devonshire who at one time owned the Ecton mines) very rich. The main areas of interest are around Ecton where the old spoil banks and the old engine house still remain.

Ecology
The river has been noted as being important for European bullhead and lamprey. In the underground section of the river where it resurfaces at Ilam, the bullhead have no pigmentation. It was previously a stronghold for white-clawed crayfish, but most of the river's population were wiped out by crayfish plague in 2008. Crayfish have been noted in the tributaries of the Manifold and it is hoped that re-colonisation can be achieved by the surviving upstream crayfish.

title: River Manifold


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User:JeremyR

Nothing to see here. Move along.

title: User:JeremyR


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List of dams and reservoirs in Germany

These are dams and reservoirs in Germany.

The German word Talsperre (literally: valley barrier) may mean dam, but it is often used to include the associated reservoir as well. The reservoirs are often separately given names ending in -see, -teich or -speicher which are the German words for "lake", "pond" and "reservoir", but in this case all may also be translated as "reservoir". The more specific word for the actual dam is Staumauer and for the lake is Stausee.

Baden-Württemberg

 * Kleine Kinzig Dam
 * Nagold Dam
 * Schluchsee - highest reservoir lake in Germany and largest lake in the Black Forest
 * Schwarzenbach Dam

Bavaria

 * Ellertshäuser See
 * Großer Brombachsee
 * Forggensee
 * Frauenau Dam
 * Sylvenstein Dam
 * Altmühlsee
 * Rothsee
 * Hahnenkammsee

Brandenburg

 * Spremberg Reservoir

Hesse

 * Aar Dam
 * Affoldern Reservoir
 * Antrift Dam
 * Diemelsee (reservoir)
 * Driedorf Reservoir
 * Edersee

Lower Saxony

 * Ecker Dam
 * Grane Dam
 * Innerste Dam
 * Oder Dam
 * Oderteich
 * Oker Dam
 * Söse Dam
 * Wendebach Dam

North Rhine-Westphalia

 * Aabach Dam
 * Agger Dam
 * Ahauser Reservoir
 * Baldeney Reservoir
 * Bever Dam
 * Beyenburg Reservoir
 * Biggesee
 * Borchen Flood Control Basin
 * Breitenbach Dam
 * Bruchbachtal-Büderich Flood Control Basin
 * Bruch Dam
 * Dahlhausen Dam
 * Diepental Dam
 * Große Dhünn Dam
 * Dreilägerbach Dam
 * Ebbinghausen Flood Control Basin
 * Eicherscheid Flood Control Basin
 * Eiserbach Dam
 * Ennepe Dam
 * Eringerfeld Flood Control Basin
 * Eschbach Dam
 * Esmecke Reservoir
 * Fuelbecke Dam
 * Fürwigge Dam
 * Genkel Dam
 * Glingebach Dam
 * Glör Dam
 * Gollentaler Grund Flood Control Basin
 * Hengsteysee
 * Kall Dam
 * Kemnader See
 * Möhne Reservoir
 * Olef Dam
 * Perlenbach Dam
 * Rur Dam
 * Sorpe Reservoir
 * Steinbachtal Dam
 * Urft Dam
 * Wahnbach Dam
 * Wehebach Dam
 * Wupper Dam

Saxony

 * Altenberg Reservoir
 * Bautzen Reservoir
 * Borna Reservoir
 * Buschbach Flood Control Basin
 * Carlsfeld Dam
 * Cranzahl Dam
 * Dörnthaler Teich
 * Dröda Dam
 * Eibenstock Dam
 * Einsiedel Dam
 * Euba Dam
 * Falkenstein Dam
 * Forchheim Auxiliary Dam
 * Friedrichswalde-Ottendorf Flood Control Basin
 * Gottleuba Dam
 * Quitzdorf Dam
 * Sosa Dam
 * Saidenbach Dam
 * Wallroda Dam

Saxony-Anhalt

 * Ecker Dam
 * Kelbra Dam
 * Kiliansteich Dam
 * Königshütte Dam
 * Mandelholz Dam
 * Rappbode Dam, highest dam in Germany.
 * Rappbode Auxiliary Dam
 * Wendefurth Dam
 * Wippra Dam
 * Zillierbach Dam

Thuringia
In Thuringia there are 171 reservoirs. The biggest of them are:
 * Bleiloch Dam (biggest reservoir in Germany, volume: ~215 million m³), river Saale
 * Deesbach Forebay (height 42.5 m; volume ~3.2 million m³), river Lichte
 * Haselbach Reservoir (~25 million m³), flooded opencast mining area
 * Hohenwarte Reservoir (volume: ~182 million m³), river Saale
 * Leibis-Lichte Dam (height 102.5 m; volume ~32,4 million m³), river Lichte
 * Neustadt Dam - Thuringia's oldest dam
 * Schmalwasser Dam (~21,2 million m³), river Schmalwasser
 * Schönbrunn Dam (~23,2 million m³), river Schleuse
 * Zeulenroda Dam (~30,4 million m³), river Weida

title: List of dams and reservoirs in Germany


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Standard atmosphere

Standard atmosphere may refer to:

 * A standard reference value for air pressure:
 * Atmosphere (unit), an approximation of the value at sea level
 * Atmospheric pressure, other reference values
 * One of various static atmospheric models of how atmospheric pressure, density, and temperature vary with altitude, such as:
 * The U.S. Standard Atmosphere, a series of models that give values for pressure, density, and temperature over a range of altitudes
 * The International Standard Atmosphere (ISA), an international standard model, defining typical atmospheric properties with altitude, at mid-latitude

title: Standard atmosphere


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Talk:Oscar Wilde

Featured picture scheduled for POTD
Hello! This is to let editors know that File:Oscar Wilde by Napoleon Sarony. Three-quarter-length photograph, seated.jpg, a featured picture used in this article, has been selected as the English Wikipedia's picture of the day (POTD) for October 16, 2024. A preview of the POTD is displayed below and can be edited at Template:POTD/2024-10-16. For the greater benefit of readers, any potential improvements or maintenance that could benefit the quality of this article should be done before its scheduled appearance on the Main Page. (170th birthday) If you have any concerns, please place a message at Wikipedia talk:Picture of the day. Thank you! Adam Cuerden (talk)Has about 8.6% of all FPs. 18:27, 1 December 2023 (UTC)

O'Fflahertie versus O'Flahertie
The sources cited in the notes section of this article, which say that "O'Fflahertie" is the correct spelling, seem to be in the minority when compared to how the vast majority of sources spell the middle name.

