Questions tagged [brownian-motion]

Brownian motion is a stochastic process, continuous in space and time, used in several domains in physics. It is the motion followed by a point which velocity is a white Gaussian noise. This tag sould be used for questions concerning the properties of Brownian motion, white Gaussian noise and physical models using these concepts, like Langevin equations. It should not be used for questions about discrete random walks.

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How loud is the thermal motion of air molecules?

In other words, given a magical room with walls that produce no vibration and transmit zero vibration from the outside, and nothing on the inside except room temperature air, what would be the noise ...
endolith's user avatar
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1 vote
1 answer
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Fokker-Planck equation for overdamped motion: how to define the average velocity

Consider the Langevin equation in the overdamped regime, $$ 0 = -\gamma \dot{\mathbf{x}} -\nabla U(\mathbf{x}) +\boldsymbol{\eta}(t) \, $$ where $\boldsymbol{\eta}$ is the usual white-noise term, $U$...
Quillo's user avatar
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17 votes
6 answers
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Which experiments prove atomic theory?

Which experiments prove atomic theory? Sub-atomic theories: atoms have: nuclei; electrons; protons; and neutrons. That the number of electrons atoms have determines their relationship with other ...
9 votes
2 answers
1k views

What type of non-differentiable continuous paths contribute for the path integral in quantum mechanics

Consider the path integral for a 1D particle subjected to a potential $V(x)$ in imaginary time $$ \int_{x(0)=x_0}^{x(T)=x_T} [dx] \, e^{- \int_0^T d\tau \left[\frac{1}{2}\dot{x}^2 + V(x(\tau))\right]}...
user11881's user avatar
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2 votes
1 answer
902 views

What is meant with overdamped motion?

I'm learning about Brownian motion. I use the approximation of overdamped motion. I read that the average acceleration is $0$ then, but I don't really understand the concept. So, what does overdamped ...
Caspertijmen1's user avatar
2 votes
1 answer
720 views

Physical meaning of the power spectrum: information it gives about the frequency content of a noise

Consider a stationary random variable $F(t)$ representing the random force on a Brownian particle in a fluid. Suppose the autocorrelation function is given by $$\langle F(0)F(t)\rangle=Ce^{-\gamma|t|}$...
SRS's user avatar
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2 votes
1 answer
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Understanding mean rate of change in Brownian motion

I found a nice discussion of Brownian motion in the Feynman lectures, reproduced online here: https://www.feynmanlectures.caltech.edu/I_41.html Feynman considers a particle undergoing a Brownian ...
Andrew Steane's user avatar
13 votes
3 answers
3k views

What is the difference between solutions of the diffusion equation with an imaginary diffusion coefficent and the wave equation's?

The diffusion equation of the form: $$ \frac{\partial u(x,t)}{\partial t} = D\frac{\partial ^2u(x,t)}{\partial x^2} $$ If one chooses a real value for $D$, the solutions are usually decaying with ...
Bloodworth's user avatar
10 votes
2 answers
517 views

Is the Navier-Stokes equation valid in $d=2$ spatial dimensions?

In this article, the authors study the time behaviour of the velocity-velocity correlation function of a particle in a gas. If the gas is immersed in $d$ spatial dimensions, they find that $$ C(t)=\...
Quillo's user avatar
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1 vote
1 answer
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Correlation of position and velocity in Brownian motion

There are two definitions of the term "Brownian motion": a physical science definition based on how things such as Brownian particles move, and a mathematical definition as a certain ...
Andrew Steane's user avatar
1 vote
1 answer
425 views

How to improve this simple Brownian motion simulation by adding viscosity?

