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{{About|the physical phenomenon|the stochastic process|Wiener process|the sports team|Brownian Motion (Ultimate)|the mobility model|Random walk}}
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[[File:Brownian motion large.gif|thumb|This is a simulation of the Brownian motion of a big particle (dust particle) that collides with a large set of smaller particles (molecules of a gas) which move with different velocities in different random directions.]]
[[File:Brownianmotion5particles150frame.gif|thumb|This is a simulation of the Brownian motion of 5 particles (yellow) that collide with a large set of 800 particles. The yellow particles leave 5 blue trails of random motion and one of them has a red velocity vector.]]
[[Image:Translational motion.gif|right]]
[[Image:Brownian hierarchical.svg|thumb|Three different views of Brownian motion, with 32 steps, 256 steps, and 2048 steps denoted by progressively lighter colors]]
[[Image:Wiener process 3d.png|thumb|A single realisation of three-dimensional Brownian motion for times 0 ≤ ''t'' ≤ 2]]
'''Brownian motion''' or '''pedesis''' (from {{lang-el|πήδησις}} {{IPA|/pɛɖeːsɪs/}} "leaping"<!--not to be confused with πέδησις "braking"-->) is the random motion of [[particle]]s suspended in a [[fluid]] (a [[liquid]] or a [[gas]]) resulting from their collision with the quick [[atom]]s or [[molecule]]s in the gas or liquid. The term "Brownian motion" can also refer to the mathematical model used to describe such random movements, which is often called a [[particle|particle theory]].<ref>{{cite book |last=Mörters |first=Peter |first2=Yuval |last2=Peres |title=Brownian Motion |date=25 May 2008 |url=http://www.stat.berkeley.edu/~peres/bmbook.pdf |accessdate=25 May 2008}}</ref>


This [[transport phenomena|transport phenomenon]] is named after the botanist [[Robert Brown (botanist)|Robert Brown]]. In 1827, while looking through a microscope at particles found in pollen grains in water, he noted that the particles moved through the water but was not able to determine the mechanisms that caused this motion. [[Atomic theory|Atoms and molecules]] had long been theorized as the constituents of matter, and many decades later, [[Albert Einstein]] published [[Annus Mirabilis papers|a paper in 1905]] that explained in precise detail how the motion that Brown had observed was a result of the pollen being moved by individual water molecules. This explanation of Brownian motion served as definitive confirmation that atoms and molecules actually 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" (Einstein had received the award five years earlier "for his services to theoretical physics" with specific citation of [[photoelectric effect|different research]]). 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.
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The mathematical model of Brownian motion has numerous real-world applications. For instance, [[Stock market]] fluctuations are often cited, although [[Benoit Mandelbrot]] rejected its applicability to stock price movements in part because these are discontinuous.<ref>{{cite book |title=The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward |last=Mandelbrot |first=B. |last2=Hudson |first2=R. |location=New York |publisher=Basic Books |year=2004 |isbn=0-465-04355-0 }}</ref>
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Brownian motion is among the simplest of the continuous-time [[stochastic process|stochastic (or probabilistic) processes]], and it is a [[limit (mathematics)|limit]] of both simpler and more complicated stochastic processes (see [[random walk]] and [[Donsker's theorem]]). This [[Universality (dynamical systems)|universality]] is closely related to the universality of the [[normal distribution]]. In both cases, it is often mathematical convenience rather than the accuracy of the models that motivates their use. This is because Brownian motion, whose time derivative is everywhere infinite, is an idealised approximation to actual random physical processes, which always have a finite time scale.
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==History==
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[[Image:PerrinPlot2.svg|thumb|left|Reproduced from the book of [[Jean Baptiste Perrin]], ''Les Atomes'', three tracings of the motion of colloidal particles of radius 0.53&nbsp;µm, as seen under the microscope, are displayed. Successive positions every 30 seconds are joined by straight line segments (the mesh size is 3.2&nbsp;µm).<ref>Perrin, 1914, [http://www.archive.org/stream/atomsper00perruoft#page/115/mode/1up p. 115]</ref>]]


The Roman [[Lucretius]]'s scientific poem "[[On the Nature of Things]]" (c. 60 BC) has a remarkable description of Brownian motion of dust particles. He uses this as a proof of the existence of atoms:
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<blockquote>"Observe what happens when sunbeams are admitted into a building and shed light on its shadowy places. You will see a multitude of tiny particles mingling in a multitude of ways... their dancing is an actual indication of underlying movements of matter that are hidden from our sight... It originates with the atoms which move of themselves [i.e., spontaneously]. Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies. So the movement mounts up from the atoms and gradually emerges to the level of our senses, so that those bodies are in motion that we see in sunbeams, moved by blows that remain invisible."</blockquote>
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Although the mingling motion of dust particles is caused largely by air currents, the glittering, tumbling motion of small dust particles is, indeed, caused chiefly by true Brownian dynamics.
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[[Jan Ingenhousz]] had described the irregular motion of [[coal]] [[dust]] particles on the surface of [[ethanol|alcohol]] in 1785 — nevertheless the discovery is often credited to the botanist [[Robert Brown (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.
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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. [[Albert Einstein]] (in one of his [[Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen|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.
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==Einstein's theory==
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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.<ref>{{cite web|url=http://users.physik.fu-berlin.de/~kleinert/files/eins_brownian.pdf |title=Investigations on the Theory of the Brownian Movement |author=Einstein, Albert |publisher=Dover Publications |year=1956 |origyear=Republication of the original 1926 translation |accessdate=2013-12-25 |format=PDF}}</ref> 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.<ref>{{cite web|url=http://www.csun.edu/~dchoudhary/Physics-Year_files/ed_diss.pdf |title=The Collected Papers of: Albert Einstein, Volume 2, The Swiss Years: Writings, 1900-1909 |publisher=Princeton University Press |year=1989 |accessdate=2013-12-25 |format=PDF}}</ref> 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 [[Avogadro constant|Avogadro's 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 mass of a mole of the gas by Avogadro's number.