I cannot find a free version of the Wildean article cited ("How Did Oscar Wilde Spell His Name?" by Donald Mead, Jan 2020), so I cannot examine the evidence provided by that article myself. I also cannot find a copy of his birth certificate to compare the spelling. Additionally, the Soethby listing ('Confessions of Tastes, Habits and Convictions', authorial manuscript, 1877) even spells the name as O'Flahertie in its item description, although the handwriting within the manuscript does seem to have the O'Fflahertie spelling.

I think this requires further research before the article makes any definitive statements on the definite spelling of his name. Yttrbium (talk) 16:28, 11 December 2023 (UTC)

 * We did research and discuss this in February & March this year. Old thread is at Talk:Oscar_Wilde/Archive_7. DuncanHill (talk) 19:59, 11 December 2023 (UTC)

title: Talk:Oscar Wilde


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Talk:Karl Pearson

Source of material
Much of the material for this article was taken from http://www.wikipedia.org/wiki/Karl_Pearson. I requested permission to use it for Wikipedia and received the following response on Jan. 8, 2003:

 * Hi Sheldon,

 * The informations is in the public domain, and this department is proud of the achievements of Karl Pearson and other pioneering statisticians. It would be good if you used some of this information for his biographical entry in Wikpedia, provided you acknowledge the source, and/or link to our web page you cited.

 * Cheers
 * Tim

 * Dr Tim Downie
 * Dept. Statistical Science
 * University College London
 * Gower Street, London WC1E 6BT
 * Ph:+44 (0)20 7679 1855
 * Fax:+44 (0)20 7383 4703
 * Fax:+44 (0)20 7383 4703

For the record, the URL cited above should probably be http://www.ucl.ac.uk/Stats/department/pearson.html. Schutz 07:52, 28 March 2006 (UTC)

Very nice article. Michael Hardy 02:07 Apr 3, 2003 (UTC)

Wife/Mother confusion
At the beginning of the article, Pearson's mother's name is Maria Pearson, née Sharpe. Later in the article Pearson's wife is named Maria Sharpe. Was ist los? 64.12.116.6 19:19, 8 September 2005 (UTC)Bruce Partington

I'll check this. She was his wife (and no relative of Granville Sharp, who even spelt his name differently).

Pearson's mother was Frances ('Fanny') Smith - who, funnily, was also Florence Nightingale's mother.

I'm looking into whether
 * (a) Pearson's father had an illegitimate child (b.1849)
 * (b) his mother was in the Retreat (mental hospital) in York, c.1885.

There seems some evidence in favour of both of the above - does anyone have any info on this?
 * Johnbibby 21:41, 28 October 2006 (UTC)

 * Johnbibby 21:41, 28 October 2006 (UTC)

It might be a good idea when mentioning his family to link yo his sons wiki page Egon Pearson — Preceding unsigned comment added by Gmdean2015 (talk • contribs) 15:23, 30 June 2019 (UTC)

 * Thank you - we did already mention and link Egon Pearson, but not in the family section. I've moved some text around to make things (I hope) clearer. DuncanHill (talk) 15:29, 30 June 2019 (UTC)

BBC
What if I'm not interested in knowing the BBC's opinion of Karl Pearson? Can I be sure that he was "cold" because the BBC claims that as one of his characteristics? 205.188.116.6 17:50, 19 September 2005 (UTC)James Moriarty, Professor of Mathematics

In this strange and surprising world, it is not beyond the realm of possibility that some individuals who are employed by the BBC may exhibit characteristics that seem cold and calculating. Lestrade 17:41, 22 September 2005 (UTC)Lestrade

Dead, yet has made recent controversy?
I am confused by the statement in the introduction, "Recently his advocacy of eugenics has become controversial.". This makes it sound as if he is alive still today when obviously it isn't the case. Also, I think any comments regarding his views on eugenics would need a source of citation. -- (A.szczep) 16:54, 28 October 2006 (UTC)

I noticed this and thought it sounded odd. But when you think about it, it makes sense! (Unlike some pages: "The more I look, the less I see ..." ?Lewis Carroll's Gubbermunzy??
 * Johnbibby 21:36, 28 October 2006 (UTC)

 * I disagree. I think the article definitely needs someone to copy edit it.
 * -- (A.szczep) 12:19, 1 November 2006 (UTC)
 * It is pretty clear that he advocated some form of eugenics. He was very close to Francis Galton. I'm not sure why his "advocacy of eugenics has recently become controversial". Maybe he was just forgotten for a long time? Intangible 03:37, 4 November 2006 (UTC)

 * I think it might be better to say: "KP advocated eugenics, and the study of eugenics has recently become controversial"
 * Johnbibby 13:15, 4 November 2006 (UTC)
 * The study of eugenics has been pretty much controversial since its inception. Intangible 16:35, 4 November 2006 (UTC)

 * And if you don't know what Pearson advocated eugenics then you don't know a thing about Pearson. Eugenics has only become really controversial again since the mid 1980s (Human Genome Project re-sparked interest in it). Pearson was a eugenicist but not of the American or German sort — the British eugenicists were of a somewhat different stripe. --24.147.86.187

 * I think it a misrepresentation to put the "Politics and eugenics" and "Awards" sections (what he is least known for) before the "Contributions to statistics" section (what he is most known for) - perhaps "Einstein and Pearson's work" as well. After all, he would not have a Wikipedia page without his enduring work in statistics and, as order of presentation suggests order of importance, the present order seems to stand the enduring accomplishments of the man upside down. That is, once the obligatory biographical information is out of the way, his accomplishments should follow in order of importance - chronological order of occurrence notwithstanding. — Preceding unsigned comment added by 76.88.1.215 (talk) 01:46, 26 November 2011 (UTC)

