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Brownian motion

From Wikipedia, the free encyclopedia
Random motion of particles suspended in a fluid

2-dimensional random walk of a silveradatom on an Ag(111) surface[1]
Simulation of the Brownian motion of a large particle, analogous to a dust particle, that collides with a large set of smaller particles, analogous to molecules of a gas, which move with different velocities in different random directions

Brownian motion is the random motion ofparticles suspended in a medium (aliquid or agas).[2] The traditional mathematical formulation of Brownian motion is that of theWiener process, which is often called Brownian motion, even in mathematical sources.

This motion pattern typically consists ofrandom fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid atthermal equilibrium, defined by a giventemperature. Within such a fluid, there exists no preferential direction of flow (as intransport phenomena). More specifically, the fluid's overalllinear andangular momenta remain null over time. Thekinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid'sinternal energy (theequipartition theorem).[3]

This motion is named after the Scottish botanistRobert Brown, who first described the phenomenon in 1827, while looking through a microscope atpollen of the plantClarkia pulchella immersed in water. In 1900, the French mathematicianLouis Bachelier modeled the stochastic process now called Brownian motion in his doctoral thesis, The Theory of Speculation (Théorie de la spéculation), prepared under the supervision ofHenri Poincaré. Then, in 1905, theoretical physicistAlbert Einstein publisheda paper in which he modelled the motion of the pollen particles as being moved by individual watermolecules, making one of his first major scientific contributions.[4]

The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. This explanation of Brownian motion served as convincing evidence thatatoms and molecules exist and was further verified experimentally byJean Perrin in 1908. Perrin was awarded theNobel Prize in Physics in 1926 "for his work on the discontinuous structure of matter".[5]

Themany-body interactions that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. Consequently, only probabilistic models applied tomolecular populations can be employed to describe it.[6] Two such models of thestatistical mechanics, due to Einstein and Smoluchowski, are presented below. Another, pure probabilistic class of models is the class of thestochastic process models. There exist sequences of both simpler and more complicated stochastic processes which converge (in thelimit) to Brownian motion (seerandom walk andDonsker's theorem).[7][8]

History

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Reproduced from the book ofJean 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).[9]

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

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

Although the mingling, tumbling motion of dust particles is caused largely by air currents, the glittering, jiggling motion of small dust particles is caused chiefly by trueBrownian dynamics; Lucretius "perfectly describes and explains the Brownian movement by a wrong example".[10]

The discovery of this phenomenon is credited to the botanistRobert Brown in 1827. Brown was studying plant reproduction when he observedpollen grains of the plantClarkia pulchella in water under a microscope. These grains contain minute particles on the order of 1/4000th of an inch (6 microns) in size. He observed these particles 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.[11][12]

The mathematics of much of stochastic analysis including the mathematics of Brownian motion was introduced byLouis Bachelier in 1900 in his PhD thesis "The theory of speculation", in which he presented an analysis of the stock and option markets. However this work was largely unknown until the 1950s.[13][14]: 33 

Albert Einstein (in one of his1905 papers) provided an explanation of Brownian motion in terms of atoms and molecules at a time when their existence was still debated. Einstein proved the relation between the probability distribution of a Brownian particle and thediffusion equation.[14]: 33  These equations describing Brownian motion were subsequently verified by the experimental work ofJean Baptiste Perrin in 1908, leading to his Nobel prize.[15]Norbert Wiener gave the first complete and rigorous mathematical analysis in 1923, leading to the underlying mathematical concept being called aWiener process.[14]

The instantaneous velocity of the Brownian motion can be defined asv = Δx/Δt, whenΔt <<τ, whereτ is the momentum relaxation time. In 2010, the instantaneous velocity of a Brownian particle (a glass microsphere trapped in air withoptical tweezers) was measured successfully. The velocity data verified theMaxwell–Boltzmann velocity distribution, and the equipartition theorem for a Brownian particle.[16]

Statistical mechanics theories

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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 themean squared displacement of a Brownian particle, while the second part consists in relating the diffusion coefficient to measurable physical quantities.[17] In this way Einstein was able to determine the size of atoms, and how many atoms there are in a mole, or themolecular weight in grams, of a gas.[18] In accordance toAvogadro'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 theAvogadro 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 themolar mass of the gas by theAvogadro constant.

