text:
Conditions of Learning

Conditions of Learning, by Robert M. Gagné, was originally published in 1965 by Holt, Rinehart and Winston and describes eight kinds of learning and nine events of instruction. This theory of learning involved two steps. The theory stipulates that there are several different types or levels of learning. The significance of these classifications is that each different type requires different types of instruction. Gagné identifies five major categories of learning: verbal information, intellectual skills, cognitive strategies, motor skills and attitudes. Different internal and external conditions are necessary for each type of learning. For example, for cognitive strategies to be learned, there must be a chance to practice developing new solutions to problems; to learn attitudes, the learner must be exposed to a credible role model or persuasive arguments.

Gagné suggests that learning tasks for intellectual skills can be organized in a hierarchy according to complexity: stimulus recognition, response generation, procedure following, use of terminology, discrimination, concept formation, rule application, and problem solving. The primary significance of the hierarchy is to identify prerequisites that should be completed to facilitate learning at each level. Prerequisites are identified by doing a task analysis of a learning/training task. Learning hierarchies provide a basis for the sequencing of instruction.

In addition, the theory outlines nine instructional events and corresponding cognitive processes: These events should satisfy or provide the necessary conditions for learning and serve as the basis for designing instruction and selecting appropriate media (Gagné, Briggs & Wager, 1992). Application
 * 1) Gaining attention (reception)
 * 2) Informing learners of the objective (expectancy)
 * 3) Stimulating recall of prior learning (retrieval)
 * 4) Presenting the stimulus (selective perception)
 * 5) Providing learning guidance (semantic encoding)
 * 6) Eliciting performance (responding)
 * 7) Providing feedback (reinforcement)
 * 8) Assessing performance (retrieval)
 * 9) Enhancing retention and transfer (generalization)

While Gagné's theoretical framework covers all aspects of learning, the focus of the theory is on intellectual skills. The theory has been applied to the design of instruction in all domains (Gagné & Driscoll, 1988). In its original formulation (Gagné, 1 962), special attention was given to military training settings. Gagné (1987) addresses the role of instructional technology in learning.

title: Conditions of Learning


text:
Jools Holland

Julian Miles Holland (born 24 January 1958) is an English pianist, bandleader, singer, composer and television presenter. He was an original member of the band Squeeze and has worked with many artists including Marc Almond, Joss Stone, Jayne County, Tom Jones, José Feliciano, Sting, Eric Clapton, Mark Knopfler, George Harrison, David Gilmour, Ringo Starr, Bono, Rod Stewart and Ruby Turner.

From 1982 until 1987, he co-presented the Channel 4 music programme The Tube. Since 1992, he has hosted Later... with Jools Holland, a music-based show aired on BBC2, on which his annual show Hootenanny is based. Holland is a published author and appears on television shows besides his own. He regularly hosted the programme Jools Holland on BBC Radio 2. In 2004 he collaborated with Tom Jones on an album of traditional R&B music. He achieved his first UK number one album in 2024 with Swing Fever, a collaboration with Rod Stewart.

Early life and education
Holland was born on 24 January 1958 in Blackheath, South East London. At the age of eight, he could play the piano fluently by ear. By his early teens he was appearing regularly in many of the pubs in South East London and the East End Docks.

Holland was educated at Shooters Hill Grammar School in southeast London, from which he was expelled for damaging a teacher's Triumph Herald.

Career
Holland began his career as a session musician. His first studio session was with Wayne County & the Electric Chairs in 1976 on their track "Fuck Off".

Holland was a founding member of the British pop band Squeeze, formed in March 1974, in which he played keyboards until 1980, through its first three albums, the eponymous Squeeze, Cool for Cats and Argybargy, before pursuing his solo career.

Holland began issuing solo records in 1978, his first EP being Boogie Woogie '78. He continued his solo career through the early 1980s, releasing an album and several singles between 1981 and 1984. He branched out into TV, co-presenting the Newcastle-based TV music show The Tube with Paula Yates. Holland used the phrase, "be there, or be an ungroovy fucker" in one early evening TV trailer for the show, live across two channels, causing him to be suspended from the show for six weeks. He referred to this in his sitcom The Groovy Fellers with Rowland Rivron. Holland also appeared as a guest host on MTV.

In 1983, Holland played an extended piano solo on The The's re-recording of "Uncertain Smile" for the album Soul Mining. In 1985, Squeeze (which had continued in Holland's absence through to 1982) unexpectedly regrouped including Holland as their keyboard player. Holland remained in the band until 1990, at which point he again departed to resume his solo career as a musician and a TV host.

In 1987, Holland formed the Jools Holland Big Band, which consisted of himself and for the show Gilson Lavis from Squeeze, which gradually grew and was renamed as Jools Holland's Rhythm and Blues Orchestra. In May 2022, it was a 17-piece orchestra and included singers Louise Marshall, Ruby Turner and Holland's daughter Mabel Ray, as well as his younger brother, singer-songwriter and keyboard player, Christopher Holland.

Between 1988 and 1990 Holland performed and co-hosted along with David Sanborn during the two seasons of the music performance programme Sunday Night on NBC late-night television. Since 1992, he has presented the music programme Later... with Jools Holland, plus an annual New Year's Eve Hootenanny.

In 1996, Holland signed a recording contract with Warner Bros. Records, and his records are now marketed through Rhino Records. On 29 November 2002, Holland was in the ensemble of musicians who performed at the Concert for George, which celebrated the music of George Harrison. In January 2005 Holland and his band performed with Eric Clapton as the headline act of the Tsunami Relief Cardiff.

On BBC Radio 2 Holland regularly hosted the programme Jools Holland, a mix of live and recorded music and general chat, featuring studio guests, along with members of his orchestra.

In March 2023, Jimmy Barnes announced the formation of supergroup The Barnestormers, featuring Barnes, Chris Cheney, Slim Jim Phantom, Jools Holland and Kevin Shirley. A self-titled album was released on 26 May 2023.

Personal life
As a teenager, Holland lived with his grandparents, which he mentioned anecdotally in a 2020 episode of Rhod Gilbert's Growing Pains.

Holland has a son, George, and daughter, Rose, with his former partner Mary Leahy. On 30 August 2005, Holland married Christabel McEwen, his girlfriend of 15 years and daughter of artist Rory McEwen. The couple have a daughter, Mabel, and McEwen has a son, Frederick Lambton, Viscount Lambton, by her former marriage to Ned Lambton, the 7th Earl of Durham.

Holland lives in Westcombe Park, south east London, where he had his studio, Helicon Mountain, built to his design and inspired by Portmeirion, the setting for the 1960s TV series The Prisoner. He also owns a manor house near the medieval Cooling Castle in Kent.

He appeared on the cover of Railway Modeller magazine in January 2019. In the attic of his house, Holland has spent ten years building a 100 ft model railway. It is full of miniature buildings and landscapes that stretch from Berlin to London. He started with photographs and paintings from early 1960s London. According to The Daily Telegraph, "In the evenings, he builds some trains and buildings before switching on some music, pouring a glass of wine and switching on the trains to watch them move around the room."

He received an OBE in 2003 in the Queen's Birthday Honours list, for services to the British music industry as a television presenter and musician. In September 2006, Holland was appointed a Deputy Lieutenant for Kent. Holland was appointed an honorary fellow of Canterbury Christ Church University at a ceremony held at Canterbury Cathedral on 30 January 2009. On 1 February 2011 he was appointed honorary colonel of 101 (City of London) Engineer Regiment. Holland has been the President of the British Watch & Clock Makers Guild since 2018, and an honorary liveryman of the Worshipful Company of Plumbers since 2019.

In June 2006, Holland performed in Southend for HIV/AIDS charity Mildmay, and in early 2007 he performed at Wells and Rochester Cathedrals to raise money for maintaining cathedral buildings. He is also patron of Drake Music.

A fan of the 1960s TV series The Prisoner, in 1987 Holland demonstrated his love of the series and starred in a spoof documentary, The Laughing Prisoner, with Stephen Fry, Terence Alexander and Hugh Laurie. Much of it was shot on location in Portmeirion, with archive footage of Patrick McGoohan. It featured musical selections by Siouxsie and the Banshees, Magnum and XTC. Holland performed a number towards the end of the programme.

Holland was an interviewer for The Beatles Anthology TV project, and appeared in the 1997 film Spiceworld as a musical director.

In 2009, Holland commissioned TV series Bangla Bangers (Chop Shop) to create a replica of the Rover JET1 for personal use.

Writing
His 2007 autobiography, Barefaced Lies and Boogie-Woogie Boasts, was BBC Radio 4 "Book of the Week" in the week beginning 8 October 2007 and was read by Holland.