Copyedits
Sorry it took five edits, but there was a lot of copyediting to do! Yellow-bellied sapsucker (talk) 23:22, 3 January 2008 (UTC)

Comment on Pearson in Gosset article
Over on Wikipedia's page for William Sealy Gosset (not on the page for Pearson), we find: "Gosset was a friend of both Pearson and Fisher, an achievement, for each had a massive ego and a loathing for the other." Does anyone have any direct reference for this? Personally, I have no notion what so ever, one way or the other. Surely, such a strongly-worded assessment as "massive ego" and "loathing" calls for some referenced materials. Also, if Pearson really were such a person, if such can be documented, should not this remark pertaining to a difficult nature also be included on the page for Pearson? David Corliss (talk) 04:13, 13 January 2008 (UTC)

Such comments are general for both Pearson and Fisher. (Even Egon Pearson's biography recognises it for KP.) No problem about including it on both pages.
 * Johnbibby (talk) 12:17, 15 January 2008 (UTC)

Did he visit Vienna?
I think the article is in error here - Not sure he ever visited Vienna - Porter says he was due to do so, to visit Gee, but changed his mind
 * Johnbibby (talk) 20:48, 14 August 2008 (UTC)

Was he a Quaker?
I know that lots of sources state that KP was brought up as a Quaker. But what is the _evidence_? The article also states he was a Quaker 'until 1889'. Again - what is the evidence?
 * Johnbibby (talk) 20:50, 14 August 2008 (UTC)

Position of the Family Memorial Image in the Article
The image which shows the Pearson Family Memorial at the end of the article, is misplaced in my opinion. And i think it should be placed along side peorson family section. However, I do not know how to change the position of a picture. Would someone kindly adjust the position of the image.Sajjad Arif (talk) 00:54, 14 November 2010 (UTC)

 * Done. But as it says on my user page, I'm blind, so I don't know if it looks any good. If it looks horrendous now, somebody will surely come by and fix it. Graham 87 03:28, 14 November 2010 (UTC)

 * I've removed it completely, as it was a poor quality photo of an unremarkable Victorian obelisk-style memorial in Crambe churchyard to Pearsons named Richard, Sarah, Thomas (all three died in 1859) and Elizabeth (died 1882), none of whom are mentioned in this article. Qwfp (talk) 08:30, 14 November 2010 (UTC)

Pearson's Product Moment Correlation Coefficient test
Surely Setion 6, namely the section on Pearson's contributions to statistics, should mention how the Pearson product moment correlation test, a test of correlation for use with parametric statistics, was developed by and named after Pearson? Vorbee (talk) 17:09, 18 June 2017 (UTC)

External links modified
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title: Talk:Karl Pearson


text:
Havelland (district)

Havelland is a district or county in Brandenburg, Germany. It is bounded by (from the north and clockwise) the districts of Ostprignitz-Ruppin and Oberhavel, the city-state of Berlin, the district of Potsdam-Mittelmark, the city of Brandenburg and the state of Saxony-Anhalt (districts of Jerichower Land and Stendal).

History
The district was established in 1993 by merging the former districts of Nauen and Rathenow.

Geography
Geographically the Havelland refers to the land either side of the Havel river and the area surrounded by the large "U" bend in its course between the city of Berlin and its confluence with the Elbe river. The district mainly consists of the areas north of the Havel river.

Coat of arms
The red eagle is the heraldic animal of Brandenburg. The head of the eagle is part of the coat of arms of Rathenow. The swan is often used as a symbol for the Havelland region.

title: Havelland (district)


text:
Talk:Flemish Brabant

Double links
But are double-links annoying ? Links are for the user's ease of use, and searching for Leuven in the list (which entails to chose the right list first ;-) is a bit annoying. Plus, it is a general convention to have the first reference to something be a link. In this case, on the reference you're fighting on. --FvdP

-- Precisely: that is why I added the link. S.

OK guys, good point. Let the link live! :-) D.D. 14:19 Jan 8, 2003 (UTC)

-

Title of article
The title of this article must be changed! There simply is no province in Belgium called "Flemish Brabant". It is called Vlaams-Brabant and that name would best be used in all other languages as well, to avoid confusion. It is one thing for English-speaking people to translate terms that are very difficult to write and/or pronounce, but it is quite another for Flemings to force-feed the British and Americans with names they will never find in official documents. The only native speakers of English who come into contact with this province (by relocating there, or visiting it for a conference) are likely to encounter, and use, the real name. —Preceding unsigned comment added by 81.165.120.4 (talk • contribs)
 * "Flemish Brabant" is a very commonly used name for the province in English. I think there's good reason for using a name which immediately identifies to the English speaker its location within the Flemish region, rather than "Vlaams Brabant" which is less helpful to those who have no knowledge of Dutch / Flemish.
 * Note also that the English pages of the official website of the province refer to it as "Flemish Brabant". --David Edgar 12:11, 29 January 2007 (UTC)

Inconsistency
In the list of political parties, sp.a (Socialistische Partij Anders) is translated as "Socialist Party – Different", while on the page wiki about this party it is translated as "Socialist Party Differently" (Socialistische Partij Anders). Sebasst.f (talk) 13:41, 12 March 2017 (UTC)

title: Talk:Flemish Brabant


text:
Cyberdog

Cyberdog was an OpenDoc-based Internet suite of applications, developed by Apple Computer for the Mac OS line of operating systems. It was introduced as a beta in February 1996 and abandoned in March 1997. The last version, Cyberdog 2.0, was released on April 28, 1997. It worked with later versions of System 7 as well as the Mac OS 8 and Mac OS 9 operating systems.

Cyberdog derived its name from a cartoon in The New Yorker captioned "On the Internet, nobody knows you're a dog."

Cyberdog 1

 * Apple released the first beta version of Cyberdog on February 16, 1996.
 * Apple released Cyberdog 1.0 on May 13, 1996.
 * Apple released Cyberdog 1.2 on December 4, 1996.