[[File:Diffusion of Brownian particles.svg|thumb|300px|The characteristic bell-shaped curves of the diffusion of Brownian particles. The distribution begins as a [[Dirac delta function]], indicating that all the particles are located at the origin at time t=0, and for increasing times they become flatter and flatter until the distribution becomes uniform in the asymptotic time limit.]]
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The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval.{{citation needed|date=October 2012}} 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<sup>21</sup> collisions per second.<ref name=Chandrasekhar43>{{cite journal |first=S. |last=Chandrasekhar |title=Stochastic problems in physics and astronomy |journal=[[Reviews of Modern Physics]] |volume=15 |issue=1 |pages=1–89 |year=1943 |doi=10.1103/RevModPhys.15.1 |mr=0008130 |bibcode = 1943RvMP...15....1C }}</ref> Thus Einstein was led to consider the collective motion of Brownian particles.{{citation needed|date=October 2012}} He showed that if ρ(''x'', ''t'') is the density of Brownian particles at point ''x'' at time ''t'', then ρ satisfies the diffusion equation:
== Through the late 1970s Replica Ray Ban Aviators ==


:<math>\frac{\partial\rho}{\partial t}=D\frac{\partial^2\rho}{\partial x^2},</math>
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where ''D'' is the [[mass diffusivity]].
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Assuming that all the particles start from the origin at the initial time t=0, the diffusion equation has the solution
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:<math>\rho(x,t)=\frac{\rho_0}{\sqrt{4\pi Dt}}e^{-\frac{x^2}{4Dt}}.</math>
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This expression allowed Einstein to calculate the [[Moment (mathematics)|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
  <li>[https://fratpoet.com/activity/p/26500/ https://fratpoet.com/activity/p/26500/]</li>
 
 
:<math>\overline{x^2}=2Dt.</math>
  <li>[http://www.mdj.cze.as/forum.php?mod=viewthread&tid=569848 http://www.mdj.cze.as/forum.php?mod=viewthread&tid=569848]</li>
 
 
This 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.<ref name=Einstein>A. Einstein,  ''Investigations of the Theory of Brownian Movement'' (Dover, 1956).</ref> 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.<ref>{{cite book|last=Lavenda|first=Bernard H.|title=Nonequilibrium Statistical Thermodynamics|year=1985|publisher=John Wiley & Sons Inc.|isbn=0-471-90670-0|page=20}}</ref>
</ul>
 
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 Avogadro's number and therefore the size of molecules. Einstein analyzed a dynamic equilibrium being established between opposing forces. The beauty of his argument is that the final result does not depend upon which forces are involved in setting up the dynamic equilibrium. In his original treatment, Einstein considered an osmotic pressure experiment, but the same conclusion can be reached in other ways. Consider, for instance, particles suspended in a viscous fluid in a gravitational field. Gravity tends to make the particles settle, whereas diffusion acts to homogenize them, driving them into regions of smaller concentration. Under the action of gravity, a particle acquires a downward speed of ''v'' = μ''mg'', where ''m'' is the mass of the particle, ''g'' is the acceleration due to gravity, and μ is the particle's [[Einstein relation (kinetic theory)|mobility]] in the fluid. [[Sir George Stokes, 1st Baronet|George Stokes]] had shown that the mobility for a spherical particle with radius ''r'' is <math>\mu=\tfrac{1}{6\pi\eta r}</math>, where η is the [[dynamic viscosity]] of the fluid. In a state of dynamic equilibrium, the particles are distributed according to the barometric distribution
 
:<math>\rho=\rho_0e^{-\frac{mgh}{k_BT}},</math>
 
where ρ−ρ<sub>0</sub> is the difference in density of particles separated by a height difference of ''h'', ''k<sub>B</sub>'' is Boltzmann's constant (namely, the ratio of the universal gas constant, ''R'', to Avogadro's number, ''N''), and ''T'' is the absolute temperature. It is Avogadro's number that is to be determined.
 
[[File:Brownian motion gamboge.jpg|thumb|The equilibrium distribution for particles of [[gamboge]] shows the tendency for granules to move to regions of lower concentration when affected by gravity.]]
Dynamic equilibrium is established because the more that particles are pulled down by gravity, the greater is the tendency for the particles to migrate to regions of lower concentration. The flux is given by [[Fick's laws of diffusion|Fick's law]],
 
:<math>J=-D\frac{d\rho}{dh},</math>
 
where ''J'' = ρ''v''. Introducing the formula for ρ, we find that
 
:<math>v=\frac{Dmg}{k_BT}.</math>
 
In a state of dynamical equilibrium, this speed must also be equal to ''v'' = μ''mg''. Notice that both expressions for ''v'' are proportional to ''mg'', reflecting how the derivation is independent of the type of forces considered. Equating these two expressions yields a formula for the diffusivity:
 
:<math>\frac{\overline{x^2}}{2t}=D=\mu k_BT=\frac{\mu RT}{N}=\frac{RT}{6\pi\eta rN}.</math>
 
Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of Boltzmann's constant as ''k<sub>B</sub>'' = ''R / N'', and the fourth equality follows from Stokes' 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'', Avogadro's number ''N'' can be determined.
 
