What is Brownian motion: Definition and 98 Discussions

Brownian motion, or pedesis (from Ancient Greek: πήδησις /pɛ̌ːdɛːsis/ "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).This pattern of motion typically consists of random fluctuations in a particle's position inside a fluid sub-domain, followed by a relocation to another sub-domain. Each relocation is followed by more fluctuations within the new closed volume. This pattern describes a fluid at thermal equilibrium, defined by a given temperature. Within such a fluid, there exists no preferential direction of flow (as in transport phenomena). More specifically, the fluid's overall linear and angular momenta remain null over time. The kinetic energies of the molecular Brownian motions, together with those of molecular rotations and vibrations, sum up to the caloric component of a fluid's internal energy (the Equipartition theorem).
This motion is named after the botanist Robert Brown, who first described the phenomenon in 1827, while looking through a microscope at pollen of the plant Clarkia pulchella immersed in water. In 1905, almost eighty years later, theoretical physicist Albert Einstein published a paper where he modeled the motion of the pollen particles as being moved by individual water molecules, making one of his first major scientific contributions. This explanation of Brownian motion served as convincing evidence that atoms and molecules exist and was further verified experimentally by Jean Perrin in 1908. Perrin was awarded the Nobel Prize in Physics in 1926 "for his work on the discontinuous structure of matter". 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.
The many-body interactions that yield the Brownian pattern cannot be solved by a model accounting for every involved molecule. In consequence, only probabilistic models applied to molecular populations can be employed to describe it. Two such models of the statistical mechanics, due to Einstein and Smoluchowski are presented below. Another, pure probabilistic class of models is the class of the stochastic process models. There exist sequences of both simpler and more complicated stochastic processes which converge (in the limit) to Brownian motion (see random walk and Donsker's theorem).

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  1. theegyptian

    I Brownian motion and the resultant of forces

    Hello, Why isn't the forces resultant on a "large" molecule (by small molecule: water for example) zero? The reason for this Brownian motion is the thermal agitation of the water molecule. If we talk about white and Gaussian noise in electronics (due to the thermal agitation of the electrons)...
  2. A

    I Is Brownian motion a purely classical phenomenon or is it also quantm?

    A water molecule is as tiny as 0.3 Angstrom. I would expect that quantum effects play a role. I'm wondering if its Brownian motion in a fluid is determined only by classical thermodynamics or if its collisional processes must take into account also quantum scatterings or other effects like...
  3. S

    I Brownian motion of charged particles?

    If we have charged particles having Brownian motion, would this motion be associated with (or produce) heat or electricity? Would it produce electromagnetic radiation (and if it would produce it, what type of radiation in the electromagnetic spectrum)? Could there be Brownian motion of charged...
  4. D

    I Brownian motion and continuity

    It is said often that in 1905 Einstein “mathematically proved” the existence of atoms. More precisely, he worked out a mathematical atomic model to explain the random motion of granules in water (Brownian motion). According to that mathematical model, if the atoms were infinitely small and...
  5. T

    A Brownian Motion (Langevin equation) correlation function

    So the Langevin equation of Brownian motion is a stochastic differential equation defined as $$m {d \textbf{v} \over{dt} } = - \lambda \textbf{v} + \eta(t)$$ where the noise function eta has correlation function $$\langle \eta_i(t) \eta_j(t') \rangle=2 \lambda k_B T \delta_{ij} \delta(t -...
  6. dedocta

    A Does gravity affect Brownian Motion?

    I know passive diffusion rates behave differently on the International Space Station relative to Earth (video of a contained flame experiment burning up there.) However, does the random walk of pollen particles etc. have slowed velocity in comparison to that on Earth? Has been bugging me for a...
  7. iVenky

    I Are quantum computers required to be cold to reduce Brownian motion?

    I understand that based on what I have read online quantum computers are required to be close to absolute zero because it introduces less error. Is it because brownian motion due to thermal agitation of molecules reduces with temperature?
  8. SchroedingersLion

    Brownian Motion - "no inertia"

    Greetings, I currently work my way through Langevin Dynamics which, in a certain limit, becomes Brownian Motion. I refer to this brief article on Wikipedia: https://en.wikipedia.org/wiki/Brownian_dynamics I understand the general LD equation given there. In order to obtain Brownian Dynamics...
  9. Einstein's PhD thesis

    Einstein's PhD thesis

  10. H

    What is the inverse of the covariance operator in Brownian motion?