Cyberdog 2
Apple released a first alpha version on December 21, 1996, with new features such as frames, cookies and animated GIF support.

Apple also released Cyberdog 2.0 with Mac OS 8.0, allowing Mac Runtime for Java to be utilized and also had minor bugs with OpenDoc fixed.

Overview
Cyberdog included email and news readers, a web browser and address book management components, as well as drag and drop FTP. OpenDoc allowed these components to be reused and embedded in other documents by the user. For instance, a "live" Cyberdog web page could be embedded in a presentation program, one of the common demonstrations of OpenDoc.

A serious problem with the OpenDoc project that Cyberdog depended on, was that it was part of a very acrimonious competition between OpenDoc consortium members and Microsoft. The members of the OpenDoc alliance were all trying to obtain traction in a market rapidly being dominated by Microsoft Office and Internet Explorer. At the same time, Microsoft used the synergy between the OS and applications divisions of the company to make it effectively mandatory that developers adopt the competing Microsoft Object Linking and Embedding (OLE) technology. OpenDoc was forced to create an interoperability layer in order to allow developers to use it, and this added a great technical burden to the project.

An offspring of Cyberdog called Subwoofer had been developed in parallel and was aimed at providing software developers with a simple library for integrating web communication protocols into applications. The project was completed after the cancellation of Cyberdog and released at the MacHack 1997 conference by Sari Harrison and Frédéric Artru. Subwoofer evolved into the URL Access library shipped with Mac OS 8.6.

Cancellation
OpenDoc had several hundred developers signed up, but the timing was poor. Apple Computer was rapidly losing money at the time. Before long, OpenDoc was scrapped, with Steve Jobs noting that they "put a bullet through (CyberDog's) head", and most of the team was laid off in March 1997. Other sources noted that Microsoft hired away three ClarisWorks developers who were responsible for OpenDoc integration into ClarisWorks.

AppleShare IP Manager from versions 5.0 to 6.2 relied on OpenDoc, but AppleShare IP 6.3, the first Mac OS 9 compatible version (released in 1999), eliminated the reliance on OpenDoc. Apple officially relinquished the last trademark on the name OpenDoc on June 11, 2005.

OpenDoc had a large memory footprint for the time, and since the OS/2 (Warp 4) versions of OpenDoc were behind schedule, Cyberdog only ran on Macintosh. Moreover, saved documents were not viewable from applications which did not support OpenDoc's Bento format. After Apple terminated Cyberdog along with the rest of OpenDoc, Cyberdog's web browser component grew outdated as web standards evolved.

Cyberdog was once positioned as a replacement for the earlier, discontinued, Apple Open Collaboration Environment.

title: Cyberdog


text:
Potsdam-Mittelmark

Potsdam-Mittelmark is a Kreis (district) in the western part of Brandenburg, Germany. Its neighbouring administrative units are (clockwise from the north) the district of Havelland, the free cities of Brandenburg and Potsdam, the state of Berlin, the district of Teltow-Fläming, and the districts of Wittenberg, Anhalt-Bitterfeld and Jerichower Land in Saxony-Anhalt.

Geography
The district includes the southern banks of the Havel river and the northern parts of the Fläming (a wooded hill chain). There are three nature parks in the district: High Fläming Nature Park, Nuthe-Nieplitz Nature Park and Westhavelland Nature Park.

History
The district was created in 1993 by merging the previous districts of Belzig, Brandenburg-Land and Potsdam-Land.

title: Potsdam-Mittelmark


text:
Wiener equation

A simple mathematical representation of Brownian motion, the Wiener equation, named after Norbert Wiener, assumes the current velocity of a fluid particle fluctuates randomly:

 * $$\mathbf{v} = \frac{d\mathbf{x}}{dt} = g(t),$$

where v is velocity, x is position, d/dt is the time derivative, and g(t) may for instance be white noise.

Since velocity changes instantly in this formalism, the Wiener equation is not suitable for short time scales. In those cases, the Langevin equation, which looks at particle acceleration, must be used.

title: Wiener equation


text:
Langevin equation

In physics, a Langevin equation (named after Paul Langevin) is a stochastic differential equation describing how a system evolves when subjected to a combination of deterministic and fluctuating ("random") forces. The dependent variables in a Langevin equation typically are collective (macroscopic) variables changing only slowly in comparison to the other (microscopic) variables of the system. The fast (microscopic) variables are responsible for the stochastic nature of the Langevin equation. One application is to Brownian motion, which models the fluctuating motion of a small particle in a fluid.

Brownian motion as a prototype
The original Langevin equation describes Brownian motion, the apparently random movement of a particle in a fluid due to collisions with the molecules of the fluid,

$$m\frac{\mathrm d\mathbf{v}}{\mathrm d t}=-\lambda \mathbf{v}+\boldsymbol{\eta}\left( t\right).$$

Here, $$\mathbf{v}$$ is the velocity of the particle, $$\lambda$$ is its damping coefficient, and $$m$$ is its mass. The force acting on the particle is written as a sum of a viscous force proportional to the particle's velocity (Stokes' law), and a noise term $$\boldsymbol{\eta}\left( t\right)$$ representing the effect of the collisions with the molecules of the fluid. The force $$\boldsymbol{\eta}\left( t\right)$$ has a Gaussian probability distribution with correlation function

$$\left\langle \eta_{i}\left( t\right)\eta_{j}\left( t'\right) \right\rangle =2\lambda k_\text{B}T\delta _{i,j} \delta \left(t-t'\right) ,$$ where $$k_\text{B}$$ is Boltzmann's constant, $$T$$ is the temperature and $$\eta_i\left( t\right)$$ is the i-th component of the vector $$\boldsymbol{\eta}\left( t\right)$$. The $\delta$-function form of the time correlation means that the force at a time $$t$$ is uncorrelated with the force at any other time. This is an approximation: the actual random force has a nonzero correlation time corresponding to the collision time of the molecules. However, the Langevin equation is used to describe the motion of a "macroscopic" particle at a much longer time scale, and in this limit the $$\delta$$-correlation and the Langevin equation becomes virtually exact.