The type of dynamical equilibrium proposed by Einstein was not new. It had been pointed out previously by [[J. J. Thomson]]<ref name=Electricity>"Electricity and Matter" (Yale University Press, New Haven, 1904), pp 80-83</ref> 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".<ref name=Electricity/>
 
An identical expression to Einstein's formula for the diffusion coefficient was also found by [[Walther Nernst]] in 1888<ref>Zeit. Phys. Chem. '''9''' (1888) 613.</ref> 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' law|Stokes's law]]. He writes <math>k'=p_0/k</math> for the diffusion coefficient ''k′'', where <math>p_0</math> 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.<ref>Leveugle, J. (2004). La Relativite', Poincaré' et Einstein, Planck, Hilbert Paris: L'Harmattan p. 181.</ref> 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.<ref>Townsend, J.E.S. (1915). "Electricity in Gases". Oxford: Clarendon Press. p. 254.</ref>
 
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.<ref>See P. Clark 1976, p. 97</ref> 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.<ref>See P. Clark 1976 for this whole paragraph</ref>
 
==Intuitive metaphor==
Consider a large balloon of 100 metres in diameter. Imagine this large balloon in a football stadium. The balloon is so large that it lies on top of many members of the crowd. Because they are excited, these fans hit the balloon at different times and in different directions with the motions being completely random. In the end, the balloon is pushed in random directions, so it should not move on average. Consider now the force exerted at a certain time. We might have 20 supporters pushing right, and 21 other supporters pushing left, where each supporter is exerting equivalent amounts of force. In this case, the forces exerted towards the left and the right are imbalanced in favor of the left; the balloon will move slightly to the left. This type of imbalance exists at all times, and it causes random motion of the balloon. If we look at this situation from far above, so that we cannot see the supporters, we see the large balloon as a small object animated by erratic movement.
{| style="width:320px; float:right; border:1px solid #ccc; background:#f9f9f9; font-size:88%; line-height:1.5em;"
|-
| [[File:Brownian Motion.ogv]]
|-
| Brownian motion model of the trajectory of a particle of dye in water.
|}
 
Consider the particles emitted by Brown's pollen grain moving randomly in water: we know that a water molecule is about 0.1 by 0.2&nbsp;nm in size, whereas the particles which Brown observed were of the order of  a few [[micrometre]]s in size (these are not to be confused with the actual pollen particle which is about 100 micrometres). So a particle from the pollen may be likened to the balloon, and the water molecules to the fans, except that in this case the balloon is surrounded by fans. The Brownian motion of a particle in a liquid is thus due to the instantaneous imbalance in the combined forces exerted by collisions of the particle with the much smaller liquid molecules (which are in random [[temperature|thermal]] motion) surrounding it.
 
An [http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/brownian/brownian.html animation of the Brownian motion concept] is available as a [[Java applet]].
 
==Theory==
 
===Smoluchowski model===
[[Marian Smoluchowski|Smoluchowski]]'s theory of Brownian motion<ref>{{cite book |last=Smoluchowski |first=M. |year=1906 |journal=<!--old, abbreviated text: Bull. Int. De l'Acad. Des Sci. De Cracovie-->Bulletin International de l'Academie des Sciences de Cracovie |page=202 |url=https://archive.org/stream/bulletininternat1906pols#page/202/mode/2up |title=<!--Polish:-->O średniej drodze cząsteczek gazu i o związku jej z teoryą dyfuzyi <!--French:-->(Sur le chemin moyen parcouru par les molécules d'un gaz et sur son rapport avec la théorie de la diffusion) <!--French:-->Mémoire présenté par M. Lad. Natanson, m. t. |trans_title=On the average path of the gas molecules and its relationship with the theory of diffusion (Submission by Mr. Lad. Natanson, m. t.)}} {{link note|note=This is from a scanned copy of the text that came from Archive.org. [https://archive.org/details/bulletininternat1906pols Main page here]}}</ref> 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: <math>\overline{(\Delta x)^2}</math>. However, when he relates it to a particle of mass ''m'' moving at a velocity ''u'' which is the result of a frictional force governed by Stokes's law, he finds
:<math>\overline{(\Delta x)^2}=2Dt=t\frac{32}{81}\frac{mu^2}{\pi\mu a}=t\frac{64}{27}\frac{\frac{1}{2}mu^2}{3\pi\mu a},</math>
where μ is the viscosity coefficient, and ''a'' is the radius of the particle. Associating the kinetic energy <math>mu^2/2</math> 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."<ref>{{Cite journal |last=Sommerfeld |first=A. |date=15 November 1917|journal=Phys. Zeit.<!--
 
Zeitschrift für Physik? [Journal of Physics]? That didn't exist yet (plus its abbreviation is Phys. Z.) Physikalische Zeitschrift, probably--><!--
 
Another lead, here: "Sommerfeld, A. (1917). Zum Andenken an Marian von Smoluchowski. Physikalische Zeitschrift, No. 22, 18, 533–539." is possibly the full citation
I found that at http://link.springer.com/chapter/10.1007%2F978-1-4613-0179-0_64. It's behind a paywall, though...
 