Releases
• 1978 "Boogie Woogie '78" (EP)

• 1981 Jools Holland and His Millionaires

• 1984 Jools Holland Meets Rock 'A' Boogie Billy (US release only)

• 1990 World of His Own

• 1991 The Full Complement

• 1992 "Together Again" (single with Sam Brown)

• 1992 The A–Z Geographer's Guide to the Piano

• 1994 Solo Piano

• 1994 Live Performance

• 1996 Sex & Jazz & Rock & Roll

• 1997 Lift the Lid

• 1998 Best Of

• 1999 Sunset Over London

• 2000 Hop the Wag

• 2001 Small World Big Band

• 2001 Jools Holland's Big Band Rhythm & Blues

• 2002 SWBB Volume Two: More Friends

• 2003 Jack O the Green (SWBB Friends 3)

• 2004 Tom Jones & Jools Holland

• 2005 Beatroute

• 2005 Swinging the Blues, Dancing the Ska

• 2006 Moving Out to the Country

• 2007 Best of Friends

• 2008 The Collection

• 2008 The Informer (with Ruby Turner)

• 2008 "The Informer" (single with Ruby Turner)

• 2009 "I Went By" (single with Louise Marshall)

• 2010 Rockinghorse

• 2011 Finding the Keys: The Best of Jools Holland

• 2012 The Golden Age of Song

• 2014 Sirens of Song (UK No. 25)

• 2015 Jools & Ruby (with Ruby Turner)

• 2016 Piano

• 2017 As You See Me Now (with José Feliciano)

• 2018 A Lovely Life to Live (with Marc Almond)

• 2021 Pianola. Piano & Friends

Film and television
• 1981 Otway & Barrett Live

• 1981 Urgh! A Music War

• 1982 Police: Around the World

• 1982–1987 The Tube (Host for 121 editions)

• 1983 Rebellious Jukebox: Compere

• 1984 The Young Ones: punk (episode entitled "Cash")

• 1985 Walking to New Orleans (Jools Holland in New Orleans)

• 1987 Eat the Rich: Sun Reporter

• 1987 Filthy Rich & Catflap: Strip Show Pianist (Episode No. 1.3)

• 1987 The Laughing Prisoner: No. 7

• 1987 French and Saunders (Episode 1.5)

• 1988 Sunday Night: Host (unknown episodes)

• 1989-1990 Juke Box Jury: Host (unknown episodes)

• 1989 The Groovy Fellers Himself, 6 episodes

• 1990-1992 Jools Holland's Happening (Noel Gay Productions 37 episodes (BSB 1990-1991); 12 episodes (Channel 4, 1991-1992). Entire series believed lost )

• 1991 Mr Roadrunner (Jools Holland in Memphis)

• 1994 There's No Business...: Pianist (uncredited)

• 1994–1995 Don't Forget Your Toothbrush

• 1995 The Beatles Anthology

• 1997 Spice World: Musical Director

• 1997 Name That Tune: Host and Pianist

• 1998 Beat Route: Round the World with Jools Holland: Host and Pianist

• 2001 Astley's Way: Tribute to composer Edwin Astley

• 2003 Jools' History of the Piano: Presenter

• 2007 Fairport@Forty: Interview

• 2007 Top Gear: Star in a reasonably priced car.

• 2009 Chop Shop Rover Concept: The Jet 1 Car : Customer

• 2012 Jools Holland – London Calling: Presenter

• 2014 The Life of Rock with Brian Pern as himself

• 2023 Little Trains & Big Names with Pete Waterman as himself

Current television programmes

 * 1992–present Later... with Jools Holland
 * 1993–present Hootenanny
 * 2020–present Celebrity Gogglebox with Vic Reeves

Books

 * "Rolling Stones": A Life on the Road (with Dora Loewenstein), Viking/Allen Lane (1998) (ISBN 0-670-88051-5)
 * Beat Route: Journeys Through Six Counties, Weidenfeld & Nicolson (1998) (ISBN 0-575-06700-4)
 * Ray Charles: Man and Music, (with Michael Lydon), Payback Press (1999) (ISBN 0-86241-929-8)
 * Hand That Changed Its Mind,  International Music Publications (2007) (ISBN 1-84328-645-9)
 * Barefaced Lies and Boogie-woogie Boasts, Penguin Books (2007) (ISBN 9780718149154)

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

title: Jools Holland


text:
Talk:Eunuch (court official)

Convention or scientific basis?
I propose a change in the very first lines. My proposal is:

''A euneuch is conventionally regarded as a male who has been castrated. Although we find that castration may have served special social functions at times, historical record, archeology, as well as etymology have not been able to give the conventional notion of euneuchs as castrated males a substantial enough basis.''

The primarily Byzantine title has equivalents in a number of empires, confederations, and kingdoms. The ordination of the euneuch assuredly has prescribed castration in certain examples. As the intersection of many fields are involved, ranging from anthropology, sociology, hermeneutics, archeology to the history of politics, theology etcetera, it has become clear that the topic of the euneuchs is a controversal one.

At least the first sentence as it stands is wrong. At the time of euneuchs in the Byzantine era, a male could become castrated for a number of reasons, without that implying the castrated person to be a euneuch. The title was not derogatory; although the notion of Scythians as the most euneuchal of societies, may have meant the Scythian peoples valued marriage less, or practiced the economy of pleasure and inheritance differently than the Hellenes and Romans. --Xactnorge (talk) 22:55, 24 February 2024 (UTC)

Removal of templates
This talk page was previously at Talk:Eunuch, where an article split occurred, and it seems to have retained the templates that were justified for a talk page at that location. I removed the template notforum, which I do not think is necessary on this talk page, like it was on Talk:Eunuch. This removal seems self-evident to me (click on the link if you are unfamiliar with the template), so I will not go into it further, unless asked. I also removed the templates WikiProject LGBT studies and WikiProject Sexuality. These were placed when the page was at "Talk:Eunuch" instead of "Talk:Eunuch (court official)". In my opinion, this article should be re-evaluated for inclusion in these WikiProjects before the templates are put back. I am not dismissing the importance of this article. I just think that it should be included in WikiProjects that are appropriate for its current content, rather than the content that the eunuch article had before the article split. While it is possible that the LGBT and Sexuality WikiProjects might be interested in this new article, I think that sociological and historical WikiProjects would probably be more interested in it. I left the talkheader template, even though it is probably more suited for more controversial articles and/or articles that get more traffic. You may remove it, if you want.

I previously added the message box that explains the situation about the talk page archives. You can modify that, too. I used substitution on the template to prevent it from being a burden on Wikipedia's severs, so it is kind of messy. I put HTML comments before and after the code in an attempt to make editing of this page easier. The spacing between talkheader and the message box is intentional. If the spacing is removed, the boxes are so close together that they look like a single item. Also, I know that the message box has a redundant link. I just added it for the convenience of anyone looking for information about the archived discussions. You may remove it, if you like. If you want to discuss something with me for some reason, please use my talk page to alert me rather than just replying here, as I do not watch this article and my editing on Wikipedia is sporadic. -- Kjkolb (talk) 23:05, 24 August 2012 (UTC)

title: Talk:Eunuch (court official)


text:
Croquembouche

A croquembouche or croque-en-bouche is a French dessert consisting of choux pastry puffs piled into a cone and bound with threads of caramel. In Italy and France, it is often served at weddings, baptisms and First Communions.

Etymology
The name comes from the French phrase croque en bouche, meaning '[something that] crunches in the mouth'.

Presentation
A croquembouche is composed of (usually cream-filled) choux piled into a cone and bound with spun sugar. It may also be decorated with other confectionery, such as sugared almonds, chocolate, and edible flowers. Sometimes it is covered in macarons or ganache.

History
The invention of the croquembouche is often attributed to Antonin Carême, who includes it in his 1815 cookbook Le Pâtissier royal parisien, but it is mentioned as early as 1806, in André Viard's culinary encyclopedia Le Cuisinier Impérial, and Antoine Beauvilliers' 1815 L'Art du Cuisinier. In Viard's encyclopedia and other early texts (e.g., Grimod de La Reynière's, Néo-physiologie du gout), it is included in lists of entremets—elaborate dishes, both savory and sweet, that were served between courses during large banquets.

Records
On 6 March 2009, alumni of the Pune-based Maharashtra State Institute of Hotel Management and Catering Technology entered the Limca Book of Records after creating India's biggest croquembouche. It was recorded as 4.5 m tall.

title: Croquembouche


text:
Gibbs paradox

In statistical mechanics, a semi-classical derivation of entropy that does not take into account the indistinguishability of particles yields an expression for entropy which is not extensive (is not proportional to the amount of substance in question). This leads to a paradox known as the Gibbs paradox, after Josiah Willard Gibbs, who proposed this thought experiment in 1874‒1875. The paradox allows for the entropy of closed systems to decrease, violating the second law of thermodynamics. A related paradox is the "mixing paradox". If one takes the perspective that the definition of entropy must be changed so as to ignore particle permutation, in the thermodynamic limit, the paradox is averted.

Illustration of the problem
Gibbs considered the following difficulty that arises if the ideal gas entropy is not extensive. Two containers of an ideal gas sit side-by-side. The gas in container #1 is identical in every respect to the gas in container #2 (i.e. in volume, mass, temperature, pressure, etc). Accordingly, they have the same entropy S. Now a door in the container wall is opened to allow the gas particles to mix between the containers. No macroscopic changes occur, as the system is in equilibrium. But if the formula for entropy is not extensive, the entropy of the combined system will not be 2S. In fact, the particular non-extensive entropy quantity considered by Gibbs predicts additional entropy (more than 2S). Closing the door then reduces the entropy again to S per box, in apparent violation of the Second Law of Thermodynamics.

As understood by Gibbs, and reemphasized more recently, this is a misuse of Gibbs' non-extensive entropy quantity. If the gas particles are distinguishable, closing the doors will not return the system to its original state – many of the particles will have switched containers. There is a freedom in defining what is "ordered", and it would be a mistake to conclude that the entropy has not increased. In particular, Gibbs' non-extensive entropy quantity for an ideal gas is not intended for situations where the number of particles changes.

