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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24 views

Work done by the drift term of Ornstein–Uhlenbeck process

Consider a particle obeying the Ornstein–Uhlenbeck process: $$ dx_{t}=\theta (\mu -x_{t})\,dt+\sigma \,dW_{t}, $$ where $x_t$ is the position of the particle at time $t$, $W_t$ denotes the Wiener ...
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Interaction energy of two brownian spherical particle in liquid [closed]

Let us consider two hard sphere in finite volume. Their motion is Brownian. What can we say about interaction energy? Is it less then $kT$? I know that we can describe this system by Langevin ...
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Canceled term in Brownian Motion's mean squared distance derivation starting from Langevin's equation

In Feynmann's Lectures on Physics, chapter 41, part 4, during the derivation of the Brownian expression for the mean square distance, Feynmann starts off with Langevin Equation, $$ m\frac{d^2 x}{dt^2}...
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Random walks applied to Brownian motion

[...] There we discovered that the mean square of the distance from one end to the other of the chain of random steps, which was the intensity of the light, is the sum of the intensities of the ...
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2answers
84 views

Why do we interpret the first term of the Fokker-Planck equation as drift?

With the derivation of the Fokker-Planck equation we get: $$\frac{\partial}{\partial t}P(x,t)=-\frac{\partial}{\partial x}(A(x,t)P(x,t))+\frac{1}{2}\frac{\partial^2}{\partial x^2}(B(x,t)P(x,t))$$ We ...
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25 views

Is brownian motion a good model for the movement of a particle in water?

I am not very knowledgable with respect to physics, I come from the math SE. I was wondering about Brownian motion and how close is the model to the phenomenon that started it all: the movement of a ...
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38 views

Unit strength Wiener process

I am not much aware of Brownian motion and/or the Wiener process. Recently I have been reading a research paper where they mention a variable $dW$ as a "unit strength Wiener process". My question is ...
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Cast Caldeira-Leggett master equation to Lindblad form

Consider a Brownian motion particle, whose motion is described by $\frac{d}{dt}\rho_{S}=-\frac{i}{\hbar}[H_{S},\rho_{S}]+\sum_{i,j}a_{i,j}(F_{i}\rho_{S}F_{j}^{\dagger}-\frac{1}{2}\{F_{j}^{\dagger}F_{...
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48 views

Counterterms in quantum brownian motion

In the part "Quantum Brownian motion" of the book, The theroy of open quantum systems written by Breuer, the author investigates on the Caldeira-Leggett model: The Hamiltonian of the particle is $H_{...
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Derive properties of fluids using Monte Carlo method on brownian motion

Given a particle inside a fluid, it's known that its movement will be unpredictable due to the random collisions with the particles of the fluid. However, the distance from the origin of motion will ...
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1answer
67 views

What formula can serve as an approximate estimate of the time taken for the smell of a perfume to reach somebody?

I am in an attempt to calculate the time required for the smell of a bottle of perfume to reach a person's nose $10$m away. Real life experience tells me that it takes several seconds. I tried to work ...
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Question about the Markovian property of the velocity of a Brownian particle following Langevin equation

I'am now studying Langevin model and Fokker-Planck equation with the lecture notes by Borghini Topics in Nonequilibrium Physics (NB: PDF). On page 92, he talks about the Markovian property of the ...
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1answer
43 views

Rand-walk/Brownian-motion on 2D lattice [closed]

I started to learn stochastic processes this year. Only had two classes, but I already have some problem. We learned about Einstein's and Langevin's description of Brownian-motion and now I need to ...
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1answer
53 views

Fractional derivatives in a QFT Lagrangian

There are is at least one question asking about fractional powers of fields in QFT (and why they're not expected to occur), and several others asking about the physical relevance of fractional ...
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Duality while studying properties of an ensemble

Was Einstein the first to propose that observing an ensemble of $N$ particles for time interval of $dt$ is same as observing a single particle of ensemble for time interval of $Ndt$?
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52 views

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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170 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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1answer
61 views

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
147 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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52 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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55 views

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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67 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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35 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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143 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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225 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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298 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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104 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
88 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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90 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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155 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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83 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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222 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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2k 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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92 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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152 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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1answer
133 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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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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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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109 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
188 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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86 views

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
262 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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333 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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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
82 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
51 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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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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105 views

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}, \{...