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Beauty

Beauty is commonly described as a feature of objects that makes them pleasurable to perceive. Such objects include landscapes, sunsets, humans and works of art. Beauty, art and taste are the main subjects of aesthetics, one of the major branches of philosophy. As a positive aesthetic value, it is contrasted with ugliness as its negative counterpart.

One difficulty in understanding beauty is that it has both objective and subjective aspects: it is seen as a property of things but also as depending on the emotional response of observers. Because of its subjective side, beauty is said to be "in the eye of the beholder". It has been argued that the ability on the side of the subject needed to perceive and judge beauty, sometimes referred to as the "sense of taste", can be trained and that the verdicts of experts coincide in the long run. This suggests the standards of validity of judgments of beauty are intersubjective, i.e. dependent on a group of judges, rather than fully subjective or objective.

Conceptions of beauty aim to capture what is essential to all beautiful things. Classical conceptions define beauty in terms of the relation between the beautiful object as a whole and its parts: the parts should stand in the right proportion to each other and thus compose an integrated harmonious whole. Hedonist conceptions see a necessary connection between pleasure and beauty, e.g. that for an object to be beautiful is for it to cause disinterested pleasure. Other conceptions include defining beautiful objects in terms of their value, of a loving attitude toward them or of their function.

Overview
Beauty, together with art and taste, is the main subject of aesthetics, one of the major branches of philosophy. Beauty is usually categorized as an aesthetic property besides other properties, like grace, elegance or the sublime. As a positive aesthetic value, beauty is contrasted with ugliness as its negative counterpart. Beauty is often listed as one of the three fundamental concepts of human understanding besides truth and goodness.

Objectivists or realists see beauty as an objective or mind-independent feature of beautiful things, which is denied by subjectivists. The source of this debate is that judgments of beauty seem to be based on subjective grounds, namely our feelings, while claiming universal correctness at the same time. This tension is sometimes referred to as the "antinomy of taste". Adherents of both sides have suggested that a certain faculty, commonly called a sense of taste, is necessary for making reliable judgments about beauty. David Hume, for example, suggests that this faculty can be trained and that the verdicts of experts coincide in the long run.

Beauty is mainly discussed in relation to concrete objects accessible to sensory perception. It has been suggested that the beauty of a thing supervenes on the sensory features of this thing. It has also been proposed that abstract objects like stories or mathematical proofs can be beautiful. Beauty plays a central role in works of art and nature.

An influential distinction among beautiful things, according to Immanuel Kant, is that between adherent beauty (pulchritudo adhaerens) and free beauty (pulchritudo vaga). A thing has adherent beauty if its beauty depends on the conception or function of this thing, unlike free or absolute beauty. Examples of adherent beauty include an ox which is beautiful as an ox but not beautiful as a horse or a photograph which is beautiful, because it depicts a beautiful building but that lacks beauty generally speaking because of its low quality.

Objectivism and subjectivism
Judgments of beauty seem to occupy an intermediary position between objective judgments, e.g. concerning the mass and shape of a grapefruit, and subjective likes, e.g. concerning whether the grapefruit tastes good. Judgments of beauty differ from the former because they are based on subjective feelings rather than objective perception. But they also differ from the latter because they lay claim on universal correctness. This tension is also reflected in common language. On the one hand, we talk about beauty as an objective feature of the world that is ascribed, for example, to landscapes, paintings or humans. The subjective side, on the other hand, is expressed in sayings like "beauty is in the eye of the beholder".

These two positions are often referred to as objectivism (or realism) and subjectivism. Objectivism is the traditional view, while subjectivism developed more recently in western philosophy. Objectivists hold that beauty is a mind-independent feature of things. On this account, the beauty of a landscape is independent of who perceives it or whether it is perceived at all. Disagreements may be explained by an inability to perceive this feature, sometimes referred to as a "lack of taste". Subjectivism, on the other hand, denies the mind-independent existence of beauty. Influential for the development of this position was John Locke's distinction between primary qualities, which the object has independent of the observer, and secondary qualities, which constitute powers in the object to produce certain ideas in the observer. When applied to beauty, there is still a sense in which it depends on the object and its powers. But this account makes the possibility of genuine disagreements about claims of beauty implausible, since the same object may produce very different ideas in distinct observers. The notion of "taste" can still be used to explain why different people disagree about what is beautiful, but there is no objectively right or wrong taste, there are just different tastes.

The problem with both the objectivist and the subjectivist position in their extreme form is that each has to deny some intuitions about beauty. This issue is sometimes discussed under the label "antinomy of taste". It has prompted various philosophers to seek a unified theory that can take all these intuitions into account. One promising route to solve this problem is to move from subjective to intersubjective theories, which hold that the standards of validity of judgments of taste are intersubjective or dependent on a group of judges rather than objective. This approach tries to explain how genuine disagreement about beauty is possible despite the fact that beauty is a mind-dependent property, dependent not on an individual but a group. A closely related theory sees beauty as a secondary or response-dependent property. On one such account, an object is beautiful "if it causes pleasure by virtue of its aesthetic properties". The problem that different people respond differently can be addressed by combining response-dependence theories with so-called ideal-observer theories: it only matters how an ideal observer would respond. There is no general agreement on how "ideal observers" are to be defined, but it is usually assumed that they are experienced judges of beauty with a fully developed sense of taste. This suggests an indirect way of solving the antinomy of taste: instead of looking for necessary and sufficient conditions of beauty itself, one can learn to identify the qualities of good critics and rely on their judgments. This approach only works if unanimity among experts was ensured. But even experienced judges may disagree in their judgments, which threatens to undermine ideal-observer theories.

Conceptions
Various conceptions of the essential features of beautiful things have been proposed but there is no consensus as to which is the right one.

Classical
The "classical conception" (see Classicism) defines beauty in terms of the relation between the beautiful object as a whole and its parts: the parts should stand in the right proportion to each other and thus compose an integrated harmonious whole. On this account, which found its most explicit articulation in the Italian Renaissance, the beauty of a human body, for example, depends, among other things, on the right proportion of the different parts of the body and on the overall symmetry. One problem with this conception is that it is difficult to give a general and detailed description of what is meant by "harmony between parts" and raises the suspicion that defining beauty through harmony results in exchanging one unclear term for another one. Some attempts have been made to dissolve this suspicion by searching for laws of beauty, like the golden ratio.

18th century philosopher Alexander Baumgarten, for example, saw laws of beauty in analogy with laws of nature and believed that they could be discovered through empirical research. As of 2003, these attempts have failed to find a general definition of beauty and several authors take the opposite claim that such laws cannot be formulated, as part of their definition of beauty.

Hedonism
A very common element in many conceptions of beauty is its relation to pleasure. Hedonism makes this relation part of the definition of beauty by holding that there is a necessary connection between pleasure and beauty, e.g. that for an object to be beautiful is for it to cause pleasure or that the experience of beauty is always accompanied by pleasure. This account is sometimes labeled as "aesthetic hedonism" in order to distinguish it from other forms of hedonism. An influential articulation of this position comes from Thomas Aquinas, who treats beauty as "that which pleases in the very apprehension of it". Immanuel Kant explains this pleasure through a harmonious interplay between the faculties of understanding and imagination. A further question for hedonists is how to explain the relation between beauty and pleasure. This problem is akin to the Euthyphro dilemma: is something beautiful because we enjoy it or do we enjoy it because it is beautiful? Identity theorists solve this problem by denying that there is a difference between beauty and pleasure: they identify beauty, or the appearance of it, with the experience of aesthetic pleasure.

Hedonists usually restrict and specify the notion of pleasure in various ways in order to avoid obvious counterexamples. One important distinction in this context is the difference between pure and mixed pleasure. Pure pleasure excludes any form of pain or unpleasant feeling while the experience of mixed pleasure can include unpleasant elements. But beauty can involve mixed pleasure, for example, in the case of a beautifully tragic story, which is why mixed pleasure is usually allowed in hedonist conceptions of beauty.

Another problem faced by hedonist theories is that we take pleasure from many things that are not beautiful. One way to address this issue is to associate beauty with a special type of pleasure: aesthetic or disinterested pleasure. A pleasure is disinterested if it is indifferent to the existence of the beautiful object or if it did not arise owing to an antecedent desire through means-end reasoning. For example, the joy of looking at a beautiful landscape would still be valuable if it turned out that this experience was an illusion, which would not be true if this joy was due to seeing the landscape as a valuable real estate opportunity. Opponents of hedonism usually concede that many experiences of beauty are pleasurable but deny that this is true for all cases. For example, a cold jaded critic may still be a good judge of beauty because of her years of experience but lack the joy that initially accompanied her work. One way to avoid this objection is to allow responses to beautiful things to lack pleasure while insisting that all beautiful things merit pleasure, that aesthetic pleasure is the only appropriate response to them.

Others
G. E. Moore explained beauty in regard to intrinsic value as "that of which the admiring contemplation is good in itself". This definition connects beauty to experience while managing to avoid some of the problems usually associated with subjectivist positions since it allows that things may be beautiful even if they are never experienced.

Another subjectivist theory of beauty comes from George Santayana, who suggested that we project pleasure onto the things we call "beautiful". So in a process akin to a category mistake, one treats one's subjective pleasure as an objective property of the beautiful thing. Other conceptions include defining beauty in terms of a loving or longing attitude toward the beautiful object or in terms of its usefulness or function. In 1871, functionalist Charles Darwin explained beauty as result of accumulative sexual selection in "The Descent of Man and Selection in Relation to Sex".

