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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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Diffusion 2D on a surface : diffusion coefficient and surface friction

We have a particle that is diffusing actively (meaning that the source of energy is a motor; the diffusion is like a Brownian motion, the only difference is that the diffusion coefficient is much ...
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2answers
145 views

The mean of Langevin equation

I have a very basic question regarding the mean of the Langevin equation. So we have an equation of the form: $$\dot{v}(t)=-\beta v(t)+ \xi (t)$$ Where $\xi (t)$ is a Gaussian white noise with an ...
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Inverse of a matrix in a Path Integral

Good morning! I can't make sense of an inverse of a matrix appearing in a calculation for a Wiener Path Integral. In discretized form: $$\int \prod_{i=1}^N \frac{dx_i}{\sqrt{\pi \epsilon}} e^{-\frac{1}...
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1answer
86 views

Brownian dynamics simulations in confined geometries [closed]

I am currently trying to implement a 2D Brownian dynamics simulation in confined geometries (corrugated channels, of the form $A\cos(2 \pi x) \ + B\ $ in this case). The concept is to compute the ...
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35 views

How long does it take for oil to coalesce in water?

I was studying the process of coalescence in emulsions. We considered $N$ bubbles of liquid 1 floating in liquid 2. The result we derived, is that if there are some dissipative forces (diffusion) the ...
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Question about the autocorrelation function of the fluctuating force in the Langevin model for Brownian motion

According to the Langevin model, we have, for the motion of Brownian particles, $$\frac{dv}{dt} = -M\gamma v + \zeta(t)$$ with $\zeta(t)$ the random force acting on the particle due to fluctuations. ...
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57 views

Statistics of 1D discrete random walks

I have already asked this question in Math.SE. Let $P(n)$ be a probability distribution on the integers. Suppose a random walker starts off at the origin and, at every positive integer time, takes a ...
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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 ...
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What is the decoherence rate and the thermal de Broglie wavelength in quantum Brownian motion?

I know that when the thermal de Broglie wavelength is on the order of the interparticle distance, the gas must be treated as a Fermi gas or a Bose gas, depending on the nature of the gas particles. I ...
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0answers
26 views

Why the distribution of Fluctuationg force in brownian motion has gaussian distribution?

I am reading the Zwanzig's book and I have a confusion about the average of the fluctuating force and its distribution. As it says $F(t)$ is a random variable that means it has a probability ...
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1answer
50 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'...
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1answer
80 views

Derivation of diffusion equation from Fokker-Planck equation

I need your help, could you please explain me the sentence "The diffusion equation is the Fokker-Planck equation for the Brownian motion". I have tried to use some assumption and transform a ...
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2answers
278 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 ...
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1answer
59 views

Smoluchowski theory of Brownian Motion

I am studing Brownian motion, in particular I am reading the book "Brownian Motion, Fluctuation, Dynamics and Application" by Mazo. Now I am dealing with Smoluchowski theory, but I am having some ...
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1answer
75 views

Brownian motion from two gaussian noise processes

Consider some brownian motion for which we obtained the following solution for the langevin equations $$ u\left(t\right)=e^{-\alpha t}\int_{0}^{t}e^{\alpha s}\left(\xi\left(s\right)-\xi'\left(s\right)...
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1answer
73 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,...
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1answer
64 views

Proof that the Wiener process is non-differentiable

I'm working through a proof showing that the Wiener process is non differentiable given as follows I am not quite sure where the 2 in front of the integral stems from though. Any help would be ...
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1answer
60 views

Long time tails in Brownian motion

Rings, D., et al. Theory of hot Brownian motion. Soft Matter 7.7 (2011): 3441-3452, doi:10.1039/C0SM00854K. In this paper the author has mentioned that vorticity diffusion is disregarded due to it's ...
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1answer
141 views

Why is the correlation function of fluctuation force in Brownian motion related to a delta function?