The paradox is averted by assuming the indistinguishability (at least effective indistinguishability) of the particles in the volume. This results in the extensive Sackur–Tetrode equation for entropy, as derived next.

Calculating the entropy of ideal gas, and making it extensive
In classical mechanics, the state of an ideal gas of energy U, volume V and with N particles, each particle having mass m, is represented by specifying the momentum vector p and the position vector x for each particle. This can be thought of as specifying a point in a 6N-dimensional phase space, where each of the axes corresponds to one of the momentum or position coordinates of one of the particles. The set of points in phase space that the gas could occupy is specified by the constraint that the gas will have a particular energy:
 * $$U=\frac{1}{2m}\sum_{i=1}^{N} (p_{ix}^2 + p_{iy}^2 + p_{iz}^2)$$

and be contained inside of the volume V (let's say V is a cube of side X so that V = X3):
 * $$0 \le x_{ij} \le X$$

for $$i = 1...N$$ and $$j = 1, 2, 3$$

The first constraint defines the surface of a 3N-dimensional hypersphere of radius (2mU)1/2 and the second is a 3N-dimensional hypercube of volume VN. These combine to form a 6N-dimensional hypercylinder. Just as the area of the wall of a cylinder is the circumference of the base times the height, so the area φ of the wall of this hypercylinder is:

\phi(U,V,N) = V^N \left(\frac{2\pi^{\frac{3N}{2}}(2mU)^{\frac{3N-1}{2}}}{\Gamma(3N/2)}\right)(1) $$

The entropy is proportional to the logarithm of the number of states that the gas could have while satisfying these constraints. In classical physics, the number of states is infinitely large, but according to quantum mechanics it is finite. Before the advent of quantum mechanics, this infinity was regularized by making phase space discrete. Phase space was divided up in blocks of volume h3N. The constant h thus appeared as a result of a mathematical trick and thought to have no physical significance. However, using quantum mechanics one recovers the same formalism in the semi-classical limit, but now with h being the Planck constant. One can qualitatively see this from Heisenberg's uncertainty principle; a volume in N phase space smaller than h3N (h is the Planck constant) cannot be specified.

To compute the number of states we must compute the volume in phase space in which the system can be found and divide that by h3N. This leads us to another problem: The volume seems to approach zero, as the region in phase space in which the system can be is an area of zero thickness. This problem is an artifact of having specified the energy U with infinite accuracy. In a generic system without symmetries, a full quantum treatment would yield a discrete non-degenerate set of energy eigenstates. An exact specification of the energy would then fix the precise state the system is in, so the number of states available to the system would be one, the entropy would thus be zero.

When we specify the internal energy to be U, what we really mean is that the total energy of the gas lies somewhere in an interval of length $$\delta U$$ around U. Here $$\delta U$$ is taken to be very small, it turns out that the entropy doesn't depend strongly on the choice of $$\delta U$$ for large N. This means that the above "area" φ must be extended to a shell of a thickness equal to an uncertainty in momentum $$\delta p = \delta\left(\sqrt{2 m U}\right) = \sqrt{\frac{m}{2 U}}\delta U$$, so the entropy is given by:
 * $$\left.\right.

S=k\,\ln(\phi \delta p/h^{3N}) $$ where the constant of proportionality is k, the Boltzmann constant. Using Stirling's approximation for the Gamma function which omits terms of less than order N, the entropy for large N becomes:

S = k N \ln \left[ V \left(\frac UN \right)^{\frac 32}\right]+ {\frac 32}kN\left( 1+ \ln\frac{4\pi m}{3h^2}\right) $$

This quantity is not extensive as can be seen by considering two identical volumes with the same particle number and the same energy. Suppose the two volumes are separated by a barrier in the beginning. Removing or reinserting the wall is reversible, but the entropy increases when the barrier is removed by the amount

\delta S = k \left[ 2N \ln(2V) - N\ln V - N \ln V \right] = 2 k N \ln 2 > 0 $$ which is in contradiction to thermodynamics if you re-insert the barrier. This is the Gibbs paradox.

The paradox is resolved by postulating that the gas particles are in fact indistinguishable. This means that all states that differ only by a permutation of particles should be considered as the same state. For example, if we have a 2-particle gas and we specify AB as a state of the gas where the first particle (A) has momentum p1 and the second particle (B) has momentum p2, then this state as well as the BA state where the B particle has momentum p1 and the A particle has momentum p2 should be counted as the same state.

For an N-particle gas, there are N! states which are identical in this sense, if one assumes that each particle is in a different single particle state. One can safely make this assumption provided the gas isn't at an extremely high density. Under normal conditions, one can thus calculate the volume of phase space occupied by the gas, by dividing Equation 1 by N!. Using the Stirling approximation again for large N, ln(N!) ≈ N ln(N) − N, the entropy for large N is:

S = k N \ln \left[ \left(\frac VN\right) \left(\frac UN \right)^{\frac 32}\right]+kN\left( {\frac 52}+ {\frac 32}\ln\frac{4\pi m}{3h^2}\right) $$ which can be easily shown to be extensive. This is the Sackur–Tetrode equation.

Mixing paradox
A closely related paradox to the Gibbs paradox is the mixing paradox. The Gibbs paradox is a special case of the "mixing paradox" which contains all the salient features. The difference is that the mixing paradox deals with arbitrary distinctions in the two gases, not just distinctions in particle ordering as Gibbs had considered. In this sense, it is a straightforward generalization to the argument laid out by Gibbs. Again take a box with a partition in it, with gas A on one side, gas B on the other side, and both gases are at the same temperature and pressure. If gas A and B are different gases, there is an entropy that arises once the gases are mixed, the entropy of mixing. If the gases are the same, no additional entropy is calculated. The additional entropy from mixing does not depend on the character of the gases; it only depends on the fact that the gases are different. The two gases may be arbitrarily similar, but the entropy from mixing does not disappear unless they are the same gas – a paradoxical discontinuity.

This "paradox" can be explained by carefully considering the definition of entropy. In particular, as concisely explained by Edwin Thompson Jaynes, definitions of entropy are arbitrary.

As a central example in Jaynes' paper points out, one can develop a theory that treats two gases as similar even if those gases may in reality be distinguished through sufficiently detailed measurement. As long as we do not perform these detailed measurements, the theory will have no internal inconsistencies. (In other words, it does not matter that we call gases A and B by the same name if we have not yet discovered that they are distinct.) If our theory calls gases A and B the same, then entropy does not change when we mix them. If our theory calls gases A and B different, then entropy does increase when they are mixed. This insight suggests that the ideas of "thermodynamic state" and of "entropy" are somewhat subjective.

The differential increase in entropy (dS) as a result of mixing dissimilar gases, multiplied by the temperature (T), equals the minimum amount of work we must do to restore the gases to their original separated state. Suppose that two gases are different, but that we are unable to detect their differences. If these gases are in a box, segregated from one another by a partition, how much work does it take to restore the system's original state after we remove the partition and let the gases mix?

None – simply reinsert the partition. Even though the gases have mixed, there was never a detectable change of state in the system, because by hypothesis the gases are experimentally indistinguishable.

As soon as we can distinguish the difference between gases, the work necessary to recover the pre-mixing macroscopic configuration from the post-mixing state becomes nonzero. This amount of work does not depend on how different the gases are, but only on whether they are distinguishable.

This line of reasoning is particularly informative when considering the concepts of indistinguishable particles and correct Boltzmann counting. Boltzmann's original expression for the number of states available to a gas assumed that a state could be expressed in terms of a number of energy "sublevels" each of which contain a particular number of particles. While the particles in a given sublevel were considered indistinguishable from each other, particles in different sublevels were considered distinguishable from particles in any other sublevel. This amounts to saying that the exchange of two particles in two different sublevels will result in a detectably different "exchange macrostate" of the gas. For example, if we consider a simple gas with N particles, at sufficiently low density that it is practically certain that each sublevel contains either one particle or none (i.e. a Maxwell–Boltzmann gas), this means that a simple container of gas will be in one of N! detectably different "exchange macrostates", one for each possible particle exchange.

Just as the mixing paradox begins with two detectably different containers, and the extra entropy that results upon mixing is proportional to the average amount of work needed to restore that initial state after mixing, so the extra entropy in Boltzmann's original derivation is proportional to the average amount of work required to restore the simple gas from some "exchange macrostate" to its original "exchange macrostate". If we assume that there is in fact no experimentally detectable difference in these "exchange macrostates" available, then using the entropy which results from assuming the particles are indistinguishable will yield a consistent theory. This is "correct Boltzmann counting".

It is often said that the resolution to the Gibbs paradox derives from the fact that, according to the quantum theory, like particles are indistinguishable in principle. By Jaynes' reasoning, if the particles are experimentally indistinguishable for whatever reason, Gibbs paradox is resolved, and quantum mechanics only provides an assurance that in the quantum realm, this indistinguishability will be true as a matter of principle, rather than being due to an insufficiently refined experimental capability.

