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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30 views

Why is Wiener process a homogenous but not a stationary stochastic process?

I found the claim that wiener process is an example of a stochastic process which is homogeneous but not stationary in the book 'The Theory of Open Quantum Systems' by Breuer and Petruccione. (Section ...
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20 views

Brownian motion and the heat equation

Einstein showed that the Brownian motion provides a solution to the heat equation. As written here, the relation between the brownian motion and the heat equation can be shown by the taylor series. ...
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28 views

Diffusion of a particle between two immiscible liquids

I am trying to find a model, or construct my own to describe the diffusion of a particle between layers of immiscible fluids with different densities. The particle size is much larger than the sizes ...
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1answer
31 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_{\...
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82 views

How does the Brownian motion of air molecules compare to the threshold of human hearing as a function of frequency?

This fantastic question essentially asks what is the noise floor of air? Both the answer given on that thread and the value stated by Microsoft are around -23 or -24 dBSPL. However, overall loudness ...
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40 views

Derivation of Evolution Operator on finding Hu-Paz-Zhang (HPZ) Quantum Brownian Motion (QBM) Master Equation

I am trying to understand this paper by Hu, Paz, and Zhang about exact master equation of QBM in general environment. In the paper they used influence functional method introduce by Feynman and Vernon ...
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1answer
43 views

Small time solution to Fokker-Planck equation

In reference to this note, a specific Focker-Planck equation with initial condition $W(\rho, t=0)=\delta(\rho-1)$ have the solution $$W\left(\rho,t\right)=\dfrac{e^{-\frac{t}{4}}}{\sqrt{\pi}t^{\frac{3}...
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1answer
84 views

Diffusion of Ink in Water

I am investigating the diffusion of ink in water. A drop of blue ink is dropped to the center of a round plate of radius $R$. Say the drop of ink has an initial radius of $r=r_0$ (the very edge of the ...
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39 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 ...
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18 views

Determining probability density function in Brownian motion simulation

Theoretically, the probability density function of brownian particle would be a function satisfy the diffusion equation in the form of: \begin{equation} \rho=\frac{N}{\sqrt{4\pi Dt}}e^{\frac{x^2}{4Dt}}...
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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 ...
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198 views

Is the Feynman's path integral a density?

The Feynman-Kac path integral formula is used to solve parabolic equations related to stochastic processes. Considering the probabilistic expression, the solution is indeed not a density. However, ...
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Unbounded time derivative of the brownian motion and Newtonian dynamical models

The second order dynamics model $dx_s = v_s ds$, $d v_s = dw_s$ where $w_s$ is s standard Brownian motion is an example of Langevin dynamics. Note that the Brownian motion in this case models a white ...
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91 views

Can we deduce that particles behave as Brownian motions if the collection obeys the Einstein model?

The density dynamics of a continuum of particles with the dynamics $$dx^i_s = d w^i_s,$$ where $dw^i_s$, $0 \leq s$, $i \in \mathcal{N}$ is a standard Brownian motion, are given by the diffusion PDE $$...
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1answer
98 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 ...
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21 views

What's the meaning of the osmotic velocity?

To describe a random-walking particle, we can use the stochastic differential equation \begin{equation} dx(t)= b\Big(x(t),t \Big) dt + dw(t) , \end{equation} which is also known as the Langevin ...
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26 views

Why is the osmotic force virtual?

In Einstein's treatment of Brownian motion he argues that (in equilibrium) there is an "osmotic pressure force" $\vec O$ that counterbalances the effect of gravity $\vec K$. This allows him to derive ...
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Is simulating 3D brownian motion in axially symmetrical 2D sensible?

I'm writing a Monte Carlo particle trajectory simulation program. There are fluid forces like the Stokes drag force involved, as well as Brownian motion. For runtime reasons, the program should be ...
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2answers
70 views

Easiest way to roughly explain Brownian motion?

All easy explanations of Brownian motion that I have found are all totally wrong in that they just essentially say something like "motion of the pollen is being moved by individual water molecules" ...
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67 views

Brownian Motion

I’m currently interested in learning some topics about the Brownian motion and the random walk (in general, from a pure statistical and probabilistic way). For that, I would like to ask you if you ...
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1answer
44 views

What does convergence to equilibrium for the Fokker-Planck equation mean?

I am a math major who recently started to study thermodynamics seriously. I have some confusing points while studying it, so I'd appreciate it if you'd correct me and give me some answers. (1) As ...
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19 views

How does Brownian motion relate to the size of atoms, and how can other phenomeon be estimated with it?

To begin, I recognise this question is similar to questions/answers already on this site. However, I am curious to how specifically we can use Brownian motion / models to estimate and hypothesise ...
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1answer
70 views

Where can I find Einstein's proof of the existence of atoms?

as the question states, where i can i find einstein's proof of the existence of atoms, and also, what math pre-requisites do i need to understand it deeply enough to be able to replicate it.
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53 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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28 views

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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80 views

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
175 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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30 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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49 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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55 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_{...
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1answer
99 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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1answer
44 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 ...
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1answer
70 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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55 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 ...
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1answer
86 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
98 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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0answers
30 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 $Ndt$?
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1answer
88 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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2answers
228 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
82 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
203 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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0answers
84 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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1answer
88 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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1answer
76 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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36 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
192 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
343 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
309 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 ...