The characteristic bell-shaped curves of the diffusion of Brownian particles. The distribution begins as aDirac delta function, indicating that all the particles are located at the origin at timet = 0. Ast increases, the distribution flattens (though remains bell-shaped), and ultimately becomes uniform in the limit that time goes to infinity.

The first part of Einstein's argument was to determine how far a Brownian particle travels in a given time interval.[4] 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.[2]

He regarded the increment of particle positions in timeτ{\displaystyle \tau } in a one-dimensional (x) space (with the coordinates chosen so that the origin lies at the initial position of the particle) as arandom variable (q{\displaystyle q}) with someprobability density functionφ(q){\displaystyle \varphi (q)} (i.e.,φ(q){\displaystyle \varphi (q)} is the probability density for a jump of magnitudeq{\displaystyle q}, i.e., the probability density of the particle incrementing its position fromx{\displaystyle x} tox+q{\displaystyle x+q} in the time intervalτ{\displaystyle \tau }). Further, assuming conservation of particle number, he expanded thenumber densityρ(x,t+τ){\displaystyle \rho (x,t+\tau )} (number of particles per unit volume aroundx{\displaystyle x}) at timet+τ{\displaystyle t+\tau } in aTaylor series,ρ(x,t+τ)=ρ(x,t)+τ∂ρ(x,t)∂t+⋯=∫−∞∞ρ(x−q,t)φ(q)dq=Eq[ρ(x−q,t)]=ρ(x,t)∫−∞∞φ(q)dq−∂ρ∂x∫−∞∞qφ(q)dq+∂2ρ∂x2∫−∞∞q22φ(q)dq+⋯=ρ(x,t)⋅1−0+∂2ρ∂x2∫−∞∞q22φ(q)dq+⋯{\displaystyle {\begin{aligned}\rho (x,t+\tau )={}&\rho (x,t)+\tau {\frac {\partial \rho (x,t)}{\partial t}}+\cdots \\[2ex]={}&\int _{-\infty }^{\infty }\rho (x-q,t)\,\varphi (q)\,dq=\mathbb {E} _{q}{\left[\rho (x-q,t)\right]}\\[1ex]={}&\rho (x,t)\,\int _{-\infty }^{\infty }\varphi (q)\,dq-{\frac {\partial \rho }{\partial x}}\,\int _{-\infty }^{\infty }q\,\varphi (q)\,dq+{\frac {\partial ^{2}\rho }{\partial x^{2}}}\,\int _{-\infty }^{\infty }{\frac {q^{2}}{2}}\varphi (q)\,dq+\cdots \\[1ex]={}&\rho (x,t)\cdot 1-0+{\cfrac {\partial ^{2}\rho }{\partial x^{2}}}\,\int _{-\infty }^{\infty }{\frac {q^{2}}{2}}\varphi (q)\,dq+\cdots \end{aligned}}}where the second equality is by definition ofφ{\displaystyle \varphi }. Theintegral 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 oddmoments) vanish because of space symmetry. What is left gives rise to the following relation:∂ρ∂t=∂2ρ∂x2⋅∫−∞∞q22τφ(q)dq+higher-order even moments.{\displaystyle {\frac {\partial \rho }{\partial t}}={\frac {\partial ^{2}\rho }{\partial x^{2}}}\cdot \int _{-\infty }^{\infty }{\frac {q^{2}}{2\tau }}\varphi (q)\,dq+{\text{higher-order even moments.}}}Where the coefficient after theLaplacian, the second moment of probability of displacementq{\displaystyle q}, is interpreted asmass diffusivityD:D=∫−∞∞q22τφ(q)dq.{\displaystyle D=\int _{-\infty }^{\infty }{\frac {q^{2}}{2\tau }}\varphi (q)\,dq.}Then the density of Brownian particlesρ at pointx at timet satisfies thediffusion equation:∂ρ∂t=D⋅∂2ρ∂x2,{\displaystyle {\frac {\partial \rho }{\partial t}}=D\cdot {\frac {\partial ^{2}\rho }{\partial x^{2}}},}