I've written a 0th order Brownian motion simulator to envision how a particle of smoke might appear to move under a microscope. There will be missing $\sqrt{2}$'s and $\frac{\pi}{2}$'s since I haven'...
uhoh's user avatar
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2 answers
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Assumption of equipartion theorem in Langevin equation

To show Einstein's diffusion relation, one can develop the mean square displacement from the Langevin equation as shown in https://en.wikipedia.org/wiki/Equipartition_theorem#Brownian_motion In this ...
Learning from masters's user avatar
24 votes
7 answers
4k views

Does diffusion cause the bottle to move to the left?

There is a solution of solute and water inside the bottle, placed on a smooth horizontal surface with no friction, with the density of the solute greater than the density of the water, and the ...
dan's user avatar
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19 votes
7 answers
7k views

How does Brownian motion prove the existence of atoms?

I have heard many people say that the existence of atoms is proven by Brownian motion. Now, I understand how an atomic theory would suggest the existence of Brownian motion. However, who is to say ...
dts's user avatar
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11 votes
4 answers
3k views

Diffusion coefficient for asymmetric (biased) random walk

I want to obtain a Fokker-Planck like equation by taking the continuous limit of a discrete asymmetric random walk. Let the probability of taking a step to the right be $p$, and the probability of ...
SarthakC's user avatar
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10 votes
2 answers
612 views

Is Thibado’s Graphene Brownian Capacitor Charger Perpetual Motion of the Second Kind?

In Fluctuation-induced current from freestanding graphene (peer-reviewed version on Phys. Rev. E, note: behind a paywall) Thiabado, et al, report the extraction of work from brownian motion. The ...
James Bowery's user avatar
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9 votes
5 answers
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Sufficient and necessary conditions on random walk to obtain standard diffusion equation

In the simplest random walk model that is generally considered, the probability of the finding the particle at time $t$ in $x$, $P(x,t)$ is given by, $$ P(x,t) = \frac{1}{2}\big[ P(x-a, t-\tau) + P(x+...
user35952's user avatar
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6 votes
2 answers
448 views

How can I include variable particle number in a Brownian dynamics simulation?

I programmed a Brownian dynamics simulation in two dimensions. (Coarse-grained proteins on surfaces with interaction potentials i.e. patchy particles.) Now I want to allow particles to leave or enter ...
DW147's user avatar
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6 votes
4 answers
2k views

Brownian motion in a box

It is often said that the Brownian motion for a particle in a box, thus a finite domain, is described by a uniform probability distribution in the longtime limit. One may easily imagine this maybe ...
user929304's user avatar
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5 votes
2 answers
546 views

What does this observation of instantaneous velocity in Brownian particles mean?

I read this artice: Physicists Prove Einstein Wrong with Observation of Instantaneous Velocity in Brownian Particles “We’ve now observed the instantaneous velocity of a Brownian particle,” says ...
Amir Rezaei's user avatar
4 votes
2 answers
600 views

Brownian motion moving nano/micro coils inside a magnetic field

Following experimental setup. We take copper coils which are small enough to be subject to brownian motion. We combine those coils with some other material to make them about as heavy as the liquid ...
pZombie's user avatar
  • 359
4 votes
1 answer
686 views

Active Matter Systems

Active matter is composed of large numbers of active "agents", each of which consumes energy in order to move or to exert mechanical forces. Due to the energy consumption, these systems are ...
Joe's user avatar
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4 votes
1 answer
539 views

What is the velocity in the Langevin equation?

The Langevin equation is a stochastic differential equation for the velocity of one degree of freedom performing Brownian motion. It is supposed to describe the motion of a big particle at a much ...
GiorgioP-DoomsdayClockIsAt-90's user avatar
3 votes
1 answer
162 views

Density density correlations of a simple Brownian particle [closed]

Suppose, I have a particle satisfying the equation \begin{equation} \frac{dX}{dt}=\eta(t) \end{equation} Where $\langle \eta(t)\eta(t')\rangle=\delta(t-t')$. I can now define a density like $\rho(x,...
Sumit Sinha's user avatar
3 votes
1 answer
179 views

Does an electron undergo a form of Brownian motion in the vacuum?