Another common feature of the Langevin equation is the occurrence of the damping coefficient $$\lambda$$ in the correlation function of the random force, which in an equilibrium system is an expression of the Einstein relation.

Mathematical aspects
A strictly $$\delta$$-correlated fluctuating force $$\boldsymbol{\eta}\left( t\right)$$ is not a function in the usual mathematical sense and even the derivative $$\mathrm d\mathbf{v}/\mathrm{d}t$$ is not defined in this limit. This problem disappears when the Langevin equation is written in integral form $$m\mathbf{v} = \int^t \left( -\lambda \mathbf{v} + \boldsymbol{\eta}\left( t\right)\right)\mathrm{d}t.$$

Therefore, the differential form is only an abbreviation for its time integral. The general mathematical term for equations of this type is "stochastic differential equation".

Another mathematical ambiguity occurs for Langevin equations with multiplicative noise, which refers to noise terms that are multiplied by a non-constant function of the dependent variables, e.g., $$\left|\boldsymbol{v}(t)\right| \boldsymbol{\eta}(t)$$. If a multiplicative noise is intrinsic to the system, its definition is ambiguous, as it is equally valid to interpret it according to Stratonovich- or Ito- scheme (see Itô calculus). Nevertheless, physical observables are independent of the interpretation, provided the latter is applied consistently when manipulating the equation. This is necessary because the symbolic rules of calculus differ depending on the interpretation scheme. If the noise is external to the system, the appropriate interpretation is the Stratonovich one.

Generic Langevin equation
There is a formal derivation of a generic Langevin equation from classical mechanics. This generic equation plays a central role in the theory of critical dynamics, and other areas of nonequilibrium statistical mechanics. The equation for Brownian motion above is a special case.

An essential step in the derivation is the division of the degrees of freedom into the categories slow and fast. For example, local thermodynamic equilibrium in a liquid is reached within a few collision times, but it takes much longer for densities of conserved quantities like mass and energy to relax to equilibrium. Thus, densities of conserved quantities, and in particular their long wavelength components, are slow variable candidates. This division can be expressed formally with the Zwanzig projection operator. Nevertheless, the derivation is not completely rigorous from a mathematical physics perspective because it relies on assumptions that lack rigorous proof, and instead are justified only as plausible approximations of physical systems.

Let $$A=\{A_i\}$$ denote the slow variables. The generic Langevin equation then reads $$\frac{\mathrm{d}A_{i}}{\mathrm{d}t}=k_\text{B}T\sum\limits_{j}{\left[ {A_{i},A_{j}}\right] \frac{{\mathrm{d}}\mathcal{H}}}-\sum\limits_{j}{\lambda _{i,j}\left( A\right) \frac{\mathrm{d}\mathcal{H}}+}\sum\limits_{j}{\frac{\mathrm{d}{\lambda _{i,j}\left(A\right) }}}+\eta _{i}\left( t\right).$$

The fluctuating force $$\eta_i\left( t\right)$$ obeys a Gaussian probability distribution with correlation function $$\left\langle {\eta _{i}\left( t\right) \eta _{j}\left( t'\right) }\right\rangle =2\lambda _{i,j}\left( A\right) \delta \left( t-t'\right).$$

This implies the Onsager reciprocity relation $$\lambda_{i,j}=\lambda_{j,i}$$ for the damping coefficients $$\lambda$$. The dependence $$\mathrm{d}\lambda_{i,j}/\mathrm{d}A_j$$ of $$\lambda$$ on $$A$$ is negligible in most cases. The symbol $$\mathcal{H}=-\ln\left(p_0\right)$$ denotes the Hamiltonian of the system, where $$p_0\left(A\right)$$ is the equilibrium probability distribution of the variables $$A$$. Finally, $$[A_i,A_j]$$ is the projection of the Poisson bracket of the slow variables $$A_i$$ and $$A_j$$ onto the space of slow variables.

In the Brownian motion case one would have $$\mathcal{H}=\mathbf{p}^2/\left(2mk_\text{B}T\right)$$, $$A=\{\mathbf{p}\}$$ or $$A=\{\mathbf{x}, \mathbf{p}\}$$ and $$[x_i, p_j]=\delta_{i,j}$$. The equation of motion $$\mathrm{d}\mathbf{x}/\mathrm{d}t=\mathbf{p}/m$$ for $$\mathbf{x}$$ is exact: there is no fluctuating force $$\eta_x$$ and no damping coefficient $$\lambda_{x,p}$$.

Thermal noise in an electrical resistor
There is a close analogy between the paradigmatic Brownian particle discussed above and Johnson noise, the electric voltage generated by thermal fluctuations in a resistor. The diagram at the right shows an electric circuit consisting of a resistance R and a capacitance C. The slow variable is the voltage U between the ends of the resistor. The Hamiltonian reads $$\mathcal{H} = E / k_\text{B}T = CU^2 / (2k_\text{B}T)$$, and the Langevin equation becomes $$\frac{\mathrm{d}U}{\mathrm{d}t} =-\frac{U}{RC} + \eta \left( t\right),\;\;\left\langle \eta \left( t\right) \eta \left( t'\right)\right\rangle = \frac{2k_\text{B}T}{RC^{2}}\delta \left(t-t'\right).$$

This equation may be used to determine the correlation function $$\left\langle U\left(t\right) U\left(t'\right) \right\rangle = \frac{k_\text{B}T}{C} \exp \left(-\frac{\left| t - t'\right| } {RC}\right) \approx 2Rk_\text{B}T \delta \left( t - t'\right),$$ which becomes white noise (Johnson noise) when the capacitance $C$ becomes negligibly small.