    in fact the answer is given in the book (written by philippe Martin). we have $$ (\tau_1| A^{-1} | \tau_2) = 2D \ min(\tau_1 ,\tau_2) = 2D(\tau_1 \theta (\tau_2 -\tau_1)+\tau_2 \theta (\tau_1 -\tau_2))$$ So $$-1/2D \frac{d^2}{d\tau_1^2} (\tau_1| A^{-1} | \tau_2) = \delta( \tau_1 - \tau_2) $$...
  11. J

    I Calculating the expected value of the square of an integral of Brownian Motion

    For a standard one-dimensional Brownian motion W(t), calculate: $$E\bigg[\Big(\frac{1}{T}\int\limits_0^TW_t\, dt\Big)^2\bigg]$$I can't figure out how the middle term simplifies. $$ \mathsf E\left(\int_0^T W_t\mathrm dt\right)^2 = \mathsf E\left[T^2W_T^2\right] - 2T\mathsf E\left[W_T\int_0^T...
  12. Lunct

    Is Brownian Motion actually random or does it follow determinism?

    If you have enough information could you not determine with certainty the movements of pollen particles in water? In other words, if you were able to measure the movements the particles, then repeat the exact same experiment, with all things controlled, would the particles move in the same way...
  13. J

    Vibration Energy Harvester from Brownian motion

    Hi all, been a while since I studied physics but I saw something that I found strange, This device: https://researchfrontiers.uark.edu/good-vibrations/ from University of Arkansas A free floating graphene sheet extracting energy from brownian motion and converting that to electric current...
  14. H

    Equation for Brownian Motion Trajectory

    I am trying to understand how one can simulate the trajectory of a Brownian particle as a function of time. I am only able to do it with the assumption that I can simply generate random values of x and then take the cumulative sums of these values to get the trajectory of the Brownian particle...
  15. D

    Brownian motion demonstration in class

    Hello, Einstein evaluated the size of an atom by analysing a brownian motion, assuming the size of the molecules is a a factor. In order to demonstrate this concept, I want to put some powders with different sizes of grains on a vibrator and watch how a ball moves. What is the expected result...
  16. SlowThinker

    B Brownian motion from virtual photons?

    This a really simple question: If I have, say, 2 ions close to one another, and measure their repulsion very precisely, is the force constant, or is it a series of little pushes caused by individual virtual photons? I know there are many misunderstandings about virtual particles, but I'm not...
  17. Avi Nandi

    Brownian Motion: Mass and size of Brownian particle

    For observation of Brownian motion, the mass and the size of the Brownian particle should be very small. Within what range the size and mass of the Brownian particle should lie?? Can a particle with small mass and bigger size and vice versa can undergo Brownian motion??
  18. It's me

    Relationship between density and probability in diffusion

    Homework Statement Consider the diffusion of a drop of ink in a water vase. The density of the ink is ## \rho (\vec{r}, t) ##, and the probability ##P(\vec{r}, t)## obeys the diffusion equation. What is the relationship between ##\rho (\vec{r}, t)## and ##P(\vec{r}, t)##? Homework...
  19. E

    What probability density is used in Brownian motion?

    Homework Statement I have a free Brownian particle and its coordinate is given as a function of time: And its first moment, or mean, is given as But what kind of probability density was used to calculate this first moment? Homework Equations I know that the first moment is calculated...
  20. Andy Resnick

    Q: Fokker-Planck (Brownian motion) for undergrads?

    All, I'm looking for a reference to help guide one of my students- a motivated physics undergrad. I would like him to work through a derivation of the mean-squared displacement of a particle undergoing free Brownian motion (free diffusion) and then for a particle held in an optical trap. All...
  21. M

    Langevin and Einstein formulas of Brownian motion

    Homework Statement Demonstrate that Eq. (1.1) will convert to the Einstein relation Eq. (1.2) in the limit of t→∞ when we assume ξ=6πaμ. Conversely, show that Eq. (1.1) will yield <x2> ~ t2 in the limit of t→0. Confirm the consistency of the principle of equipartition of energy. Homework...
  22. M

    Thermodynamics: Air pressure at 3000m above sea level

    Hello all! Can you please provide some guidance with this problem? 1. Homework Statement Calculate the air pressure at 3000m above sea level assuming that the molecular weight of air is 29 and the ambient temperature is constant against height. Homework Equations Stokes-Einstein equation. In...
  23. S

    Neglecting brownian motion for certain particle diameters?