--> |page= 535 |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}{{full|date=November 2012}}</ref>
 
Smoluchowski<ref>{{cite book |last=Smoluchowski |first=M. |year=1906 |journal=<!--old, abbreviated text: Bull. Int. De l'Acad. Des Sci. De Cracovie-->Bulletin International de l'Academie des Sciences de Cracovie |page=577 |url=https://archive.org/stream/bulletininternat1906pols#page/577/mode/2up |title=<!--Polish-->Zarys teoryi kinetycznej ruchu Brownai roztworów mętnych. <!--French:-->(Essai d'une théorie cinétique du mouvement Brownien et des milieux troubles). <!--French:-->Mémoire présenté par M. Lad. Natanson m. t. |trans_title=Test of a kinetic theory of Brownian motion and turbid media. (Submission by Mr Lad. Natanson m. t.)}} {{link note|note=This is from a scanned copy of the text that came from Archive.org. [https://archive.org/details/bulletininternat1906pols Main page here]}}</ref> 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. In order to do so, he uses, unknowingly, the [[ballot theorem]], first proved by [[William Allen Whitworth|W.A. Whitworth]] in 1887.<ref>{{Cite document |last=Whithworth |first= W. A.|year=1965 |title=Choice and Chance|publisher=Hafner Pub. Co. |postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}</ref> The ballot theorem states that if a candidate A scores ''m'' votes and candidate B scores ''n''−''m'' that the probability throughout the counting that A will have more votes than B is
 
:<math>\frac{[m-(n-m)]}{[m+n-m]} = \frac{2m-n}{n},</math>
 
no matter how large the total number of votes ''n'' may be. In other words, if one candidate has an edge on the other candidate he will tend to keep that edge even though there is nothing favoring either candidate on a ballot extraction.
 
If the probability of ''m'' gains and ''n''−''m'' losses follows a [[binomial distribution]],
 
:<math>P_{m,n}=\binom{n}{m}2^{-n},</math>
 
with equal ''a priori'' probabilities of 1/2, the mean total gain is
 
:<math>\overline{2m-n}=\sum_{m=\frac{n}{2}}^n (2m-n)P_{m,n}=\frac{n n!}{2^n \left [ \left (\frac{n}{2}
\right )!
\right ]^2}.</math>
If ''n'' is large enough so that Stirling's approximation can be used in the form
:<math>n!\approx\left(\frac{n}{e}
\right)^n\sqrt{2\pi n},</math>
then the expected total gain will be{{citation needed|date=July 2012}}
:<math>\overline{2m-n}\approx\sqrt{\frac{2n}{\pi}},</math>
showing that it increases as the square root of the total population.
 
Suppose that a Brownian particle of mass ''M'' is surrounded by lighter particles of mass ''m'' which are traveling at a speed ''u''. Then, reasons Smoluchowski, in any collision between a surrounding and Brownian particles, the velocity transmitted to the latter will be ''mu''/''M''. This ratio is of the order of 10<sup>−7</sup>&nbsp;cm/s. But we also have to take into consideration that in a gas there will be more than 10<sup>16</sup> collisions in a second, and even greater in a liquid where we expect that there will be 10<sup>20</sup> 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<sup>8</sup> to 10<sup>10</sup> collisions in one second, then velocity of the Brownian particle may be anywhere between 10 to 1000&nbsp;cm/s. Thus, even though there are equal probabilities for forward and backward collisions there will be a net tendency to keep the Brownian particle in motion, just as the ballot theorem predicts.
 
These orders of magnitude are not exact because they don't take into consideration the velocity of the Brownian particle, ''U'', which depends on the collisions that tend to accelerate and decelerate it. The larger ''U'' is, the greater will be the collisions that will retard it so that the velocity of a Brownian particle can never increase without limit. Could a 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, <math>MU^2/2</math>, will be equal, on the average, to the kinetic energy of the surrounding fluid particle, <math>mu^2/2</math>.
 
In 1906 Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion.<ref>{{cite journal |last=Smoluchowski |first=M. |year=1906 |title=Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen |journal=[[Annalen der Physik]] |volume=326 |issue=14 |pages=756–780 |doi=10.1002/andp.19063261405 |bibcode=1906AnP...326..756V}}</ref> The model assumes collisions with ''M''&nbsp;{{Unicode|≫}}&nbsp;''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<sub>R</sub>'' is the number of collisions from the right and ''N<sub>L</sub>'' the number of collisions from the left then after ''N'' collisions the particle's velocity will have changed by Δ''V''(2''N<sub>R</sub>−''N''). The [[multiplicity (mathematics)|multiplicity]] is then simply given by:
 
:<math> \binom{N}{N_R} = \frac{N!}{N_R!(N-N_R)!}</math>
 
and the total number of possible states is given by 2<sup>''N''</sup>. Therefore the probability of the particle being hit from the right ''N<sub>R</sub>'' times is:
 
:<math>P_N(N_R)=\frac{N!}{2^NN_R!(N-N_R)!}</math>
 
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 cannot be made. For example, the assumption that on average there 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.
 
==Modeling using differential equations==
The equations governing Brownian motion relate slightly differently to each of the two definitions of ''Brownian motion'' given at the start of this article.
 