Critical dynamics
The dynamics of the order parameter $$\varphi$$ of a second order phase transition slows down near the critical point and can be described with a Langevin equation. The simplest case is the universality class "model A" with a non-conserved scalar order parameter, realized for instance in axial ferromagnets, $$\begin{align} \frac{\partial}{\partial t}\varphi{\left(\mathbf{x},t\right)} &= -\lambda\frac{\delta\mathcal{H}}{\delta\varphi} + \eta{\left(\mathbf{x},t\right)},\\[2ex] \mathcal{H} & =\int d^{d}x \left[ \frac{1}{2} r_0 \varphi^2 + u \varphi^4 + \frac{1}{2} \left(\nabla\varphi\right)^2 \right], \\[2ex] \left\langle \eta{\left(\mathbf{x},t\right)} \,\eta{\left(\mathbf{x}',t'\right)}\right\rangle &= 2 \lambda \,\delta{\left(\mathbf{x}-\mathbf{x}'\right)} \; \delta{\left(t-t'\right)}. \end{align}$$ Other universality classes (the nomenclature is "model A",..., "model J") contain a diffusing order parameter, order parameters with several components, other critical variables and/or contributions from Poisson brackets.

Harmonic oscillator in a fluid
$$ m\frac{dv}{dt} = -\lambda v + \eta (t)-k x $$

A particle in a fluid is described by a Langevin equation with a potential energy function, a damping force, and thermal fluctuations given by the fluctuation dissipation theorem. If the potential is quadratic then the constant energy curves are ellipses, as shown in the figure. If there is dissipation but no thermal noise, a particle continually loses energy to the environment, and its time-dependent phase portrait (velocity vs position) corresponds to an inward spiral toward 0 velocity. By contrast, thermal fluctuations continually add energy to the particle and prevent it from reaching exactly 0 velocity. Rather, the initial ensemble of stochastic oscillators approaches a steady state in which the velocity and position are distributed according to the Maxwell–Boltzmann distribution. In the plot below (figure 2), the long time velocity distribution (blue) and position distributions (orange) in a harmonic potential ($ U = \frac{1}{2} k x^2 $ ) is plotted with the Boltzmann probabilities for velocity (green) and position (red). In particular, the late time behavior depicts thermal equilibrium.

Trajectories of free Brownian particles
Consider a free particle of mass $$m$$ with equation of motion described by $$ m \frac{d \mathbf{v}}{dt} = -\frac{\mathbf{v}}{\mu} + \boldsymbol{\eta}(t), $$ where $$\mathbf{v} = d\mathbf{r}/dt$$ is the particle velocity, $$\mu$$ is the particle mobility, and $$\boldsymbol{\eta}(t) = m \mathbf{a}(t)$$ is a rapidly fluctuating force whose time-average vanishes over a characteristic timescale $$t_c$$ of particle collisions, i.e. $$\overline{\boldsymbol{\eta}(t)} = 0$$. The general solution to the equation of motion is $$ \mathbf{v}(t) = \mathbf{v}(0) e^{-t/\tau} + \int_0^t \mathbf{a}(t') e^{-(t-t')/\tau} dt', $$ where $$\tau = m\mu$$ is the correlation time of the noise term. It can also be shown that the autocorrelation function of the particle velocity $$\mathbf{v}$$ is given by

$$ \begin{align} R_{vv}(t_1,t_2) & \equiv \langle \mathbf{v}(t_1) \cdot \mathbf{v}(t_2) \rangle \\ & = v^2(0) e^{-(t_1+t_2)/\tau} + \int_0^{t_1} \int_0^{t_2} R_{aa}(t_1',t_2') e^{-(t_1+t_2-t_1'-t_2')/\tau} dt_1' dt_2' \\ & \simeq v^2(0) e^{-|t_2-t_1|/\tau} + \left[\frac{3k_\text{B}T}{m} - v^2(0)\right] \Big[e^{-|t_2-t_1|/\tau} - e^{-(t_1+t_2)/\tau}\Big], \end{align} $$ where we have used the property that the variables $$\mathbf{a}(t_1')$$ and $$\mathbf{a}(t_2')$$ become uncorrelated for time separations $$t_2'-t_1' \gg t_c$$. Besides, the value of $\lim_{t \to \infty} \langle v^2 (t) \rangle = \lim_{t \to \infty} R_{vv}(t,t)$ is set to be equal to $$3k_\text{B}T/m$$ such that it obeys the equipartition theorem. If the system is initially at thermal equilibrium already with $$v^2(0) = 3 k_\text{B} T/m$$, then $$ \langle v^2(t) \rangle = 3 k_\text{B} T/m$$ for all $$t$$, meaning that the system remains at equilibrium at all times.

The velocity $$\mathbf{v}(t)$$ of the Brownian particle can be integrated to yield its trajectory $$\mathbf{r}(t)$$. If it is initially located at the origin with probability 1, then the result is $$ \mathbf{r}(t) = \mathbf{v}(0) \tau \left(1-e^{-t/\tau}\right) + \tau \int_0^t \mathbf{a}(t') \left[1 - e^{-(t-t') / \tau}\right] dt'.$$

Hence, the average displacement $\langle \mathbf{r}(t) \rangle = \mathbf{v}(0) \tau \left(1-e^{-t/\tau}\right)$ asymptotes to $$\mathbf{v}(0) \tau$$ as the system relaxes. The mean squared displacement can be determined similarly:

$$ \langle r^2(t) \rangle = v^2(0) \tau^2 \left(1 - e^{-t/\tau}\right)^2 - \frac{3k_\text{B}T}{m} \tau^2 \left(1 - e^{-t/\tau}\right) \left(3 - e^{-t/\tau}\right) + \frac{6k_\text{B}T}{m} \tau t. $$

This expression implies that $$\langle r^2(t \ll \tau) \rangle \simeq v^2(0) t^2$$, indicating that the motion of Brownian particles at timescales much shorter than the relaxation time $$\tau$$ of the system is (approximately) time-reversal invariant. On the other hand, $$\langle r^2(t \gg \tau) \rangle \simeq 6 k_\text{B} T \tau t/m = 6 \mu k_\text{B} T t = 6Dt$$, which indicates an irreversible, dissipative process.