Greco-Roman tradition
The classical Greek noun that best translates to the English-language words "beauty" or "beautiful" was κάλλος, kallos, and the adjective was καλός, kalos. This is also translated as "good" or "of fine quality" and thus has a broader meaning than mere physical or material beauty. Similarly, kallos was used differently from the English word beauty in that it first and foremost applied to humans and bears an erotic connotation. The Koine Greek word for beautiful was ὡραῖος, hōraios, an adjective etymologically coming from the word ὥρα, hōra, meaning "hour". In Koine Greek, beauty was thus associated with "being of one's hour". Thus, a ripe fruit (of its time) was considered beautiful, whereas a young woman trying to appear older or an older woman trying to appear younger would not be considered beautiful. In Attic Greek, hōraios had many meanings, including "youthful" and "ripe old age". Another classical term in use to describe beauty was pulchrum (Latin).

Beauty for ancient thinkers existed both in form, which is the material world as it is, and as embodied in the spirit, which is the world of mental formations. Greek mythology mentions Helen of Troy as the most beautiful woman. Ancient Greek architecture is based on this view of symmetry and proportion.

Pre-Socratic
In one fragment of Heraclitus's writings (Fragment 106) he mentions beauty, this reads: "To God all things are beautiful, good, right..." The earliest Western theory of beauty can be found in the works of early Greek philosophers from the pre-Socratic period, such as Pythagoras, who conceived of beauty as useful for a moral education of the soul. He wrote of how people experience pleasure when aware of a certain type of formal situation present in reality, perceivable by sight or through the ear and discovered the underlying mathematical ratios in the harmonic scales in music. The Pythagoreans conceived of the presence of beauty in universal terms, which is, as existing in a cosmological state, they observed beauty in the heavens. They saw a strong connection between mathematics and beauty. In particular, they noted that objects proportioned according to the golden ratio seemed more attractive.

Classical period
The classical concept of beauty is one that exhibits perfect proportion (Wolfflin). In this context, the concept belonged often within the discipline of mathematics. An idea of spiritual beauty emerged during the classical period, beauty was something embodying divine goodness, while the demonstration of behaviour which might be classified as beautiful, from an inner state of morality which is aligned to the good.

The writing of Xenophon shows a conversation between Socrates and Aristippus. Socrates discerned differences in the conception of the beautiful, for example, in inanimate objects, the effectiveness of execution of design was a deciding factor on the perception of beauty in something. By the account of Xenophon, Socrates found beauty congruent with that to which was defined as the morally good, in short, he thought beauty coincident with the good.

Beauty is a subject of Plato in his work Symposium. In the work, the high priestess Diotima describes how beauty moves out from a core singular appreciation of the body to outer appreciations via loved ones, to the world in its state of culture and society (Wright). In other words, Diotoma gives to Socrates an explanation of how love should begin with erotic attachment, and end with the transcending of the physical to an appreciation of beauty as a thing in itself. The ascent of love begins with one's own body, then secondarily, in appreciating beauty in another's body, thirdly beauty in the soul, which cognates to beauty in the mind in the modern sense, fourthly beauty in institutions, laws and activities, fifthly beauty in knowledge, the sciences, and finally to lastly love beauty itself, which translates to the original Greek language term as auto to kalon. In the final state, auto to kalon and truth are united as one. There is the sense in the text, concerning love and beauty they both co-exist but are still independent or, in other words, mutually exclusive, since love does not have beauty since it seeks beauty. The work toward the end provides a description of beauty in a negative sense.

Plato also discusses beauty in his work Phaedrus, and identifies Alcibiades as beautiful in Parmenides. He considered beauty to be the Idea (Form) above all other Ideas. Platonic thought synthesized beauty with the divine. Scruton (cited: Konstan) states Plato states of the idea of beauty, of it (the idea), being something inviting desirousness (c.f seducing), and, promotes an intellectual renunciation (c.f. denouncing) of desire. For Alexander Nehamas, it is only the locating of desire to which the sense of beauty exists, in the considerations of Plato.

Aristotle defines beauty in Metaphysics as having order, symmetry and definiteness which the mathematical sciences exhibit to a special degree. He saw a relationship between the beautiful (to kalon) and virtue, arguing that "Virtue aims at the beautiful."

Roman
In De Natura Deorum Cicero wrote: "the splendour and beauty of creation", in respect to this, and all the facets of reality resulting from creation, he postulated these to be a reason to see the existence of a God as creator.

Western Middle Ages
In the Middle Ages, Catholic philosophers like Thomas Aquinas included beauty among the transcendental attributes of being. In his Summa Theologica, Aquinas described the three conditions of beauty as: integritas (wholeness), consonantia (harmony and proportion), and claritas (a radiance and clarity that makes the form of a thing apparent to the mind).

In the Gothic Architecture of the High and Late Middle Ages, light was considered the most beautiful revelation of God, which was heralded in design. Examples are the stained glass of Gothic Cathedrals including Notre-Dame de Paris and Chartres Cathedral.

St. Augustine said of beauty "Beauty is indeed a good gift of God; but that the good may not think it a great good, God dispenses it even to the wicked."

Renaissance
Classical philosophy and sculptures of men and women produced according to the Greek philosophers' tenets of ideal human beauty were rediscovered in Renaissance Europe, leading to a re-adoption of what became known as a "classical ideal". In terms of female human beauty, a woman whose appearance conforms to these tenets is still called a "classical beauty" or said to possess a "classical beauty", whilst the foundations laid by Greek and Roman artists have also supplied the standard for male beauty and female beauty in western civilization as seen, for example, in the Winged Victory of Samothrace. During the Gothic era, the classical aesthetical canon of beauty was rejected as sinful. Later, Renaissance and Humanist thinkers rejected this view, and considered beauty to be the product of rational order and harmonious proportions. Renaissance artists and architects (such as Giorgio Vasari in his "Lives of Artists") criticised the Gothic period as irrational and barbarian. This point of view of Gothic art lasted until Romanticism, in the 19th century. Vasari aligned himself to the classical notion and thought of beauty as defined as arising from proportion and order.

Age of Reason
The Age of Reason saw a rise in an interest in beauty as a philosophical subject. For example, Scottish philosopher Francis Hutcheson argued that beauty is "unity in variety and variety in unity". He wrote that beauty was neither purely subjective nor purely objective—it could be understood not as "any Quality suppos'd to be in the Object, which should of itself be beautiful, without relation to any Mind which perceives it: For Beauty, like other Names of sensible Ideas, properly denotes the Perception of some mind; ... however we generally imagine that there is something in the Object just like our Perception."

Immanuel Kant believed that there could be no "universal criterion of the beautiful" and that the experience of beauty is subjective, but that an object is judged to be beautiful when it seems to display "purposiveness"; that is, when its form is perceived to have the character of a thing designed according to some principle and fitted for a purpose. He distinguished "free beauty" from "merely adherent beauty", explaining that "the first presupposes no concept of what the object ought to be; the second does presuppose such a concept and the perfection of the object in accordance therewith." By this definition, free beauty is found in seashells and wordless music; adherent beauty in buildings and the human body.

The Romantic poets, too, became highly concerned with the nature of beauty, with John Keats arguing in Ode on a Grecian Urn that:

 * Beauty is truth, truth beauty, —that is all
 * Ye know on earth, and all ye need to know.

Western 19th and 20th century
In the Romantic period, Edmund Burke postulated a difference between beauty in its classical meaning and the sublime. The concept of the sublime, as explicated by Burke and Kant, suggested viewing Gothic art and architecture, though not in accordance with the classical standard of beauty, as sublime.

The 20th century saw an increasing rejection of beauty by artists and philosophers alike, culminating in postmodernism's anti-aesthetics. This is despite beauty being a central concern of one of postmodernism's main influences, Friedrich Nietzsche, who argued that the Will to Power was the Will to Beauty.

In the aftermath of postmodernism's rejection of beauty, thinkers have returned to beauty as an important value. American analytic philosopher Guy Sircello proposed his New Theory of Beauty as an effort to reaffirm the status of beauty as an important philosophical concept. He rejected the subjectivism of Kant and sought to identify the properties inherent in an object that make it beautiful. He called qualities such as vividness, boldness, and subtlety "properties of qualitative degree" (PQDs) and stated that a PQD makes an object beautiful if it is not—and does not create the appearance of—"a property of deficiency, lack, or defect"; and if the PQD is strongly present in the object.

Elaine Scarry argues that beauty is related to justice.

Beauty is also studied by psychologists and neuroscientists in the field of experimental aesthetics and neuroesthetics respectively. Psychological theories see beauty as a form of pleasure. Correlational findings support the view that more beautiful objects are also more pleasing. Some studies suggest that higher experienced beauty is associated with activity in the medial orbitofrontal cortex. This approach of localizing the processing of beauty in one brain region has received criticism within the field.

Philosopher and novelist Umberto Eco wrote On Beauty: A History of a Western Idea (2004) and On Ugliness (2007). The narrator of his novel The Name of the Rose follows Aquinas in declaring: "three things concur in creating beauty: first of all integrity or perfection, and for this reason, we consider ugly all incomplete things; then proper proportion or consonance; and finally clarity and light", before going on to say "the sight of the beautiful implies peace". Mike Phillips has described Umberto Eco's On Beauty as "incoherent" and criticized him for focusing only on Western European history and devoting none of his book to Eastern European, Asian, or African history. Amy Finnerty described Eco's work On Ugliness favorably.