Assuming thatN particles start from the origin at the initial timet = 0, the diffusion equation has the solution:[19]ρ(x,t)=N4πDtexp⁡(−x24Dt).{\displaystyle \rho (x,t)={\frac {N}{\sqrt {4\pi Dt}}}\exp {\left(-{\frac {x^{2}}{4Dt}}\right)}.}This expression (which is anormal distribution with the meanμ=0{\displaystyle \mu =0} and varianceσ2=2Dt{\displaystyle \sigma ^{2}=2Dt} usually called Brownian motionBt{\displaystyle B_{t}}) allowed Einstein to calculate themoments 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 byE[x2]=2Dt.{\displaystyle \mathbb {E} {\left[x^{2}\right]}=2Dt.}This equation expresses the mean squared displacement in terms of the time elapsed and the diffusivity. From this expression Einstein argued that the displacement of a Brownian particle is not proportional to the elapsed time, but rather to its square root.[17] 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.[20]

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

In his original treatment, Einstein considered anosmotic 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 ofv =μmg, wherem is the mass of the particle,g is the acceleration due to gravity, andμ is the particle'smobility in the fluid.George Stokes had shown that the mobility for a spherical particle with radiusr isμ=16πηr{\displaystyle \mu ={\tfrac {1}{6\pi \eta r}}}, whereη is thedynamic viscosity of the fluid. In a state of dynamic equilibrium, and under the hypothesis of isothermal fluid, the particles are distributed according to thebarometric distributionρ=ρoexp⁡(−mghkBT),{\displaystyle \rho =\rho _{o}\,\exp \left({-{\frac {mgh}{k_{\text{B}}T}}}\right),}whereρ −ρo is the difference in density of particles separated by a height difference, ofh=z−zo{\displaystyle h=z-z_{o}},kB is theBoltzmann constant (the ratio of theuniversal gas constant,R, to theAvogadro constant,NA), andT is theabsolute temperature.

Perrin examined the equilibrium (barometric distribution) of granules (0.6microns) ofgamboge, a viscous substance, under the microscope. The granules move against gravity to regions of lower concentration. The relative change in density observed in 10 microns of suspension is equivalent to that occurring in 6 km of air.

Dynamic equilibrium is established because the more that particles are pulled down bygravity, the greater the tendency for the particles to migrate to regions of lower concentration. The flux is given byFick's law,J=−Ddρdh,{\displaystyle J=-D{\frac {d\rho }{dh}},}whereJ =ρv. Introducing the formula forρ, we find thatv=DmgkBT.{\displaystyle v={\frac {Dmg}{k_{\text{B}}T}}.}

In a state of dynamical equilibrium, this speed must also be equal tov =μmg. Both expressions forv are proportional tomg, reflecting that the derivation is independent of the type of forces considered. Similarly, one can derive an equivalent formula for identicalcharged particles of chargeq in a uniformelectric field of magnitudeE, wheremg is replaced with theelectrostatic forceqE. Equating these two expressions yields theEinstein relation for the diffusivity, independent ofmg orqE or other such forces:E[x2]2t=D=μkBT=μRTNA=RT6πηrNA.{\displaystyle {\frac {\mathbb {E} {\left[x^{2}\right]}}{2t}}=D=\mu k_{\text{B}}T={\frac {\mu RT}{N_{\text{A}}}}={\frac {RT}{6\pi \eta rN_{\text{A}}}}.}Here the first equality follows from the first part of Einstein's theory, the third equality follows from the definition of theBoltzmann constant askB =R /NA, and the fourth equality follows from Stokes's formula for the mobility. By measuring the mean squared displacement over a time interval along with the universal gas constantR, the temperatureT, the viscosityη, and the particle radiusr, the Avogadro constantNA can be determined.