For the first step to derive fluctuation-dissipation theorem, I find $$\langle F(t)F(t')\rangle=2B\delta(t-t')$$ where $B$ is a constant, and $F(t)$ is a random fluctuating force with Gaussian ...
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34 views

Autocorrelation function question

I am a little confused about two things in the context of non-Markovian Langevin equation. Let $C(t) \equiv \big<A(t)A^*(0)\big>$ where A is a phase variable. Why is it that $\frac{dA(t)}{dt} =...
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9 views

Is there some literature where diffusing molecules have to encounter a reflecting boundary which is absorbing at some places?

I am familiar with the concept of diffusion with reflecting and absorbing boundary in 1D. My question is, is there a way by which we can write Green's equation for a boundary, in 2D, that is primarily ...
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939 views

What's the difference between Brownian motion and diffusion?

I find it hard to see the differences between Brownian motion and diffusion. As I understand, both are represented by the diffusion equation – am I right? And if I'm not, how is Brownian ...
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62 views

Terminal velocity Vs Relaxation time

I'm surely trapped in a misunderstanding. Consider an experimental situation where a molecule is dropped into water, and imagine that the viscous force $\eta v$ perfectly compensates the ...
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0answers
85 views

Brownian motion of a particle varying with time

I was thinking about a Brownian particle executing random motion in a liquid. Is there any time dependence if the particle travel some distance say $r$? well, obviously it is time dependence. i think ...
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27 views

Approximate model for stochastic process - Brownian + Anomalous diffusion

I have a molecular system where the diffusion process is governed by two independent degrees of freedom - centre of mass motion and internal motion. In this context, we generally model the diffusive ...
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1answer
87 views

Feynman Lectures Vol I 41-4: Find the number of collisions received by a water molecule per second

In The Feynman Lectures on Physics Vol. I Ch. 41: The Brownian Movement, $\S4$ The random walk we are told: The reader may easily verify that the number of collisions a single molecule of water ...
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1answer
17 views

Rotational diffusion - why isn't $\hat n(t)=\hat n(0) \; \forall t$?

Consider the rotational Langevin equation in the absence of an external force: $$\frac{d \hat n(t)}{dt} =\vec{\xi}(t) \times \hat n(t)$$ where $\vec \xi(t)$ is a Gaussian white noise and $\hat n(t) \...
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0answers
18 views

Thermalization of cold Rb atoms in a dipole trap

I wish to know if the dynamics of cold atoms in optical dipole trap can be modeled as Brownian motion (and if so, what is the bath/thermalization mechanism?), and thus, subsequently can be simulated ...
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1answer
40 views

Will a Brownian Particle hit an infinite wall in 3-D geometry with probability 1?

I know that in 3-D the probability of recurring a given point is zero for a Brownian particle. Given an infinite absorbing wall/plane the probability of ultimately getting absorbed, for a Brownian ...
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0answers
26 views

How to solve absorbing boundary conditions for Brownian motion in 3D?

I am trying to solve the absorbing boundary conditions for a molecule moving with drift and Brownian motion for 3D motion. For this I have searched for some literature and got one method, i.e., Method ...
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13 views

Tilting ratchet; need for time varying force?

With a tilting ratchet (a form of Brownian ratchet) it is often said (e.g. here) that the force applied is periodic with mean zero. As I understand it we get motion in the case of a constant force. I ...
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81 views

Movement of a random walk in the limit (a particle in diffusion)

I asked this question in Math Exchange and MathOverflow and obtained no answer. This question may lack of mathematical rigorous, but I would like to understand why this type of reasoning is sometimes ...
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1answer
153 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|}$...
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Interpretation of quantum superposition and classical Brownian motion

In the standard, Copenhagen interpretation of quantum mechanics, the usual ontology assigned to the phenomenon that repeated measurements of a quantum mechanical observable yielding different results ...
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1answer
176 views