Chinese philosophy
Chinese philosophy has traditionally not made a separate discipline of the philosophy of beauty. Confucius identified beauty with goodness, and considered a virtuous personality to be the greatest of beauties: In his philosophy, "a neighborhood with a ren man in it is a beautiful neighborhood." Confucius's student Zeng Shen expressed a similar idea: "few men could see the beauty in some one whom they dislike." Mencius considered "complete truthfulness" to be beauty. Zhu Xi said: "When one has strenuously implemented goodness until it is filled to completion and has accumulated truth, then the beauty will reside within it and will not depend on externals."

Human attributes

The word "beauty" is often used as a countable noun to describe a beautiful woman.

The characterization of a person as "beautiful", whether on an individual basis or by community consensus, is often based on some combination of inner beauty, which includes psychological factors such as personality, intelligence, grace, politeness, charisma, integrity, congruence and elegance, and outer beauty (i.e. physical attractiveness) which includes physical attributes which are valued on an aesthetic basis.

Standards of beauty have changed over time, based on changing cultural values. Historically, paintings show a wide range of different standards for beauty.

A strong indicator of physical beauty is "averageness". When images of human faces are averaged together to form a composite image, they become progressively closer to the "ideal" image and are perceived as more attractive. This was first noticed in 1883, when Francis Galton overlaid photographic composite images of the faces of vegetarians and criminals to see if there was a typical facial appearance for each. When doing this, he noticed that the composite images were more attractive compared to any of the individual images. Researchers have replicated the result under more controlled conditions and found that the computer-generated, mathematical average of a series of faces is rated more favorably than individual faces. It is argued that it is evolutionarily advantageous that sexual creatures are attracted to mates who possess predominantly common or average features, because it suggests the absence of genetic or acquired defects.

Since the 1970's there has been increasing evidence that a preference for beautiful faces emerges early in infancy, and is probably innate, and that the rules by which attractiveness is established are similar across different genders and cultures.

A feature of beautiful women which has been explored by researchers is a waist–hip ratio of approximately 0.70. As of 2004, physiologists had shown that women with hourglass figures were more fertile than other women because of higher levels of certain female hormones, a fact that may subconsciously condition males choosing mates. In 2008, other commentators have suggested that this preference may not be universal. For instance, in some non-Western cultures in which women have to do work such as finding food, men tend to have preferences for higher waist-hip ratios.

Exposure to the thin ideal in mass media, such as fashion magazines, directly correlates with body dissatisfaction, low self-esteem, and the development of eating disorders among female viewers. Further, the widening gap between individual body sizes and societal ideals continues to breed anxiety among young girls as they grow, highlighting the dangerous nature of beauty standards in society.

Western concept
A study using Chinese immigrants and Hispanic, Black and White American citizens found that their ideals of female beauty were not significantly different. Participants in the study rated Asian and Latina women as more attractive than White and Black women, and it was found that Asian and Latina women had more of the attributes that were considered attractive for women. Exposure to Western media did not influence or improve the Asian men's ratings of White women.

One study found that East Asian women in the United States are closer to the ideal figure promoted in Western media, and that East Asian women conform to both Western and Eastern influences in the United States. East Asian men were found to be more impacted by Western beauty ideals then East Asian women, in the United States. East Asian men felt as though their bodies were not large enough and therefore deviated from the Western norm. East Asian men and white Western women were found to have the highest levels of body dissatisfaction in the United States. A study of African American and South Asian women found that some had internalized a white beauty ideal that placed light skin and straight hair at the top.

Eurocentric standards for men include tallness, leanness, and muscularity, which have been idolized through American media, such as in Hollywood films and magazine covers.

In of the United States, African Americans have historically been subjected to beauty ideals that often do not reflect their own appearance, which can lead to issues of low self-esteem. African-American philosopher Cornel West elaborates that, "much of black self-hatred and self-contempt has to do with the refusal of many black Americans to love their own black bodies-especially their black noses, hips, lips, and hair." According to Patton (2006), the stereotype of African-American women's inferiority (relative to other races of women) maintains a system of oppression based on race and gender that operates to the detriment of women of all races, and also black men. In the 1960s, the black is beautiful cultural movement sought to dispel the notion of a Eurocentric concept of beauty.

Much criticism has been directed at models of beauty which depend solely upon Western ideals of beauty, as seen, for example, in the Barbie franchise. Criticisms of Barbie are often centered around concerns that children consider Barbie a role model of beauty and will attempt to emulate her. One of the most common criticisms of Barbie is that she promotes an unrealistic idea of body image for a young woman, leading to a risk that girls who attempt to emulate her will become anorexic.

As of 1998, these criticisms of the lack of diversity in such franchises as the Barbie model of beauty in Western culture, had led to a dialogue to create non-exclusive models of Western ideals in body type for young girls who do not match the thinness ideal that Barbie represents. Mattel responded to these criticisms.

In East Asian cultures, familial pressures and cultural norms shape beauty ideals. A 2017 experimental study concluded that Asian cultural idealization of "fragile" girls was impacting Asian American women's lifestyle, eating, and appearance choices.

Effects on society
Researchers have found that good-looking students get higher grades from their teachers than students with an ordinary appearance. Some studies using mock criminal trials have shown that physically attractive "defendants" are less likely to be convicted—and if convicted are likely to receive lighter sentences—than less attractive ones (although the opposite effect was observed when the alleged crime was swindling, perhaps because jurors perceived the defendant's attractiveness as facilitating the crime). Studies among teens and young adults, such as those of psychiatrist and self-help author Eva Ritvo show that skin conditions have a profound effect on social behavior and opportunity.

How much money a person earns may also be influenced by physical beauty. One study found that people low in physical attractiveness earn 5 to 10 percent less than ordinary-looking people, who in turn earn 3 to 8 percent less than those who are considered good-looking. In the market for loans, the least attractive people are less likely to get approvals, although they are less likely to default. In the marriage market, women's looks are at a premium, but men's looks do not matter much. The impact of physical attractiveness on earnings varies across races, with the largest beauty wage gap among black women and black men.

Conversely, being very unattractive increases the individual's propensity for criminal activity for a number of crimes ranging from burglary to theft to selling illicit drugs.

Discrimination against others based on their appearance is known as lookism.

title: Beauty


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Talk:Bretwalda

Early comments
I also created a page (erroneously) Bretwaldas which is really just a placeholder for the list of them. I'll merge it with this. sjc

The British, originally were called Breta, Bretta. The Saxon language word walda (modern High German Verwalter) is translated into English language as administrator. The German language word Wald (English forest) is connected also. user:H.J.

Further cognate moved here from article: --Henrygb 09:22, 7 October 2005 (UTC)
 * It may also be of interest to some researchers to know that 'walda' is stikingly similar to the Latvian and ancient Baltic term for overlord, 'valda'. This may suggest a Baltic descent of the original (or one of the original) 'bretwalda' or 'British overlords'.

I've just expanded the list to include all the Mercian overlords of England, rather than a select two.TharkunColl 01:14, 28 December 2005 (UTC)

 * I've done a substantial rewrite on this as I don't think the existing article was very clear. I'm actually in favour of removing most of the kings on the list of overlords, as I think it is likely to confuse readers into thinking that there was some sort of continuity when there most definitely was not. If no one has any comments on this then I'll remove most of that list in a few days. Harthacanute 16:12, 7 February 2006 (UTC)

During the period of Mercian hegemenoy over England, mid 7th to mid 9th century, the overlordship stabilised with the kings of Mercia. This is attested by numerous charters (indeed, the title "king of the English", Rex Anglorum, was first used by them. This should not be confused with the earler period. Both Bede and the West Saxon chronicle exhibit a marked anti-Mercian bias, but this is corrected by the charters, coinage, and other sources. TharkunColl 16:28, 7 February 2006 (UTC)

 * Basically, what I don't like is the dates on the list. It makes it seem that there was one clear overlord at all times. Is this what you're advocating? Harthacanute 16:57, 7 February 2006 (UTC)

Not for the earlier period. But during the Mercian hegemony, the overlordship is coterminous with their reigns as kings of Mercia, because it was Mercia itself as a political entity that was dominant. TharkunColl 17:27, 7 February 2006 (UTC)

 * I accept your points on Mercian hegemony, although I'd raise Simon Keynes intepretation of there being 3 stages of Mercian hegemony, with an initial phase under Penda (who's missing from the list) and Wulfhere, a second phase under Aethelbald and Offa and a third phase under Coenwulf, Ceolwulf and Beornwulf. (See, inter alia, his article in Blackwell on Mercia). What I dislike is the image this page is presenting, which is that there was some continuity of rule in Anglo-Saxon England. I've made similar points on the Kings of England discussion. On this I think we'll have to agree to differ, and, provided the warnings regarding use of the term Bretwalda remain, I'm happy to leave the list of overlords on the page. Harthacanute 17:45, 7 February 2006 (UTC)

Harthacanute, your opinions about the title of Bretwalda are peculiar, to say the least. I've noticed that you have removed its mention in a number of articles, which you defend either by claiming that "no evidence for this title in the 7th century", or that Bede did not have this title in mind when he spoke of men huius primus imperium (H.E. 2.5).This interpretation flies in the face of authorities like Sir Frank Stenton, who believed that this title did exist (Anglo-Saxon Englsnd, pp. 33f), & more recent writers like D.P. Kirby (e.g., The Earliest English Kings, pp. 17f). Do you have any secondary sources to quote in defense for your personal opinion? Or should I assume this is the result of your original research? -- llywrch 06:15, 18 March 2006 (UTC)