The type of dynamical equilibrium proposed by Einstein was not new. It had been pointed out previously byJ. J. Thomson[21] in his series of lectures at Yale University in May 1903 that the dynamic equilibrium between the velocity generated by aconcentration 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".[21]

An identical expression to Einstein's formula for the diffusion coefficient was also found byWalther Nernst in 1888[22] in which he expressed the diffusion coefficient as the ratio of theosmotic pressure to the ratio of thefrictional 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 byStokes's law. He writesk′=po/k{\displaystyle k'=p_{o}/k} for the diffusion coefficientk′, wherepo{\displaystyle p_{o}} is the osmotic pressure andk is the ratio of the frictional force to the molecular viscosity which he assumes is given by Stokes's formula for the viscosity. Introducing theideal gas law per unit volume for the osmotic pressure, the formula becomes identical to that of Einstein's.[23] 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 themean free path.[24]

Confirming Einstein's formula experimentally proved difficult.Initial attempts byTheodor Svedberg in 1906 and 1907 were critiqued by Einstein and by Perrin as not measuring a quantity directly comparable to the formula.Victor Henri in 1908 took cinematographic shots through a microscope and found quantitative disagreement with the formula but again the analysis was uncertain.[25] Einstein's predictions were finally confirmed in a series of experiments carried out by Chaudesaigues in 1908 and Perrin in 1909.[26][11] The confirmation of Einstein's theory constituted empirical progress for thekinetic theory of heat. In essence, Einstein showed that the motion can be predicted directly from the kinetic model ofthermal equilibrium. The importance of the theory lay in the fact that it confirmed the kinetic theory's account of thesecond law of thermodynamics as being an essentially statistical law.[27]

Brownian motion model of the trajectory of a particle of dye in water

Smoluchowski model

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Smoluchowski's theory of Brownian motion[28] starts from the same premise as that of Einstein and derives the same probability distributionρ(x,t) for the displacement of a Brownian particle along thex in timet. He therefore gets the same expression for the mean squared displacement:E[(Δx)2]{\displaystyle \mathbb {E} {\left[(\Delta x)^{2}\right]}}. However, when he relates it to a particle of massm moving at a velocityu which is the result of a frictional force governed by Stokes's law, he findsE[(Δx)2]=2Dt=t3281mu2πμa=t642712mu23πμa,{\displaystyle \mathbb {E} {\left[(\Delta x)^{2}\right]}=2Dt=t{\frac {32}{81}}{\frac {mu^{2}}{\pi \mu a}}=t{\frac {64}{27}}{\frac {{\frac {1}{2}}mu^{2}}{3\pi \mu a}},}whereμ is the viscosity coefficient, anda is the radius of the particle. Associating the kinetic energymu2/2{\displaystyle mu^{2}/2} with the thermal energyRT/N, the expression for the mean squared displacement is64/27 times that found by Einstein. The fraction 27/64 was commented on byArnold Sommerfeld in his necrology on Smoluchowski: "The numerical coefficient of Einstein, which differs from Smoluchowski by 27/64 can only be put in doubt."[29]

Smoluchowski[30] attempts to answer the question of why a Brownian particle should be displaced by bombardments of smaller particles when the probabilities for striking it in the forward and rear directions are equal.If the probability ofm gains andn −m losses follows abinomial distribution,Pm,n=(nm)2−n,{\displaystyle P_{m,n}={\binom {n}{m}}2^{-n},}with equala priori probabilities of 1/2, the mean total gain isE[2m−n]=∑m=n2n(2m−n)Pm,n=nn!2n+1[(n2)!]2.{\displaystyle \mathbb {E} {\left[2m-n\right]}=\sum _{m={\frac {n}{2}}}^{n}(2m-n)P_{m,n}={\frac {nn!}{2^{n+1}\left[\left({\frac {n}{2}}\right)!\right]^{2}}}.}

Ifn is large enough so that Stirling's approximation can be used in the formn!≈(ne)n2πn,{\displaystyle n!\approx \left({\frac {n}{e}}\right)^{n}{\sqrt {2\pi n}},}then the expected total gain will be[citation needed]E[2m−n]≈n2π,{\displaystyle \mathbb {E} {\left[2m-n\right]}\approx {\sqrt {\frac {n}{2\pi }}},}showing that it increases as the square root of the total population.

Suppose that a Brownian particle of massM is surrounded by lighter particles of massm which are traveling at a speedu. Then, reasons Smoluchowski, in any collision between a surrounding and Brownian particles, the velocity transmitted to the latter will bemu/M. This ratio is of the order of10−7 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 between10–1000 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 do not take into consideration the velocity of the Brownian particle,U, which depends on the collisions that tend to accelerate and decelerate it. The largerU 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,MU2/2{\displaystyle MU^{2}/2}, will be equal, on the average, to the kinetic energy of the surrounding fluid particle,mu2/2{\displaystyle mu^{2}/2}.

