Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

0
votes
0answers
32 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 ...
1
vote
0answers
17 views

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_{...
1
vote
1answer
35 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_{...
0
votes
1answer
29 views

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 ...
1
vote
1answer
62 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 ...
1
vote
0answers
43 views

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 ...
1
vote
1answer
25 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 ...
-1
votes
0answers
44 views

Assuming standard pressure and temperature, how far would a given air molecule move in 1 hour

I'm trying to calculate how far a given Air Molecule would actually traverse in a room in 1 hour assuming standard states (i.e. thermal equilibrium and approximately standard temperature and pressure.)...
1
vote
1answer
38 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 ...
3
votes
0answers
28 views

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 $Ndt$?
0
votes
0answers
39 views

Derivation of classical Fokker-Planck equation

For Fokker-Planck equation in Ornstein-Uhlenbeck process, we have $$\frac{\partial p(x,t)}{\partial t}=\gamma\frac{\partial xp(x,t)}{\partial x}+D\frac{\partial^2p(x,t)}{\partial x^2}$$ However I ...
0
votes
1answer
32 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 ...
2
votes
2answers
157 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 ...
1
vote
1answer
43 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}...
1
vote
1answer
108 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 ...
1
vote
0answers
39 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 ...
0
votes
1answer
40 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. ...
0
votes
1answer
61 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 ...
15
votes
7answers
4k 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 ...
1
vote
0answers
32 views

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 ...
1
vote
0answers
28 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 ...
1
vote
1answer
108 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'...
2
votes
1answer
111 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 ...
5
votes
2answers
291 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 ...
0
votes
1answer
71 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 ...
1
vote
1answer
79 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)...
3
votes
1answer
77 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,...
1
vote
1answer
82 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 ...
4
votes
1answer
67 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 ...
1
vote
1answer
178 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 ...
0
votes
0answers
36 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} =...
0
votes
0answers
10 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 ...
0
votes
1answer
1k 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 ...
0
votes
1answer
74 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 ...
1
vote
0answers
117 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 ...
0
votes
0answers
29 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 ...
1
vote
1answer
103 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 ...
0
votes
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) \...
0
votes
0answers
20 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 ...
2
votes
0answers
46 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 ...
0
votes
0answers
90 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 ...
1
vote
1answer
162 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|}$...
1
vote
0answers
70 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 ...
2
votes
1answer
218 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 $L^2$. ...
0
votes
1answer
269 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 ...
2
votes
0answers
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 ...
2
votes
1answer
59 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 ...
0
votes
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 ...
3
votes
0answers
45 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 + \...
3
votes
0answers
102 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}, \{...