 * This does indeed fly in the face of Frank Stenton who, seminal as his work may have been in the 1940s, was trying to convey a particular view of the past in his Anglo-Saxon England, with things moving steadily from small free Germanic kingdoms, through confederacy and in to the kingdom of England. His use of Bretwalda fits into this teleological approach. However, the view is now substantially dated. I'm essentially running with Simon Keynes on this:
 * "Bede's list is best understood as the product of personal reflection on his part. It is likely, in the same way, that the chronicler's use of the term 'Bretwalda' did not represent Ecgberht's succession to a recognised office, with powers and responsibilities particular to itself, but rather a flight of fancy, important to the chronicler but of no real importance in the unfolding course of political development." (Keynes in Blackwell Encyclopaedia of Anglo-Saxon England, (1999), p74)
 * Essentially, no one is denying the existence of overlordship, but the word Bretwalda holds particular connotations that are problematic. I haven't removed its mention in any article, but changed the text so that it is clear that the earliest record we have of the word Bretwalda being used in this context is in a late 9th-century chronicle. Hope that clarifies my editing. Harthacanute 21:50, 19 March 2006 (UTC)

 * Thanks for responding, Harthacanute. I looked again at your edits, & you are correct & I was mistaken: you have not removed any content. I apologize for saying that.

 * However, I am still concerned about you emphasis on one authority's opinion about this matter: Simon Keynes may be correct, but the best way to handle this in accordance to NPOV would be to explain Keynes' stance, & how it is opposed to Sir Frank's. (As a digression, re-reading ASE I failed to note a clear sense of teleological import in his discussion of Bretwalda. On one hand he writes it is a term from "encomiastic poetry", & on the other he states that the title "should not be regarded as a barabrous imitation of imperial dignity, nor can it express ... the supremacy of an English overlord over British kings." His understanding of the term is more complex than you give him credit for.)

 * As a last note, can you suggest one of Keynes' writings in which he explains his views on Bretwalda? Print encyclopedias focus more on packing as much information into one article than on explaining the intricacies at length. -- llywrch 06:56, 20 March 2006 (UTC)

 * Yes, on re-reading my edit I guess I'm a little harsh on Stenton - I appreciate he does have a complex and nuanced understanding of the situation. However, I still stand that his view is now substantially dated. The best exposition of Simon Keynes' view that I could find is in the New Cambridge Medieval History; I've copied the main (rather long!) paragraph here for convenience. It also deals with his concern over the use of the term Heptarchy, which I share. He goes on in the article to outline his approach. I'd be happy to incorporate a historiographical element into the Bretwalda page; perhaps you could add some other interpretations to my subpage and that can serve the basis of such a section? Harthacanute 23:33, 20 March 2006 (UTC)

 * Reading that paragraph, I'm a little surprised that Kyenes is arguing against an interpretation of Bretwalda' that I never seriously considered was viable; had you asked me before this discussion, I would had expressed a concept of Bretwalda very close to his, that it was an attribute not a formal title. One analogy would be "Tallest boy in class": one could identify a person whom this title fit most years -- yet there is no formal title that is transferred from one individual to another. Another analogy would be "most powerful warlord" -- which is the implication Bretwalda carried.

 * (Keynes' opinion actually supports the deletion of the Bretwalda infoboxes that appear at the bottom of a number of pages. Personally, I've never been happy about them because they implied this was a formal title with implied rights & powers, but I haven't had either the time or the amunition to campaign for this.)

 * I'll see what I can find on the historiography of the term. Accounts of how informed opinion has evolved in regards to topics is one area Wikipedia has always been weak in -- not British history alone. -- llywrch 17:05, 22 March 2006 (UTC)

Yes, I have a somewhat similar view regarding the infoboxes. I think they're okay, though, in that anyone who doesn't know what Bretwalda means will click on the link and find this page which deals with (or soon will!) all the issues. Harthacanute 20:56, 22 March 2006 (UTC)

Historiographical focus
In line with what's just been said, I've toned down the language a bit in the article and given the views of Stenton and Keynes, though these could easily be expanded on. More controversially, I've also removed the list of Mercian kings from the page. It seems more and more like this page is going to be about the term Bretwalda and how it was used and applied, while having the list of Mercian kings goes back to this idea of assuming there is a fixed title and that there must be an incumbent. Increasingly it seems like we need a page on Anglo-Saxon overlordship, or perhaps more generally Anglo-Saxon kingship, which could cover the "Mercian supremacy" and all the other intracacies of early kingship. I'll make the page myself if I find time. Harthacanute 21:24, 2 April 2006 (UTC)

 * Were any of Egbert's successors called Bretwalda? They weren't even overlords, unlike the Mercians, who were only excluded because of West Saxon bias. TharkunColl 21:59, 2 April 2006 (UTC)

 * That's a pretty good point, and I think you're right. If we are going for this as a page about who the title was attributed to, then I think the changes I'm about to make will make sense. Perhaps there is also scope for mentioning more of the nuances of Bede and the Chronicle on this page - i.e. why the Mercian kings are passed over. Harthacanute 23:23, 2 April 2006 (UTC)

King of the Saxons
I've added a short section listing the English rulers to which the Welsh annalists gave the title "King of the Saxons". I think this is important as it casts doubt on the idea that the Mercian kings were recognized overlords of the English kings from 658 till 829. I have not changed the list of Mercian rulers, but I have renamed that section "Mercian rulers" rather than "Mercian overlords". This is not a terribly good heading, but I think it needs some heading that avoids giving the impression that the Mercian kings before Ethelbald had a clear authority over the other English kings. Vortimer (talk) 05:28, 29 May 2009 (UTC)

Non-bretwaldas
I have removed the lists of those not described as 'bretwalda' in any source, and given each of the kings listed his full regnal dates in place of the dates forming a continuous sequence. As the article itself indicates, academic consensus has long since rejected the old notion that there was a concrete, generally recognised and more or less continuous position of supreme overlord. The true substance of the term is highly dubious, and it probably amounts to little or nothing more than self-proclaimed or retrospectively perceived 'bragging rights' for the duration of a (usually brief) ascendancy. It is a deeply problematic interpretative conceit of one historian, borrowed and rebranded two and a half centuries later by another, not a solidly attested institutional reality. It is absurd to try to contrive a 'real' list of bretwaldas, compensating for the anti-Mercian chauvinism of the Northumbrian and West Saxon sources, since the reality of the concept of 'bretwalda' outside those same sources is so questionable. Similarly, even as an interpretation of Bede's original conception, the idea that the kings referred to by Bede can be fitted into a continuous sequence, each one smoothly ceding the supposed 'post' to the next, is without foundation in evidence. The disappearance of one fragile hegemony did not necessarily imply the immediate establishment of another. Zburh (talk) 00:10, 3 September 2009 (UTC)

 * As a matter of fact, Bede didn't describe anyone as "Bretwalda" either - he was writing in Latin, and listed those who held imperium, as he called it. Furthermore, some Mercian rulers did indeed assume Latin titles that could be translated as bretwalda - Æthelbald for example, who called himself "King of Britain", plus Offa and Coenwulf with similar titles. I've restored the most powerful of the Mercian rulers. I accept your other point about dates though. ðarkun coll 19:14, 6 September 2009 (UTC)

Obsession with Mercia
The fact that powerful Mercians are excluded from the bretwalda and imperium lists is repeated at least three times in different places in this article (and that while it remains rather unclear and overly laconic on other points). The impression is almost as if the main author of this article is a Mercian nationalist! These bits should be merged into one, and so should the repetitions of the argument that bretwalda-ship is a literary West Saxon invention, which are also dispersed all over the article like a refrain in a song.--91.148.159.4 (talk) 00:06, 25 December 2009 (UTC)

Chosen High King of Britain
Common re-write article. Bretwalda means high king of Britain. Walda means chosen,elected .Ethymologicaly related.Modern Scandinavian 'val' -election. Val-halla - Hall of the Chosen. Bret-walda chosen of britain something like Irish Ari-Ra high king. Edelward (talk) 21:46, 19 February 2010 (UTC)

 * If you are looking for Swedish cognates, välde is more relevant.--91.148.159.4 (talk) 22:27, 31 March 2010 (UTC)

An Obvious Derivation
Everybody seems to have missed an obvious possibility. It seems to me that Bretwalda is derived from Old Welsh. The first syllable is clearly from the ancient name of the island and the second is derived from 'gwladwr', ie 'ruler', which would be subject to lenition following an adjective or a noun that is used as an adjective, giving 'wladwr'. —Preceding unsigned comment added by 94.196.1.11 (talk) 20:11, 8 March 2010 (UTC)