In 1906, Smoluchowski published a one-dimensional model to describe a particle undergoing Brownian motion.[31] The model assumes collisions withM ≫m whereM is the test particle's mass andm 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 ofΔV. IfNR is the number of collisions from the right andNL the number of collisions from the left then afterN collisions the particle's velocity will have changed byΔV(2NR −N). Themultiplicity is then simply given by:(NNR)=N!NR!(N−NR)!{\displaystyle {\binom {N}{N_{\text{R}}}}={\frac {N!}{N_{\text{R}}!(N-N_{\text{R}})!}}}and the total number of possible states is given by2N. Therefore, the probability of the particle being hit from the rightNR times is:PN(NR)=N!2NNR!(N−NR)!{\displaystyle P_{N}(N_{\text{R}})={\frac {N!}{2^{N}N_{\text{R}}!(N-N_{\text{R}})!}}}

As a result of its simplicity, Smoluchowski's 1D model can only qualitatively describe Brownian motion. For a realistic particle undergoing Brownian motion in a fluid, many of the assumptions don't apply. For example, the assumption that on average there are 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 possibleΔVs instead of always just one in a realistic situation.

Langevin equation

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Main article:Langevin equation

Thediffusion equation yields an approximation of the time evolution of theprobability density function associated with the position of the particle going under a Brownian movement under the physical definition. The approximation becomes valid on timescales much larger than the timescale of individual atomic collisions, since it does not include a term to describe the acceleration of particles during collision. The time evolution of the position of the Brownian particle over all time scales described using theLangevin equation, an equation that involves a random force field representing the effect of thethermal fluctuations of the solvent on the particle.[16] At longer times scales, where acceleration is negligible, individual particle dynamics can be approximated usingBrownian dynamics in place ofLangevin dynamics.

Astrophysics: star motion within galaxies

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Instellar dynamics, a massive body (star,black hole, etc.) can experience Brownian motion as it responds togravitational forces from surrounding stars.[32] The rms velocityV of the massive object, of massM, is related to the rms velocityv⋆{\displaystyle v_{\star }} of the background stars byMV2≈mv⋆2{\displaystyle MV^{2}\approx mv_{\star }^{2}}wherem≪M{\displaystyle m\ll M} is the mass of the background stars. The gravitational force from the massive object causes nearby stars to move faster than they otherwise would, increasing bothv⋆{\displaystyle v_{\star }} andV.[32] The Brownian velocity ofSgr A*, thesupermassive black hole at the center of theMilky Way galaxy, is predicted from this formula to be less than 1 km s−1.[33]

Mathematics

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Main article:Wiener process
An animated example of a Brownian motion-likerandom walk on a 2D surface with periodic boundary conditions. In thescaling limit, random walk approaches the Wiener process according toDonsker's theorem.

Inmathematics, Brownian motion is described by theWiener process, a continuous-timestochastic process named in honor ofNorbert Wiener. It is one of the best knownLévy processes (càdlàg stochastic processes withstationaryindependent increments) and occurs frequently in pure and applied mathematics,economics andphysics.

A single realisation of three-dimensional Brownian motion for times0 ≤t ≤ 2

The Wiener processWt is characterized by four facts:[34]

  1. W0 = 0
  2. Wt isalmost surely continuous
  3. Wt has independent increments
  4. Wt−Ws∼N(0,t−s){\displaystyle W_{t}-W_{s}\sim {\mathcal {N}}(0,t-s)}(for0≤s≤t{\displaystyle 0\leq s\leq t}).

N(μ,σ2){\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})} denotes thenormal distribution withexpected valueμ andvarianceσ2. The condition that it has independent increments means that if0≤s1<t1≤s2<t2{\displaystyle 0\leq s_{1}<t_{1}\leq s_{2}<t_{2}} thenWt1−Ws1{\displaystyle W_{t_{1}}-W_{s_{1}}} andWt2−Ws2{\displaystyle W_{t_{2}}-W_{s_{2}}} are independent random variables. In addition, for somefiltrationFt{\displaystyle {\mathcal {F}}_{t}},Wt{\displaystyle W_{t}} isFt{\displaystyle {\mathcal {F}}_{t}}measurable for allt≥0{\displaystyle t\geq 0}.