Non-extensive entropy of two ideal gases and how to fix it
In this section, we present in rough outline a purely classical derivation of the non-extensive entropy for an ideal gas considered by Gibbs before "correct counting" (indistinguishability of particles) is accounted for. This is followed by a brief discussion of two standard methods for making the entropy extensive. Finally, we present a third method, due to R. Swendsen, for an extensive (additive) result for the entropy of two systems if they are allowed to exchange particles with each other.

Setup
We will present a simplified version of the calculation. It differs from the full calculation in three ways:
 * 1) The ideal gas consists of particles confined to one spatial dimension.
 * 2) We keep only the terms of order $$n \log(n)$$, dropping all terms of size n or less, where n is the number of particles. For our purposes, this is enough, because this is where the Gibbs paradox shows up and where it must be resolved. The neglected terms play a role when the number of particles is not very large, such as in computer simulation and nanotechnology. Also, they are needed in deriving the Sackur–Tetrode equation.
 * 3) The subdivision of phase space into units of the Planck constant (h) is omitted. Instead, the entropy is defined using an integral over the "accessible" portion of phase space. This serves to highlight the purely classical nature of the calculation.

We begin with a version of Boltzmann's entropy in which the integrand is all of accessible phase space:
 * $$S = k \ln\Omega = k \ln \oint\limits_{H(\vec p, \vec q)=E} (d\vec p\,)^n (d\vec q\,)^n$$

The integral is restricted to a contour of available regions of phase space, subject to conservation of energy. In contrast to the one-dimensional line integrals encountered in elementary physics, the contour of constant energy possesses a vast number of dimensions. The justification for integrating over phase space using the canonical measure involves the assumption of equal probability. The assumption can be made by invoking the ergodic hypothesis as well as the Liouville's theorem of Hamiltonian systems.

(The ergodic hypothesis underlies the ability of a physical system to reach thermal equilibrium, but this may not always hold for computer simulations (see the Fermi–Pasta–Ulam–Tsingou problem) or in certain real-world systems such as non-thermal plasmas.)

Liouville's theorem assumes a fixed number of dimensions that the system 'explores'. In calculations of entropy, the number dimensions is proportional to the number of particles in the system, which forces phase space to abruptly change dimensionality when particles are added or subtracted. This may explain the difficulties in constructing a clear and simple derivation for the dependence of entropy on the number of particles.

For the ideal gas, the accessible phase space is an (n − 1)-sphere (also called a hypersphere) in the n-dimensional $$\vec v$$ space:
 * $$E=\sum_{j=1}^n \frac 1 2 mv_j^2\,,$$

To recover the paradoxical result that entropy is not extensive, we integrate over phase space for a gas of $$n$$ monatomic particles confined to a single spatial dimension by $$0<x<\ell$$. Since our only purpose is to illuminate a paradox, we simplify notation by taking the particle's mass and the Boltzmann constant equal to unity: $$ m = k = 1$$. We represent points in phase-space and its x and v parts by n and 2n dimensional vectors:
 * $$\vec\xi = [x_1,..., x_n, v_1,...,v_n]=[\vec x, \vec v\,]$$  where    $$\vec x = [x_1,..., x_n]$$   and   $$\vec v = [v_1,...,v_n]\,.$$

To calculate entropy, we use the fact that the (n-1)-sphere, $$\Sigma v_j^2=R^2\, ,$$ has an (n − 1)-dimensional "hypersurface volume" of
 * $$\tilde A_n(R)=\frac{n\pi^{n/2}}{(n/2)!} R^{n-1}\,.$$

For example, if n = 2, the 1-sphere is the circle $$\tilde A_2(R)=2\pi R$$, a "hypersurface" in the plane. When the sphere is even-dimensional (n odd), it will be necessary to use the gamma function to give meaning to the factorial; see below.

Gibbs paradox in a one-dimensional gas
Gibbs paradox arises when entropy is calculated using an $$n$$ dimensional phase space, where $$n$$ is also the number of particles in the gas. These particles are spatially confined to the one-dimensional interval $$\ell^n$$. The volume of the surface of fixed energy is
 * $$\Omega_{E,\ell} = \left( \int dx_1 \int dx_2 ...\int dx_n\right)\left(\underbrace{\int dv_1 \int dv_2 ...\int dv_n}_{\Sigma v_i^2 = 2E}\right)$$

The subscripts on $$\Omega$$ are used to define the 'state variables' and will be discussed later, when it is argued that the number of particles, $$n$$ lacks full status as a state variable in this calculation. The integral over configuration space is $$\ell^n$$. As indicated by the underbrace, the integral over velocity space is restricted to the "surface area" of the n − 1 dimensional hypersphere of radius $$\sqrt{2E}$$, and is therefore equal to the "area" of that hypersurface. Thus
 * $$\Omega_{E,\ell} = \ell^n \frac{n\pi^{n/2}}{(n/2)!} (2E)^{\frac{n-1}{2}}$$

 * {| class="toccolours collapsible collapsed" width="50%" style="text-align:left"x

! Click to view the algebraic steps
 * We begin with:
 * $$\Omega_{E,\ell} = \ell^n \frac{n\pi^{n/2}}{(n/2)!} (2E)^{\frac{n-1}{2}}$$
 * $$\Omega_{E,\ell} = \ell^n \frac{n\pi^{n/2}}{(n/2)!} (2E)^{\frac{n-1}{2}}$$

\ln\Omega_{E,\ell} = \ln\left(\ell^n n\pi^{n/2}   (2E)^{\frac{n-1}{2}}  \right) - \ln\left[  (n/2)! \right] $$ Both terms on the right hand side have dominant terms. Using the Stirling approximation for large M, $$\ln M! \approx M\ln M$$ $$-M+\ln\sqrt{2\pi M} $$, we have:

\ln\left(\ell^n n\pi^{n/2}   (2E)^{\frac{n-1}{2}}  \right) = \underbrace{\ln\left(\ell^n E^\frac{n}{2}\right)}_{\text{important}}  + \underbrace{ \ln\left( \frac{n (2\pi)^\frac{n}{2} }{\sqrt {2E}} \right)}_{\text{drop}} $$

\begin{align} - \ln [(n/2)! ] &\approx  -\frac n 2 \ln \frac n 2 + \frac n 2 -\ln \sqrt{n\pi}\\ &= \underbrace{-\frac n 2 \ln n}_{\text{keep}} + \underbrace{\frac n 2 \ln 2 +\frac n 2 -\ln \sqrt{n\pi}}_{\text{drop}} \\ \end{align} $$

Terms are neglected if they exhibit less variation with a parameter, and we compare terms that vary with the same parameter. Entropy is defined with an additive arbitrary constant because the area in phase space depends on what units are used. For that reason it does not matter if entropy is large or small for a given value of E. We instead to seek how entropy varies with E, i.e., we seek $$\partial S/\partial E$$:
 * An expression such as $$E^\frac 1 2 $$ is much less important than an expression like$$E^\frac n 2 $$
 * An expression like $$\pi^n $$ is much less important than an expression like $$n^n $$. Note that the logarithm is not a strongly increasing function.  The neglect of terms proportional to n compared with terms proportional to n ln n is only justified if n is extremely large.

Combining the important terms:
 * $$\begin{align}

\ln\Omega_{E,\ell} &= \ln\left(\ell^n E^\frac{n}{2}\right)  -\frac n 2 \ln n\\ &= n\ln\ell + \frac n 2 \ln\left(\frac E n\right)\\ \end{align}$$

 * }

After approximating the factorial and dropping the small terms, we obtain
 * $$\begin{align}

\ln\Omega_{E,\ell} &\approx n\ln\ell + n \ln\sqrt{\frac E n} + \text{const.}\\ &= \underbrace{n\ln\frac{\ell}{n} + n \ln\sqrt{\frac E n}}_{\text{extensive}} + \,n\ln n +\text{const.}\\ \end{align}$$

In the second expression, the term $$n\ln n$$ was subtracted and added, using the fact that $$\ln\ell-\ln n = \ln (\ell /n)$$. This was done to highlight exactly how the "entropy" defined here fails to be an extensive property of matter. The first two terms are extensive: if the volume of the system doubles, but gets filled with the same density of particles with the same energy, then each of these terms doubles. But the third term is neither extensive nor intensive and is therefore wrong.

The arbitrary constant has been added because entropy can usually be viewed as being defined up to an arbitrary additive constant. This is especially necessary when entropy is defined as the logarithm of a phase space volume measured in units of momentum-position. Any change in how these units are defined will add or subtract a constant from the value of the entropy.

