title: Marco Asensio

text: After being scouted by Real Madrid and FC Barcelona, Asensio made his senior debut with Mallorca's reserves in the 2013–14 season in the Tercera División, despite still being a junior; he made his first competitive appearance for the first team on 27 October 2013, playing the last six minutes in a 1–3 away loss against Recreativo de Huelva in the Segunda División.


title: Marco Asensio

text: Again from the bench, Asensio played the last six minutes in a 0–0 home draw against CD Lugo. After impressing in his first games, he was promoted to the first team by manager José Luis Oltra.


title: Marco Asensio

text: Asensio scored his first professional goal on 16 March 2014, netting the first in a 2–0 home win over CD Tenerife. He was made a starter under new coach Valeri Karpin, scoring against CA Osasuna, Deportivo Alavés and UE Llagostera within a month.


title: Marco Asensio

text: On 24 November 2014, Real Madrid reached an agreement in principle to sign Asensio. On 5 December the deal was officially announced, with the player (who had been a supporter of the club as a child) signing a six-year deal for a €3.9 million fee and remaining with the "Bermellones" on loan until the end of the campaign. According to Real Madrid president Florentino Pérez, Spanish tennis player Rafael Nadal played a crucial role in the transfer of Marco Asensio to Real Madrid.


title: Marco Asensio

text: On 20 August 2015, after making the whole pre-season with Real Madrid, Asensio was loaned to RCD Espanyol in La Liga. He played his first game in the competition on 19 September, featuring 86 minutes in a 3–2 success at Real Sociedad, and he ended his spell at the Estadi Cornellà-El Prat with 12 overall assists.


title: Marco Asensio

text: Returning to the Santiago Bernabéu Stadium for 2016–17, Asensio's first competitive appearance took place on 9 August, as he played the full 120 minutes and scored a 25-meter effort in a 3–2 win against fellow Spaniards Sevilla FC in the UEFA Super Cup. He made his first league start 12 days later, netting the second goal in a 3–0 success at Real Sociedad.


title: Marco Asensio

text: Asensio featured 23 times over his first season, scoring three times as the club won the national championship for the first time since 2012. He added as many goals in that year's UEFA Champions League, including one in the final against Juventus (4–1) to give the Spaniards their 12th title in the latter competition.


title: Marco Asensio

text: Asensio scored his first goal of the 2017–18 season with a 25-yard finish in a 3–1 win over Barcelona at the Camp Nou, for the Supercopa de España, He repeated the feat with a similarly spectacular effort in the second leg, in a 5–1 aggregate triumph.


title: Marco Asensio

text: On 28 September 2017, Asensio extended his contract until 2023. On 18 February 2018, in a match where he contributed with a brace to a 5–3 away win over Real Betis, he scored Real Madrid's 6,000th goal in the Spanish top division.


title: Marco Asensio

text: Asensio made 11 appearances while scoring one goal during the 2017–18 Champions League, and the club won its third consecutive and 13th overall title in the competition.


title: Marco Asensio

text: On 24 July 2019, during a preseason match against Arsenal, he tore his ACL. On 19 June 2020, in a 3–0 win over Valencia, he came on for Federico Valverde in 74th minute, scored the second goal with his first touch, and provided the assist to Karim Benzema for the third goal. He returned for the final stretch of the season, scoring three goals in nine games, as Real Madrid won the 2019–20 La Liga.


title: Marco Asensio

text: On 22 September 2021, he scored his first senior hat-trick against his former club Mallorca, which ended in a 6–1 victory.


title: Marco Asensio

text: Being of both Dutch and Spanish descent, he chose to represent Spain although he received invitation from the Royal Dutch Football Association. After being a regular for Spain at youth level, Asensio made his debut for the under-21 team on 26 March 2015, coming on as a late substitute for goalscorer Gerard Deulofeu in a 2–0 friendly win over Norway in Cartagena. In July, he was part of the victorious under-19 side at the UEFA European Championship in Greece, where he scored both goals in a semi-final victory over France in Katerini, in the 88th minute and in added time.


title: Marco Asensio

text: On 17 May 2016, Asensio and Espanyol teammate Pau López were called up to the full side for a friendly against Bosnia and Herzegovina. He made his debut on the 29th, starting in the 3–1 win in Switzerland.


title: Marco Asensio

text: Selected in the squad for the 2017 European Under-21 Championship by manager Albert Celades, Asensio scored a hat-trick in his debut in the competition, helping to a 5–0 group stage routing of Macedonia. He reached with his teammates the final of the tournament, losing 1–0 against Germany.


title: Marco Asensio

text: Back with the seniors, he was included in Julen Lopetegui's squad for the 2018 FIFA World Cup, making his debut in the competition on 20 June by replacing Real Madrid teammate Lucas Vázquez for the final ten minutes of the 1–0 group stage win against Iran.


title: Marco Asensio

text: Asensio scored his first goal for Spain on 11 September 2018, playing the entire 6–0 home rout of Croatia for the UEFA Nations League and also being directly involved in the play that led to Lovre Kalinić's own goal.


title: Marco Asensio

text: In July 2021, he was included as overage player in the 22-player squad of the under-23 team for the 2020 Summer Olympics. In the semi-finals of the tournament, Asensio scored the winning goal against hosts Japan in extra time on the 115th minute to send Spain to the final, which they lost 2–1 to defending champions Brazil.


title: Marco Asensio

text: Asensio's father, Gilberto (a Basque who spent his childhood in Essen, Germany), was also a footballer. Also an attacking midfielder, he represented Barakaldo CF as a youth; Marco's elder brother Igor played for Platges de Calvià, as a defender.


title: Marco Asensio

text: Asensio's goal celebration in the 2017 Champions League final was dedicated to his family: "I told them that if I scored then I'd run to where they were. I knew where they were and that is what I did."


title: Trotskyism

text: Trotskyism is the political ideology and branch of Marxism developed by Ukrainian-Russian revolutionary Leon Trotsky and some other members of the Left Opposition and Fourth International. Trotsky self-identified as an orthodox Marxist, a revolutionary Marxist, and Bolshevik–Leninist, a follower of Marx, Engels, and 3L: Vladimir Lenin, Karl Liebknecht, and Rosa Luxemburg. He supported founding a vanguard party of the proletariat, proletarian internationalism, and a dictatorship of the proletariat (as opposed to the "dictatorship of the bourgeoisie", which Marxists argue defines capitalism) based on working-class self-emancipation and mass democracy. Trotskyists are critical of Stalinism as they oppose Joseph Stalin's theory of socialism in one country in favour of Trotsky's theory of permanent revolution. Trotskyists criticize the bureaucracy and anti-democratic current developed in the Soviet Union under Stalin.


title: Trotskyism

text: Vladimir Lenin and Trotsky, despite their ideological disputes, were close personally prior to the London congress of social democrats in 1903 and during the First World War. Lenin and Trotsky were close ideologically and personally during the Russian Revolution and its aftermath, and Trotskyists and some others call Trotsky its "co-leader". Trotsky was the Red Army's paramount leader in the Revolutionary period's direct aftermath. Trotsky initially opposed some aspects of Leninism but eventually concluded that unity between the Mensheviks and Bolsheviks was impossible and joined the Bolsheviks. Trotsky played a leading role with Lenin in the October Revolution. Assessing Trotsky, Lenin wrote: "Trotsky long ago said that unification is impossible. Trotsky understood this and from that time on, there has been no better Bolshevik."


title: Trotskyism

text: In 1927, Trotsky was purged from the Communist Party and Soviet politics. In October, by order of Stalin, Trotsky was removed from power and, in November, expelled from the All-Union Communist Party (Bolsheviks). He was exiled to Alma-Ata (now Almaty) in January 1928 and then expelled from the Soviet Union in February 1929. As the head of the Fourth International, Trotsky continued in exile to oppose what he termed the degenerated workers' state in the Soviet Union. On 20 August 1940, Trotsky was attacked in Mexico City by Ramón Mercader, a Spanish-born NKVD agent, and died the next day in a hospital. His murder is considered a political assassination. Almost all Trotskyists within the All-Union Communist Party (Bolsheviks) were executed in the Great Purges of 1937–1938, effectively removing all of Trotsky's internal influence in the Soviet Union. Nikita Khrushchev had come to power as head of the Communist Party in Ukraine, signing lists of other Trotskyists to be executed. Trotsky and the party of Trotskyists were still recognized as enemies of the USSR during Khrushchev's rule of the Soviet Union from 1956.