There are some questions on StackExchange such as this one Brownian Motion in Vacuum asking about Brownian motion in the vacuum. There are related papers such as this one: https://arxiv.org/abs/quant-...
Andrew Steane's user avatar
3 votes
1 answer
778 views

Lagrangian description of Brownian motion?

I'm interested in the existence of a Lagrangian field theory description of Bronwnian motion, does such a thing exist? Given a particle of some spin $\sigma$, which has a Lagrangian associated with ...
Arthur Suvorov's user avatar
3 votes
3 answers
932 views

Langevin Equation - Stochastic Differential Equation. What are the subtleties?

I am trying to find out the motion of a particle in 3D governed by the Langevin equation, numerically. Anyway, the Langevin equation is given by $$m \ddot{x} = -(6\pi a\nu) \dot{x} + F_b $$ where $...
Candy Man's user avatar
  • 425
2 votes
1 answer
64 views

Basic doubt regarding Markov Processes

Take the Langevin equation for the position of a particle in Brownian motion. $$ m\frac{d^2x}{dt^2} = -\gamma\frac{dx}{dt} + \eta(t) $$ My professor wrote this as the following in the class: $$ \lim_{\...
CondensedChatter's user avatar
2 votes
3 answers
276 views

Second order brownian motion $\ddot{x}(t) = \xi(t)$

I'd like to solve for the pdf of position $$P(x,t) = \Big\langle \delta\Big(x-\int_0^t dt_1 \int_0^{t_1}dt_2 \xi(t_2)\Big)\Big\rangle $$ for the second order Brownian motion given by a Langevin-type ...
kevinkayaks's user avatar
2 votes
0 answers
175 views

Overdamped Fokker-Planck equation with general force field (gradient + curl)

I am looking for a general formulation of the Fokker-Planck equation for diffusing particles in a general force field $F = -\nabla U + \nabla\times A$ in the overdamped regime (Smoluchowski equation). ...
LukeWasinahurry's user avatar
2 votes
1 answer
150 views

What is a real world example of noise excitation in the dynamics of macro objects (other than to model sensor noise)?

The literature on stochastic processes (Ornstein–Uhlenbeck, Langevin) is not very clear as to the motivation behind using the Brownian motion or other types of noise in the dynamics. Are there any ...
kbakshi314's user avatar
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2 votes
2 answers
411 views

Is there any shortcoming of the Langevin equation which is solved by its generalization?

The ordinary Langevin equation describing the velocity $v(t)$ of a Brownian particle of mass $M$ in a fluid bath in equilibrium at a fixed temperature reads $$M\frac{dv}{dt}=-M\gamma v(t)+\zeta(t)+F_{\...
SRS's user avatar
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1 vote
2 answers
414 views

Brownian Motion in Vacuum

The term Brownian Motion is defined by Wikipedia as "random motion of particles suspended in a" liquid or gas. Thus it is not defined for the vacuum. It is explained as interaction between ...
B M's user avatar
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1 vote
0 answers
79 views

An elementary random walk model to incorporate non-Gaussianity

I am preparing a talk for young students to introduce heterogenous dynamics in complex fluids and give them a flavour of non-Gaussianity in displacements which are defined by, $$ \alpha (t) = \frac{\...
user35952's user avatar
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1 vote
3 answers
2k views

Force causing diffusion

I was curious if there was an equation describing the force acting on a particle (say, sitting in a fluid) that causes it to diffuse. If so, does it include the diffusion coefficient D? Based on ...
GnomeSort's user avatar
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0 votes
0 answers
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Connecting the diffusion coefficient in 2-dimensions and 3-dimensions?

Say the diffusion coefficient of the concentration of a particle in a fluid in 3-dimensions is $D_{3\textrm{d}}$. Can we estimate the diffusion coefficient of the same particle in the same fluid, in a ...
a06e's user avatar
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