Collision rate of Brownian particles with a surface

Let us consider a 3D box of volume $V$, containing $N$ identical Brownian particles. The diffusion coefficient of the particles is noted $D$. Inside this box there is a square surface of area $L^2$. ...
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1answer
232 views

Brownian motion and equilibrium

I would like to know if when you consider a system in which you have Brownian motion if it is considered a system in equilibrium or far from equilibrium and why. i.e., is Brownian motion considered as ...
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39 views

Thermal fluctuations in orientation of point particles

I am modeling group of point particles with 6 degrees of freedom each - 3 positional degrees of freedom and 3 orientational degrees of freedom. So, each particle has 3 position coordinates and a unit ...
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1answer
47 views

Brownian Ratchet with mechanism in vacuum

So first of all I want to state that I indeed understand Feynman's reasoning as to why the Brownian ratched fails. (At least what's written about it in Wikipedia.) I want to consider the following ...
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1answer
50 views

What is the difference between these two equations for the probability density function of Brownian motion?

I have been seeing two different versions of the density function everywhere. One involves Dt as the diffusion coefficient: $$ f(x) = 1/\sqrt{4πDt} \exp(-x^2/(4Dt)) $$ Whereas the other seems more ...
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43 views

Continuous Measurement equations

In a physics text, "Quantum Measurement Theory and it's Applications" by Kurt Jacobs, it describes the idea of a "continuous measurement" (measurement taking place over time $T$): $$dy = x_{true}dt + \...
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Quantum Brownian Motion - Calculation of moments [closed]

The master equation of quantum brownian motion is derived as \begin{equation}\frac{d}{dt} \hat{\rho_s}(t) = -i[\hat{H_S} + \frac{1}{2}M\tilde{\Omega}^2 \hat{X}^2 , \hat{\rho_s}(t)] -i\gamma[\hat{X}, \{...
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215 views

Derive Schrodinger Equation From Brownian Motion

I read the following text form physics forum. How do you formally derive Schrodinger equation this way? (The Feynman-Kac formula) A Wiener process represents Brownian motion. Brownian motion has two ...
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2answers
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Why have scientists accepted that Brownian motion is caused by collisions with water molecules?

Colloid (sol) particles have a diameter between 1 and 1000 nanometers, approximately. If a sol particle, 275 nm in diameter, were compared to a water molecule, 0.275 nm in diameter, it would be 275 / ...
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0answers
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Jacobian for underdamped limit

The underdamped langevin equation for harmonically bound particle is given as: $m\overset{..}{x}+\zeta\overset{.}{x}+k\overset{.}{x}=\eta(t)$. Suppose I know Probability distribution function of ...
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3answers
606 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 ...
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1answer
230 views

The underdamped langevin equation in two dimension

Suppose I have a system which is a parabola in polar coordinates. That is there is a minima of the potential at r=R. The potentially is spherically symmetric. How do I write the langevin equation of ...
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1answer
217 views

Renormalization and Brownian Motion

I've been reading about the renormalization group in QFT from Peskin & Schroeder, and wanted to consolidate understanding of "irrelevant operator" by connecting it to something more intuitive, aka ...
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2answers
246 views

Power spectral density of freely diffusing particle versus trapped particle?

Recently I was exposed to optical tweezer experiment in which we pulled an axonal membrane (of a chicken embryo neuronal cell) using an optically trapped 2.4 $\mu$m diameter polystyrene bead. A lot of ...
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1answer
461 views

Effect of System Potential on Quantum Master Equation

The usual microscopic derivation of master equation is done with the total Hamiltonian being the sum of the system Hamiltonian, environment and the coupling one. Suppose, the system itself is in a ...
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1answer
167 views

Intuition for using Brownian motion to solve Laplace and heat equations

I'm a mathematics student with zero physical intuition. In my course, we learned that Brownian motion can be used to construct the solutions to certain PDEs, including Laplace's equation and the heat ...