We are not here to come up with our own theories on the origin of words, we are here to report upon the views of scholars. I also doubt that it is an "obvious possibility" that the Old English word is derived from Brythonic (which would incidentally make it one of the very, very, few words of Brythonic origin in Old, Middle and even Modern English) hence it hasn't ever been seriously suggested. The last element is a common element in Old English and is the ancestor of the modern "wielder" but in Old English meant a "ruler" or "owner", thus more "wield" in the sense of "to wield power" than "to hold an object". The "-walda" element really isn't debated, it is the first element that is. "Brytan" could either be "broad" (and is a very common element) or "Britain" (which would make it unique). We don't need to prescribe a Welsh source to an Old English word when there is a more obvious etymology. If it is so obvious that "-walda" is from "gwladwr" then why is a term similar to brytanwalda (brython-gwladwr? prydein-gwladwr?) not found in any form of Welsh? Also, "gwladwr" is Modern Welsh and means "countryman", "rustic" or "peasant" (not quite the same thing as a powerful ruler!) and is from "gwlad" ("country"). I know many people find it sad that there aren't many words in English that derive from Welsh but we shouldn't make up theories to try to prove there are actually more of them; especially when there are far more likely theories as to the origin (see Occam's Razor). Once again however we are here to cite sources from the research of others not do our own original research. Sigurd Dragon Slayer (talk) 13:16, 7 August 2011 (UTC)

What of Wledig, a title used by the British kings of the Dark Age? We find for instance Corotic Wledic of Strathcylde, Cunneda Wledic of Gwynedd, and even Emrys Wledig for Ambrosius. It seems to have meant "Holder of Lands" John D. Croft (talk) 16:11, 9 August 2011 (UTC)

 * Croft, "holder of lands" seems the closest & also implies the lowest level of abstraction: what did those words mean originally? "Bret" is close to both "breit" (wide, overall) & "Brett" (board) in modern German (or in the Middle Ages, the "Saxon" part of Anglo-Saxon) & the order of those consonants & vowel hasn't changed in Germanica too much during almost 2K yr. "Walda" is close to German "Wald" (woods, forest), which much of the land was in the Middle Ages & which looked & meant similar in Old Norse & Old German. "Covering or over all the woods & forests" or more to the point "holder of lands" seems most accurate. BubbleDine (talk) 13:12, 30 January 2012 (UTC)

Removed sentences...
"Bretwëlde was a title in 11th and 12th centuries used in Ireland to denote the "high king". Apparently part of the Gaelic lexicon, this spelling does suggest a Germanic (Saxon) origin, as the 'w' is most likely pronounced as a 'v'."

I have removed the above sentences as they are most likely nonsense (I cannot find any reference to them anywhere other than on the internet on pages that are referencing this article) and other than that it is still ridiculous as if it was part of the lexicon of Gaelic it would have to be a borrowing from brytanwalda but not for the reason cited. The "w" in Old English (as it was originally in all Germanic languages... see Proto-Germanic) was pronounced as the "w" in English (that is the "w" in say "water" or "woman") not as the "w" in High German and most other modern Germanic languages (which are indeed like the "v" in English words like "very" or "vacant" (which are incidentally mostly borrowing from other languages). Sigurd Dragon Slayer (talk) 13:34, 7 August 2011 (UTC)

Removed sentences (part deux)
"Bretwëlde was a term used in 11th and 12th century Ireland to denote the 'high king'. Part of the Gaelic lexicon, this spelling suggests a Germanic (Saxon) origin, as the 'w' is most likely pronounced as a 'v'. It is also possible that the term derives from the Welsh Brit Gweldig, the term for a ruler of Britain."

Removed the same again. I have never seen such a word in any Irish annal. "Bretwëlde" is not "part of the Gaelic lexicon", we do not have such a word in our language. Fergananim (talk) 06:11, 4 December 2012 (UTC)

Possible connection between Brytt and modern danish Bred
As a dane, I immedietly thought that brytenwalda ment bredvælde, or "broad rule", which is close to a suggested possible translation. I find no mention of this connection anywhere in the article, but though I would leave this note to anyone with real linguistics skills to find (in case nobody had considered it before). — Preceding unsigned comment added by 212.130.79.38 (talk) 13:08, 7 January 2015 (UTC)

Relation to the habits of the European Germanic Tribes to choose a leader
Tacitus already tells that for Germanic tribes in times of war a leader will be chosen. It is very likely that the one with the most votes would be called Bretwalda: The Broadly Elected, or in today's German: Den breit Gewahldeten. The reason that the term is only mentioned for the first time in 9th century is the illiteracy of the Anglo-saxons when they arrive. Habits don't change quickly and although some Celtic words were adopted, it is not likely that in particular the most important role of the clan is a non-Anglo-Saxon word. Let alone a Welsh word where Welsh itself is an Anglo-Saxon word meaning foreigner. The original inhabitants of the Britton island were considered as the foreigners by the Anglo-Saxons! Which is even more a reason to not have the highest and elected noble give a Welsh title. Sorry for that, but id does not make sense. Lz89z1 (talk) 09:17, 16 September 2017 (UTC)

Relation to Finn Folcwalding
Bretwalda and Folcwalding relate to each other. Latter meaning "chosen by the people". Lz89z1 (talk) 19:00, 16 October 2018 (UTC)

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

title: Talk:Bretwalda


text:
Talk:Brandenburg

History
I had added a list of rulers in Brandenburg in Germania from the time after Charlemagne till 1134 Albert I of Brandenburg (Albrecht the Baer).

Someone erased the list of more than 100 office holders ,writing something about German apologists.

As it now states in the article, it seems that Slavs came , took over and only Slavs lived there till 1134, when Albrecht the Baer brought in Germans. This has been a popular theory by English speakers for many years.

It is false.

Does anyone have an explatation, why this widespread theory with English speakers exists ?

Why does anyone want to suppress the history of Brandenburg (or Eastern Germania) now in the year 2001, by "omitting" early records?

Is that not falsifying ?user:H.J.

If you want any sympathy for your dubious "all of this really belongs to Germany" cause, perhaps you can start by not twisting the history of ancient and medieval Europe to show that Celts, Slavs, and Baltic peoples are really Germans -- something you constantly imply. Moreover, you might just take a look at naming conventions and the format for articles written by most of the Wiki-contibutors. If you did, you'd notice that throwing in chunks of unrelated facts does not good wiki make. That's why people remove lot's of your stuff. If you want to add every member of the house of Brandenburg-Prussia, why don't you do something like and then add /Individual Ruler Name, where {=[ ??

It's not that bloody hard. JHK

[personal attack removed] The so-called Germans who lived there when the Slavs invaded were nothing of the sort -- at least not in the way you mean. They were most likely a combination of Franks, Saxons, and perhaps a few other tributary people brought in to guard the march. You can's say the Franks were German. It's not the same. If you insist on it, then you also have to accept that they were French, which reduces the "Germans owned Prussia first" argument to the irrelevant. Oh wait...that's right...the French are really German, so really, it's ok for the Kaiser to just kind of absorb France into the Empire...JHK

Aren't the Franks usually considered to be German? They spoke a Germanic language, hailed from the area, and were called such by contemporaries. Of course the ancient Germans are not the same people as the modern Germans, but still, it seems somewhat odd to protest the title.

Even if the following is true, it doesn't belong in an encyclopedia article on the history of Brandenburg, so I've moved it here:

"The country was known as the March of Brandenburg since after 900 AD were a margrave had supreme ruling powers. Other parts of the empire had the same arrangements and the marquis of western France was a margrave as well. A strange phenomena appeared in the last century. There have been numerous books and articles written by historical story tellers with the purpose to gain advantage, by attempting to show that there was an enormous border, a hugh dividing line between the German people and the inhabitants of eastern Europe. Anyone honestly looking at the recorded facts of that time will have to dismiss this as false. This hugh dividing line did not become reality until a thousand years later."

Our goal is to write an encyclopedia. Accusing everyone who disagrees with you of not honestly looking at the facts is unhelpful, at best. Vicki Rosenzweig

Soo... Someone mentioned that cities should be known by name of the time... why It's called Brandenburg in XI century thenm and not BRENNA?!? szopen

Berlin-Brandenburg Merging
Any info on the Berlin-Brandenburg merging plan ? -- Bact 15:21, 2 August 2005 (UTC)

, talking breifly about the name of the new state: Brandenburg-Berlin, Berlin-Brandenburgh, or Prussia. -- Bact 15:35, 2 August 2005 (UTC)

Difficult to understand the decission of many voters in Brandenburg not to marge with Berlin. That would have created the 6th largest Land in Germany by GDP, a large economy which would have attracted investment, ana d combined population of 6 million. It was an insane decission which costs billions of euros. A conbined Land of Berlin-Brandenburg would save money (one Government instead of two) and atract money (large market of 6 million people) A huge opportunity was lost to create thousands of jobs. I hope the merger will take place anyway in the next future because it makes sense.--83.37.194.250 (talk) 01:10, 18 October 2011 (UTC)

"Slavic: Brennabor"
What is 'Slavic', in this context? Either elaborate on why the words "Slavic: Brennabor" should be in this article - and please give sources - or leave it out. This is simply nonsensical to one not actually familiar with the Gdansk/Danzig pissing-up-the-wall-contest, and doesn't belong in an 'encyclopedia'. Space Cadet, what does this have to do with NPOV? Innocent if somewhat dreary articles such as 'Brandenburg' should not be mere pawns in a far broader conflict between Polish chauvanists and German chauvanists. Colonel Mustard 11:47, 11 February 2006 (UTC)

"Invasion of Western Slavic Lands by Germany"
Tacitus has never been to the so called "Germania", his sources were second hand at best, and the purpose of writing the book was political not ethnographic, so I would not call it "recording". Ptolemy's map leaves too much for speculation, and he never visited the lands either. There was no Germania in the 7th century when the Slavic people arrived. The Germanic tribes lft in 3rd century. Saxon Mark and Nordmark were established in the 10th century. Frankish kingdom of Charlmagne barely reached northern Elbe. Check your info and sources. Space Cadet 23:37, 8 November 2006 (UTC)
 * Provide sources that are better than Tacitus and Ptolemy, then. -- Matthead discuß!    O       18:27, 12 November 2006 (UTC)
 * Ptolemy and Tacitus are no "sources", so you provide some. And what about my other points?Space Cadet 21:52, 12 November 2006 (UTC)