title: Trotskyism

text: Trotsky's Fourth International was established in the French Third Republic in 1938 when Trotskyists argued that the Comintern or Third International had become irretrievably "lost to Stalinism" and thus incapable of leading the international working class to political power. In contemporary English language usage, an advocate of Trotsky's ideas is often called a "Trotskyist". A Trotskyist may be called a "Trotskyite" or "Trot", especially by a critic of Trotskyism.


title: Trotskyism

text: According to Trotsky, his program could be distinguished from other Marxist theories by five key elements:


title: Trotskyism

text: On the political spectrum of Marxism, Trotskyists are usually considered to be on the left. In the 1920s, they called themselves the Left Opposition, although today's left communism is distinct and usually non-Bolshevik. The terminological disagreement can be confusing because different versions of a left-right political spectrum are used. Anti-revisionists consider themselves the ultimate leftists on a spectrum from communism on the left to imperialist capitalism on the right. However, given that Stalinism is often labelled rightist within the communist spectrum and left communism leftist, anti-revisionists' idea of the left is very different from that of left communism. Despite being Bolshevik-Leninist comrades during the Russian Revolution and Russian Civil War, Trotsky and Stalin became enemies in the 1920s and, after that, opposed the legitimacy of each other's forms of Leninism. Trotsky was highly critical of the Stalinist USSR for suppressing democracy and the lack of adequate economic planning.


title: Trotskyism

text: In 1905, Trotsky formulated his theory of permanent revolution, which later became a defining characteristic of Trotskyism. Until 1905, some revolutionaries claimed that Marx's theory of history posited that only a revolution in a European capitalist society would lead to a socialist one. According to this position, a socialist revolution could not occur in a backward, feudal country such as early 20th-century Russia when it had such a small and almost powerless capitalist class.


title: Trotskyism

text: The theory of permanent revolution addressed how such feudal regimes were to be overthrown and how socialism could be established given the lack of economic prerequisites. Trotsky argued that only the working class could overthrow feudalism and win the peasantry's support in Russia. Furthermore, he argued that the Russian working class would not stop there. They would win their revolution against the weak capitalist class, establish a workers' state in Russia and appeal to the working class in the advanced capitalist countries worldwide. As a result, the global working class would come to Russia's aid, and socialism could develop worldwide.


title: Trotskyism

text: Revolutions in Britain in the 17th century and in France in 1789 abolished feudalism and established the essential requisites for the development of capitalism. Trotsky argued that these revolutions would not be repeated in Russia.


title: Trotskyism

text: In "Results and Prospects", written in 1906, Trotsky outlines his theory in detail, arguing: "History does not repeat itself. However much one may compare the Russian Revolution with the Great French Revolution, the former can never be transformed into a repetition of the latter." In the French Revolution of 1789, France experienced what Marxists called a "bourgeois-democratic revolution"—a regime was established wherein the bourgeoisie overthrew the existing French feudalistic system. The bourgeoisie then moved towards establishing a regime of democratic parliamentary institutions. However, while democratic rights were extended to the bourgeoisie, they were not generally extended to a universal franchise. The freedom for workers to organize unions or to strike was not achieved without considerable struggle.


title: Trotskyism

text: Trotsky argues that countries like Russia had no "enlightened, active" revolutionary bourgeoisie which could play the same role, and the working class constituted a tiny minority. By the time of the European revolutions of 1848, "the bourgeoisie was already unable to play a comparable role. It did not want and was not able to undertake the revolutionary liquidation of the social system that stood in its path to power."


title: Trotskyism

text: The theory of permanent revolution considers that in many countries that are thought under Trotskyism to have not yet completed a bourgeois-democratic revolution, the capitalist class opposes the creation of any revolutionary situation. They fear stirring the working class into fighting for its revolutionary aspirations against their exploitation by capitalism. In Russia, the working class, although a small minority in a predominantly peasant-based society, was organised in vast factories owned by the capitalist class and into large working-class districts. During the Russian Revolution of 1905, the capitalist class found it necessary to ally with reactionary elements such as the essentially feudal landlords and, ultimately, the existing Czarist Russian state forces. This was to protect their ownership of their property—factories, banks, etc.—from expropriation by the revolutionary working class.


title: Trotskyism

text: Therefore, according to the theory of permanent revolution, the capitalist classes of economically backward countries are weak and incapable of carrying through revolutionary change. As a result, they are linked to and rely on the feudal landowners in many ways. Thus Trotsky argues that because a majority of the branches of industry in Russia originated under the direct influence of government measures—sometimes with the help of government subsidies—the capitalist class was again tied to the ruling elite. The capitalist class were subservient to European capital.


title: Trotskyism

text: The theory of permanent revolution further considers that the peasantry as a whole cannot take on the task of carrying through the revolution because it is dispersed in small holdings throughout the country and forms a heterogeneous grouping, including the rich peasants who employ rural workers and aspire to landlordism as well as the poor peasants who aspire to own more land. Trotsky argues: "All historical experience [...] shows that the peasantry are absolutely incapable of taking up an independent political role".


title: Trotskyism

text: Trotskyists differ on the extent to which this is true today. However, even the most orthodox tend to recognise in the late twentieth century a new development in the revolts of the rural poor: the self-organising struggles of the landless, along with many other struggles that in some ways reflect the militant united, organised struggles of the working class, which to various degrees do not bear the marks of class divisions typical of the heroic peasant struggles of previous epochs. However, orthodox Trotskyists today still argue that the town- and city-based working-class struggle is central to the task of a successful socialist revolution linked to these struggles of the rural poor. They argue that the working class learns of the necessity to conduct a collective struggle, for instance, in trade unions, arising from its social conditions in the factories and workplaces; and that the collective consciousness it achieves as a result is an essential ingredient of the socialist reconstruction of society.


title: Trotskyism

text: Trotsky himself argued that only the proletariat or working class were capable of achieving the tasks of that bourgeois revolution. In 1905, the working class in Russia, a generation brought together in vast factories from the relative isolation of peasant life, saw the result of its labour as a vast collective effort, also seeing the only means of struggling against its oppression in terms of a collective effort, forming workers councils (soviets) in the course of the revolution of that year. In 1906, Trotsky argued:


title: Trotskyism

text: For instance, the Putilov Factory numbered 12,000 workers in 1900 and, according to Trotsky, 36,000 in July 1917.


title: Trotskyism

text: Although only a tiny minority in Russian society, the proletariat would lead a revolution to emancipate the peasantry and thus "secure the support of the peasantry" as part of that revolution, on whose support it will rely. However, to improve their conditions, the working class must create a revolution of their own, which would accomplish the bourgeois revolution and establish a workers' state.


title: Trotskyism

text: According to classical Marxism, a revolution in peasant-based countries such as Russia ultimately prepares the ground for capitalism's development since the liberated peasants become small owners, producers, and traders. This leads to the growth of commodity markets, from which a new capitalist class emerges. Only fully developed capitalist conditions prepare the basis for socialism.


title: Trotskyism

text: Trotsky agreed that a new socialist state and economy in a country like Russia would not be able to hold out against the pressures of a hostile capitalist world and the internal pressures of its backward economy. Trotsky argued that the revolution must quickly spread to capitalist countries, bringing about a socialist revolution that must spread worldwide. In this way, the revolution is "permanent", moving out of necessity first, from the bourgeois revolution to the workers’ revolution and from there uninterruptedly to European and worldwide revolutions.


title: Trotskyism

text: An internationalist outlook of permanent revolution is found in the works of Karl Marx. The term "permanent revolution" is taken from a remark of Marx in his March 1850 Address: "it is our task", Marx said:


title: Trotskyism

text: According to Trotsky, the term "Trotskyism" was coined by Pavel Milyukov (sometimes transliterated as Paul Miliukoff), the ideological leader of the Constitutional Democratic Party (Kadets) in Russia. Milyukov waged a bitter war against Trotskyism "as early as 1905".


title: Trotskyism

text: Trotsky was elected chairman of the St. Petersburg Soviet during the Russian Revolution of 1905. He pursued a policy of proletarian revolution at a time when other socialist trends advocated a transition to a "bourgeois" (capitalist) regime to replace the essentially feudal Romanov state. This year, Trotsky developed the theory of permanent revolution, as it later became known (see below). In 1905, Trotsky quotes from a postscript to a book by Milyukov, "The Elections to the Second State Duma", published no later than May 1907:


title: Trotskyism

text: Milyukov suggests that the mood of the "democratic public" was in support of Trotsky's policy of the overthrow of the Romanov regime alongside a workers' revolution to overthrow the capitalist owners of industry, support for strike action and the establishment of democratically elected workers' councils or "soviets".


