Contents 1 History
2
2.1 Einstein's theory 2.2 Smoluchowski model 2.3 Other physics models using partial differential equations 2.4 Astrophysics: star motion within galaxies 3 Mathematics 3.1 Statistics 3.2 Lévy characterisation 3.3 Riemannian manifold 4 Narrow escape 5 See also 6 References 7 Further reading 8 External links History[edit]
The Roman Lucretius's scientific poem "On the Nature of Things" (c. 60
BC) has a remarkable description of
Reproduced from the book of Jean Baptiste Perrin, Les Atomes, three tracings of the motion of colloidal particles of radius 0.53 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 µm).[2] "Observe what happens when sunbeams are admitted into a building and
shed light on its shadowy places. You will see a multitude of tiny
particles mingling in a multitude of ways... their dancing is an
actual indication of underlying movements of matter that are hidden
from our sight... It originates with the atoms which move of
themselves [i.e., spontaneously]. Then those small compound bodies
that are least removed from the impetus of the atoms are set in motion
by the impact of their invisible blows and in turn cannon against
slightly larger bodies. So the movement mounts up from the atoms and
gradually emerges to the level of our senses, so that those bodies are
in motion that we see in sunbeams, moved by blows that remain
invisible."
Although the mingling 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.
While
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. The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval.[6][citation needed] Classical mechanics is unable to determine this distance because of the enormous number of bombardments a Brownian particle will undergo, roughly of the order of 1014 collisions per second.[1] Thus Einstein was led to consider the collective motion of Brownian particles[citation needed]. He regarded the increment of particle positions in unrestricted one dimensional (x) domain as a random variable ( Δ displaystyle Delta or x, under coordinate transformation so that the origin lies at the initial position of the particle) with some probability density function φ ( Δ ) displaystyle varphi (Delta ) . Further, assuming conservation of particle number, he expanded the density (number of particles per unit volume) change in a Taylor series, ρ ( x , t ) + τ ∂ ρ ( x ) ∂ t + ⋯ = ρ ( x , t + τ ) = ∫ − ∞ + ∞ ρ ( x + Δ , t ) ⋅ φ ( Δ ) d Δ = ρ ( x , t ) ⋅ ∫ − ∞ + ∞ φ ( Δ ) d Δ + ∂ ρ ∂ x ⋅ ∫ − ∞ + ∞ Δ ⋅ φ ( Δ ) d Δ + ∂ 2 ρ ∂ x 2 ⋅ ∫ − ∞ + ∞ Δ 2 2 ⋅ φ ( Δ ) d Δ + ⋯ = ρ ( x , t ) ⋅ 1 + 0 + ∂ 2 ρ ∂ x 2 ⋅ ∫ − ∞ + ∞ Δ 2 2 ⋅ φ ( Δ ) d Δ + ⋯ displaystyle begin aligned rho (x,t)+tau frac partial rho (x) partial t +cdots =rho (x,t+tau )= &int _ -infty ^ +infty rho (x+Delta ,t)cdot varphi (Delta ),mathrm d Delta \= &rho (x,t)cdot int _ -infty ^ +infty varphi (Delta ),dDelta + frac partial rho partial x cdot int _ -infty ^ +infty Delta cdot varphi (Delta ),mathrm d Delta \& + frac partial ^ 2 rho partial x^ 2 cdot int _ -infty ^ +infty frac Delta ^ 2 2 cdot varphi (Delta ),mathrm d Delta +cdots \= &rho (x,t)cdot 1+0+ frac partial ^ 2 rho partial x^ 2 cdot int _ -infty ^ +infty frac Delta ^ 2 2 cdot varphi (Delta ),mathrm d Delta +cdots end aligned where the second equality in the first line is by definition of φ displaystyle varphi . The integral in the first term is equal to one by the definition of probability, and the second and other even terms (i.e. first and other odd moments) vanish because of space symmetry. What is left gives rise to the following relation: ∂ ρ ∂ t = ∂ 2 ρ ∂ x 2 ⋅ ∫ − ∞ + ∞ Δ 2 2 τ ⋅ φ ( Δ ) d Δ + higher-order even moments displaystyle frac partial rho partial t = frac partial ^ 2 rho partial x^ 2 cdot int _ -infty ^ +infty frac Delta ^ 2 2,tau cdot varphi (Delta ),mathrm d Delta + text higher-order even moments Where the coefficient before the Laplacian, the second moment of probability of displacement Δ displaystyle Delta , is interpreted as mass diffusivity D: D = ∫ − ∞ + ∞ Δ 2 2 τ ⋅ φ ( Δ ) d Δ displaystyle D=int _ -infty ^ +infty frac Delta ^ 2 2,tau cdot varphi (Delta ),mathrm d Delta Then the density of Brownian particles ρ at point x at time t satisfies the diffusion equation: ∂ ρ ∂ t = D ⋅ ∂ 2 ρ ∂ x 2 , displaystyle frac partial rho partial t =Dcdot frac partial ^ 2 rho partial x^ 2 , Assuming that N particles start from the origin at the initial time t=0, the diffusion equation has the solution ρ ( x , t ) = N 4 π D t e − x 2 4 D t . displaystyle rho (x,t)= frac N sqrt 4pi Dt e^ - frac x^ 2 4Dt . This expression (which is a normal distribution with the mean μ = 0 displaystyle mu =0 and variance σ 2 = 2 D t displaystyle sigma ^ 2 =2Dt usually called