===Mathematics===
{{Main|Wiener process}}
[[File:2D Random Walk 400x400.ogv|thumb|right|300px|An animated example of a Brownian motion-like [[random walk]] on a [[torus]]. In the [[scaling limit]], random walk approaches the Wiener process according to [[Donsker's theorem]].]]
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 process]]es ([[càdlàg]] stochastic processes with [[stationary process|stationary]] [[statistical independence|independent]] increments) and occurs frequently in pure and applied mathematics, [[economy|economics]] and [[physics]].
 
The Wiener process ''W<sub>t</sub>'' is characterised by four facts:
#''W''<sub>0</sub> = 0
#''W<sub>t</sub>'' is [[almost surely]] continuous
#''W<sub>t</sub>'' has independent increments
#<math>W_t-W_s\sim \mathcal{N}(0,t-s)</math> (for <math>0 \leq s \le t</math>).
<math>\mathcal{N}(\mu, \sigma^2)</math> denotes the [[normal distribution]] with [[expected value]] μ and [[variance]] σ<sup>2</sup>. The condition that it has independent increments means that if <math>0 \leq s_1 \leq t_1 \leq s_2 \leq t_2</math> then <math>W_{t_1}-W_{s_1}</math> and <math>W_{t_2}-W_{s_2}</math> are independent random variables.
 
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 (probability theory)|martingale]] with ''W''<sub>0</sub> = 0 and [[quadratic variation]] <math>[W_t, W_t] = t</math>.
 
A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent <math>\mathcal{N}(0, 1)</math> random variables. This representation can be obtained using the [[Karhunen&ndash;Loève theorem]].
 
The Wiener process can be constructed as the [[scaling limit]] of a [[random walk]], or other discrete-time stochastic processes with stationary independent increments. This is known as [[Donsker's theorem]]. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed [[neighborhood (mathematics)|neighborhood]] of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it is [[scale invariance|scale invariant]].
 
The time evolution of the position of the Brownian particle itself can be described approximately by a [[Langevin equation]], an equation which involves a random force field representing the effect of the [[thermal fluctuations]] of the solvent on the Brownian particle. On long timescales, the mathematical Brownian motion is well described by a Langevin equation. On small timescales, [[inertia]]l effects are prevalent in the Langevin equation.  However the mathematical ''Brownian motion'' is exempt of such inertial effects. Note that inertial effects have to be considered in the Langevin equation, otherwise the equation becomes singular.{{Clarify|date=April 2010}} 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.{{Clarify|date=April 2010}}
 
===Physics===
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 [[Langevin equation|short]] timescales.
 
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.
 
The displacement of a particle undergoing Brownian motion is obtained by solving the [[diffusion equation]] under appropriate boundary conditions and finding the [[root mean square|rms]] of the solution. This shows that the displacement varies as the square root of the time (not linearly), which explains why previous experimental results concerning the velocity of Brownian particles gave nonsensical results. A linear time dependence was incorrectly assumed.
 
At very short time scales, however, the motion of a particle is dominated by its inertia and its displacement will be linearly dependent on time: Δ''x'' = ''v''Δ''t''. So the instantaneous velocity of the Brownian motion can be measured as v = Δx/Δt, when Δt << τ, where τ is the momentum relaxation time. In 2010, the instantaneous velocity of a Brownian particle (a glass microsphere trapped in air with an optical tweezer) was measured successfully.<ref>{{cite journal | last = Li | first = Tongcang | last2 = Kheifets | first2 = Simon | last3 = Medellin | first3 = David | last4 = Raizen | first4 = Mark | title = Measurement of the instantaneous velocity of a Brownian particle | journal = Science | volume = 328 | issue = 5986 | pages = 1673–1675 | date = June 2010 | doi = 10.1126/science.1189403 | bibcode = 2010Sci...328.1673L  | pmid = 20488989 | registration = yes}}</ref> The velocity data verified the Maxwell-Boltzmann velocity distribution, and the equipartition theorem for a Brownian particle.
The Brownian motion can be modeled by a random walk.<ref>
{{Cite book |
last = Weiss  |
first = G. H. |
title = Aspects and applications of the random walk |
year= 1994 |
publisher= North Holland}}</ref> Random walks in porous media or fractals are anomalous.<ref>
{{Cite book |
last1 =  Ben-Avraham |
first1 = D. |
last2 =  Havlin |
first2 = S. |
authorlink2 = Shlomo Havlin |
title = Diffusion and reaction in disordered systems |
year= 2000 |
publisher= Cambridge University Press |
url= http://havlin.biu.ac.il/Shlomo%20Havlin%20books_d_r.php }}</ref>
 
In the general case, Brownian motion is a [[Markov process|non-Markov random process]] and described by [[Stochastic calculus|stochastic integral equations]].<ref name="mor-skr2a">{{cite journal |first=A. N. |last=Morozov |first2=A. V. |last2=Skripkin |title=Spherical particle Brownian motion in viscous medium as non-Markovian random process |journal=Physics Letters A |year=2011 |volume=375 |issue=46 |pages=4113–4115 |doi=10.1016/j.physleta.2011.10.001 |bibcode=2011PhLA..375.4113M |subscription=yes}}</ref>
 
==Lévy characterisation==
The French mathematician [[Paul Lévy (mathematician)|Paul Lévy]] proved the following theorem, which gives a necessary and sufficient condition for a continuous '''R'''<sup>''n''</sup>-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.
 