title: Trotskyism

text: During his leadership of the Russian revolution of 1905, Trotsky argued that once it became clear that the Tsar's army would not come out in support of the workers, it was necessary to retreat before the armed might of the state in as good an order as possible. In 1917, Trotsky was again elected chairman of the Petrograd soviet, but this time soon came to lead the Military Revolutionary Committee, which had the allegiance of the Petrograd garrison and carried through the October 1917 insurrection. Stalin wrote:


title: Trotskyism

text: As a result of his role in the Russian Revolution of 1917, the theory of permanent revolution was embraced by the young Soviet state until 1924.


title: Trotskyism

text: The Russian revolution of 1917 was marked by two revolutions: the relatively spontaneous February 1917 revolution and the 25 October 1917 seizure of power by the Bolsheviks, who had gained the leadership of the Petrograd soviet.


title: Trotskyism

text: Before the February 1917 Russian revolution, Lenin had formulated a slogan calling for the "democratic dictatorship of the proletariat and the peasantry", but after the February revolution, through his April Theses, Lenin instead called for "all power to the Soviets". Nevertheless, Lenin continued to emphasise (as did Trotsky) the classical Marxist position that the peasantry formed a basis for the development of capitalism, not socialism.


title: Trotskyism

text: Also, before February 1917, Trotsky had not accepted the importance of a Bolshevik-style organisation. Once the February 1917 Russian revolution had broken out, Trotsky admitted the importance of a Bolshevik organisation and joined the Bolsheviks in July 1917. Although many, like Stalin, saw Trotsky's role in the October 1917 Russian revolution as central, Trotsky wrote that without Lenin and the Bolshevik Party, the October revolution of 1917 would not have taken place.


title: Trotskyism

text: As a result, since 1917, Trotskyism as a political theory has been fully committed to a Leninist style of democratic centralist party organisation, which Trotskyists argue must not be confused with the party organisation as it later developed under Stalin. Trotsky had previously suggested that Lenin's method of organisation would lead to a dictatorship. However, it is essential to emphasise that after 1917, orthodox Trotskyists argue that the loss of democracy in the Soviet Union was caused by the failure of the revolution to spread internationally and the consequent wars, isolation, and imperialist intervention, not the Bolshevik style of organisation.


title: Trotskyism

text: Lenin's outlook had always been that the Russian revolution would need to stimulate a Socialist revolution in Western Europe so that this European socialist society would come to the aid of the Russian revolution and enable Russia to advance towards socialism. Lenin stated:


title: Trotskyism

text: This outlook matched Trotsky's theory of permanent revolution precisely. Trotsky's permanent revolution had foreseen that the working class would not stop at the bourgeois democratic stage of the revolution but proceed towards a workers' state, as happened in 1917. The Polish Trotskyist Isaac Deutscher maintains that in 1917, Lenin changed his attitude toward Trotsky's theory of Permanent Revolution, and after the October revolution, it was adopted by the Bolsheviks.


title: Trotskyism

text: In "The Stalin School of Falsification", Trotsky argues that what he calls the "legend of Trotskyism" was formulated by Grigory Zinoviev and Lev Kamenev in collaboration with Stalin in 1924 in response to the criticisms Trotsky raised of Politburo policy. Orlando Figes argues: "The urge to silence Trotsky, and all criticism of the Politburo, was in itself a crucial factor in Stalin's rise to power".


title: Trotskyism

text: During 1922–1924, Lenin suffered a series of strokes and became increasingly incapacitated. In a document dictated before his death in 1924 while describing Trotsky as "distinguished not only by his exceptional abilities—personally he is, to be sure, the most able man in the present Central Committee" and also maintaining that "his non-Bolshevik past should not be held against him", Lenin criticized him for "showing excessive preoccupation with the purely administrative side of the work" and also requested that Stalin be removed from his position of General Secretary, but his notes remained suppressed until 1956. Zinoviev and Kamenev broke with Stalin in 1925 and joined Trotsky in 1926 in what was known as the United Opposition.


title: Trotskyism

text: In 1926, Stalin allied with Nikolai Bukharin, who led the campaign against "Trotskyism". In "The Stalin School of Falsification", Trotsky quotes Bukharin's 1918 pamphlet, "From the Collapse of Czarism to the Fall of the Bourgeoisie", which was re-printed in 1923 by the party publishing house, Proletari. Bukharin explains and embraces Trotsky's theory of permanent revolution in this pamphlet: "The Russian proletariat is confronted more sharply than ever before with the problem of the international revolution ... The grand total of relationships which have arisen in Europe leads to this inevitable conclusion. Thus, the permanent revolution in Russia is passing into the European proletarian revolution". Yet it is common knowledge, Trotsky argues, that three years later in 1926 "Bukharin was the chief and indeed the sole theoretician of the entire campaign against 'Trotskyism', summed up in the struggle against the theory of the permanent revolution."


title: Trotskyism

text: Trotsky wrote that the Left Opposition grew in influence throughout the 1920s, attempting to reform the Communist Party, but in 1927 Stalin declared "civil war" against them:


title: Trotskyism

text: The defeat of the European working class led to further isolation in Russia and further suppression of the Opposition. Trotsky argued that the "so-called struggle against 'Trotskyism' grew out of the bureaucratic reaction against the October Revolution [of 1917]". He responded to the one-sided civil war with his "Letter to the Bureau of Party History" (1927), contrasting what he claimed to be the falsification of history with the official history of just a few years before. He further accused Stalin of derailing the Chinese revolution and causing the massacre of the Chinese workers:


title: Trotskyism

text: Trotsky was sent into internal exile, and his supporters were jailed. For instance, Victor Serge first "spent six weeks in a cell" after a visit at midnight, then 85 days in an inner GPU cell, most of it in solitary confinement. He details the jailings of the Left Opposition. However, the Left Opposition worked secretly within the Soviet Union. Trotsky was eventually exiled to Turkey and moved to France, Norway and finally Mexico.


title: Trotskyism

text: After 1928, the various Communist Parties worldwide expelled Trotskyists from their ranks. Most Trotskyists defend the economic achievements of the planned economy in the Soviet Union during the 1920s and 1930s, despite the "misleadership" of the Soviet bureaucracy and what they claim to be the loss of democracy. Trotskyists claim that in 1928 inner party democracy and soviet democracy, which was at the foundation of Bolshevism, had been destroyed within the various Communist Parties. Anyone who disagreed with the party line was labelled a Trotskyist and even a fascist.


title: Trotskyism

text: In 1937, Stalin again unleashed what Trotskyists say was a political terror against their Left Opposition and many of the remaining Old Bolsheviks (those who had played vital roles in the October Revolution in 1917) in the face of increased opposition, particularly in the army.


title: Trotskyism

text: Trotsky founded the International Left Opposition in 1930. It was meant to be an opposition group within the Comintern, but anyone who joined or was suspected of joining the ILO was immediately expelled from the Comintern. The ILO, therefore, concluded that opposing Stalinism from within the communist organizations controlled by Stalin's supporters had become impossible, so new organizations had to be formed. In 1933, the ILO was renamed the International Communist League (ICL), which formed the basis of the Fourth International, founded in Paris in 1938.


title: Trotskyism

text: Trotsky said that only the Fourth International, based on Lenin's theory of the vanguard party, could lead the world revolution and that it would need to be built in opposition to the capitalists and the Stalinists.


title: Trotskyism

text: Trotsky argued that the defeat of the German working class and the coming to power of Adolf Hitler in 1933 was due in part to the mistakes of the Third Period policy of the Communist International and that the subsequent failure of the Communist Parties to draw the correct lessons from those defeats showed that they were no longer capable of reform and a new international organisation of the working class must be organised. The transitional demand tactic had to be a key element.


title: Trotskyism

text: At the time of the founding of the Fourth International in 1938, Trotskyism was a mass political current in Vietnam, Sri Lanka and slightly later Bolivia. There was also a substantial Trotskyist movement in China which included the founding father of the Chinese communist movement, Chen Duxiu, amongst its number. Wherever Stalinists gained power, they prioritised hunting down Trotskyists and treated them as the worst enemies.


title: Trotskyism

text: The Fourth International suffered repression and disruption through the Second World War. Isolated from each other and faced with political developments quite unlike those anticipated by Trotsky, some Trotskyist organizations decided that the Soviet Union could no longer be called a degenerated workers' state and withdrew from the Fourth International. After 1945, Trotskyism was smashed as a mass movement in Vietnam and marginalised in many other countries.