B t displaystyle B_ t ) allowed Einstein to calculate the moments directly. The first moment is seen to vanish, meaning that the Brownian particle is equally likely to move to the left as it is to move to the right. The second moment is, however, non-vanishing, being given by x 2 ¯ = 2 D t . displaystyle overline x^ 2 =2,D,t. 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.[4] 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.[7] 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 = μmg, where m is the mass of the particle, g is the acceleration due to gravity, and μ is the particle's mobility in the fluid. George Stokes had shown that the mobility for a spherical particle with radius r is μ = 1 6 π η r displaystyle mu = tfrac 1 6pi eta r , where η is the dynamic viscosity of the fluid. In a state of dynamic equilibrium, and under the hypothesis of isothermal fluid, the particles are distributed according to the barometric distribution ρ = ρ 0 e − m g h k B T , displaystyle rho =rho _ 0 e^ - frac mgh k_ B T , where ρ−ρ0 is the difference in density of particles separated by
a height difference of h, kB is
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 law, J = − D d ρ d h , displaystyle J=-D frac drho dh , where J = ρv. Introducing the formula for ρ, we find that v = D m g k B T . displaystyle v= frac Dmg k_ B T . In a state of dynamical equilibrium, this speed must also be equal to v = μmg. 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: x 2 ¯ 2 t = D = μ k B T = μ R T N = R T 6 π η r N . displaystyle frac overline x^ 2 2t =D=mu k_ B T= frac mu RT N = frac RT 6pi eta rN . 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 kB = R / N, and the fourth equality follows from Stokes's
formula for the mobility. By measuring the mean squared displacement
over a time interval along with the universal gas constant R, the
temperature T, the viscosity η, and the particle radius r, 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[8] 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".[8]
An identical expression to Einstein's formula for the diffusion
coefficient was also found by
k ′ = p 0 / k displaystyle k'=p_ 0 /k for the diffusion coefficient k′, where p 0 displaystyle p_ 0 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.[10] The use of
Smoluchowski model[edit] Smoluchowski's theory of Brownian motion[14] 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: ( Δ x ) 2 ¯ displaystyle overline (Delta x)^ 2 . 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 ( Δ x ) 2 ¯ = 2 D t = t 32 81 μ 2 π μ a = t 64 27 1 2 μ 2 3 π μ a , displaystyle 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 3pi mu a , where μ is the viscosity coefficient, and a is the radius of the particle. Associating the kinetic energy μ 2 / 2 displaystyle mu ^ 2 /2 with the thermal energy RT/N, the expression for the mean squared
displacement is 64/27 times that found by Einstein. The fraction 27/64
was commented on by
[ m − ( n − m ) ] [ m + n − m ] = 2 m − n n , displaystyle frac [m-(n-m)] [m+n-m] = frac 2m-n n , 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, P m , n = ( n m ) 2 − n , displaystyle P_ m,n = binom n m 2^ -n , with equal a priori probabilities of 1/2, the mean total gain is 2 m − n ¯ = ∑ m = n 2 n ( 2 m − n ) P m , n = n n ! 2 n [ ( n 2 ) ! ] 2 . displaystyle overline 2m-n =sum _ m= frac n 2 ^ n (2m-n)P_ m,n = frac nn! 2^ n left[left( frac n 2 right)!right]^ 2 . If n is large enough so that Stirling's approximation can be used in the form n ! ≈ ( n e ) n 2 π n , displaystyle n!approx left( frac n e right)^ n sqrt 2pi n , then the expected total gain will be[citation needed] 2 m − n ¯ ≈ 2 n π , displaystyle overline 2m-n approx sqrt frac 2n pi , showing that it increases as the square root of the total population. Suppose that a Brownian particle of mass M is surrounded by lighter particles of mass m which are traveling at a speed u. Then, reasons Smoluchowski, in any collision between a surrounding and Brownian particles, the velocity transmitted to the latter will be mu/M. This ratio is of the order of 10−7 cm/s. But we also have to take into consideration that in a gas there will be more than 1016 collisions in a second, and even greater in a liquid where we expect that there will be 1020 collision in one second. Some of these collisions will tend to accelerate the Brownian particle; others will tend to decelerate it. If there is a mean excess of one kind of collision or the other to be of the order of 108 to 1010 collisions in one second, then velocity of the Brownian particle may be anywhere between 10 and 1000 cm/s. Thus, even though there are equal probabilities for forward and backward collisions there will be a net tendency to keep the Brownian particle in motion, just as the ballot theorem predicts. These orders of magnitude are not exact because they don't take into consideration the velocity of the Brownian particle, U, which depends on the collisions that tend to accelerate and decelerate it. The larger U is, the greater will be the collisions that will retard it so that the velocity of a Brownian particle can never increase without limit. Could such a process occur, it would be tantamount to a perpetual motion of the second type. And since equipartition of energy applies, the kinetic energy of the Brownian particle, M U 2 / 2 displaystyle MU^ 2 /2 , will be equal, on the average, to the kinetic energy of the surrounding fluid particle, m u 2 / 2 displaystyle mu^ 2 /2 . In 1906 Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion.[18] The model assumes collisions with M ≫ m where M is the test particle's mass and m the mass of one of the individual particles composing the fluid. It is assumed that the particle collisions are confined to one dimension and that it is equally probable for the test particle to be hit from the left as from the right. It is also assumed that every collision always imparts the same magnitude of ΔV. If NR is the number of collisions from the right and NL the number of collisions from the left then after N collisions the particle's velocity will have changed by ΔV(2NR − N). The multiplicity is then simply given by: ( N N R ) = N ! N R ! ( N − N R ) ! displaystyle binom N N_ R = frac N! N_ R !(N-N_ R )! and the total number of possible states is given by 2N. Therefore, the probability of the particle being hit from the right NR times is: P N ( N R ) = N ! 2 N N R ! ( N − N R ) ! displaystyle P_ N (N_ R )= frac N! 2^ N N_ R !(N-N_ R )! As a result of its simplicity, Smoluchowski's 1D model can only
qualitatively describe Brownian motion. For a realistic particle
undergoing
v ⋆ displaystyle v_ star of the background stars by M V 2 ≈ m v ⋆ 2 displaystyle MV^ 2 approx mv_ star ^ 2 where m ≪ M displaystyle mll M is the mass of the background stars. The gravitational force from the massive object causes nearby stars to move faster than they otherwise would, increasing both v ⋆ displaystyle v_ star and V.[20] The Brownian velocity of Sgr A*, the supermassive black hole at the center of the Milky Way galaxy, is predicted from this formula to be less than 1 km s−1.[21] Mathematics[edit] Main article: Wiener process Play media 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,
A single realisation of three-dimensional
The
W0 = 0 Wt is almost surely continuous Wt has independent increments W t − W s ∼ N ( 0 , t − s ) displaystyle W_ t -W_ s sim mathcal N (0,t-s) (for 0 ≤ s ≤ t displaystyle 0leq sleq t ). N ( μ , σ 2 ) displaystyle mathcal N (mu ,sigma ^ 2 ) denotes the normal distribution with expected value μ and variance σ2. The condition that it has independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 displaystyle 0leq s_ 1 <t_ 1 leq s_ 2 <t_ 2 then W t 1 − W s 1 displaystyle W_ t_ 1 -W_ s_ 1 and W t 2 − W s 2 displaystyle W_ t_ 2 -W_ s_ 2 are independent random variables.
An alternative characterisation of the
[ W t , W t ] = t displaystyle [W_ t ,W_ t ]=t .
A third characterisation is that the
N ( 0 , 1 ) displaystyle mathcal N (0,1) random variables. This representation can be obtained using the
Karhunen–Loève theorem.
The
X is a
X is a martingale with respect to P (and its own natural filtration); and for all 1 ≤ i, j ≤ n, Xi(t)Xj(t) −δijt is a martingale with respect to P (and its own natural filtration), where δij denotes the Kronecker delta. Riemannian manifold[edit]
This section may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (June 2011) (Learn how and when to remove this template message) The infinitesimal generator (and hence characteristic operator) of a
A displaystyle mathcal A in local coordinates xi, 1 ≤ i ≤ m, is given
by ½ΔLB, where ΔLB is the
Δ L B = 1 det ( g ) ∑ i = 1 m ∂ ∂ x i ( det ( g ) ∑ j = 1 m g i j ∂ ∂ x j ) , displaystyle Delta _ mathrm LB = frac 1 sqrt det(g) sum _ i=1 ^ m frac partial partial x_ i left( sqrt det(g) sum _ j=1 ^ m g^ ij frac partial partial x_ j right), where [gij] = [gij]−1 in the sense of the inverse of a square matrix. Narrow escape[edit] 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[edit] Brownian bridge: a
References[edit] ^ a b Feynman, R. (1964). "The Brownian Movement". The Feynman
Lectures of Physics, Volume I. pp. 41-1.
^ Perrin, Jean (1914). Atoms. p. 115.
^ Mandelbrot, B.; Hudson, R. (2004). The (Mis)behavior of Markets: A
Further reading[edit] 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" (PDF). Philosophical
Magazine. 4: 161–173. Also includes a subsequent defense by
Brown of his original observations, Additional remarks on active
molecules.
Chaudesaigues, M. (1908). "Le mouvement brownien et la formule
d'Einstein" [
See also Perrin's book "Les Atomes" (1914). von Smoluchowski, M. (1906). "Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen". Annalen der Physik. 21 (14): 756–780. Bibcode:1906AnP...326..756V. doi:10.1002/andp.19063261405. Svedberg, T. (1907). Studien zur Lehre von den kolloiden Losungen. 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[edit] Wikimedia Commons has media related to Brownian motion.
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