Recovering Boltzmann statistics
If the external potential is conservative and the noise term derives from a reservoir in thermal equilibrium, then the long-time solution to the Langevin equation must reduce to the Boltzmann distribution, which is the probability distribution function for particles in thermal equilibrium. In the special case of overdamped dynamics, the inertia of the particle is negligible in comparison to the damping force, and the trajectory $$x(t)$$ is described by the overdamped Langevin equation

$$ \lambda \frac{dx}{dt} = - \frac{\partial V(x)}{\partial x} + \eta(t)\equiv - \frac{\partial V(x)}{\partial x}+\sqrt{2 \lambda k_\text{B} T} \frac{dB_t}{dt},$$

where $$\lambda$$ is the damping constant. The term $$\eta(t)$$ is white noise, characterized by $$\left\langle\eta(t) \eta(t')\right\rangle = 2 k_\text{B} T \lambda \delta(t-t')$$ (formally, the Wiener process). One way to solve this equation is to introduce a test function $$f$$ and calculate its average. The average of $$ f(x(t))$$ should be time-independent for finite $$x(t)$$, leading to

$$ \frac{d}{dt} \left\langle f(x(t))\right\rangle = 0,$$

Itô's lemma for the Itô drift-diffusion process $$ dX_t = \mu_t \, dt + \sigma_t \, dB_t $$ says that the differential of a twice-differentiable function $f(t, x)$ is given by $$ df = \left(\frac{\partial f}{\partial t} + \mu_t\frac{\partial f}{\partial x} + \frac{\sigma_t^2}{2}\frac{\partial^2 f}{\partial x^2}\right)dt + \sigma_t\frac{\partial f}{\partial x}\,dB_t. $$

Applying this to the calculation of $$\langle f(x(t)) \rangle$$ gives $$ \left\langle -f'(x)\frac{\partial V}{\partial x} + k_\text{B} T f''(x) \right\rangle=0.$$

This average can be written using the probability density function $$p(x)$$; $$ \begin{align} & \int \left( -f'(x)\frac{\partial V}{\partial x}p(x) + {k_\text{B} T} f''(x)p(x) \right) dx \\ = &\int \left( -f'(x)\frac{\partial V}{\partial x} p(x) - {k_\text{B} T} f'(x)p'(x) \right) dx \\ = & \; 0 \end{align}$$ where the second term was integrated by parts (hence the negative sign). Since this is true for arbitrary functions $$f$$, it follows that $$ \frac{\partial V}{\partial x} p(x) + {k_\text{B} T} p'(x) = 0,$$ thus recovering the Boltzmann distribution $$ p(x) \propto \exp \left( {-\frac{ V(x)}{k_\text{B} T}}\right).$$

Equivalent techniques
In some situations, one is primarily interested in the noise-averaged behavior of the Langevin equation, as opposed to the solution for particular realizations of the noise. This section describes techniques for obtaining this averaged behavior that are distinct from—but also equivalent to—the stochastic calculus inherent in the Langevin equation.

Fokker–Planck equation
A Fokker–Planck equation is a deterministic equation for the time dependent probability density $$P\left(A,t\right)$$ of stochastic variables $$A$$. The Fokker–Planck equation corresponding to the generic Langevin equation described in this article is the following: $$\frac{\partial P\left(A,t\right)}{\partial t} = \sum_{i,j} \frac{\partial}{\partial A_i} \left(-k_\text{B}T\left[A_i,A_j\right] \frac{\partial\mathcal{H}}{\partial A_j} + \lambda_{i,j} \frac{\partial\mathcal{H}}{\partial A_j} + \lambda_{i,j}\frac{\partial}{\partial A_j}\right) P\left(A,t\right).$$ The equilibrium distribution $$P(A) = p_0(A) = \text{const}\times\exp(-\mathcal{H})$$ is a stationary solution.

Klein–Kramers equation
The Fokker–Planck equation for an underdamped Brownian particle is called the Klein–Kramers equation. If the Langevin equations are written as $$\begin{align} \dot{\mathbf{r}} &= \frac{\mathbf{p}}{m} \\ \dot{\mathbf{p}} &= -\xi \, \mathbf{p} - \nabla V(\mathbf{r}) + \sqrt{2 m \xi k_{\mathrm{B}} T} \boldsymbol{\eta}(t), \qquad \langle \boldsymbol{\eta}^{\mathrm{T}}(t) \boldsymbol{\eta}(t') \rangle = \mathbf{I} \delta(t-t') \end{align}$$ where $$\mathbf{p}$$ is the momentum, then the corresponding Fokker–Planck equation is $$\frac{\partial f}{\partial t} + \frac{1}{m} \mathbf{p} \cdot \nabla_{\mathbf{r}} f = \xi \nabla_{\mathbf{p}} \cdot \left( \mathbf{p} \, f \right) + \nabla_{\mathbf{p}} \cdot \left( \nabla V(\mathbf{r}) \, f \right) + m \xi k_{\mathrm{B}} T \, \nabla_{\mathbf{p}}^2 f$$ Here $$\nabla_{\mathbf{r}}$$ and $$\nabla_{\mathbf{p}}$$ are the gradient operator with respect to $r$ and $p$, and $$\nabla_{\mathbf{p}}^2$$ is the Laplacian with respect to $p$.