An alternative characterisation of the Wiener process is the so-calledLévy characterisation that says that the Wiener process is an almost surely continuousmartingale withW0 = 0 andquadratic variation[Wt,Wt]=t{\displaystyle [W_{t},W_{t}]=t}.

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

The Wiener process can be constructed as thescaling limit of arandom walk, or other discrete-time stochastic processes with stationary independent increments. This is known asDonsker's theorem. Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixedneighborhood of the origin infinitely often) whereas it is not recurrent in dimensions three and higher. Unlike the random walk, it isscale invariant.A d-dimensionalGaussian free field has been described as "a d-dimensional-time analog of Brownian motion."[35]

Statistics

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The Brownian motion can be modeled by arandom walk.[36]

In the general case, Brownian motion is aMarkov process and described bystochastic integral equations.[37]

Lévy characterisation

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The French mathematicianPaul Lévy proved the following theorem, which gives a necessary and sufficient condition for a continuousRn-valued stochastic processX to actually ben-dimensional Brownian motion. Hence, Lévy's condition can actually be used as an alternative definition of Brownian motion.

LetX = (X1, ...,Xn) be a continuous stochastic process on aprobability space(Ω, Σ,P) taking values inRn. Then the following are equivalent:

  1. X is a Brownian motion with respect toP, i.e., the law ofX with respect toP is the same as the law of ann-dimensional Brownian motion, i.e., thepush-forward measureX∗(P) isclassical Wiener measure onC0([0, ∞);Rn).
  2. both
    1. X is amartingale with respect toP (and its ownnatural filtration); and
    2. for all1 ≤i,j ≤n,Xi(t)Xj(t) −δijt is a martingale with respect toP (and its ownnatural filtration), whereδij denotes theKronecker delta.

Spectral content

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The spectral content of a stochastic processXt{\displaystyle X_{t}} can be found from thepower spectral density, formally defined asS(ω)=limT→∞1TE{|∫0TeiωtXtdt|2},{\displaystyle S(\omega )=\lim _{T\to \infty }{\frac {1}{T}}\mathbb {E} \left\{\left|\int _{0}^{T}e^{i\omega t}X_{t}dt\right|^{2}\right\},}whereE{\displaystyle \mathbb {E} } stands for theexpected value. The power spectral density of Brownian motion is found to be[38]SBM(ω)=4Dω2.{\displaystyle S_{BM}(\omega )={\frac {4D}{\omega ^{2}}}.}whereD is thediffusion coefficient ofXt. For naturally occurring signals, the spectral content can be found from the power spectral density of a single realization, with finite available time, i.e.,S(1)(ω,T)=1T|∫0TeiωtXtdt|2,{\displaystyle S^{(1)}(\omega ,T)={\frac {1}{T}}\left|\int _{0}^{T}e^{i\omega t}X_{t}dt\right|^{2},}which for an individual realization of a Brownian motion trajectory,[39] it is found to have expected valueμBM(ω,T){\displaystyle \mu _{BM}(\omega ,T)}μBM(ω,T)=4Dω2[1−sin⁡(ωT)ωT]{\displaystyle \mu _{\text{BM}}(\omega ,T)={\frac {4D}{\omega ^{2}}}\left[1-{\frac {\sin \left(\omega T\right)}{\omega T}}\right]}andvarianceσBM2(ω,T){\displaystyle \sigma _{\text{BM}}^{2}(\omega ,T)}[39]σS2(f,T)=E{(ST(j)(f))2}−μS2(f,T)=20D2f4[1−(6−cos⁡(fT))2sin⁡(fT)5fT+(17−cos⁡(2fT)−16cos⁡(fT))10f2T2].{\displaystyle \sigma _{S}^{2}(f,T)=\mathbb {E} \left\{\left(S_{T}^{(j)}(f)\right)^{2}\right\}-\mu _{S}^{2}(f,T)={\frac {20D^{2}}{f^{4}}}\left[1-{\Big (}6-\cos \left(fT\right){\Big )}{\frac {2\sin \left(fT\right)}{5fT}}+{\frac {{\Big (}17-\cos \left(2fT\right)-16\cos \left(fT\right){\Big )}}{10f^{2}T^{2}}}\right].}