Two standard ways to make the classical entropy extensive
As discussed above, an extensive form of entropy is recovered if we divide the volume of phase space, $$\Omega_{E,\ell}$$, by n!. An alternative approach is to argue that the dependence on particle number cannot be trusted on the grounds that changing $$n$$ also changes the dimensionality of phase space. Such changes in dimensionality lie outside the scope of Hamiltonian mechanics and Liouville's theorem. For that reason it is plausible to allow the arbitrary constant to be a function of $$n$$. Defining the function to be, $$f(n)=-\frac32 n\ln n$$, we have:
 * $$\begin{align}

S = \ln\Omega_{E,\ell} &\approx n\ln\ell + n \ln\sqrt{E} + \text{const.}\\ &= n\ln\ell + n \ln\sqrt{E} + f(n)\\ \ln\Omega_{E,\ell,n}&\approx n\ln\frac{\ell}{n} + n \ln\sqrt{\frac E n} +\text{const.},\\ \end{align}$$ which has extensive scaling:
 * $$ S(\alpha E,\alpha\ell,\alpha n) = \alpha\, S(E,\ell,n)$$

Swendsen's particle-exchange approach
Following Swendsen, we allow two systems to exchange particles. This essentially 'makes room' in phase space for particles to enter or leave without requiring a change in the number of dimensions of phase space. The total number of particles is $$N$$:
 * $$n_A$$ particles have coordinates $$0<x<\ell_A$$.
 * The total energy of these particles is $$E_A$$
 * $$n_B$$ particles have coordinates $$0<x<\ell_B$$.
 * The total energy of these particles is $$E_B$$
 * The system is subject to the constraints, $$E_A+E_B=E$$ and $$n_A+n_B=N$$

Taking the integral over phase space, we have:
 * $$\begin{align}

\Omega_{E,\ell,N} &= \left( \underbrace{\int dx_? ... \int dx_?}_{n_A\;\text{terms}} \underbrace{\int dx_?...\int dx_N}_{n_B\;\text{terms}}\right)\left(\underbrace{\int dv_1 \int dv_2 ...\int dv_N}_{\Sigma v^2 = 2E_A\;\text{or}\;\Sigma v^2 = 2E_B}\right) \\ &=\left(\ell_A\right)^{n_A}\left(\ell_B\right)^{n_B} \left(\underbrace{\frac{N!}{n_A!n_B!}}_{\text{combination}}\right) \left(\underbrace{\frac{n_A\pi^{n_A/2}}{(n_A/2)!} (2E_A)^{\frac{n_A-1}{2}}}_{n_A-\text{sphere}}\right) \left(\underbrace{\frac{n_B\pi^{n_B/2}}{(n_B/2)!} (2E_B)^{\frac{n_B-1}{2}}}_{n_B-\text{sphere}}\right) \end{align}$$

The question marks (?) serve as a reminder that we may not assume that the first nA particles (i.e. 1 through nA) are in system A while the other particles (nB through N) are in system B. (This is further discussed in the next section.)

Taking the logarithm and keeping only the largest terms, we have: $$S =\ln \Omega_{E,\ell,N} \approx n_A\ln\left(\frac{n_A}{\ell_A}\sqrt{\frac{E_A}{\ell_A}}\right)+ n_B\ln\left(\frac{n_B}{\ell_B}\sqrt{\frac{E_B}{\ell_B}}\right)+ N\ln N + \text{const.}$$

This can be interpreted as the sum of the entropy of system A and system B, both extensive. And there is a term, $$N\ln N$$, that is not extensive.

Visualizing the particle-exchange approach in three dimensions
The correct (extensive) formulas for systems A and B were obtained because we included all the possible ways that the two systems could exchange particles. The use of combinations (i.e. N particles choose NA) was used to ascertain the number of ways N particles can be divided into system A containing nA particles and system B containing nB particles. This counting is not justified on physical grounds, but on the need to integrate over phase space. As will be illustrated below, phase space contains not a single nA-sphere and a single nB-sphere, but instead
 * $$ {N \choose n_A} = \frac{N!}{n_A!n_B!}$$

pairs of n-spheres, all situated in the same (N + 1)-dimensional velocity space. The integral over accessible phase space must include all of these n-spheres, as can be seen in the figure, which shows the actual velocity phase space associated with a gas that consists of three particles. Moreover, this gas has been divided into two systems, A and B.

If we ignore the spatial variables, the phase space of a gas with three particles is three dimensional, which permits one to sketch the n-spheres over which the integral over phase space must be taken. If all three particles are together, the split between the two gases is 3|0. Accessible phase space is delimited by an ordinary sphere (2-sphere) with a radius that is either $$\sqrt{2E_1}$$ or $$\sqrt{2E_2}$$ (depending which system has the particles).

If the split is 2|1, then phase space consists of circles and points. Each circle occupies two dimensions, and for each circle, two points lie on the third axis, equidistant from the center of the circle. In other words, if system A has 2 particles, accessible phase space consists of 3 pairs of n-spheres, each pair being a 1-sphere and a 0-sphere:
 * $$ v_1^2+v_2^2=2E_A \qquad v_3^2 = 2E_B$$
 * $$ v_2^2+v_3^2=2E_A \qquad v_1^2 = 2E_B$$
 * $$ v_3^2+v_1^2=2E_A \qquad v_2^2 = 2E_B$$

Note that
 * $$ {3 \choose 2} = 3.$$

title: Gibbs paradox


text:
Bill Bradley

William Warren Bradley (born July 28, 1943) is an American politician and former professional basketball player. He served three terms as a Democratic U.S. senator from New Jersey (1979–1997). He ran for the Democratic Party's nomination for president in the 2000 election, which he lost to Vice President Al Gore.

Bradley was born and raised in Crystal City, Missouri, a small town 45 mi south of St. Louis. He excelled at basketball from an early age. He did well academically and was an all-county and all-state basketball player in high school. He was offered 75 college scholarships, but declined them all to attend Princeton University. He won a gold medal as a member of the 1964 Olympic basketball team and was the Most Outstanding Player of the 1965 NCAA Tournament, when Princeton finished third. After graduating in 1965, he attended Oxford on a Rhodes Scholarship where he was a member of Worcester College, delaying a decision for two years on whether or not to play in the National Basketball Association (NBA).

While at Oxford, Bradley played one season of professional basketball in Europe and eventually decided to join the New York Knicks in the 1967–68 season, after serving six months in the Air Force Reserve. He spent his entire ten-year professional basketball career playing for the Knicks, winning NBA titles in 1970 and 1973. Retiring in 1977, he ran for a seat in the United States Senate the following year, from his adopted home state of New Jersey. He was re-elected in 1984 and 1990, left the Senate in 1997, and was an unsuccessful candidate for the 2000 Democratic presidential nomination.

Bradley is the author of seven non-fiction books, most recently We Can All Do Better, and hosts a weekly radio show, American Voices, on Sirius Satellite Radio. He is a corporate director of Starbucks and a partner at investment bank Allen & Company in New York City. Bradley is a member of the ReFormers Caucus of Issue One. He also serves on that group's advisory board.

Bradley is a member of both the American Academy of Arts and Sciences and the American Philosophical Society. In 2008 Bradley was inducted into the New Jersey Hall of Fame.

Early life
Bradley was born on July 28, 1943, in Crystal City, Missouri, the only child of Warren (June 22, 1901 – October 1, 1994), who despite leaving high school after a year had become a bank president, and Susan "Susie" Crowe (June 12, 1909 – November 30, 1995), a teacher and former high school basketball player. Politicians and politics were standard dinner-table topics in Bradley's childhood, and he described his father as a "solid Republican" who was an elector for Thomas E. Dewey in the 1948 presidential election. An active Boy Scout, he became an Eagle Scout and member of the Order of the Arrow.

Bradley began playing basketball at the age of nine. He was a star at Crystal City High School, where he scored 3,068 points in his scholastic career, was twice named All-American, and was elected to the Missouri Association of Student Councils. He received 75 college scholarship offers, although he applied to only five schools and only scored a 485 out of 800 on the Verbal portion of the SAT, which—despite being likely in the top third of all test takers that year—normally would have caused selective schools like Princeton University to reject him.

Bradley's basketball ability benefited from his height—5 ft in the seventh grade, 6 ft in the eighth grade, and his adult size of 6 ft by the age of 15—and unusually wide peripheral vision, which he worked to improve by focusing on faraway objects while walking. During his high school years, Bradley maintained a rigorous practice schedule, a habit he carried through college. He would work on the court for "three and a half hours every day after school, nine to five on Saturday, one-thirty to five on Sunday, and, in the summer, about three hours a day. He put ten pounds of lead slivers in his sneakers, set up chairs as opponents and dribbled in a slalom fashion around them, and wore eyeglass frames that had a piece of cardboard taped to them so that he could not see the floor, for "a good dribbler never looks at the ball."

College career
Bradley was considered to be the top high school basketball player in the country. He initially chose to attend Duke in the fall of 1961. However, after breaking his foot in the summer of 1961 during a baseball game and thinking about his college decision outside of basketball, Bradley decided to enroll at Princeton due to its record in preparing students for government or United States Foreign Service work. He had been awarded a scholarship at Duke, but not at Princeton; the Ivy League does not allow its members to award athletic scholarships, and he was disqualified from receiving financial aid because of his family's wealth.

Bradley wore #42 in honor of childhood hero Dick Kazmaier, who had won the Heisman Trophy at Princeton. He was so superior to the rest of the freshman team that coach Eddie Donovan chose lineups by saying "You, you, you, you, and Bradley". Bradley averaged more than 30 points per game for the freshman team, at one point making 57 consecutive free throws, breaking a record set by a member of the NBA's Syracuse Nationals. The following year, as a sophomore, he was a varsity starter in Butch van Breda Kolff's first year as coach of the Tigers.

In his sophomore year Bradley scored 40 points in an 82–81 loss to St. Joseph's and was named to The Sporting News All-American first team in early 1963. The coach of the St. Louis Hawks believed he was ready to play professional basketball. The AP and United Press International polls both put Bradley on the second team, establishing him as the top sophomore player in the country; Bradley also hit .316 as a first baseman for the baseball team. The following year The Sporting News again named him to its All-American team as its only junior, and as its player of the year. At the Olympic basketball trials in April 1964, Bradley played guard instead of his usual forward position but was still a top performer. He was one of three chosen unanimously for the Olympic team, the youngest chosen, and the only undergraduate. The Olympic team won its sixth consecutive gold medal.