title: Trotskyism

text: The International Secretariat of the Fourth International (ISFI) organised an international conference in 1946 and then World Congresses in 1948 and 1951 to assess the expropriation of the capitalists in Eastern Europe and Yugoslavia, the threat of a Third World War and the tasks of revolutionaries. The Eastern European Communist-led governments, which came into being after World War II without a social revolution, were described by a resolution of the 1948 congress as presiding over capitalist economies. By 1951, the Congress had concluded that they had become "deformed workers' states". As the Cold War intensified, the ISFI's 1951 World Congress adopted theses by Michel Pablo that anticipated an international civil war. Pablo's followers considered that the Communist Parties, under pressure from the real workers' movement, could escape Stalin's manipulations and follow a revolutionary orientation.


title: Trotskyism

text: The 1951 Congress argued that Trotskyists should start to conduct systematic work inside those Communist Parties, followed by the majority of the working class. However, the ISFI's view that the Soviet leadership was counter-revolutionary remained unchanged. The 1951 Congress argued that the Soviet Union took over these countries because of the military and political results of World War II and instituted nationalized property relations only after its attempts at placating capitalism failed to protect those countries from the threat of incursion by the West.


title: Trotskyism

text: Pablo began expelling many people who disagreed with his thesis and did not want to dissolve their organizations within the Communist Parties. For instance, he expelled most of the French section and replaced its leadership. As a result, the opposition to Pablo eventually rose to the surface, with the Open Letter to Trotskyists of the World, by Socialist Workers Party leader James P. Cannon.


title: Trotskyism

text: The Fourth International split in 1953 into two public factions. Several sections of the International established the International Committee of the Fourth International (ICFI) as an alternative centre to the International Secretariat, in which they felt a revisionist faction led by Michel Pablo had taken power and recommitted themselves to the Lenin-Trotsky Theory of the Party and Trotsky's theory of Permanent Revolution. From 1960, led by the U.S Socialist Workers Party, many ICFI sections began the reunification process with the IS, but factions split off and continued their commitment to the ICFI. Today, national parties committed to the ICFI call themselves the Socialist Equality Party.


title: Trotskyism

text: The Bolivian Trotskyist party (, POR) became a mass party in the late 1940s and early 1950s and, together with other groups, played a central role during and immediately after the period termed the Bolivian National Revolution.


title: Trotskyism

text: In Brazil, as an officially recognised platform or faction of the PT until 1992, the Trotskyist Movimento Convergência Socialista (CS), which founded the United Socialist Workers' Party (PSTU) in 1994, saw a number of its members elected to national, state and local legislative bodies during the 1980s. The Socialism and Liberty Party (PSOL) presidential candidate in the 2006 general elections, Heloísa Helena, is a Trotskyist member of the Workers Party of Brazil (PT), a legislative deputy in Alagoas and in 1999 was elected to the Federal Senate. Expelled from the PT in December 2003, she helped found PSOL, in which various Trotskyist groups play a prominent role.


title: Trotskyism

text: In Argentina, the Workers' Revolutionary Party (Partido Revolucionario de los Trabajadores, PRT) lay in the merger of two leftist organizations in 1965, the Revolutionary and Popular Amerindian Front (, FRIP) and Worker's Word . In 1968, the PRT adhered to the Fourth International, based in Paris. That same year, a related organisation was founded in Argentina, the ERP (People's Revolutionary Army), which became South America's strongest rural guerrilla movement during the 1970s. The PRT left the Fourth International in 1973. During the Dirty War, the Argentine military regime suppressed both the PRT and the ERP. ERP commander Roberto Santucho was killed in July 1976. Owing to the ruthless repression, PRT showed no signs of activity after 1977. During the 1980s in Argentina, the Trotskyist party founded in 1982 by Nahuel Moreno, MAS (, Movement Toward Socialism), claimed to be the "largest Trotskyist party" in the world before it broke into many different fragments in the late 1980s, including the present-day MST, PTS, Nuevo MAS, IS, PRS, FOS, etc. In 1989, an electoral front with the Communist Party and Christian nationalist groups called ("United Left") retrieved 3.49% of the vote, representing 580,944 voters. Today, the Workers' Party in Argentina has an electoral base in Salta Province in the far north, particularly in the city of Salta itself; and has become the third political force in the provinces of Tucumán, also in the north; and Santa Cruz, in the south.


title: Trotskyism

text: Venezuelan president Hugo Chávez declared himself a Trotskyist during the swearing-in of his cabinet two days before his inauguration on 10 January 2007. Venezuelan Trotskyist organizations do not regard Chávez as a Trotskyist, with some describing him as a bourgeois nationalist while others consider him an honest revolutionary leader who made significant mistakes due to him lacking a Marxist analysis.


title: Trotskyism

text: In China, various left opposition groups in the late 1920s sought to engage Trotsky against the Comintern policy of support for the Kuomintang. In 1931, at Trotsky's urging, the various factions united in the Communist League of China, adopting Trotsky's document "The Political Situation in China and the Task of the Bolshevik-Leninist Opposition". Prominent members include Chen Duxiu, Wang Fanxi and Chen Qichang. The League was persecuted by the Nationalist government and by the Chinese Communist Party.


title: Trotskyism

text: In 1939, Ho Chi Minh, then a Comintern agent in southern China, reported that "everyone united to fight the Japanese except the Trotskyists. These traitors . . . adopted the ‘resolution’: ‘In the war against the Japanese, our position is clear: those who wanted the war and have illusions about the Kuomintang government, those concretely have committed treason. The union between the Communist Party and the Kuomintang is nothing but conscious treason’. And other ignominies of this kind." The Trotskyists were to be "crushed." In 1949, the Revolutionary Communist Party of China (; RCP) fled to Hong Kong. Since 1974, the party has been legally active as October Review, its official publication.


title: Trotskyism

text: In French Indochina during the 1930s, Vietnamese Trotskyism, led by Tạ Thu Thâu, was a significant current, particularly in Saigon, Cochinchina. In 1929, in the French Left Opposition , Ta Thu Thau condemned the Comintern for leading Chinese Communists (in 1927) to "the graveyard" through its support for the Kuomintang. The "'Sun Yat-sen-ist' synthesis of democracy, nationalism and socialism" was "a kind of nationalist mysticism." In Indochina, it could only obscure "the concrete class relationships, and the real, organic liaison between the indigenous bourgeoisie and French imperialism," in the light of which the call for independence is "mechanical and formalistic." "A revolution based on the organisation of the proletarian and peasant masses is the only one capable of liberating the colonies ... The question of independence must be bound up with that of the proletarian socialist revolution."


title: Trotskyism

text: For a period in the 1930s, Ta Thu Thau's Struggle group, centred around the newspaper La Lutte, was sufficiently strong to induce "Stalinists" (members of the then Indochinese Communist Party) to collaborate with the Trotskyists in support of labour and peasant struggles, and in the presentation of a common Workers Slate for Saigon municipal, and Cochinchina Council, elections. Ta Thu Thau was captured and executed by the Communist-front "Viet Minh" in September 1945. Many, if not most, of his fellow "luttuers" were subsequently killed, caught between the Viet Minh and the French effort at colonial reconquest.


title: Trotskyism

text: In Sri Lanka, a group of Trotskyists (known as the "T Group"), including South Asia's pioneer Trotskyist, Philip Gunawardena, who had been active in Trotskyist politics in Europe, and his colleague N. M. Perera, were instrumental in the foundation of the Lanka Sama Samaja Party (LSSP) in 1935. It expelled its pro-Moscow wing in 1940, becoming a Trotskyist-led party. In 1942, following the escape of the leaders of the LSSP from a British prison, a unified Bolshevik–Leninist Party of India, Ceylon and Burma (BLPI) was established in India, bringing together the many Trotskyist groups in the subcontinent. The BLPI was active in the Quit India Movement and the labour movement, capturing the second oldest union in India. Its high point was when it led the strikes which followed the Bombay Mutiny.


title: Trotskyism

text: After the war, the Sri Lanka section split into the Lanka Sama Samaja Party and the Bolshevik Samasamaja Party (BSP). In the general election of 1947, the LSSP became the main opposition party, winning ten seats, the BSP winning a further 5. It joined the Trotskyist Fourth International after fusion with the BSP in 1950 and led a general strike (Hartal) in 1953.


title: Trotskyism

text: In 1964, the LSSP joined a coalition government with Sirimavo Bandaranaike, with three members, NM Perera, Cholomondeley Goonewardene, and Anil Moonesinghe, brought into the new cabinet. This led to the expulsion of the party from the Fourth International. A section of the LSSP split to form the LSSP (Revolutionary) and joined the Fourth International after the LSSP proper was expelled. The LSSP (Revolutionary) later split into factions led by Bala Tampoe and Edmund Samarakkody. Another faction, the "Sakthi" Group, led by V. Karalasingham, rejoined the LSSP in 1966.