    Hi, Imagine we have few particles with different diameters in fluid flow. so, can we neglect brownian forces at certain particle diameters? Actually I'm looking for a criteria in which we can neglect the effect of brownian forces in fluid flow. Thanks :-)
  24. P

    MHB Some questions about Brownian Motion and Birth-Death in Markov chain

    Hi, I need urgent answers. Basically, I don't have background in Markov and I don't need to learn it now actually. But I have to solve the questions below somehow. If somebody can give detailed answers to the questions below (From beginning to the final solution with explanations), then I will...
  25. N

    Evaluation of an integral

    I am graduate student in engineering. In course of my research I have encountered an integral of this form ##\int_{t'}^{t} e ^{-b t_1} dt_1 \int_{t'}^{t_1 } e ^{b t_2} dt_2 \int_{t'}^{t_2 } e ^{-b t_3} dt_3 \int_{t'}^{t_3} e ^{b t_4} dt_4 ... \int_{t'}^{t_{n-1}} e ^{b t_n} dt_n ## I am trying...
  26. A

    The Standard Deviation of a Brownian Force

    I'm trying to understand the derivation of the expression for the random Brownian force on a particle in a medium with coefficient of viscosity η. It turns out it is gaussian over some timescale, with a standard deviation that depends on the temperature and the viscosity. I'd like to read a...
  27. E

    Sufficient conditions for a Brownian motion

    I'm given a probability measure ##\mathbb P## on ##\Omega = \{f\in C([0,1],\mathbb R): \enspace f(0)=0\}## and told that ##\mathbb P## satisfies i.i.d. increments. I'm interested in the weakest additional conditions that will ensure that ##\mathbb P## describes a Brownian motion, i.e. that...
  28. Warpspeed13

    Can Brownian Motion Suspend Larger Particles in a Denser Fluid?

    I was reading about ferrofluid and I was wondering how you would go about calculating the maximum size of a particle that could be suspended by Brownian motion in a fluid? Can a denser fluid suspend larger particles?
  29. S

    Variance of Geometric Brownian motion?

    I am trying to derive the Probability distribution of Geometric Brownian motion, and I don't know how to find the variance. start with geometric brownian motion dX=\mu X dt + \sigma X dB I use ito's lemma working towards the solution, and I get this. \ln X = (\mu - \frac{\sigma...
  30. Telemachus

    Brownian motion, correlation functions

    Hi. I have a doubt on the derivation of the correlation function for the velocity and position in Brownian motion from the Langevin equation. I have that for a brownian particle: ##\displaystyle v(t)=v_0e^{-\frac{\gamma}{m}t}+\frac{1}{m}\int_0^{t}dse^{-\frac{\gamma}{m}(t-s)}\xi(s)## (1)...
  31. M

    Geometric Brownian Motion

    Hi, I am trying to answer the following question: Consider a geometric Brownian motion S(t) with S(0) = S_0 and parameters μ and σ^2. Write down an approximation of S(t) in terms of a product of random variables. By taking the limit of the expectation of these compute the expectation of S(t)...
  32. I

    Where do i learn about brownian motion?

    not sure if this is the right section to post this question, but i was wondering in what class i would learn about brownian motion. I took undergraduate thermodynamics and i feel that it could be mathematically described with the material already covered in my thermo class, but is there a class...
  33. C

    Statements regarding Brownian motion experiment

    Homework Statement In an experiment to demonstrate Brownian motion in a gas, a brightly illuminated cell containing smoke is viewed under a microscope. The observer sees a large number of bright specks undergoing random motion. Which one of the following statements about this experiment...
  34. HakimPhilo

    Is Quantum Brownian Motion Linked to Virtual Particle Frequencies?

    Hello EVERYBODY! :) I have a question: Does the frequency of virtual particles (apparition/disappearance) match with nucleus random path - known as Quantum Browniαn Motion? Thank you in advance! ;)
  35. X

    How to approach this Brownian Motion Problems

    1. Problem is in the Attachment This link will help your figure these question...
  36. L

    Difficulties in understanding Brownian motion.?

    I hope you can bear with me as I warm up to my question regarding Brownian motion. I am currently studying physics on my own and am watching a series of lectures on Quantum Theory. Obviously I am just some guy with an interest in physics and I have no clue. Apparently, Brownian motion is the...
  37. M

    MHB Brownian Motion: Martingale Property

    Hi! I need some help at the following exercise... Let B be a typical brownian motion with μ>0 and x ε R. X_{t}:=x+B_{t}+μt, for each t>=0, a brownian motion with velocity μ that starts at x. For r ε R, T_{r}:=inf{s>=0:X_{s}=r} and φ(r):=exp(-2μr). Show that M_{t}:=φ(X_{t}) for t>=0 is...
  38. S

    What significance does brownian motion has to do with coal explosion?