As a senior and team captain in the 1964–1965 season, Bradley became a household name. Only the third tallest on his team, but called "easily the No. 1 player in college basketball today", "the best amateur basketball player in the United States", and "The White Oscar Robertson", he scored 41 points before fouling out of the game in an 80–78 loss to Michigan and their star player Cazzie Russell in the 1964 ECAC Holiday Basketball semi-final at Madison Square Garden, then led Princeton to the NCAA Final Four after defeating heavy favorite Providence and Jimmy Walker by 40 points. The team then lost to Michigan in the semifinals, but Bradley scored a record 58 points in the consolation game to lead the team to victory against Wichita State and earn himself the Final Four MVP. In total, Bradley scored 2,503 points at Princeton, averaging 30.2 points per game. He was awarded the 1965 James E. Sullivan Award, presented annually to the United States' top amateur athlete, the first basketball player to win the honor, and the second Princeton student to win the award, after runner Bill Bonthron in 1934.

Bradley holds a number of Ivy League career records, including total and average points (1,253/29.83, respectively), and free throws made and attempted (409/468, 87.4%). Ivy League season records he holds similarly include total and average points (464/33.14, 1964) and most free throws made (153 in 170 attempts, 90.0%, 1962–1963). Bradley also holds the career point record at Princeton and many other school records, including the top ten slots in the category of total points scored in a game, but likely could have scored many more points if he had not insisted so often on passing the ball, in what his coaches called "Bradley's hope passes", to inferior teammates closer to the basket; he only emphasized his own scoring when Princeton was behind or, as during the Wichita State game, his teammates forced Bradley to shoot by returning passes to him. Van Breda Kolff often encouraged Bradley to be more of a "one on one" player, stating that "Bill is not hungry. At least ninety percent of the time, when he gets the ball, he is looking for a pass."

The coach described Bradley as "not the most physical player. Others can run faster and jump higher. The difference ... is self-discipline." Afraid that he was not qualified for Princeton, Bradley recalled that after almost failing freshman French and biology, he "just lived in the library". Bradley had three to four hours of classes and four hours of basketball practice daily, studied an average of seven hours each weekday, and up to 24 more hours each weekend, frequently spoke for the Fellowship of Christian Athletes around the country, and taught Sunday school at the local Presbyterian Church. When practicing he did not move from a location on the court unless he made at least ten of 13 shots, and could detect whether a basket was an inch too low from the regulation ten feet. Bradley took losses personally, outraged when other freshman players laughed and joked after a loss. His only criticism of childhood hero Wilt Chamberlain was that Chamberlain lacked a killer instinct.

Others noted that Bradley seemed to lack enemies despite great athletic, academic, and social success. Classmate Larry Lucchino described Bradley as having an "aura ... of near-idolatry". All 15 Princeton University eating clubs asked him to join; Bradley chose Cottage Club. Fans shouted "Don't touch God!" when opposing players' bodies hit his on court. Roommates helped answer dozens of letters each week asking for autographs, mementos, and public appearances. Each year improving from mediocre freshman grades, Bradley graduated magna cum laude after writing his senior thesis about Harry S. Truman's 1940 United States Senate campaign, titled "On That Record I Stand", and received a Rhodes Scholarship at Worcester College, Oxford. At Princeton, Bradley was taught by John William Ward. His years at Princeton was the subject of Pulitzer Prize-winning author John McPhee's January 23, 1965, article "A Sense of Where You Are" in The New Yorker, which McPhee expanded into a book of the same name. The title came from Bradley's explanation for his ability to repeatedly throw a basketball over his shoulder and into the basket while looking away from it. In 1965, Bradley received the Golden Plate Award of the American Academy of Achievement.

Professional career

Bradley's graduation year, 1965, was the last year that the NBA's territorial rule was in effect, which gave professional teams first rights to draft players who attended college within 50 miles of the team. The New York Knicks—one mile closer to Princeton than the Philadelphia 76ers—drafted Bradley as a territorial pick in the 1965 draft, but he did not sign a contract with the team immediately. While studying Politics, Philosophy, and Economics (PPE) at Oxford, he commuted to Italy to play professional basketball for Olimpia Milano, then called Simmenthal, during the 1965–66 season, where the team won a European Champions Cup (predecessor to the modern EuroLeague). Bradley was also a member of the Oxford University Basketball Club and helped lead the men's basketball team to back-to-back British University Sports Federation (B.U.S.F.) championships in 1965 and 1966 and the Amateur Basketball Association (A.B.B.A.) National Championship in 1966.

Bradley dropped out of Oxford in April 1967, two months before graduation, to enter the Air Force Reserves. (The following year, Oxford allowed Bradley to take "special exams", enabling him to graduate.) He served six months on active duty as an officer, though the requirement was four years' service. (On March 6, 1967, President Lyndon B. Johnson declared that he would issue an executive order that Selective Service deferments for post-graduate study would henceforth be limited to the medical and dental fields. )

Bradley joined the New York Knicks in December 1967, having missed the preseason and several weeks of the 1967–1968 season. He was placed in the backcourt, although he had spent his high school and college careers as a forward. Neither he nor the team did well, and in the following season, he was returned to the forward slot. Then, in his third season, the Knicks won their first NBA championship, followed by the second in the 1972–73 season, when he made the only All-Star Game appearance of his career. Over 742 NBA games – all with the Knicks – Bradley scored a total of 9,217 points, an average of 12.4 points per game, and averaged 3.4 assists per game. His best season scoring average was 16.1 points per game in the 1972–73 season, during which he also averaged a career-best 4.5 assists per game. As in college Bradley was an aggressive player, pushing and shoving to intimidate and distract opponents.

During his NBA career, Bradley used his fame on the court to explore social as well as political issues, meeting with journalists, government officials, academics, businesspeople, and social activists. He also worked as an assistant to the director of the Office of Economic Opportunity in Washington, D.C., and as a teacher in the street academies of Harlem. In 1976, he also became an author by publishing Life on the Run. Using a 20-day stretch of time during one season as the main focus of the book, he chronicled his experiences in the NBA and the people he met along the way. Bradley wrote that he was uncomfortable using his celebrity status to earn extra money endorsing products as other players did.

Retiring from basketball in 1977, he was elected to the Naismith Memorial Basketball Hall of Fame in 1983, along with teammate Dave DeBusschere. In 1984, the Knicks retired his number 24 jersey; he was the fourth player so honored by the Knicks, after Willis Reed, Walt Frazier, and DeBusschere. He is one of only two players, along with Manu Ginóbili, to have won a EuroLeague title, an NBA championship, and an Olympic gold medal.

Politics
Politics was a frequent subject of discussion in the Bradley household, and some of his relatives held local and county political offices. He majored in history at Princeton and was present in the Senate chamber when the Civil Rights Act of 1964 was passed. Van Breda Kolff and many others who knew him predicted that Bradley would be Governor of Missouri, or president, by 40. His Rhodes application stated "I can best serve mankind as a politician". Bradley spent his time at Oxford focusing on European political and economic history.

During his third year with the Knicks, Bradley told Robert Lipsyte that he regretted only focusing on school and basketball at Princeton; "perhaps considered a smart athlete" by society, "or an athlete with character, but still a particular kind of object instead of a particular human being". In Life on the Run, Bradley wrote that he had intended to only play in the NBA for four years before signing a second contract for four more. The New York Times's review of the book stated that "it does not seem ... that there was much in the way of intellectual contact" with teammates, and speculated that after basketball "Perhaps he will turn to politics at last". In 1978 Bradley said that congressman Mo Udall, himself a former professional basketball player, had told him ten years earlier that professional sports could help prepare him for politics, depending on what he did with his non-playing time. A year after the Lipsyte conversation, Bradley gave a speech to 113 top Missouri scholar-athletes. Instead of just congratulations as they expected, the NBA starter quoted Bob Dylan and Joni Mitchell in giving advice he wished he had heard in high school:

"Thousands of people who do not know me use my participation on a Sunday afternoon as an excuse for non-action, as a fix to help them escape their own everyday problems, and society's problems. The toll of providing that experience is beginning to register on me"

"Are you being subtly programmed into being a certain kind of person with a narrow range of traditional career alternatives?" Bradley asked the audience. "If so, rebel".

U.S. Senate
After four years of political campaigning for Democratic candidates around New Jersey, Bradley decided in the summer of 1977 to retire from the Knicks and run in the 1978 United States Senate election in New Jersey. He felt his time had been well-spent in "paying his dues". The seat was held by liberal Republican and four-term incumbent Clifford P. Case. Case lost the Republican primary to anti-tax conservative Jeffrey Bell, who, like Bradley, was 34 years old as the campaign season began. Bradley won the election with about 56% of the vote. During the campaign, Yale football player John Spagnola was Bradley's bodyguard and driver.