title: Trotskyism

text: In 1968, another faction of the LSSP (Revolutionary), led by Keerthi Balasooriya split, to form the Revolutionary Socialist League – more commonly known as the ""Kamkaru Mawatha" Group", after the name of their publication – and joined the International Committee of the Fourth International (ICFI). In 1987, the group changed its name to Socialist Equality Party.


title: Trotskyism

text: In 1974, a secret faction of the LSSP, allied to the Militant group in the United Kingdom, emerged. In 1977, this faction was expelled and formed the Nava Sama Samaja Party, led by Vasudeva Nanayakkara.


title: Trotskyism

text: In India, the BLPI fractured. In 1948, at the Fourth International's request, the party's rump dissolved into the Congress Socialist Party as an exercise in entryism.


title: Trotskyism

text: The French section of the Fourth International was the Internationalist Communist Party (PCI). In 1952 the party split when the Fourth International removed its Central Committee and split again when in 1953, the Fourth International itself divided. Further divisions occurred over which independence faction to support in the Algerian War.


title: Trotskyism

text: In 1967, the rump of the PCI renamed itself the "Internationalist Communist Organisation" (, OCI). It proliferated during the May 1968 student demonstrations but was banned alongside other far-left groups, such as the (Proletarian Left). Members temporarily reconstituted the group as the Trotskyist Organisation but soon obtained a state order permitting the reformation of the OCI. By 1970, the OCI was able to organise a 10,000-strong youth rally. The group also gained a strong base in trade unions. However, further splits and disintegration followed.


title: Trotskyism

text: In 2016 Jean-Luc Mélenchon, formerly of the ICO, launched the left-wing political platform (Unbowed France), subsequently endorsed by several parties, including his own Left Party and the French Communist Party. In the 2017 French Presidential Election, he received 19% in the first round. In the same election, Philippe Poutou of the New Anticapitalist Party, into which the Revolutionary Communist League () dissolved itself in 2008, won 1.20% of the vote. The only openly Trotskyist candidate, Nathalie Arthaud of Workers' Struggle (), won 0.64% of the vote.


title: Trotskyism

text: In Britain during the 1980s, the entryist Militant group operated within the Labour Party with three members of parliament and effective control of Liverpool City Council. Described by journalist Michael Crick as "Britain's fifth most important political party" in 1986, it played a prominent role in the 1989–1991 anti-poll tax movement, which was widely thought to have led to the downfall of British Prime Minister Margaret Thatcher.


title: Trotskyism

text: The most enduring of several Trotskyist parties in Britain has been the Socialist Workers Party, formerly the International Socialists (IS). Its founder Tony Cliff rejected the orthodox Trotskyist view of the USSR as a "deformed worker's state." Communist-party regimes were "state capitalist." The SWP has founded several front organisations through which they have sought to exert influence over the broader left, such as the Anti-Nazi League in the late 1970s and the Stop the War Coalition in 2001. It also allied with George Galloway and Respect, whose dissolution in 2007 caused an internal crisis in the SWP. A more serious internal crisis, leading to a significant decline in the party's membership, emerged in 2013. Allegations of rape and sexual assault made against a leading party member developed into a dispute over the practice of democratic centralism (defended by the party's international secretary Alex Callinicos).


title: Trotskyism

text: In April 2019, a rival splinter from IS made headlines when three former members of the Revolutionary Communist Party campaigned in the European Parliamentary election as candidates for the Brexit Party, and a fourth, Munira Mirza, was appointed head of the Number 10 Downing Street policy unit by the new Conservative Prime Minister Boris Johnson. The RCP's rejection of the SWP's critical engagement with the Labour Party and trade unions had morphed into embracing right-wing libertarian positions.


title: Trotskyism

text: The Socialist Party in Ireland was formed in 1990 by members who had been expelled by the Irish Labour Party's leader Dick Spring. It has had support in the Fingal electoral district and the city of Limerick. In 2018, it had three elected officials in Dáil Éireann. Paul Murphy representing Dublin West (Dáil constituency),


title: Trotskyism

text: Mick Barry representing Cork North-Central (Dáil constituency), and Ruth Coppinger representing Dublin West (Dáil constituency).


title: Trotskyism

text: In Portugal's October 2015 parliamentary election, the Left Bloc won 550,945 votes, translating into 10.19% of the expressed votes and 19 (out of 230) "deputados" (members of parliament). Although founded by several leftist tendencies, it still expresses much of the Trotskyist thought upheld and developed by its former leader, Francisco Louçã.


title: Trotskyism

text: In Turkey, there are some Trotskyist organizations, including the International Socialist Tendency's section (Revolutionary Workers' Socialist Party), Coordinating Committee for the Refoundation of the Fourth International's section (Revolutionary Workers' Party), Permanent Revolution Movement (SDH), "Socialism Magazine" (sympathizers of the International Committee of the Fourth International), and several small groups.


title: Trotskyism

text: The Fourth International derives from the 1963 reunification of the two public factions into which the Fourth International split in 1953: the International Secretariat of the Fourth International (ISFI) and some sections of the International Committee of the Fourth International (ICFI). It is often referred to as the United Secretariat of the Fourth International, the name of its leading committee before 2003. The USFI retains sections and sympathizing organizations in over 50 countries, including France's "Ligue Communiste Revolutionnaire" (LCR) and sections in Portugal, Sri Lanka, the Philippines, and Pakistan.


title: Trotskyism

text: The International Committee of the Fourth International maintains its independent organization and publishes the World Socialist Web Site.


title: Trotskyism

text: The Committee for a Workers' International (CWI) was founded in 1974 and has sections in over 35 countries. Before 1997, most organisations affiliated with the CWI sought to build an entrist Marxist wing within the large social democratic parties. The CWI has adopted a range of tactics, including working with trade unions, but in some cases working within or supporting other parties, endorsing Bernie Sanders for the 2016 U.S. Democratic Party nomination and encouraging him to run independently.


title: Trotskyism

text: In France, the LCR is rivalled by , the French section of the Internationalist Communist Union (UCI), with small sections in a handful of other countries. It focuses its activities, whether propaganda or intervention, on the industrial proletariat.


title: Trotskyism

text: The Committee for a Marxist International (CMI) founders claim they were expelled from the CWI when the CWI abandoned entryism. The CWI claims they left, and no expulsions were carried out. Since 2006, it has been known as the International Marxist Tendency (IMT). CMI/IMT groups continue the policy of entering mainstream social democratic, communist or radical parties. Currently, International Marxist Tendency (IMT) is headed by Alan Woods.


title: Trotskyism

text: The list of Trotskyist internationals shows that there are a large number of other multinational tendencies that stand in the tradition of Leon Trotsky.


title: Trotskyism

text: Trotskyism has been criticised from various directions. In 1935, Marxist–Leninist Moissaye J. Olgin argued that Trotskyism was "the enemy of the working class" and "should be shunned by anybody who has sympathy for the revolutionary movement of the exploited and oppressed the world over." The African American Marxist–Leninist Harry Haywood, who spent much time in the Soviet Union during the 1920s and 1930s, stated that although he had been somewhat interested in Trotsky's ideas when he was young, he came to see it as "a disruptive force on the fringes of the international revolutionary movement" which eventually developed into "a counter-revolutionary conspiracy against the Party and the Soviet state". He continued to put forward his following belief:


title: Trotskyism

text: Trotsky was not defeated by bureaucratic decisions or Stalin's control of the Party apparatus—as his partisans and Trotskyite historians claim. He had his day in court and finally lost because his whole position flew in the face of Soviet and world realities. He was doomed to defeat because his ideas were incorrect and failed to conform to objective conditions, as well as the needs and interests of the Soviet people.


title: Trotskyism

text: Other figures associated with Marxism–Leninism criticized Trotskyist political theory, including Régis Debray and Earl Browder.


title: Trotskyism

text: Polish philosopher Leszek Kołakowski wrote: "Both Trotsky and Bukharin were emphatic in their assurances that forced labour was an organic part of the new society."


title: Trotskyism

text: Some left communists, such as Paul Mattick, claim that the October Revolution was totalitarian from the start and, therefore, Trotskyism has no fundamental differences from Stalinism in practice or theory.