For sufficiently long realization times, the expected value of the power spectrum of a single trajectory converges to the formally defined power spectral densityS(ω){\displaystyle S(\omega )}, but its coefficient of variationγ=σ/μ{\displaystyle \gamma =\sigma /\mu } tends to5/2{\displaystyle {\sqrt {5}}/2}. This implies the distribution ofS(1)(ω,T){\displaystyle S^{(1)}(\omega ,T)} is broad even in the infinite time limit.

Riemannian manifolds

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Brownian motion on a sphere

Brownian motion is usually considered to take place inEuclidean space. It is natural to consider how such motion generalizes to more complex shapes, such assurfaces or higher dimensionalmanifolds. The formalization requires the space to possess some form of aderivative, as well as ametric, so that aLaplacian can be defined. Both of these are available onRiemannian manifolds.

Riemannian manifolds have the property thatgeodesics can be described inpolar coordinates; that is, displacements are always in a radial direction, at some given angle. Uniform random motion is then described by Gaussians along the radial direction, independent of the angle, the same as in Euclidean space.

Theinfinitesimal generator (and hencecharacteristic operator) of Brownian motion on EuclideanRn is⁠1/2⁠Δ, whereΔ denotes theLaplace operator. Brownian motion on anm-dimensionalRiemannian manifold(M,g) can be defined as diffusion onM with the characteristic operator given by⁠1/2⁠ΔLB, half theLaplace–Beltrami operatorΔLB.

One of the topics of study is a characterization of thePoincaré recurrence time for such systems.[15]

Narrow escape

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Thenarrow escape problem is a ubiquitous problem in biology, biophysics and cellular biology which has the following formulation: a Brownian particle (ion,molecule, orprotein) 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 asingular perturbation problem.