Inaccurate chart removed
I removed the earlier version of this chart because it is inaccurate. It inaccurately suggests that the Free State of Prussia included what had been Royal Prussia after 1918 (which it did not) and, even more egregiously, that the entire medieval region of Prussia was part of Brandenburg after 1945, when in fact it was divided between Poland and the Soviet Union/Russia. Also, if the purpose of the chart is to show the lands that made up the Kingdom of Prussia, there were many more besides the March of Brandenburg and the Duchy of Prussia. Marco polo 00:35, 23 June 2007 (UTC)

I restored the template/chart after editing it. Marco polo 01:09, 23 June 2007 (UTC)

Joblessness
I would suggest that you change "joblessness" to "unemployment" as it fits the register for wikipedia better. I won't change it as I'm not logged in. —Preceding unsigned comment added by 213.38.11.86 (talk) 10:06, 21 July 2008 (UTC)

Placement of administrative map of the state to the right
The adminstrative map of the state was originally in the middle of the text (under the header for "Landkreise" table). I moved it to a place between "Landkreise" and "kreisfreie Städte". But the best place for it would be to the right of both tables (and the left of the column of 3 images) - and I couldn't get it to fit there. Would somebody with better knowledge of Wiki syntax move it there, please? 212.58.193.37 (talk) 14:03, 10 June 2009 (UTC) Need for Area code

Need for Area code
Please someone include the area code for Brandenbrug.--Giraldezjota (talk) 10:59, 10 November 2010 (UTC)

vandalism
Some of the links for this site have been hijacked to go to an anti Semitic 9/11 conspiracy site. Could someone fix this I tried but it's beyond me. 201.35.125.147 (talk) 13:08, 18 February 2012 (UTC)

External links modified
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History
We happen to be missing a rather important particular section of history between the German Empire and East Germany. Widgetdog (talk) 16:33, 8 February 2016 (UTC)

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Brandenburg Gate
Does Brandenburg Gate deserve a mention in the Culture section of this article, even though it's in Berlin? Imagine Reason (talk) 18:49, 23 November 2020 (UTC)

title: Talk:Brandenburg


text:
Brownian motion

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas).

This motion pattern typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the equipartition theorem).

This motion is named after the botanist Robert Brown, who first described the phenomenon in 1827, while looking through a microscope at pollen of the plant Clarkia pulchella immersed in water. In 1900, the French mathematician Louis Bachelier modeled the stochastic process now called Brownian motion in his doctoral thesis, The Theory of Speculation (Théorie de la spéculation), prepared under the supervision of Henri Poincaré. Then, in 1905, theoretical physicist Albert Einstein published a paper where he modeled the motion of the pollen particles as being moved by individual water molecules, making one of his first major scientific contributions.

The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. This explanation of Brownian motion served as convincing evidence that atoms and molecules exist and was further verified experimentally by Jean Perrin in 1908. Perrin was awarded the Nobel Prize in Physics in 1926 "for his work on the discontinuous structure of matter".

The many-body interactions that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. Consequently, only probabilistic models applied to molecular populations can be employed to describe it. Two such models of the statistical mechanics, due to Einstein and Smoluchowski, are presented below. Another, pure probabilistic class of models is the class of the stochastic process models. There exist sequences of both simpler and more complicated stochastic processes which converge (in the limit) to Brownian motion (see random walk and Donsker's theorem).

History

The Roman philosopher-poet Lucretius' scientific poem "On the Nature of Things" (c. 60 BC) has a remarkable description of the motion of dust particles in verses 113–140 from Book II. He uses this as a proof of the existence of atoms:

"Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden from our sight... It originates with the atoms which move of themselves [i.e., spontaneously]. Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies. So the movement mounts up from the atoms and gradually emerges to the level of our senses so that those bodies are in motion that we see in sunbeams, moved by blows that remain invisible."

Although the mingling, tumbling motion of dust particles is caused largely by air currents, the glittering, jiggling motion of small dust particles is caused chiefly by true Brownian dynamics; Lucretius "perfectly describes and explains the Brownian movement by a wrong example".

While Jan Ingenhousz described the irregular motion of coal dust particles on the surface of alcohol in 1785, the discovery of this phenomenon is often credited to the botanist Robert Brown in 1827. Brown was studying pollen grains of the plant Clarkia pulchella suspended in water under a microscope when he observed minute particles, ejected by the pollen grains, executing a jittery motion. By repeating the experiment with particles of inorganic matter he was able to rule out that the motion was life-related, although its origin was yet to be explained.

The first person to describe the mathematics behind Brownian motion was Thorvald N. Thiele in a paper on the method of least squares published in 1880. This was followed independently by Louis Bachelier in 1900 in his PhD thesis "The theory of speculation", in which he presented a stochastic analysis of the stock and option markets. The Brownian motion model of the stock market is often cited, but Benoit Mandelbrot rejected its applicability to stock price movements in part because these are discontinuous.

Albert Einstein (in one of his 1905 papers) and Marian Smoluchowski (1906) brought the solution of the problem to the attention of physicists, and presented it as a way to indirectly confirm the existence of atoms and molecules. Their equations describing Brownian motion were subsequently verified by the experimental work of Jean Baptiste Perrin in 1908.

Einstein's theory
There are two parts to Einstein's theory: the first part consists in the formulation of a diffusion equation for Brownian particles, in which the diffusion coefficient is related to the mean squared displacement of a Brownian particle, while the second part consists in relating the diffusion coefficient to measurable physical quantities. In this way Einstein was able to determine the size of atoms, and how many atoms there are in a mole, or the molecular weight in grams, of a gas. In accordance to Avogadro's law, this volume is the same for all ideal gases, which is 22.414 liters at standard temperature and pressure. The number of atoms contained in this volume is referred to as the Avogadro number, and the determination of this number is tantamount to the knowledge of the mass of an atom, since the latter is obtained by dividing the molar mass of the gas by the Avogadro constant.

The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval. Classical mechanics is unable to determine this distance because of the enormous number of bombardments a Brownian particle will undergo, roughly of the order of 1014 collisions per second.

He regarded the increment of particle positions in time $$\tau$$ in a one-dimensional (x) space (with the coordinates chosen so that the origin lies at the initial position of the particle) as a random variable ($$q$$) with some probability density function $$\varphi(q)$$ (i.e., $$\varphi(q) $$ is the probability density for a jump of magnitude $$q$$, i.e., the probability density of the particle incrementing its position from $$x$$ to $$x + q$$ in the time interval $$\tau$$). Further, assuming conservation of particle number, he expanded the number density $$\rho(x,t+\tau)$$ (number of particles per unit volume around $$x$$) at time $$t + \tau$$ in a Taylor series, $$\begin{align} \rho(x, t+\tau) ={}& \rho(x,t) + \tau \frac{\partial\rho(x,t)}{\partial t} + \cdots \\[2ex] ={}& \int_{-\infty}^{\infty} \rho(x - q, t) \, \varphi(q) \, dq = \mathbb{E}_q{\left[\rho(x - q, t)\right]} \\[1ex] ={}& \rho(x, t) \, \int_{-\infty}^{\infty} \varphi(q) \, dq - \frac{\partial\rho}{\partial x} \, \int_{-\infty}^{\infty} q \, \varphi(q) \, dq + \frac{\partial^2 \rho}{\partial x^2} \, \int_{-\infty}^{\infty} \frac{q^2}{2} \varphi(q) \, dq + \cdots \\[1ex] ={}& \rho(x, t) \cdot 1 - 0 + \cfrac{\partial^2 \rho}{\partial x^2} \, \int_{-\infty}^{\infty} \frac{q^2}{2} \varphi(q) \, dq + \cdots \end{align}$$ where the second equality is by definition of $$\varphi$$. The integral in the first term is equal to one by the definition of probability, and the second and other even terms (i.e. first and other odd moments) vanish because of space symmetry. What is left gives rise to the following relation: $$\frac{\partial\rho}{\partial t} = \frac{\partial^2 \rho}{\partial x^2} \cdot \int_{-\infty}^{\infty} \frac{q^2}{2 \tau} \varphi(q) \, dq + \text{higher-order even moments.}$$ Where the coefficient after the Laplacian, the second moment of probability of displacement $$q$$, is interpreted as mass diffusivity D: $$D = \int_{-\infty}^{\infty} \frac{q^2}{2 \tau} \varphi(q) \, dq.$$ Then the density of Brownian particles $ρ$ at point $x$ at time $t$ satisfies the diffusion equation: $$\frac{\partial\rho}{\partial t} = D\cdot \frac{\partial^2\rho}{\partial x^2},$$

Assuming that N particles start from the origin at the initial time t = 0, the diffusion equation has the solution $$\rho(x,t) = \frac{N}{\sqrt{4\pi Dt}} \exp{\left(-\frac{x^2}{4Dt}\right)}.$$ This expression (which is a normal distribution with the mean $$ \mu = 0$$ and variance $$ \sigma^2 = 2Dt$$ usually called Brownian motion $$ B_t$$) allowed Einstein to calculate the moments directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to the right. The second moment is, however, non-vanishing, being given by $$\mathbb{E}{\left[x^2\right]} = 2 D t.$$ This equation expresses the mean squared displacement in terms of the time elapsed and the diffusivity. From this expression Einstein argued that the displacement of a Brownian particle is not proportional to the elapsed time, but rather to its square root. His argument is based on a conceptual switch from the "ensemble" of Brownian particles to the "single" Brownian particle: we can speak of the relative number of particles at a single instant just as well as of the time it takes a Brownian particle to reach a given point.