    Apparently : What significance does brownian motion has to do with coal explosion?[Brownian motion or pedesis is the presumably random moving of particles suspended in a fluid (a liquid or a gas) resulting from their bombardment by the fast-moving atoms or molecules in the gas or liquid.] Now...
  39. A

    Brownian motion solves Laplace's Equation?

    There seems to be a curious connection between Brownian Motion, stochastic diffusion process, and EM. http://en.wikipedia.org/wiki/Stochastic_processes_and_boundary_value_problems I was hoping to share and to have someone add some insight on on what it means that the Dirichlet boundary...
  40. I

    Proving a process is Brownian Motion

    Homework Statement Is the process \{X(t)\}_{t\geq 0}, where X(t)=\rho B_1(t)+\sqrt{1-\rho^2}B_2(t) Standard Brownian Motion? Where \rho\in(0,1), \ B_1(t) and B_2(t) are independent standard brownian motionsHomework Equations The Attempt at a Solution Obviously X(0)=0. Now let 0\leq...
  41. I

    Conditional Brownian motion

    I computed the distribution of B_s given B_t, where 0\leq s <t and \left\{B_t\right\}_{t\geq 0} is a standard brownian motion. It's normal obviously.. My question is, how do I phrase what I've done exactly? Is it that I computed the distribution of B_s over \sigma(B_t)?
  42. G

    MHB Integral Notation of an exponential Brownian motion

    Good day! I am reading the paper of Marc Yor (www.jstor.org/stable/1427477). equation (1.a) seems unfamiliar to me since the $ds$ comes first before the exponential part; $$ \int_0^t ds \exp(aB_s + bs). $$ Can you please help me clarify if there is a difference with the above notation as...
  43. O

    Integral with respect to Brownian motion.

    Suppose that \sigma(t,T) is a deterministic process, where t varies and T is a constant. We also have that t \in [0,T]. Also W(t) is a Wiener process. My First Question What is \displaystyle \ \ d\int_0^t \sigma(u,T)dW(u)? My lecture slides assert that it's equal to \sigma(t,T)dW(t) but I'm...
  44. O

    Brownian motion integration/calculus

    We all know that \int_0^t dB(s) = B(t), where B(t) is a standard Brownian Motion. However, is this identity true? \int_{t_1}^{t_2} dB(s) = B(t_2) - B(t_1)
  45. D

    2-D Brownian motion with correlated noise

    dx/dt = η(t) dy/dt = ζ(t) where <η(t)>=<ζ(t)>=0 <η(t)η(t')> = 2Dδ(t-t') <ζ(t)ζ(t')> = 2Dδ(t-t') If <η(t)ζ(t')> = 0, we have the standard 2-D diffusion equation and the analytical solution is known. If <η(t)ζ(t')> = 2Dδ(t-t'), or η(t) = ζ(t), we can transform it into a 1-D problem...
  46. W

    What is Brownian Motion (Weiner process)?

    Hi all, When random walk takes the steps at random times, and in that case the position X_t is defined as the continuum of times t≥0, isn't this concept/phenomenon/rule is Brownian motion (Weiner process)? At http://en.wikipedia.org/wiki/Wiener_process in section "Characterizations of the...
  47. C

    History Can History be Modeled as Brownian Motion?

    I read somewhere that the "path of history" measured in some way can be modeled as Brownian motion with a mean collision time. There's been several very *specific* models such as: http://onlinelibrary.wiley.com/doi/10.1002/asm.3150030303/abstract However, what I'd like to know is that...
  48. S

    Estimating Displacement of Particle in Brownian Motion

    Suppose I have a large particle of mass M that is randomly emitting small particles. The magnitude of the momenta of the small particles is \delta p (and it is equal for all of them. Each particle is launched in a random direction (in 3 spatial dimensions--although we can work with 1 dimension...
  49. Steve Zissou

    How does the Lagrangian apply to Brownian motion?

    Hello, how do we apply the idea of the Lagrangian to a Brownian motion? I guess what I mean is what is the Lagrangian functional form for a Brownian motion? Thanks
  50. T

    Exact Solution of Geometric Brownian Motion

    Hi! Probably I am just confused, but why for the exact solution of the geometric brownian motion dX_t = \mu X_t dt+\sigma X_t dW_t we have to apply Ito's lemma and manipulate the expression obtained with dlogX_t? Couldn't we directly use the espression dX_t / X_t = dlogX_t in the equation dX_t /...