See also

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References

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  1. ^Meyburg, Jan Philipp; Diesing, Detlef (2017). "Teaching the Growth, Ripening, and Agglomeration of Nanostructures in Computer Experiments".Journal of Chemical Education.94 (9):1225–1231.Bibcode:2017JChEd..94.1225M.doi:10.1021/acs.jchemed.6b01008.
  2. ^abFeynman, Richard (1964)."The Brownian Movement".The Feynman Lectures of Physics, Volume I. p. 41.
  3. ^Pathria, RK (1972). Statistical Mechanics. Pergamon Press. pp. 43–48, 73–74. ISBN 0-08-016747-0.
  4. ^abEinstein, Albert (1905)."Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen" [On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat](PDF).Annalen der Physik (in German).322 (8):549–560.Bibcode:1905AnP...322..549E.doi:10.1002/andp.19053220806.Archived(PDF) from the original on 9 October 2022.
  5. ^"The Nobel Prize in Physics 1926".NobelPrize.org. Retrieved29 May 2019.
  6. ^Tsekov, Roumen (1995). "Brownian motion of molecules: the classical theory".Ann. Univ. Sofia.88: 57.arXiv:1005.1490.Bibcode:1995AUSFC..88...57T.the behavior of a Brownian particle is quite irregular and can be described only in the frames of a statistical approach.
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  9. ^Perrin, Jean (1914).Atoms. London : Constable. p. 115.{{cite book}}: CS1 maint: publisher location (link)
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  11. ^abBrush, Stephen G. (1968)."A History of Random Processes: I. Brownian Movement from Brown to Perrin".Archive for History of Exact Sciences.5 (1):1–36.ISSN 0003-9519.
  12. ^Brown, Robert, Bennett, John J., Hardwicke, Robert (1866).The miscellaneous botanical works of Robert Brown. Vol. 1. Published for the Ray society by R. Hardwicke. pp. 463–486.
  13. ^Davis, Mark H. A.; Bachelier, Louis; Etheridge, Alison (2011).Louis Bachelier's Theory of Speculation: The Origins of Modern Finance. Princeton: Princeton University Press.ISBN 978-1-4008-2930-9.
  14. ^abcMörters, Peter; Peres, Yuval (1 January 2001).Brownian Motion (1 ed.). Cambridge University Press.doi:10.1017/cbo9780511750489.ISBN 978-0-521-76018-8.
  15. ^abGrigor’yan, Alexander (19 February 1999)."Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds".Bulletin of the American Mathematical Society.36 (2):135–249.doi:10.1090/S0273-0979-99-00776-4.ISSN 0273-0979.
  16. ^abLi, Tongcang; Kheifets, Simon; Medellin, David; Raizen, Mark (2010)."Measurement of the instantaneous velocity of a Brownian particle"(PDF).Science.328 (5986):1673–1675.Bibcode:2010Sci...328.1673L.CiteSeerX 10.1.1.167.8245.doi:10.1126/science.1189403.PMID 20488989.S2CID 45828908. Archived fromthe original(PDF) on 31 March 2011.
  17. ^abEinstein, Albert (1956) [1926].Investigations on the Theory of the Brownian Movement(PDF). Dover Publications.Archived(PDF) from the original on 9 October 2022. Retrieved25 December 2013.
  18. ^Stachel, J., ed. (1989)."Einstein's Dissertation on the Determination of Molecular Dimensions"(PDF).The Collected Papers of Albert Einstein, Volume 2. Princeton University Press.Archived(PDF) from the original on 9 October 2022.
  19. ^Kozdron, Michael J. (2008)."Brownian Motion and the Heat Equation - chpt. 3 Albert Einstein's proof of the existence of Brownian motion"(PDF).University of Regina. Archived fromthe original(PDF) on 26 November 2024. Retrieved3 November 2025.
  20. ^Lavenda, Bernard H. (1985).Nonequilibrium Statistical Thermodynamics. John Wiley & Sons. p. 20.ISBN 978-0-471-90670-4.
  21. ^abThomson, J. J. (1904).Electricity and Matter. Yale University Press. pp. 80–83.
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  27. ^See P. Clark 1976, p. 97
  28. ^Smoluchowski, M. M. (1906)."Sur le chemin moyen parcouru par les molécules d'un gaz et sur son rapport avec la théorie de la diffusion" [On the average path taken by gas molecules and its relation with the theory of diffusion].Bulletin International de l'Académie des Sciences de Cracovie (in French): 202.
  29. ^See p. 535 inSommerfeld, A. (1917). "Zum Andenken an Marian von Smoluchowski" [In Memory of Marian von Smoluchowski].Physikalische Zeitschrift (in German).18 (22):533–539.
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  31. ^von Smoluchowski, M. (1906)."Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen".Annalen der Physik (in German).326 (14):756–780.Bibcode:1906AnP...326..756V.doi:10.1002/andp.19063261405.
  32. ^abMerritt, David (2013).Dynamics and Evolution of Galactic Nuclei. Princeton University Press. p. 575.ISBN 9781400846122.OL 16802359W.
  33. ^Reid, M. J.; Brunthaler, A. (2004). "The Proper Motion of Sagittarius A*. II. The Mass of Sagittarius A*".The Astrophysical Journal.616 (2):872–884.arXiv:astro-ph/0408107.Bibcode:2004ApJ...616..872R.doi:10.1086/424960.S2CID 16568545.
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  35. ^Sheffield, Scott (9 May 2007)."Gaussian free fields for mathematicians".Probability Theory and Related Fields.139 (3–4):521–541.arXiv:math/0312099.doi:10.1007/s00440-006-0050-1.ISSN 0178-8051.
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  37. ^Morozov, A. N.; Skripkin, A. V. (2011). "Spherical particle Brownian motion in viscous medium as non-Markovian random process".Physics Letters A.375 (46):4113–4115.Bibcode:2011PhLA..375.4113M.doi:10.1016/j.physleta.2011.10.001.
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  39. ^abKrapf, Diego; Marinari, Enzo; Metzler, Ralf; Oshanin, Gleb; Xu, Xinran; Squarcini, Alessio (2018)."Power spectral density of a single Brownian trajectory: what one can and cannot learn from it".New Journal of Physics.20 (2): 023029.arXiv:1801.02986.Bibcode:2018NJPh...20b3029K.doi:10.1088/1367-2630/aaa67c.ISSN 1367-2630.S2CID 485685.

Further reading

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