title: Trotskyism

text: In the United States, Dwight Macdonald broke with Trotsky and left the Trotskyist Socialist Workers Party by raising the question of the Kronstadt rebellion, which Trotsky, as leader of the Soviet Red Army, and the other Bolsheviks had brutally repressed. He then moved towards democratic socialism and anarchism. The Lithuanian-American anarchist Emma Goldman raised a similar critique of Trotsky's role in the events around the Kronstadt rebellion. In her essay "Trotsky Protests Too Much", she says: "I admit, the dictatorship under Stalin's rule has become monstrous. That does not, however, lessen the guilt of Leon Trotsky as one of the actors in the revolutionary drama of which Kronstadt was one of the bloodiest scenes". Trotsky defended the actions of the Red Army in his essay "Hue and Cry over Kronstadt".


title: Prostate massage

text: Prostate massage is the massage or stimulation of the male prostate gland for medical purposes or sexual stimulation.


title: Prostate massage

text: The prostate takes part in the sexual response cycle, and is essential for the production of semen. Due to its proximity to the anterior rectal wall, it can be stimulated from the anterior wall of the rectum or externally via the perineum.


title: Prostate massage

text: Prostate massage is part of the digital rectal examination (DRE) routinely given to men by urologists in order to look for nodules of prostate cancer and to obtain an expressed prostatic secretion (EPS) specimen for microscopy and microbiological culture to screen for prostatitis.


title: Prostate massage

text: In the late 1990s, a small number of doctors tried prostate massage in conjunction with antibiotics for the treatment of with uncertain results. In recent trials, however, prostate massage was not shown to improve outcomes compared to antibiotics alone. As a consequence of these findings, prostate massage is not officially sanctioned in medicine for the treatment of any medical disorder today. Prostatic massage should "never" be performed on patients with , because the infection can spread elsewhere in the body if massage is performed.


title: Prostate massage

text: Once the most popular therapeutic maneuver used to treat prostatitis, it was abandoned as primary therapy in the 1960s.


title: Prostate massage

text: In the late 1990s the ineffectiveness of drug treatments for chronic prostatitis led to a brief resurgence of interest in prostate massage. In a recent trial, however, prostate massage was not shown to improve outcomes compared to antibiotics alone.


title: Prostate massage

text: Vigorous prostate massage has been documented to have injurious consequences: periprostatic hemorrhage, cellulitis, septicaemia, possible disturbance and metastasis of prostate cancer to other parts of the body, and hemorrhoidal flare-up, and rectal fissures.


title: Prostate massage

text: Electroejaculation is a procedure in which nerves are stimulated via an electric probe, which is inserted into the rectum adjacent to the prostate. The stimulus voltage stimulates nearby nerves, resulting in contraction of the pelvic muscles and ejaculation. It is most commonly encountered in animal husbandry for the purpose of collecting semen samples for testing or breeding. Some devices are used under general anesthesia on humans who have certain types of anejaculation. Electroejaculation may also be used for posthumous sperm retrieval in humans. Electroejaculation is a different procedure from manual prostate massage.


title: Prostate massage

text: Prostate massage is also used as an erotic massage for sexual stimulation, often in order to reach orgasm. The prostate is sometimes referred to as the "male G-spot" or "P-spot". Some men can achieve orgasm through stimulation of the prostate gland, such as prostate massage or receptive anal intercourse, and men who report the sensation of prostate stimulation often give descriptions similar to females' accounts of G-spot stimulation. Prostate stimulation can produce a stronger, more powerful, and "deeper" orgasm than solely penile stimulation, described by some men as more widespread, intense, and enduring, and allowing for greater feelings of ecstasy than orgasm elicited by penile stimulation only. However, all male orgasms, including those by penile stimulation, involve muscular contractions in the prostate gland. It is possible for some men to achieve orgasms through prostate stimulation alone. Stimulation may be achieved by use of one or more fingers or by using sex toys designed to bring pressure to the anterior wall of the rectum at the location of the gland.


title: Prostate massage

text: Prostate massage may be practiced between sexual partners, either as a discrete activity or during other sexual acts stimulating the penis for example. The increasing availability (online via the Internet) of purpose-designed, safe and hygienic sex toys and devices aimed at prostate stimulation may encourage sexual experimentation and incorporation into sex play. The use of a finger for anal penetration and prostate stimulation can enhance a man's orgasm or vary the sensations experienced during sexual arousal. The finger of the prostate massager is introduced into the rectum through the anus and the prostate gland is gently massaged via the anterior (front) wall of the rectum. In some individuals or for some massagers the correct location of the prostate may be slightly too deep or the finger too short to reach easily. Prostate massage can be performed individually or with the help of a partner.


title: Prostate massage

text: Prostate massage may also be used in long-term orgasm denial play, for the purpose of relieving immediate need for orgasm without impacting general feelings of arousal. For this purpose it is often referred to as milking.


title: Prostate massage

text: There are safety matters relating to prostate stimulation and anal penetration. It is strongly recommended that plenty of lubricant be used with prostate massagers to prevent rectal lining damage. A smaller instrument or finger may be introduced gradually to minimize the discomfort that some may feel. Massagers may be used with or without a condom; however, because of the bacteria found in the rectum, if a condom is not used, it is very important to clean the tool with soap before use in another orifice or by a partner. Receiving anal stimulation may cause feelings of having to defecate. More often than not, this is just a sensation that the stimulation causes and may take some getting used to.


title: Prostate massage

text: A "prostate massager" is a device for massaging the prostate gland. The shape of a prostate massager is similar to a finger, since prostate massages are traditionally given digitally (for example, via fingering). They usually have a slightly curved head to effectively massage the prostate. Lubricant is usually necessary before inserting anything into the anus and helps to avoid injury to the rectal mucosa (lining). Caution should be exercised when a prostate massager is used because of the sensitivity of the prostate. Correct use involves a medium to light repetitive massage, or circular motion—the device being used to administer the massage should not be used too vigorously or without care, since this may lead to injury.


title: Prostate massage

text: Prostate massage equipment ranges from dildos to butt plugs. When used in sexual practice, prostate massagers are commonly referred to as "prostate toys", "prostate sex toys", and "anal toys". These prostate massagers are inserted into the rectum through the anus and are intended to stimulate the prostate by simple massaging or vibrating. They are used during foreplay by many couples.


title: Prostate massage

text: Prostate dildos are similar to vaginal dildos, but they tend to be more curved, slimmer and with a softer texture. Some of the new prostate dildos on the market are driven by batteries and offer vibration at the tip; the speed or intensity of which may be changed depending on the subject's personal preference. Unlike vaginal dildos, the anal prostate massager has a flared end to prevent it from being fully inserted and 'lost' inside the rectum.


title: Prostate massage

text: Some men prefer butt plugs, which are easy to use, can be inserted freely and left in place while the man's hands are free for other sexual activities such as masturbation. Anal plugs also come in various shapes, sizes and designs and are not commonly intended to stimulate the prostate. Newer, more angled models (second generation) of prostate massagers have been developed to provide a more direct and thorough massage of the prostate gland. These new devices feature a more curved shape and are slightly longer than the originals. They commonly have a narrow neck and a flared end to avoid losing them in the rectum. While many massagers rely upon the body's own natural muscular contractions of the anal sphincter and anal wall to stimulate the prostate, some of the newer models come with vibrators built into them to increase sexual pleasure.


title: Chorizo

text: Chorizo (, from Spanish ; similar to but distinct from Portuguese ) is a type of pork cured meat originating from the Iberian Peninsula.


title: Chorizo

text: In Europe, chorizo is a fermented, cured, smoked meat, which may be sliced and eaten without cooking, or added as an ingredient to add flavor to other dishes. Elsewhere, some sausages sold as chorizo may not be fermented and cured, and require cooking before eating. Spanish and Portuguese are distinctly different products, despite both getting their smokiness and deep red color from dried, smoked, red peppers (/).


title: Chorizo

text: Iberian chorizo is eaten sliced in a sandwich, grilled, fried, or simmered in liquid, including apple cider or other strong alcoholic beverages such as . It is also used as a partial replacement for ground (minced) beef or pork.


title: Chorizo

text: The word "chorizo" probably comes from the Late Latin 'salted', via the Portuguese ; it is a doublet of the Spanish word 'sausage', which was transmitted through Italian .


title: Chorizo

text: In English, "chorizo" is usually pronounced , though sometimes the Castilian Spanish [θ] sound is used: .


title: Chorizo

text: Spanish chorizo is made from coarsely chopped pork and pork fat, seasoned with garlic, – a smoked paprika – and salt. It is generally classed as either (spicy) or (sweet), depending upon the type of used. Hundreds of regional varieties of Spanish chorizo, both smoked and unsmoked, may contain herbs, and other ingredients. For example, chorizo de Pamplona is a thicker sausage with the meat more finely ground. Among the varieties is from the La Rioja region, which has PGI protection within the EU.