The second part of Einstein's theory relates the diffusion constant to physically measurable quantities, such as the mean squared displacement of a particle in a given time interval. This result enables the experimental determination of the Avogadro number and therefore the size of molecules. Einstein analyzed a dynamic equilibrium being established between opposing forces. The beauty of his argument is that the final result does not depend upon which forces are involved in setting up the dynamic equilibrium.

In his original treatment, Einstein considered an osmotic pressure experiment, but the same conclusion can be reached in other ways.

Consider, for instance, particles suspended in a viscous fluid in a gravitational field. Gravity tends to make the particles settle, whereas diffusion acts to homogenize them, driving them into regions of smaller concentration. Under the action of gravity, a particle acquires a downward speed of $v = μmg$, where $m$ is the mass of the particle, $g$ is the acceleration due to gravity, and $μ$ is the particle's mobility in the fluid. George Stokes had shown that the mobility for a spherical particle with radius $r$ is $$\mu = \tfrac{1}{6\pi\eta r}$$, where $η$ is the dynamic viscosity of the fluid. In a state of dynamic equilibrium, and under the hypothesis of isothermal fluid, the particles are distributed according to the barometric distribution $$\rho = \rho_o \, \exp\left({-\frac{m g h}{k_\text{B} T}}\right),$$ where $ρ − ρ_{o}$ is the difference in density of particles separated by a height difference, of $$h = z - z_o$$, $k_{B}$ is the Boltzmann constant (the ratio of the universal gas constant, $R$, to the Avogadro constant, $NA$), and $T$ is the absolute temperature.

Dynamic equilibrium is established because the more that particles are pulled down by gravity, the greater the tendency for the particles to migrate to regions of lower concentration. The flux is given by Fick's law, $$J = -D \frac{d\rho}{dh},$$ where $J = ρv$. Introducing the formula for $ρ$, we find that $$v = \frac{D m g}{k_\text{B} T}.$$

In a state of dynamical equilibrium, this speed must also be equal to $v = μmg$. Both expressions for $v$ are proportional to $mg$, reflecting that the derivation is independent of the type of forces considered. Similarly, one can derive an equivalent formula for identical charged particles of charge $q$ in a uniform electric field of magnitude $E$, where $mg$ is replaced with the electrostatic force $qE$. Equating these two expressions yields the Einstein relation for the diffusivity, independent of $mg$ or $qE$ or other such forces: $$ \frac{\mathbb{E}{\left[x^2\right]}}{2t} = D = \mu k_\text{B} T = \frac{\mu R T}{N_\text{A}} = \frac{RT}{6\pi\eta r N_\text{A}}.$$ Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of the Boltzmann constant as $k_{B} = R / NA$, and the fourth equality follows from Stokes's formula for the mobility. By measuring the mean squared displacement over a time interval along with the universal gas constant $R$, the temperature $T$, the viscosity $η$, and the particle radius $r$, the Avogadro constant $NA$ can be determined.

The type of dynamical equilibrium proposed by Einstein was not new. It had been pointed out previously by J. J. Thomson in his series of lectures at Yale University in May 1903 that the dynamic equilibrium between the velocity generated by a concentration gradient given by Fick's law and the velocity due to the variation of the partial pressure caused when ions are set in motion "gives us a method of determining Avogadro's Constant which is independent of any hypothesis as to the shape or size of molecules, or of the way in which they act upon each other".

An identical expression to Einstein's formula for the diffusion coefficient was also found by Walther Nernst in 1888 in which he expressed the diffusion coefficient as the ratio of the osmotic pressure to the ratio of the frictional force and the velocity to which it gives rise. The former was equated to the law of van 't Hoff while the latter was given by Stokes's law. He writes $$k' = p_o/k$$ for the diffusion coefficient $k′$, where $$p_o$$ is the osmotic pressure and $k$ is the ratio of the frictional force to the molecular viscosity which he assumes is given by Stokes's formula for the viscosity. Introducing the ideal gas law per unit volume for the osmotic pressure, the formula becomes identical to that of Einstein's. The use of Stokes's law in Nernst's case, as well as in Einstein and Smoluchowski, is not strictly applicable since it does not apply to the case where the radius of the sphere is small in comparison with the mean free path.

At first, the predictions of Einstein's formula were seemingly refuted by a series of experiments by Svedberg in 1906 and 1907, which gave displacements of the particles as 4 to 6 times the predicted value, and by Henri in 1908 who found displacements 3 times greater than Einstein's formula predicted. But Einstein's predictions were finally confirmed in a series of experiments carried out by Chaudesaigues in 1908 and Perrin in 1909. The confirmation of Einstein's theory constituted empirical progress for the kinetic theory of heat. In essence, Einstein showed that the motion can be predicted directly from the kinetic model of thermal equilibrium. The importance of the theory lay in the fact that it confirmed the kinetic theory's account of the second law of thermodynamics as being an essentially statistical law.

Smoluchowski model
Smoluchowski's theory of Brownian motion starts from the same premise as that of Einstein and derives the same probability distribution $ρ(x, t)$ for the displacement of a Brownian particle along the $x$ in time $t$. He therefore gets the same expression for the mean squared displacement: $\mathbb{E}{\left[(\Delta x)^2\right]}$. However, when he relates it to a particle of mass $m$ moving at a velocity $u$ which is the result of a frictional force governed by Stokes's law, he finds $$\mathbb{E}{\left[(\Delta x)^2\right]} = 2Dt = t \frac{32}{81} \frac{mu^2}{\pi \mu a} = t\frac{64}{27} \frac{\frac{1}{2}mu^2}{3 \pi \mu a},$$ where $μ$ is the viscosity coefficient, and $a$ is the radius of the particle. Associating the kinetic energy $$mu^2/2$$ with the thermal energy $RT/N$, the expression for the mean squared displacement is $64/27$ times that found by Einstein. The fraction 27/64 was commented on by Arnold Sommerfeld in his necrology on Smoluchowski: "The numerical coefficient of Einstein, which differs from Smoluchowski by 27/64 can only be put in doubt."

Smoluchowski attempts to answer the question of why a Brownian particle should be displaced by bombardments of smaller particles when the probabilities for striking it in the forward and rear directions are equal. If the probability of $m$ gains and $n − m$ losses follows a binomial distribution, $$P_{m,n} = \binom{n}{m} 2^{-n},$$ with equal probabilities of 1/2, the mean total gain is $$\mathbb{E}{\left[2m-n\right]} = \sum_{m=\frac{n}{2}}^n (2m-n)P_{m,n}=\frac{n n!}{2^n \left [ \left (\frac{n}{2} \right )! \right ]^2}.$$

If $n$ is large enough so that Stirling's approximation can be used in the form $$n!\approx\left(\frac{n}{e}\right)^n\sqrt{2\pi n},$$ then the expected total gain will be $$\mathbb{E}{\left[2m - n\right]} \approx \sqrt{\frac{2n}{\pi}},$$ showing that it increases as the square root of the total population.

Suppose that a Brownian particle of mass $M$ is surrounded by lighter particles of mass $m$ which are traveling at a speed $u$. Then, reasons Smoluchowski, in any collision between a surrounding and Brownian particles, the velocity transmitted to the latter will be $mu/M$. This ratio is of the order of $cm/s$. But we also have to take into consideration that in a gas there will be more than 1016 collisions in a second, and even greater in a liquid where we expect that there will be 1020 collision in one second. Some of these collisions will tend to accelerate the Brownian particle; others will tend to decelerate it. If there is a mean excess of one kind of collision or the other to be of the order of 108 to 1010 collisions in one second, then velocity of the Brownian particle may be anywhere between $10 cm/s$. Thus, even though there are equal probabilities for forward and backward collisions there will be a net tendency to keep the Brownian particle in motion, just as the ballot theorem predicts.

These orders of magnitude are not exact because they don't take into consideration the velocity of the Brownian particle, $U$, which depends on the collisions that tend to accelerate and decelerate it. The larger $U$ is, the greater will be the collisions that will retard it so that the velocity of a Brownian particle can never increase without limit. Could such a process occur, it would be tantamount to a perpetual motion of the second type. And since equipartition of energy applies, the kinetic energy of the Brownian particle, $MU^2/2$, will be equal, on the average, to the kinetic energy of the surrounding fluid particle, $mu^2/2$.