In the Senate, Bradley acquired a reputation for being somewhat aloof and was thought of as a "policy wonk", specializing in complex reform initiatives. Among these was the 1986 overhaul of the federal tax code, co-sponsored with Dick Gephardt, which reduced the tax rate schedule to just two brackets, 15 percent and 28 percent, and eliminated many kinds of deductions. Domestic policy initiatives that Bradley led or was associated with included reform of child support enforcement; legislation concerning lead-related children's health problems; the Earned Income Tax Credit; campaign finance reform; a re-apportioning of California water rights; and federal budget reform to reduce the deficit, which included, in 1981, supporting Reagan's spending cuts but opposing his parallel tax cut package, one of only three senators to take this position. He sponsored the Freedom Support Act, an exchange program between the republics of the former Soviet Union and the United States.

Bradley was re-elected in 1984 with 65% of the vote against Montclair mayor Mary V. Mochary.

In 1987, Bradley re-introduced legislation that would return 1.3 million acres of land in the Black Hills of South Dakota to the Sioux tribe that had been illegally seized by President Ulysses S. Grant under the threat of starvation of the tribe in 1877. The legislation proposed to keep Mount Rushmore within the US Park Service and 1.3 million acres of the Black Hills to return to jurisdiction under a Sioux National Council. The legislation died in committee.

In 1988, he was encouraged to seek the Democratic nomination for president, but he declined to enter the race, saying that he would know when he was ready. In 1990, a controversy over a state income tax increase—on which he refused to take a position—and his proposal on merit pay for teachers, which led the NJEA to support his opponent, turned his once-obscure rival for the Senate, Christine Todd Whitman, into a viable candidate, and Bradley won by only a slim margin. In 1995, he announced he would not run for re-election, publicly declaring American politics "broken".

While he was a senator, Bradley walked the beaches from Cape May to Sandy Hook, a four-day, 127-mile trip each Labor Day weekend, to assess beach and ocean conditions and talk with constituents. Bradley was criticized for neglecting constituent services while in office.

Presidential candidate
Bradley ran in the 2000 presidential primaries, opposing incumbent Vice President Al Gore for his party's nomination. Bradley campaigned as the liberal alternative to Gore, taking positions to the left of Gore on a number of issues, including universal health care, gun control, and campaign finance reform. On the issue of taxes, Bradley trumpeted his sponsorship of the Tax Reform Act of 1986, which had significantly cut tax rates while abolishing dozens of loopholes. He voiced his belief that the best possible tax code would be one with low rates and no loopholes, but he refused to rule out the idea of raising taxes to pay for his health care program, calling the idea of such a pledge "dishonest".

On public education, he proposed to make over $2 billion in block grants available to each state every year. He further promised to bring 60,000 new teachers into the education system in hard-to-staff areas over ten years by offering college scholarships to anyone who agreed to become a teacher after graduating; Gore offered a similar proposal.

Bradley also made child poverty a significant issue in his campaign. He promised to address the minimum wage, expand the Earned Income Tax Credit, allow single parents on welfare to keep their child support payments, make the Dependent Care Tax Credit refundable, build support homes for pregnant teenagers, enroll 400,000 more children in Head Start, and increase the availability of food stamps.

Although Gore was considered the party favorite, Bradley received a number of high-profile endorsements, including senators Paul Wellstone, Bob Kerrey, and Daniel Patrick Moynihan; former Secretary of Labor Robert Reich; former New York City mayor Ed Koch; former Federal Reserve chairman Paul Volcker; and basketball stars Michael Jordan and Phil Jackson. Bradley and Jackson have been close friends since they were teammates playing for the New York Knicks. Jackson was a vocal supporter of Bradley's run for the presidency and often wore his campaign button in public. Jackson announced his acceptance of the position of head coach of the Los Angeles Lakers while Bradley was campaigning in California in 1999, and he was a "regular draw on the Bradley money trail" during the campaign. Bradley later called it a "great honor" to be the presenter when Jackson was inducted into the Naismith Memorial Basketball Hall of Fame in 2007.

Bradley's campaign initially had strong prospects, due to high-profile endorsements and as his fundraising efforts gave him a deep war chest. However, it floundered, in part because it was overshadowed by Senator John McCain's far more attention-gaining, but ultimately unsuccessful, campaign for the Republican nomination; McCain had stolen Bradley's "thunder" on several occasions. Bradley was much embarrassed by his two to one defeat in the Iowa caucus, despite spending heavily there, as the unions pledged their support for Gore. Bradley then lost the New Hampshire primary 53–47%, which had been viewed as a must-win state for his campaign to remain competitive. Bradley finished a distant second during each of the primaries on Super Tuesday.

On March 9, 2000, after failing to win any of the first 20 primaries and caucuses in the election process, Bradley withdrew his campaign and endorsed Gore; he ruled out the idea of running as the vice-presidential candidate and did not answer questions about possible future runs for the presidency. He said that he would continue to speak out regarding his brand of politics, calling for campaign finance reform, gun control, and increased health care insurance.

After politics

In 1999 Bradley was awarded the Honorary degree of Doctor of Laws (LL.D) by the University of North Carolina at Chapel Hill. Later in 2000, Bradley was offered the chairmanship of the United States Olympic Committee, which he turned down. In September 2002, Bradley turned down a request from New Jersey Democrats to replace Robert Torricelli on the ballot for his old Senate seat, which another former senator, Frank Lautenberg, accepted. Oxford University awarded Bradley an honorary Doctor of Civil Law (DCL) in 2003, with a citation that described him in part as "an outstandingly distinguished athlete, a weighty pillar of the Senate, and still a powerful advocate of the weak". In 2007 Bradley was awarded the Distinguished Eagle Scout Award. This award is given in recognition of community service more than 25 years after a scout first earns the Eagle badge.

In January 2004, Bradley and Gore both endorsed Howard Dean for president in the 2004 Democratic primaries. In January 2008, Bradley announced that he was supporting Barack Obama in the 2008 Democratic primary. He campaigned for Obama and appeared on political news shows as a surrogate. Bradley's name was mentioned as a possible replacement for Tom Daschle as nominee for Secretary of Health and Human Services in the Obama administration after Daschle withdrew from consideration; the position went to Kansas governor Kathleen Sebelius. He has occasionally been involved in political matters, most recently consulting the Senate Finance Committee on tax reform along with former colleague Bob Packwood.

He has worked as a corporate consultant and investment banker. He has been a managing director of Allen & Company LLC, since 2001, was chief outside advisor to McKinsey & Company's nonprofit division, the McKinsey Global Institute, from 2001 to 2004, and is a member of the board of directors of QuinStreet and the private company Raydiance. Bradley is a senior advisor to the private equity firm Catterton Partners. Bradley is also a board member of DonorsChoose.org, an online charity that connects individuals to classrooms in need. He is also the Chair of the Advisory Council for Acumen Fund, a non-profit global venture fund that uses entrepreneurial approaches to solve the problems of global poverty. Bradley is a co-chair for the advisory board of Issue One, a non-profit whose goal is to reduce the influence of money in American politics.

Bradley is a member of the board of directors of the American Committee on East-West Accord. And he has been member of the advisory board of the Peter G. Peterson Foundation.

Bradley created an autobiographical one-man show, Rolling Along, which was filmed before a live audience in a New York theater in 2022. The film debuted at the 2023 Tribeca Film Festival and launched on streaming service Max in February 2024.

Personal life
As a young man Bradley avoided women who wanted to date a celebrity. He wrote in Life on the Run that being famous had taught him what beautiful women experienced, "the unnaturalness of being a sex object". Future TV journalist Diane Sawyer was a serious girlfriend in college.

Bradley married Ernestine (née Misslbeck) Schlant, a German-born professor of comparative literature, in 1974. She has a daughter, Stephanie, from a previous marriage, and they have one daughter, Theresa Anne. Bradley and Schlant divorced in 2007. His partner since 2009 has been former LBJ Library director Betty Sue Flowers.

NBA
Source

Regular season
{| class="wikitable sortable" style="font-size:95%; text-align:right;" !Year !Team !GP !GS !MPG !FG% !FT% !RPG !APG !SPG !BPG !PPG
 * style="text-align:left;"|
 * style="text-align:left;"|New York
 * 45 || 11 || 19.4 || .416 || .731 || 2.5 || 3.0 || || || 8.0
 * style="text-align:left;"|
 * style="text-align:left;"|New York
 * 82 || 39 || 29.4 || .429 || .814 || 4.3 || 3.7 || || || 12.4
 * style="text-align:left; background:#afe6ba;"|†
 * style="text-align:left;"|New York
 * 67 || 64 || 31.3 || .460 || .824 || 3.6 || 4.0 || || || 14.5
 * style="text-align:left;"|
 * style="text-align:left;"|New York
 * 78 || 66 || 29.5 || .453 || .823 || 3.3 || 3.6 || || || 12.4
 * style="text-align:left;"|
 * style="text-align:left;"|New York
 * 78 || 78 || 35.6 || .465 || .849 || 3.2 || 4.0 || || || 15.1
 * style="text-align:left; background:#afe6ba;"|†
 * style="text-align:left;"|New York
 * style="background:#cfecec;" | 82* || style="background:#cfecec;" | 82* || 36.6 || .459 || .871 || 3.7 || 4.5 || || || 16.1
 * style="text-align:left;"|
 * style="text-align:left;"|New York
 * style="background:#cfecec;" | 82* || style="background:#cfecec;" | 82* || 34.3 || .451 || .874 || 3.1 || 3.0 || .5 || .3 || 14.0
 * style="text-align:left;"|
 * style="text-align:left;"|New York
 * 79 || 79 || 35.3 || .436 || .873 || 3.2 || 3.1 || .9 || .2 || 13.3
 * style="text-align:left;"|
 * style="text-align:left;"|New York
 * 82 || 82 || 33.0 || .433 || .878 || 2.9 || 3.0 || .8 || .2 || 11.1
 * style="text-align:left;"|
 * style="text-align:left;"|New York
 * 67 || 5 || 15.3 || .464 || .810 || 1.5 || 1.9 || .4 || .1 || 4.3
 * - class="sortbottom"
 * style="text-align:center;" colspan="2"|Career
 * 742 || 588 || 30.7 || .448 || .840 || 3.2 || 3.4 || .7 || .2 || 12.4
 * - class="sortbottom"
 * style="text-align:center;" colspan="2"|All-Star
 * 1 || 0 || 12.0 || .400 || – || 1.0 || .0 || || || 4.0
 * style="text-align:left;"|
 * style="text-align:left;"|New York
 * 67 || 5 || 15.3 || .464 || .810 || 1.5 || 1.9 || .4 || .1 || 4.3
 * - class="sortbottom"
 * style="text-align:center;" colspan="2"|Career
 * 742 || 588 || 30.7 || .448 || .840 || 3.2 || 3.4 || .7 || .2 || 12.4
 * - class="sortbottom"
 * style="text-align:center;" colspan="2"|All-Star
 * 1 || 0 || 12.0 || .400 || – || 1.0 || .0 || || || 4.0
 * 1 || 0 || 12.0 || .400 || – || 1.0 || .0 || || || 4.0