title: Chorizo

text: Chorizo is made in short or long and hard or soft varieties; leaner varieties are suited to being eaten at room temperature as an appetizer or tapas, whereas the fattier versions are generally used for cooking. A rule of thumb is that long, thin chorizos are sweet, and short chorizos are spicy, although this is not always the case.


title: Chorizo

text: Spain produces many other pork specialties, as well, such as or , cured and air-dried in a similar way. is a lean, cured meat to slice, made from the loin of the pig, which is marinated and then air-dried. is another cured sausage without the seasoning of chorizo, but flavoured with black peppercorns, instead.


title: Chorizo

text: Depending on the variety, chorizo can be eaten sliced without further cooking, sometimes sliced in a sandwich, or grilled, fried, or baked alongside other foodstuffs, and is also an ingredient in several dishes where it accompanies beans, such as or or served as a tapas, such as "Chorizo in Red Wine sauce".


title: Chorizo

text: The version of these dishes (with all the trimmings, literally sacraments) adds to chorizo other preserved meats such as (cured bacon) and (blood sausage).


title: Chorizo

text: Portuguese or (the latter usually denoting a larger or thicker version) is a distinct product and not to be confused with chorizo. It is made (at least) with pork, fat, paprika, garlic, and salt (wine and hot pepper also being common ingredients in some regions). It is then stuffed into natural casings from pig or lamb and slowly dried over smoke. The many different varieties differ in color, shape, spices and taste. White pepper, piri-piri, cumin and cinnamon are a common addition. The Portuguese varieties tend to be hotter than the Spanish ones. Many dishes of Portuguese and Brazilian cuisine make use of , including and .


title: Chorizo

text: A popular way to prepare chouriço is slicing it part-way through and cooking it over an alcohol flame at the table (sometimes called , but more commonly just "chouriço assado") in purpose-made glazed earthenware dishes with a lattice top.


title: Chorizo

text: In Johannesburg, South Africa, the many Portuguese immigrants in the 1960s from Portugal and Mozambique tended to settle in a suburb called La Rochelle (Little Portugal). Most of them either returned to Portugal or moved on to more affluent suburbs in the city, but restaurants in the area and the very well-supported annual "Lusitoland" fundraiser festival have on the menu.


title: Chorizo

text: In the heavily Portuguese counties in the US states of Rhode Island and southeastern Massachusetts, is often served with little neck clams and white beans. sandwiches on grinder rolls, with sautéed green peppers and onions, are commonly available at local delis and convenience stores. Stuffed quahogs (also known as stuffies), a Rhode Island specialty, usually include .


title: Chorizo

text: In Portugal, can be made with blood, similar to blood sausage or black pudding and is called (blood ) or morcela. Other types of include , , , , and .


title: Chorizo

text: Based on the uncooked Spanish (fresh chorizo), the Mexican versions of chorizo are made not just from fatty pork, but also from beef, venison, chicken, turkey, and even tofu, kosher, and vegan versions are made. The meat is usually ground (minced) rather than chopped, and different seasonings are used. This type is better known in Mexico and other parts of the Americas, including the border areas of the United States, and is not frequently found in Europe. Due to culinary tradition and the high cost of imported Spanish smoked paprika, Mexican chorizo is usually made with native chili peppers of the same "Capsicum annuum" species, making it spicier than Spanish and Portuguese varieties of the sausage. Mexican longaniza are typically longer and spicier than chorizo.


title: Chorizo

text: The area around Toluca specializes in "green" chorizo, made with some combination of tomatillo, cilantro, chili peppers, and garlic. Most Mexican chorizo is a deep reddish color, and is largely available in two varieties, fresh and dried, though fresh is much more common. Some of the cheapest commercial chorizos use offal stuffed in inedible plastic casing to resemble sausage links, rather than muscle meat. Before consumption, the casing is usually cut open and the sausage is fried in a pan and mashed with a fork until it resembles finely minced ground beef. A common alternative recipe does not have casings. Pork and beef are cured overnight in vinegar and chili powder. Served for breakfast, lunch, or dinner, it has the finely minced texture mentioned above, and is quite intense in flavor.


title: Chorizo

text: In Mexico, restaurants and food stands make tacos, (or ), burritos, and with cooked chorizo, and it is also a popular pizza topping. is a popular breakfast dish in Mexico and areas of the United States with Mexican populations. It is made by mixing fried chorizo with scrambled eggs. is often used in breakfast burritos, tacos, and taquitos. Another popular Mexican recipe is fried chorizo combined with pinto or black refried beans. This combination is often used in as a spread, or as a side dish where plain refried beans would normally be served. In Mexico and the southwestern United States chorizo is also used to make (or ), a popular appetizer consisting of small pieces of chorizo served with melted cheese and eaten with small corn tortillas or tortilla chips. In heavily Mexican parts of the United States, a popular filling for breakfast tacos is , or diced potatoes sautéed until soft with chorizo mixed in.


title: Chorizo

text: In Puerto Rico, Panama, and the Dominican Republic, chorizo and are considered two different types of meat. Puerto Rican chorizo is a smoked, well-seasoned sausage nearly identical to the smoked versions in Spain. Puerto Rican and Dominican have a very different taste and appearance. The seasoned meat is stuffed into a pork casing and is formed very long by hand. It is then hung to air-dry. can then be fried in oil or cooked with rice or beans. It is eaten with many different dishes.


title: Chorizo

text: In contrast to Spanish chorizo, in the United States the term generally refers to a sausage that is never dried, has a fattier filling, and is very spicy. It is most popular in areas with large Cuban, Dominican or Puerto Rican populations or near the Mexican border, especially in the Southwest near Chihuahua, Sonora, and Nuevo Leon. It is also found further north in places like Austin, Texas or Santa Fe, New Mexico, where its earliest evidence dates to well before the Wild West. It is most commonly eaten for breakfast on its own, or mixed with a local version of .


title: Chorizo

text: In Louisiana, Creole and Cajun cuisine both feature a variant of chorizo called , which is frequently used in the Creole dish of red beans and rice. As with its cousin to the west, smoking this variant is an acceptable practice in local cuisine.


title: Chorizo

text: In Ecuador, many types of sausage have been directly adopted from European or North American cuisine. All sorts of salami, either raw or smoked, are known just as salami. Most commonly known are sorts from Spanish chorizo, Italian pepperoni, and wiener sausages; wieners are the most popular. Some local specialities include , , and . , as in most Spanish-speaking countries, is basically cooked pork blood encased in pork intestine casing (black pudding in English). is a thin sausage containing almost any mixture of meat, fat, or even cartilage, smoked rather than fresh. Chorizo is a mixture of chopped pork meat, pork fat, salt, whole pepper grains, cinnamon, achiote, and other spices, which produce its characteristic deep red color. A traditional dish consists of fried egg, mashed potatoes, avocado, salad, and slices of fried chorizo.


title: Chorizo

text: In Argentina, Uruguay, Bolivia, Peru, Colombia and Venezuela, chorizo is the name for any coarse meat sausage. Spanish-style chorizo is also available, and is distinguished by the name ('Spanish chorizo'). Argentine chorizos are normally made of pork, and are not spicy hot. Some Argentine chorizos include other types of meat, typically beef. In Argentina, Bolivia, Paraguay, Uruguay, Chile, and Peru, fresh chorizo, cooked and served in a bread roll, is called a . In Colombia, chorizo is usually accompanied by .


title: Chorizo

text: In Brazil, is the word used for what in the rest of Latin America is ; meat sausages similar to the chorizos of other Latin American countries are called . Many varieties of Portuguese-style and are used in many different types of dishes, such as .


title: Chorizo

text: In Bolivia, chorizos are made of pork, fried and served with salad (tomato, lettuce, onion, boiled carrots and ), , and a slice of bread soaked with chorizo fat. Chorizo sandwiches, without , are also eaten.


title: Chorizo

text: In Goa, India, which was ruled by the Portuguese for 450 years and has a large percentage of Goan Catholics, is made from pork marinaded in a mixture of vinegar, red chilies, and spices such as garlic, ginger, cumin, turmeric, cloves, pepper, and cinnamon, which is stuffed into casings. These are enjoyed either with the local Goan Portuguese-style crusty bread, or pearl onions, or both. They are also used, cut into chunks, as the meat ingredient in rice pilaf. They can be raw (wet), smoked or cured through salting and air-drying.