In 1906 Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion. The model assumes collisions with $M ≫ m$ where $M$ is the test particle's mass and $m$ the mass of one of the individual particles composing the fluid. It is assumed that the particle collisions are confined to one dimension and that it is equally probable for the test particle to be hit from the left as from the right. It is also assumed that every collision always imparts the same magnitude of ΔV. If $N_{R}$ is the number of collisions from the right and $N_{L}$ the number of collisions from the left then after $N$ collisions the particle's velocity will have changed by $ΔV(2N_{R} − N)$. The multiplicity is then simply given by: $$ \binom{N}{N_\text{R}} = \frac{N!}{N_\text{R}!(N - N_\text{R})!}$$ and the total number of possible states is given by $2^{N}$. Therefore, the probability of the particle being hit from the right $N_{R}$ times is: $$P_N(N_\text{R}) = \frac{N!}{2^N N_\text{R}!(N-N_\text{R})!}$$

As a result of its simplicity, Smoluchowski's 1D model can only qualitatively describe Brownian motion. For a realistic particle undergoing Brownian motion in a fluid, many of the assumptions don't apply. For example, the assumption that on average occurs an equal number of collisions from the right as from the left falls apart once the particle is in motion. Also, there would be a distribution of different possible $ΔV$s instead of always just one in a realistic situation.

Other physics models using partial differential equations
The diffusion equation yields an approximation of the time evolution of the probability density function associated with the position of the particle going under a Brownian movement under the physical definition. The approximation is valid on short timescales.

The time evolution of the position of the Brownian particle itself is best described using the Langevin equation, an equation that involves a random force field representing the effect of the thermal fluctuations of the solvent on the particle. In Langevin dynamics and Brownian dynamics, the Langevin equation is used to efficiently simulate the dynamics of molecular systems that exhibit a strong Brownian component.

The displacement of a particle undergoing Brownian motion is obtained by solving the diffusion equation under appropriate boundary conditions and finding the rms of the solution. This shows that the displacement varies as the square root of the time (not linearly), which explains why previous experimental results concerning the velocity of Brownian particles gave nonsensical results. A linear time dependence was incorrectly assumed.

At very short time scales, however, the motion of a particle is dominated by its inertia and its displacement will be linearly dependent on time: $Δx = vΔt$. So the instantaneous velocity of the Brownian motion can be measured as $v = Δx/Δt$, when $Δt << τ$, where $τ$ is the momentum relaxation time. In 2010, the instantaneous velocity of a Brownian particle (a glass microsphere trapped in air with optical tweezers) was measured successfully. The velocity data verified the Maxwell–Boltzmann velocity distribution, and the equipartition theorem for a Brownian particle.

Astrophysics: star motion within galaxies
In stellar dynamics, a massive body (star, black hole, etc.) can experience Brownian motion as it responds to gravitational forces from surrounding stars. The rms velocity $V$ of the massive object, of mass $M$, is related to the rms velocity $$v_\star$$ of the background stars by $$ MV^2 \approx m v_\star^2 $$ where $$m \ll M$$ is the mass of the background stars. The gravitational force from the massive object causes nearby stars to move faster than they otherwise would, increasing both $$v_\star$$ and $V$. The Brownian velocity of Sgr A*, the supermassive black hole at the center of the Milky Way galaxy, is predicted from this formula to be less than 1 km s−1.

Mathematics

In mathematics, Brownian motion is described by the Wiener process, a continuous-time stochastic process named in honor of Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics and physics.

The Wiener process $0 ≤ t ≤ 2$ is characterized by four facts: $$\mathcal{N}(\mu, \sigma^2)$$ denotes the normal distribution with expected value $μ$ and variance $W_{t}$. The condition that it has independent increments means that if $$0 \leq s_1 < t_1 \leq s_2 < t_2$$ then $$W_{t_1}-W_{s_1}$$ and $$W_{t_2}-W_{s_2}$$ are independent random variables. In addition, for some filtration $\mathcal{F}_t$, $$W_t$$ is $$\mathcal{F}_t$$ measurable for all $t \geq 0$.
 * 1) $W_{0} = 0$ is almost surely continuous
 * 2) $W_{t}$ has independent increments
 * 3) $$W_t-W_s\sim \mathcal{N}(0,t-s)$$ (for $0 \leq s \le t$).
 * 1) $$W_t-W_s\sim \mathcal{N}(0,t-s)$$ (for $0 \leq s \le t$).

An alternative characterisation of the Wiener process is the so-called Lévy characterisation that says that the Wiener process is an almost surely continuous martingale with $W_{t}$ and quadratic variation $[W_t, W_t] = t$.

A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent $$\mathcal{N}(0, 1)$$ random variables. This representation can be obtained using the Kosambi–Karhunen–Loève theorem.

The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed neighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it is scale invariant.

The time evolution of the position of the Brownian particle itself can be described approximately by a Langevin equation, an equation which involves a random force field representing the effect of the thermal fluctuations of the solvent on the Brownian particle. On long timescales, the mathematical Brownian motion is well described by a Langevin equation. On small timescales, inertial effects are prevalent in the Langevin equation. However the mathematical Brownian motion is exempt of such inertial effects. Inertial effects have to be considered in the Langevin equation, otherwise the equation becomes singular. so that simply removing the inertia term from this equation would not yield an exact description, but rather a singular behavior in which the particle doesn't move at all.

A d-dimensional Gaussian free field has been described as "a d-dimensional-time analog of Brownian motion."

Statistics
The Brownian motion can be modeled by a random walk.

In the general case, Brownian motion is a Markov process and described by stochastic integral equations.

Lévy characterisation
The French mathematician Paul Lévy proved the following theorem, which gives a necessary and sufficient condition for a continuous $σ^{2}$-valued stochastic process $W_{0} = 0$ to actually be $n$-dimensional Brownian motion. Hence, Lévy's condition can actually be used as an alternative definition of Brownian motion.

Let $R^{n}$ be a continuous stochastic process on a probability space $X$ taking values in $X = (X_{1}, ..., X_{n})$. Then the following are equivalent:

 * 1) $(Ω, Σ, P)$ is a Brownian motion with respect to $R^{n}$, i.e., the law of $X$ with respect to $P$ is the same as the law of an $n$-dimensional Brownian motion, i.e., the push-forward measure $X$ is classical Wiener measure on $P$.
 * 2) both
 * 3) $X_{∗}(P)$ is a martingale with respect to $C_{0}([0, ∞); R^{n})$ (and its own natural filtration); and
 * 4) for all $X$, $P$ is a martingale with respect to $1 ≤ i, j ≤ n$ (and its own natural filtration), where $X_{i}(t) X_{j}(t) − δ_{ij} t$ denotes the Kronecker delta.

Spectral content
The spectral content of a stochastic process $$X_t$$ can be found from the power spectral density, formally defined as $$S(\omega)=\lim_{T\to\infty}\frac{1}{T}\mathbb{E}\left\{ \left|\int^T_0 e^{i \omega t} X_t dt \right|^2\right\}, $$ where $$\mathbb{E}$$ stands for the expected value. The power spectral density of Brownian motion is found to be $$S_{BM}(\omega) = \frac{4 D}{\omega^2}.$$ where $D$ is the diffusion coefficient of $P$. For naturally occurring signals, the spectral content can be found from the power spectral density of a single realization, with finite available time, i.e., $$S^{(1)}(\omega,T) = \frac{1}{T}\left|\int^T_0 e^{i \omega t}X_t dt\right|^2 ,$$ which for an individual realization of a Brownian motion trajectory, it is found to have expected value $$\mu_{BM}(\omega,T)$$ $$\mu_\text{BM}(\omega,T) = \frac{4 D}{\omega^2} \left[1 - \frac{\sin\left(\omega T\right)}{\omega T}\right]$$ and variance $$\sigma_\text{BM}^2(\omega,T)$$ $$\sigma_S^2(f,T) = \mathbb{E}\left\{\left(S^{(j)}_T(f)\right)^2\right\}-\mu_S^2(f,T) =\frac{20 D^2}{f^4}\left[ 1-\Big(6-\cos\left(f T\right)\Big) \frac{2\sin\left(f T\right)}{5fT} +\frac{\Big(17-\cos\left(2fT\right) - 16\cos\left(f T\right)\Big)}{10 f^2 T^2} \right].$$

For sufficiently long realization times, the expected value of the power spectrum of a single trajectory converges to the formally defined power spectral density $S(\omega)$, but its coefficient of variation $$\gamma = \sigma / \mu$$ tends to $\sqrt{5}/2$. This implies the distribution of $$S^{(1)}(\omega,T)$$ is broad even in the infinite time limit.

Riemannian manifold

The infinitesimal generator (and hence characteristic operator) of a Brownian motion on $δ_{ij}$ is easily calculated to be $X_{t}$, where $R^{n}$ denotes the Laplace operator. In image processing and computer vision, the Laplacian operator has been used for various tasks such as blob and edge detection. This observation is useful in defining Brownian motion on an $m$-dimensional Riemannian manifold $1⁄2Δ$: a Brownian motion on $M$ is defined to be a diffusion on $M$ whose characteristic operator $$\mathcal{A}$$ in local coordinates $Δ$, $(M, g)$, is given by $x_{i}$, where $1 ≤ i ≤ m$ is the Laplace–Beltrami operator given in local coordinates by $$\Delta_{\mathrm{LB}} = \frac{1}{\sqrt{\det(g)}} \sum_{i=1}^m \frac{\partial}{\partial x_i} \left(\sqrt{\det(g)} \sum_{j=1}^m g^{ij} \frac{\partial}{\partial x_j} \right),$$ where $1⁄2Δ_{LB}$ in the sense of the inverse of a square matrix.

Narrow escape
The narrow escape problem is a ubiquitous problem in biology, biophysics and cellular biology which has the following formulation: a Brownian particle (ion, molecule, or protein) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape. The narrow escape problem is that of calculating the mean escape time. This time diverges as the window shrinks, thus rendering the calculation a singular perturbation problem.

title: Brownian motion