Playoffs
{| class="wikitable sortable" style="font-size:95%; text-align:right;" !Year !Team !GP !GS !MPG !FG% !FT% !RPG !APG !SPG !BPG !PPG
 * style="text-align:left;"|1968
 * style="text-align:left;"|New York
 * 6 || 0 || 10.7 || .429 || .692 || 1.0 || .3 || || || 5.5
 * style="text-align:left;"|1969
 * style="text-align:left;"|New York
 * 10 || 10 || 41.9 || .461 || .769 || 7.3 || 4.0 || || || 16.0
 * style="text-align:left; background:#afe6ba;"|1970†
 * style="text-align:left;"|New York
 * style="background:#cfecec;" | 19* || style="background:#cfecec;" | 19* || 32.4 || .429 || .814 || 3.8 || 3.2 || || || 12.4
 * style="text-align:left;"|1971
 * style="text-align:left;"|New York
 * 12 || 12 || 30.7 || .424 || .737 || 3.4 || 3.6 || || || 10.5
 * style="text-align:left;"|1972
 * style="text-align:left;"|New York
 * style="background:#cfecec;" | 16* || style="background:#cfecec;" | 16* || 37.1 || .467 || .839 || 2.9 || 3.4 || || || 16.2
 * style="text-align:left; background:#afe6ba;"|1973†
 * style="text-align:left;"|New York
 * style="background:#cfecec;" | 17* || style="background:#cfecec;" | 17* || 34.5 || .448 || .800 || 3.4 || 2.6 || || || 14.0
 * style="text-align:left;"|1974
 * style="text-align:left;"|New York
 * 12 || 12 || 35.4 || .396 || .862 || 2.3 || 1.1 || .6 || .3 || 12.6
 * style="text-align:left;"|1975
 * style="text-align:left;"|New York
 * 3 || 3 || 29.3 || .375 || 1.000 || 3.0 || 2.0 || .7 || .0 || 6.7
 * - class="sortbottom"
 * style="text-align:center;" colspan="2"|Career
 * 95 || 89 || 33.3 || .438 || .805 || 3.5 || 2.8 || .6 || .2 || 12.9
 * 12 || 12 || 35.4 || .396 || .862 || 2.3 || 1.1 || .6 || .3 || 12.6
 * style="text-align:left;"|1975
 * style="text-align:left;"|New York
 * 3 || 3 || 29.3 || .375 || 1.000 || 3.0 || 2.0 || .7 || .0 || 6.7
 * - class="sortbottom"
 * style="text-align:center;" colspan="2"|Career
 * 95 || 89 || 33.3 || .438 || .805 || 3.5 || 2.8 || .6 || .2 || 12.9
 * 95 || 89 || 33.3 || .438 || .805 || 3.5 || 2.8 || .6 || .2 || 12.9

Published works

 * Bradley, Bill Life on the Run (Bantam Books, 1977) ISBN 0-553110551
 * Bradley, Bill Time Present, Time Past: A Memoir (Alfred A. Knopf, 1996) ISBN 978-0679444886
 * Bradley, Bill Values of the Game (Artisan, 1998) ISBN 1-57965116X
 * Bradley, Bill The Journey from Here (Artisan, 2000) ISBN 1-579651658
 * Bradley, Bill The New American Story (Random House, 2007) ISBN 978-1400065073
 * Bradley, Bill We Can All Do Better (Vanguard Press, May 8, 2012) ISBN 978-1593157296

title: Bill Bradley


text:
Particle-induced X-ray emission

Particle-induced X-ray emission or proton-induced X-ray emission (PIXE) is a technique used for determining the elemental composition of a material or a sample. When a material is exposed to an ion beam, atomic interactions occur that give off EM radiation of wavelengths in the x-ray part of the electromagnetic spectrum specific to an element. PIXE is a powerful, yet non-destructive elemental analysis technique now used routinely by geologists, archaeologists, art conservators and others to help answer questions of provenance, dating and authenticity.

The technique was first proposed in 1970 by Sven Johansson of Lund University, Sweden, and developed over the next few years with his colleagues Roland Akselsson and Thomas B Johansson.

Recent extensions of PIXE using tightly focused beams (down to 1 μm) gives the additional capability of microscopic analysis. This technique, called microPIXE, can be used to determine the distribution of trace elements in a wide range of samples. A related technique, particle-induced gamma-ray emission (PIGE) can be used to detect some light elements.

Theory
Three types of spectra can be collected from a PIXE experiment:

 * 1) X-ray emission spectrum.
 * 2) Rutherford backscattering spectrum.
 * 3) Proton transmission spectrum.

X-ray emission
Quantum theory states that orbiting electrons of an atom must occupy discrete energy levels in order to be stable. Bombardment with ions of sufficient energy (usually MeV protons) produced by an ion accelerator, will cause inner shell ionization of atoms in a specimen. Outer shell electrons drop down to replace inner shell vacancies, however only certain transitions are allowed. X-rays of a characteristic energy of the element are emitted. An energy dispersive detector is used to record and measure these X-rays.

Only elements heavier than fluorine can be detected. The lower detection limit for a PIXE beam is given by the ability of the X-rays to pass through the window between the chamber and the X-ray detector. The upper limit is given by the ionisation cross section, the probability of the K electron shell ionisation, this is maximal when the velocity of the proton matches the velocity of the electron (10% of the speed of light), therefore 3 MeV proton beams are optimal.

Proton backscattering
Protons can also interact with the nucleus of the atoms in the sample through elastic collisions, Rutherford backscattering, often repelling the proton at angles close to 180 degrees. The backscatter give information on the sample thickness and composition. The bulk sample properties allow for the correction of X-ray photon loss within the sample.

Proton transmission
The transmission of protons through a sample can also be used to get information about the sample. Channeling is one of the processes that can be used to study crystals.

Protein analysis
Protein analysis using microPIXE allow for the determination of the elemental composition of liquid and crystalline proteins. microPIXE can quantify the metal content of protein molecules with a relative accuracy of between 10% and 20%.

The advantage of microPIXE is that given a protein of known sequence, the X-ray emission from sulfur can be used as an internal standard to calculate the number of metal atoms per protein monomer. Because only relative concentrations are calculated there are only minimal systematic errors, and the results are totally internally consistent.

The relative concentrations of DNA to protein (and metals) can also be measured using the phosphate groups of the bases as an internal calibration.

Data analysis
Analysis of the data collected can be performed by the programs Dan32, the front end to gupix.

Limitations
In order to get a meaningful sulfur signal from the analysis, the buffer should not contain sulfur (i.e. no BES, DDT, HEPES, MES, MOPSO or PIPES compounds). Excessive amounts of chlorine in the buffer should also be avoided, since this will overlap with the sulfur peak; KBr and NaBr are suitable alternatives.

Advantages
There are many advantages to using a proton beam over an electron beam. There is less crystal charging from Bremsstrahlung radiation, although there is some from the emission of Auger electrons, and there is significantly less than if the primary beam was itself an electron beam.

Because of the higher mass of protons relative to electrons, there is less lateral deflection of the beam; this is important for proton beam writing applications.

Scanning
Two-dimensional maps of elemental compositions can be generated by scanning the microPIXE beam across the target.

Cell and tissue analysis
Whole cell and tissue analysis is possible using a microPIXE beam, this method is also referred to as nuclear microscopy.

Artifact analysis
MicroPIXE is a useful technique for the non-destructive analysis of paintings and antiques. Although it provides only an elemental analysis, it can be used to distinguish and measure layers within the thickness of an artifact. The technique is comparable with destructive techniques such as the ICP family of analyses.

Proton beam writing
Proton beams can be used for writing (proton beam writing) through either the hardening of a polymer (by proton induced cross-linking), or through the degradation of a proton sensitive material. This may have important effects in the field of nanotechnology.

title: Particle-induced X-ray emission