Let ''X''&nbsp;=&nbsp;(''X''<sub>1</sub>,&nbsp;...,&nbsp;''X''<sub>''n''</sub>) be a continuous stochastic process on a [[probability space]] (Ω,&nbsp;Σ,&nbsp;'''P''') taking values in '''R'''<sup>''n''</sup>. Then the following are equivalent:
 
# ''X'' is a Brownian motion with respect to '''P''', i.e., the law of ''X'' with respect to '''P''' is the same as the law of an ''n''-dimensional Brownian motion, i.e., the [[push-forward measure]] ''X''<sub>∗</sub>('''P''') is [[classical Wiener measure]] on ''C''<sub>0</sub>([0,&nbsp;+∞);&nbsp;'''R'''<sup>''n''</sup>).
# both
## ''X'' is a [[martingale (probability theory)|martingale]] with respect to '''P''' (and its own [[natural filtration]]); and
## for all 1&nbsp;≤&nbsp;''i'',&nbsp;''j''&nbsp;≤&nbsp;''n'', ''X''<sub>''i''</sub>(''t'')''X''<sub>''j''</sub>(''t'')&nbsp;&minus;''δ''<sub>''ij''</sub>''t'' is a martingale with respect to '''P''' (and its own [[natural filtration]]), where ''δ''<sub>''ij''</sub> denotes the [[Kronecker delta]].
 
==Riemannian manifold==
[[Image:BMonSphere.jpg|thumb|Brownian motion on a 2-sphere]]
{{Technical|section|date=June 2011}}
 
The [[Infinitesimal generator (stochastic processes)|infinitesimal generator]] (and hence characteristic operator) of a Brownian motion on '''R'''<sup>''n''</sup> is easily calculated to be ½Δ, where Δ denotes the [[Laplace operator]]. This observation is useful in defining Brownian motion on an ''m''-dimensional [[Riemannian manifold]] (''M'',&nbsp;''g''): a '''Brownian motion on''' ''M'' is defined to be a diffusion on ''M'' whose characteristic operator <math>\mathcal{A}</math> in local coordinates ''x''<sub>''i''</sub>, 1&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''m'', is given by ½Δ<sub>LB</sub>, where Δ<sub>LB</sub> is the [[Laplace–Beltrami operator]] given in local coordinates by
 
:<math>\Delta_{\mathrm{LB}}=\frac{1}{\sqrt{\det(g)}} \sum_{i=1}^m \frac{\partial}{\partial x_i} \left(\sqrt{\det(g)} \sum_{j=1}^m g^{ij} \frac{\partial}{\partial x_j} \right),</math>
 
where [''g''<sup>''ij''</sup>]&nbsp;=&nbsp;[''g''<sub>''ij''</sub>]<sup>&minus;1</sup> in the sense of [[Invertible matrix|the inverse of a square matrix]].
 
==Gravitational motion==
In [[stellar dynamics]], a massive body (star, [[black hole]], etc.) can experience Brownian motion as it responds to [[gravitational|gravitational forces]] from surrounding stars.<ref name=Merritt>{{cite book|last=Merritt|first=David|author-link=David Merritt|title=Dynamics and Evolution of Galactic Nuclei|year=2013|publisher=Princeton University Press |isbn=9781400846122|page=575|url=http://openlibrary.org/works/OL16802359W/Dynamics_and_Evolution_of_Galactic_Nuclei}}</ref> The rms velocity ''V'' of the massive object, of mass ''M'', is related to the rms velocity <math>v_\star</math> of the background stars by
:<math> MV^2 \approx m v_\star^2 </math>
where <math>m\ll M</math> 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 <math>v_\star</math> and ''V''.<ref name=Merritt/> The Brownian velocity of [[Sagittarius A*|Sgr A*]], the [[supermassive black hole]] at the center of the [[Milky Way galaxy]], is predicted from this formula to be less than 1&nbsp;km ''s''<sup>−1</sup>.<ref name="Reid">{{cite journal | last = Reid  | first = M. J. | last2 = Brunthaler | first2 = A. | title = The Proper Motion of Sagittarius A*. II. The Mass of Sagittarius A* | journal = The Astrophysical Journal | volume = 616| issue = 2| pages = 872–884 | date = December 2004 |bibcode = 2004ApJ...616..872R | doi = 10.1086/424960|arxiv = astro-ph/0408107}}</ref>
 
==Narrow escape==
The '''[[Narrow escape problem]]''' is a ubiquitous problem in biology, biophysics and cellular biology which has the following formulation: a Brownian particle ([[ion]], [[molecule]], or [[protein]]) is confined to a bounded domain (a compartment or a cell) by a reflecting boundary, except for a small window through which it can escape.  The narrow escape problem is that of calculating the mean escape time.  This time diverges as the window shrinks, thus rendering the calculation a [[singular perturbation]] problem.
 
==See also==
{{div col|colwidth=30em}}
* [[Brownian bridge]]: a Brownian motion that is required to "bridge" specified values at specified times
* [[Brownian covariance]]
* [[Brownian dynamics]]
* [[Brownian motion of sol particles]]
* [[Brownian motor]]
* [[Brownian noise]] ([[Martin Gardner]] proposed this name for sound generated with random intervals. It is a pun on Brownian motion and [[white noise]].)
* [[Brownian ratchet]]
* [[Brownian surface]]
* [[Brownian tree]]
* [[Rotational Brownian motion]]
* [[Complex system]]
* [[Diffusion equation]]
* [[Geometric Brownian motion]]
* [[Itō diffusion]]: a generalisation of Brownian motion
* [[Langevin equation]]
* [[Lévy arcsine law]]
* [[Local time (mathematics)]]
* [[Marangoni effect]]
* [[Mark G. Raizen]]
* [[Nanoparticle tracking analysis]]
* [[Narrow escape problem]]
* [[Osmosis]]
* [[Random walk]]
* [[Schramm–Loewner evolution]]
* [[Single particle tracking]]
* [[Surface diffusion]]: a type of constrained Brownian motion.
* [[Tyndall effect]]: physical chemistry phenomenon where particles are involved; used to differentiate between the different types of mixtures.
* [[Ultramicroscope]]
{{div col end}}
 
==References==
{{Reflist|30em}}
* [[Lester Eli Dubins]] & Gideon Schwarz (1965). ''On Continuous Martingales''. Proceedings of the National Academy of Science.
 