title: Chorizo

text: Three kinds of are found in Goa: dry, wet, and skin. Dry is aged in the sun for long periods (three months or more). Wet has been aged for about a month or less. Skin , also aged, is rare and difficult to find. It consists primarily of minced pork skin along with some of its subcutaneous fat. All three are made in variations such as hot, medium, and mild. Other variations exist, depending on the size of the links, which range from . Typically, the wet varieties tend to be longer than the dry ones.


title: Chorizo

text: Goan should be distinguished from "Goan frankfurters", which look similar to equivalents in the United States, but with a predominantly peppercorn flavor.


title: Chorizo

text: "Longaniza" (; Visayan: , , ) are Philippine chorizos flavored with indigenous spices, and may be made of chicken, beef, or even tuna. While the term "longaniza" refers to fresh sausages, it is also used in the Philippines to refer to cured sausages. Philippine are often dyed red with seeds. There are dozens of variants from various regions in the Philippines.


title: Brownian motion

text: Brownian motion, or pedesis (from "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).


title: Brownian motion

text: This pattern of motion 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).


title: Brownian motion

text: 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 1905, almost eighty years later, 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".


title: Brownian motion

text: The many-body interactions that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. In consequence, 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).


title: Brownian motion

text: 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:


title: Brownian motion

text: 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".


title: Brownian motion

text: 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.


title: Brownian motion

text: 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.


title: Brownian motion

text: 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.


title: Brownian motion

text: 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.


title: Brownian motion

text: 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 10 collisions per second.


title: Brownian motion

text: He regarded the increment of particle positions in time formula_1 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 (formula_2) with some probability density function formula_3 (i.e., formula_4 is the probability density for a jump of magnitude formula_2, i.e., the probability density of the particle incrementing its position from formula_6 to formula_7 in the time interval formula_1). Further, assuming conservation of particle number, he expanded the number density formula_9 (number of particles per unit volume around formula_6) at time formula_11 in a Taylor series,


title: Brownian motion

text: where the second equality is by definition of formula_13. 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:


title: Brownian motion

text: Where the coefficient after the Laplacian, the second moment of probability of displacement formula_2, is interpreted as mass diffusivity "D":


title: Brownian motion

text: Then the density of Brownian particles "ρ" at point "x" at time "t" satisfies the diffusion equation:


title: Brownian motion

text: Assuming that "N" particles start from the origin at the initial time "t" = 0, the diffusion equation has the solution


title: Brownian motion

text: This expression (which is a normal distribution with the mean formula_19 and variance formula_20 usually called Brownian motion formula_21) 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


title: Brownian motion

text: 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.


title: Brownian motion

text: 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.


title: Brownian motion

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


title: Brownian motion

text: 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 formula_23, 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


title: Brownian motion

text: where "ρ" − "ρ" is the difference in density of particles separated by a height difference, of formula_25, "k" is the Boltzmann constant (the ratio of the universal gas constant, "R", to the Avogadro constant, "N"), and "T" is the absolute temperature.


title: Brownian motion

text: 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,


title: Brownian motion

text: 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:


title: Brownian motion

text: 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" = "R" / "N", 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 "N" can be determined.


title: Brownian motion

text: 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".


title: Brownian motion

text: 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 formula_29 for the diffusion coefficient "k"', where formula_30 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.


title: Brownian motion

text: 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.


title: Brownian motion

text: 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: formula_31. However, when he relates it to a particle of mass "m" moving at a velocity formula_32 which is the result of a frictional force governed by Stokes's law, he finds


title: Brownian motion

text: where "μ" is the viscosity coefficient, and formula_34 is the radius of the particle. Associating the kinetic energy formula_35 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."


title: Brownian motion

text: 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.


title: Brownian motion

text: 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 10 cm/s. But we also have to take into consideration that in a gas there will be more than 10 collisions in a second, and even greater in a liquid where we expect that there will be 10 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 10 to 10 collisions in one second, then velocity of the Brownian particle may be anywhere between 10 and 1000 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.


title: Brownian motion

text: 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, formula_40, will be equal, on the average, to the kinetic energy of the surrounding fluid particle, formula_35.


title: Brownian motion

text: 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 Δ"V". If "N" is the number of collisions from the right and "N" the number of collisions from the left then after "N" collisions the particle's velocity will have changed by Δ"V"(2"N" − "N"). The multiplicity is then simply given by:


title: Brownian motion

text: and the total number of possible states is given by 2. Therefore, the probability of the particle being hit from the right "N" times is:


title: Brownian motion

text: 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 Δ"V"s instead of always just one in a realistic situation.


title: Brownian motion

text: The diffusion equation yields an approximation of the time evolution of the probability density function associated to the position of the particle going under a Brownian movement under the physical definition. The approximation is valid on short timescales.


title: Brownian motion

text: The time evolution of the position of the Brownian particle itself is best described using Langevin equation, an equation which involves a random force field representing the effect of the thermal fluctuations of the solvent on the particle.


title: Brownian motion

text: 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.


title: Brownian motion

text: 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: Δ"x" = "v"Δ"t". So the instantaneous velocity of the Brownian motion can be measured as "v" = Δ"x"/Δ"t", when Δ"t" « "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.


title: Brownian motion

text: 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 formula_44 of the background stars by


title: Brownian motion

text: where formula_46 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 formula_44 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.


title: Brownian motion

text: 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.


title: Brownian motion

text: formula_50 denotes the normal distribution with expected value "μ" and variance "σ". The condition that it has independent increments means that if formula_51 then formula_52 and formula_53 are independent random variables. In addition, for some filtration formula_54, formula_55 is formula_54 measurable for all formula_57.


title: Brownian motion

text: 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" = 0 and quadratic variation formula_58.


title: Brownian motion

text: A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent formula_59 random variables. This representation can be obtained using the Karhunen–Loève theorem.


title: Brownian motion

text: 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.


title: Brownian motion

text: 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.


title: Brownian motion

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


title: Brownian motion

text: The French mathematician Paul Lévy proved the following theorem, which gives a necessary and sufficient condition for a continuous R-valued stochastic process "X" to actually be "n"-dimensional Brownian motion. Hence, Lévy's condition can actually be used as an alternative definition of Brownian motion.


title: Brownian motion

text: Let "X" = ("X", ..., "X") be a continuous stochastic process on a probability space (Ω, Σ, P) taking values in R. Then the following are equivalent:


title: Brownian motion

text: The spectral content of a stochastic process formula_60 can be found from the power spectral density, formally defined as


title: Brownian motion

text: where formula_62 stands for the expected value. The power spectral density of Brownian motion is found to be


title: Brownian motion

text: where formula_64 is the diffusion coefficient of formula_60. 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.,


title: Brownian motion

text: which for an individual realization of a Brownian motion trajectory, it is found to have expected value formula_67


title: Brownian motion

text: For sufficiently long realization times, the expected value of the power spectrum of a single trajectory converges to the formally defined power spectral density formula_71, but its coefficient of variation formula_72 tends to formula_73. This implies the distribution of formula_74 is broad even in the infinite time limit.


title: Brownian motion

text: The infinitesimal generator (and hence characteristic operator) of a Brownian motion on R is easily calculated to be ½Δ, where Δ 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 ("M", "g"): a Brownian motion on "M" is defined to be a diffusion on "M" whose characteristic operator formula_75 in local coordinates "x", 1 ≤ "i" ≤ "m", is given by ½Δ, where Δ is the Laplace–Beltrami operator given in local coordinates by


title: Brownian motion

text: 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: Internal Revenue Service

text: The Internal Revenue Service (IRS) is the revenue service for the United States federal government, which is responsible for collecting U.S. federal taxes and administering the Internal Revenue Code, the main body of the federal statutory tax law. It is an agency of the Department of the Treasury and led by the Commissioner of Internal Revenue, who is appointed to a five-year term by the President of the United States. The duties of the IRS include providing tax assistance to taxpayers; pursuing and resolving instances of erroneous or fraudulent tax filings; and overseeing various benefits programs, including the Affordable Care Act.


title: Internal Revenue Service

text: The IRS originates from the Commissioner of Internal Revenue, a federal office created in 1862 to assess the nation's first income tax to fund the American Civil War. The temporary measure provided over a fifth of the Union's war expenses before being allowed to expire a decade later. In 1913, the Sixteenth Amendment to the U.S. Constitution was ratified authorizing Congress to impose a tax on income, and the Bureau of Internal Revenue was established. In 1953, the agency was renamed the Internal Revenue Service, and in subsequent decades underwent numerous reforms and reorganizations, most significantly in the 1990s.