In $$d$$-dimensional free space, corresponding to $$V(\mathbf{r}) = \text{constant}$$ on $$\mathbb{R}^{d}$$, this equation can be solved using Fourier transforms. If the particle is initialized at $$t = 0$$ with position $$\mathbf{r}'$$ and momentum $$\mathbf{p}'$$, corresponding to initial condition $$f(\mathbf{r}, \mathbf{p}, 0) = \delta(\mathbf{r}-\mathbf{r}')\delta(\mathbf{p}-\mathbf{p}')$$, then the solution is $$\begin{align} f(\mathbf{r}, \mathbf{p}, t) =& \frac{1}{\left(2 \pi \sigma_X \sigma_P \sqrt{1 - \beta^2}\right)^d} \times \\ & \quad\exp\left[-\frac{1}{2(1-\beta^2)} \left( \frac{|\mathbf{r} - \boldsymbol{\mu}_X|^2}{\sigma_X^2} + \frac{|\mathbf{p} - \boldsymbol{\mu}_P|^2}{\sigma_P^2} -  \frac{2 \beta (\mathbf{r} - \boldsymbol{\mu}_X) \cdot (\mathbf{p} - \boldsymbol{\mu}_P)}{\sigma_X \sigma_P} \right) \right] \end{align}$$ where $$\begin{align} &\sigma^2_X = \frac{k_{\mathrm{B}} T}{m \xi^2} \left[1 + 2 \xi t - \left(2 - e^{-\xi t}\right)^2 \right]; \qquad  \sigma^2_P = m k_{\mathrm{B}} T \left(1 - e^{-2 \xi t} \right) \\ &\beta = \frac{k_\text{B} T}{\xi \sigma_X \sigma_P} \left(1 - e^{-\xi t}\right)^2 \\ &\boldsymbol{\mu}_X = \mathbf{r}' + (m \xi)^{-1} \left(1 - e^{-\xi t} \right) \mathbf{p}' ; \qquad \boldsymbol{\mu}_P = \mathbf{p}' e^{-\xi t}. \end{align}$$ In three spatial dimensions, the mean squared displacement is $$\langle \mathbf{r}(t)^2 \rangle = \int f(\mathbf{r}, \mathbf{p}, t) \mathbf{r}^2 \, d\mathbf{r}d\mathbf{p} = \boldsymbol{\mu}_X^2 + 3 \sigma_X^2$$

Path integral
A path integral equivalent to a Langevin equation can be obtained from the corresponding Fokker–Planck equation or by transforming the Gaussian probability distribution $$P^{(\eta)}(\eta)\mathrm{d}\eta$$ of the fluctuating force $$\eta$$ to a probability distribution of the slow variables, schematically $$P(A)\mathrm{d}A = P^{(\eta)}(\eta(A))\det(\mathrm{d}\eta/\mathrm{d}A)\mathrm{d}A$$. The functional determinant and associated mathematical subtleties drop out if the Langevin equation is discretized in the natural (causal) way, where $$A(t+\Delta t)-A(t)$$ depends on $$A(t)$$ but not on $$A(t+\Delta t)$$. It turns out to be convenient to introduce auxiliary response variables $$\tilde A$$. The path integral equivalent to the generic Langevin equation then reads $$\int P(A,\tilde{A})\,\mathrm{d}A\,\mathrm{d}\tilde{A} = N\int \exp \left( L(A,\tilde{A})\right) \mathrm{d}A\,\mathrm{d}\tilde{A},$$ where $$N$$ is a normalization factor and $$L(A,\tilde{A}) = \int \sum_{i,j} \left\{ \tilde{A}_{i}\lambda_{i,j}\tilde{A}_{j}-\widetilde{A}_{i} \left \{ \delta_{i,j} \frac{\mathrm{d}A_j}{\mathrm{d}t}-k_\text{B}T\left[A_i, A_j\right]\frac{\mathrm{d}\mathcal{H}}{\mathrm{d}A_{j}}+\lambda_{i,j}\frac{\mathrm{d}\mathcal{H}}{\mathrm{d}A_j} - \frac{\mathrm{d}\lambda_{i,j}}{\mathrm{d}A_j} \right \} \right\} \mathrm{d}t.$$ The path integral formulation allows for the use of tools from quantum field theory, such as perturbation and renormalization group methods. This formulation is typically referred to as either the Martin-Siggia-Rose formalism or the Janssen-De Dominicis formalism after its developers. The mathematical formalism for this representation can be developed on abstract Wiener space.

title: Langevin equation


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Equivalent rectangular bandwidth

The equivalent rectangular bandwidth or ERB is a measure used in psychoacoustics, which gives an approximation to the bandwidths of the filters in human hearing, using the unrealistic but convenient simplification of modeling the filters as rectangular band-pass filters, or band-stop filters, like in tailor-made notched music training (TMNMT).

Approximations
For moderate sound levels and young listeners, the bandwidth of human auditory filters can be approximated by the polynomial equation:

where f is the center frequency of the filter in kHz and ERB(f) is the bandwidth of the filter in Hz. The approximation is based on the results of a number of published simultaneous masking experiments and is valid from 0.1 to 6.5 kHz.

The above approximation was given in 1983 by Moore and Glasberg, who in 1990 published another (linear) approximation:

where f is in kHz and ERB(f) is in Hz. The approximation is applicable at moderate sound levels and for values of f between 0.1 and 10 kHz.

ERB-rate scale
The ERB-rate scale, or ERB-number scale, can be defined as a function ERBS(f) which returns the number of equivalent rectangular bandwidths below the given frequency f. The units of the ERB-number scale are known ERBs, or as Cams, following a suggestion by Hartmann. The scale can be constructed by solving the following differential system of equations:

\begin{cases} \mathrm{ERBS}(0) = 0\\ \frac{df}{d\mathrm{ERBS}(f)} = \mathrm{ERB}(f)\\ \end{cases} $$

The solution for ERBS(f) is the integral of the reciprocal of ERB(f) with the constant of integration set in such a way that ERBS(0) = 0.

Using the second order polynomial approximation ($$) for ERB(f) yields:

\mathrm{ERBS}(f) = 11.17 \cdot \ln\left(\frac{f+0.312}{f+14.675}\right) + 43.0 $$

where f is in kHz. The VOICEBOX speech processing toolbox for MATLAB implements the conversion and its inverse as:

\mathrm{ERBS}(f) = 11.17268 \cdot \ln\left(1 + \frac{46.06538 \cdot f}{f + 14678.49}\right) $$

f = \frac{676170.4}{47.06538 - e^{0.08950404 \cdot \mathrm{ERBS}(f)}} - 14678.49 $$

where f is in Hz.

Using the linear approximation ($$) for ERB(f) yields:

\mathrm{ERBS}(f) = 21.4 \cdot \log_{10}(1 + 0.00437 \cdot f) $$

where f is in Hz.

title: Equivalent rectangular bandwidth