==Further reading==
* Brown, Robert (1828). "A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies." Phil. Mag. 4, 161–173. [http://sciweb.nybg.org/science2/pdfs/dws/Brownian.pdf (PDF version of original paper including a subsequent defense by Brown of his original observations, ''Additional remarks on active molecules''.)]
* {{cite journal | last1 = Chaudesaigues | first1 = M. | year = 1908 | title = Le mouvement brownien et la formule d'Einstein |trans_title = Brownian motion and Einstein's formula | language = French | url = | journal = Comptes Rendus | volume = 147 | issue = | pages = 1044–6}}
* Clark, P. (1976). 'Atomism versus thermodynamics' in ''Method and appraisal in the physical sciences'', Colin Howson (ed.). Cambridge University Press.
* Cohen, Ruben D. (1986). "[http://rdcohen.50megs.com/BrownianMotion.pdf Self Similarity in Brownian Motion and Other Ergodic Phenomena]", ''Journal of Chemical Education 63'', pp.&nbsp;933–934.
* {{cite journal | author=Einstein, A. | title=Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen | language=German | journal=Annalen der Physik  | volume=322 | pages=549–560 | year=1905 | url=http://www.physik.uni-augsburg.de/annalen/history/einstein-papers/1905_17_549-560.pdf |bibcode=1905AnP...322..549E | doi=10.1002/andp.19053220806 | issue=8}}
* {{cite book |author=Einstein, A. |title=Investigations on the Theory of Brownian Movement |location=New York |publisher=Dover |year=1956 |isbn=0-486-60304-0 |url=http://www.fisicateorica.me/repositorio/howto/artigoshistoricosordemcronologica/1905%20-Einstein%201905%20Investigations%20on%20the%20Theory%20of%20the%20Brownian%20Movement.pdf |format=PDF}}
* {{cite journal |author=Henri, V |year=1908 |title=Etudes cinematographique du mouvement brownien |trans_title=Cinematographic studies of Brownian motion |journal=Comptes Rendus |issue=146 |pages=1024–6 |language=French}}
* [[Lucretius]], 'On The Nature of Things.', translated by William Ellery Leonard. (''[http://onlinebooks.library.upenn.edu/webbin/gutbook/lookup?num=785 on-line version]'', from [[Project Gutenberg]]. See the heading 'Atomic Motions'; this translation differs slightly from the one quoted).
* Nelson, Edward, ''Dynamical Theories of Brownian Motion'' (1967). [http://www.math.princeton.edu/~nelson/books.html (PDF version of this out-of-print book, from the author's webpage.)]
* {{cite journal |author=Pearle, P.; Collett, B.; Bart, K.; Bilderback, D.; Newman, D.; Samuels, S. |year=2010 |url=http://ajp.aapt.org/resource/1/ajpias/v78/i12/p1278_s1 |title=What Brown saw and you can too |journal=Am. J. Phys. |volume=78 |issue=12 |pages=1278–1289 |bibcode=2010AmJPh..78.1278P |doi=10.1119/1.3475685 |subscription=yes}}
* [[Jean Baptiste Perrin|Perrin, J.]] (1909). "Mouvement brownien et réalité moléculaire" [Brownian movement and molecular reality]. ''[[Annales de chimie et de physique|Ann. Chim. Phys.]]'' 8ième série '''18''', 5–114.
** See also Perrin's book "Les Atomes" (1914).
* {{cite journal | author=Smoluchowski, M. | title=Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen | journal=Annalen der Physik  | volume=21 | pages=756–780 | year=1906 | url=http://gallica.bnf.fr/ark:/12148/bpt6k15328k/f770.chemindefer | doi=10.1002/andp.19063261405|bibcode = 1906AnP...326..756V  | issue=14}}
* {{cite book |last=Svedberg |first=T. |year=1907 |title=Studien zur Lehre von den kolloiden Losungen}}
* [[Thorvald N. Thiele|Theile, T. N.]]
** Danish version: "Om Anvendelse af mindste Kvadraters Methode i nogle Tilfælde, hvor en Komplikation af visse Slags uensartede tilfældige Fejlkilder giver Fejlene en ‘systematisk’ Karakter".
** French version: "Sur la compensation de quelques erreurs quasi-systématiques par la méthodes de moindre carrés" published simultaneously in ''Vidensk. Selsk. Skr. 5. Rk., naturvid. og mat. Afd.'', 12:381–408, 1880.
 
==External links==
{{Commons category|Brownian motion}}
{{wikisource|A brief account of microscopical observations made on the particles contained in the pollen of plants}}
* [http://galileo.phys.virginia.edu/classes/109N/more_stuff/Applets/brownian/applet.html Brownian motion java simulation]
* [https://www.youtube.com/watch?v=oOnD_S2jq4U Brownian motion to simulate stock prices in Excel] (YouTube video)
* [http://tritemio.github.io/PyBroMo/ A single-molecule brownian motion diffusion simulator]
* [http://xxx.imsc.res.in/abs/physics/0412132 Article for the school-going child]
* [https://web.archive.org/web/20010222031055/http://www.bun.kyoto-u.ac.jp/~suchii/einsteinBM.html Einstein on Brownian Motion]
* [http://arxiv.org/abs/0705.1951 Brownian Motion, "Diverse and Undulating"]
* [http://physerver.hamilton.edu/Research/Brownian/index.html Discusses history, botany and physics of Brown's original observations, with videos]
* [http://www.gizmag.com/einsteins-prediction-finally-witnessed/16212/ "Einstein's prediction finally witnessed one century later"] : a test to observe the velocity of Brownian motion
* [http://cafemath.fr/mathblog/article.php?page=BrownianMotion.php "Café math : brownian motion (Part 1)"] : A blog article describing brownian motion (definition, symmetries, simulation)
* [http://scitation.aip.org/content/aapt/journal/ajp/81/3/10.1119/1.4772632 "Finite different algorithm to simulate the Brownian motion of a particle"] {{subscription required}}
 
{{Fractals}}
{{Einstein}}
 
{{DEFAULTSORT:Brownian Motion}}
[[Category:Stochastic processes]]
[[Category:Variants of random walks]]
[[Category:Fractals]]
[[Category:Statistical mechanics]]
[[Category:Colloidal chemistry]]
[[Category:Robert Brown (botanist)]]
[[Category:Albert Einstein]]

Revision as of 20:05, 23 February 2014

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The evening sweetest moments originated from the performers giving thanks. Toning down the waterworks, we instead got solemn, heartfelt and genuinely moving. Best Actress winner Catherine Walker (The Talk From the Town) declaring her love for Buy Timberland Online Australia Best Actor's Declan Conlon (The House), Mullen thanking husband Sean McGinley whose "love was so strong, even when they had to work apart." And most touchingly, Best Supporting Actor, Aaron Monaghan (Conversations on the Homecoming), thanking the judges "for putting a smile on my mother

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Only the most nave can fail to notice how "it's the money, stupid" that determines clout. No matter what in: politics, business, sport. As well as in Irish racing, the money is HRI. Actually it is the Dept of Agriculture, but since HRI gets to decide who gets what on a daytoday basis, it's the administrative body, stupid.

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That afternoon we went for a big hunt along the fringe of some pine forest, once agin sign and prints were aplenty after a few hours and heading back, Balty saw a red that we must have walked Ugg Boots Sale Auburn past bedded down, go smashing with the forest. The hunt ended up being 8.5km so a total of 14 km covered during the day hunting on foot, this showed as balty ended up getting lost at the end as he am far behind.

The use of using national team names and nicknames, player names on the site and within the game is purely to describe the nature of the content and game being provided in this website. No official or commercial connection to the Cricket Australia, the ECB, The Ashes organising committee, the ICC or anyone player is implied or intended. This is purely a fundraising effort for local cricket clubs

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Because for some reason, it just doesn't seem real. You see an image of an airplane, gnatsized from the backdrop of New York City, because it careens into the WTC and sets off a fireball, and your mind shuts down. It wants to think that this is special effects, the plane really didn't strike that building, but instead passed behind it Replica Ray Ban Aviators as pyrotechnics were trigger. The bodies of people hurling themselves from the windows. those are mannequins, or computer graphics. The mind wants to believe these things, because it's simply not wired to process the magnitude of the tragedy. It's like attempting to count the stars in the sky while measuring their distance from Earth. You can quote numbers, but they're meaningless. Up to 50,000 people routinely make use of the WTC. To say nothing of people in and around the immediate area. A persons mind. it Barbour International Australia reads these numbers, and they don't /click/.

The 19 points Yorkshire took from this match mean that they leapfrog Hampshire into second place. With Kent also winning at Grace Road the final two rounds of matches look tasty. Evidently of it, Yorkshire have the easiest runin from the top four, and in two weeks' time today's events may seem to have been crucial.

Through the late 1970s, however, the deadly scourge had been eliminated from the face of the world thanks to mass vaccination campaigns that protected millions and left all of them with a small scar on their upper arm. With nowhere to hide in the natural world are the virus only host was beaten into extinction. and also the other in Russia. Absent a catastrophic lab accident, deliberate release or the genetic reengineering of the virus, smallpox won't ever again spread death and misery across the globe.

But MSNBC wimped out. It appears as though it caved into McCain complaints, which will anger liberals. It looks like it opted for dullness. David Gregory? FOX news didn get to be the numberone cable news network when you are dull and balanced, but when you are Oakley Jawbone controversial and obnoxious. MSNBC should have learned from FOX if it wants ratings, which it apparently doesn not as long as it thinks it is New Balance 996 "The Place for Politics" and not for biased sportstalkradio type ranting.

Jones said Jacobs served as associate pastor at St. Anthony Parish around the North Side from 1975 to 1979 so that as an associate pastor at St. Philip the Apostle Parish on the East Side from 1979 to 1981. Following studies at Boston College from 1981 to 1986, he taught philosophy in the Pontifical College Josephinum on the Far North Side until 1993. Also, he was a priest in residence at St. Paul the Apostle Parish in Westerville from 1986 to 1989.

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