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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Ito-Stratonovich drift term for spatial white noise
Suppose I have a Langevin equation with multiplicative noise of the form
$$ \dot{x} = f(x) + g(x)\eta(t) $$
where $ \eta(t) $ is a Gaussian white noise with zero average, unit strength, and delta ...
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Full translation of the paper written by M Smoluchowski in 1906 "Zur kinetischen Theorie der Brownschen Molekular Bewegung und der Suspensionen"
May I ask if the full translation of the classic paper on Brownian motion and SDE written by M Smoluchowski in 1906 "Zur kinetischen Theorie der Brownschen Molekular Bewegung und der Suspensionen&...
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On the Fokker-Planck equation: deriving the transition PDF for small times
I report below (part of) page 73 of the book The Fokker-Planck Equation, by H. Risken
We now derive an expression for the transition probability density for small $\tau$
\begin{equation}\tag{1}
p(x,t+...
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Geometric Brownian Motion versus Ornstein-Uhlenbeck process
The Geometric Brownian Motion model is a continuous-time stochastic process in which a particle move according to a random fluctuations (Wiener process) and a drift term.
The corresponding stochastic ...
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Changing sign at a Ornstein-Uhlenbeck process: mean, variance and likelihood
I am working with a multivariate Ornstein-Uhlenbeck process and its statistical properties (likelihood, expected values and variance).
The Ornstein-Uhlenbeck process can be described as a random walk ...
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Derivation of Heisenberg-Langevin equations for an atom coupled to a reservoir of harmonic oscillators
I have been working on the paper of A. Lezama on the "Numerical investigation of the quantum fluctuations..." (2008) and been trying to replicate their derivation and simulation. However, I ...
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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,
$$
P(x,t) = \frac{1}{2}\big[ P(x-a, t-\tau) + P(x+...
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Corollary of Wiener process and the the appearance of $\sqrt{t}$
One of the properties of a Wiener process is given by (taken from https://en.wikipedia.org/wiki/Wiener_process),
A corollary useful for simulation is that we can write, for $t_1 < t_2$:
$$W_{t_2} =...
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Mean square displacement of brownian particle at small times
Starting from the Langevin equation for a 1D Brownian particle, the mean square displacement (MSD) is calculated:
$$\langle r^2 \rangle = \frac{2k_BT}{ζ}[t+τ_B(e^{-t/τ_B}-1)]$$
When $t\rightarrow \...
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Use equipatition theorem when studying Brownian motion [duplicate]
Not a physicist so excuse my ignorance.
I am currently studying introductory topics on Brownian mechanics. Utilizing the Langevin equation for a Brownian particle submerged in a fluid with no external ...
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If charged particles have Brownian motion, would this motion be associated with (or produce) heat or electricity?
If we have charged particles having Brownian motion, would this motion be associated with (or produce) heat or electricity? Would it produce electromagnetic radiation (and if it would produce it, what ...
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An invitation to brownian traps to capture or destroy viruses and bacteria
I'm interested for general feedback about modern brownian traps for viruses and bacteria. In my rememberings a related reference is a column added to the article in Spanish [1] (I don't remember well ...
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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,
$$
\alpha (t) = \frac{\...
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Generalized random walk process: Fokker-Planck equation
The diffusion equation
$$
\frac{\partial p(x,t)}{\partial t} = D \frac{\partial^2 p(x,t)}{\partial x^2}
$$
can be derived from a simple random walk on a line.
For example, if the probability of ...
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Mean square speed from mean square displacement?
Consider a distribution of particles in space with a mean square displacement defined in each spatial direction $q$ as $\langle d_q^2 \rangle = \langle |q(t) - q_0|^2 \rangle$, where $q_0$ is the ...
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Brownian particle in two heat baths
A Brownian particle of mass $m$ in a heat bath at temperature $T_{1}$ can be described by:
$$\frac{\mathrm dv}{\mathrm dt} = -\frac{\gamma}{m}v + \frac{1}{m}\xi .$$
However, if I assume that a ...
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How to explain Bernoulli's principle with Brownian motion?
Air pressure is generated by Brownian motion pushing against solid objects. The integration of all molecule collisions with the boundary is then the air pressure pushing against that object.
But can ...
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Random walk in finite VERSUS infinite space: Probability density functions and their interpretation
I am studying the probability density function of a random walk in a confined geometry (2D-BOX). I am also comparing this probability density function to its equivalent in infinite two-dimensional ...
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Properties of random-walk in infinite and finite two-dimensional space: probability of two particles being in the same location at time t
I have been told that one of the property of the continuous-time random walk in two dimensions is that:
$$\int_{Z} \, G(z, t | p_1) \, G(z, t | p_2) \,dz = \,G(p_1,p_2,2t)$$
where ...
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How can we experimentally confirm that atoms/molecules in a solid actually "move"?
The atoms in a solid are so attracted to each other that they "vibrate" and don't move past each other.
How do scientists "measure" that atomic vibration in a solid (let's say at ...
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Where can I get a good quality video (preferably slow motion / high frame rate) of Brownian motion particles for tracking their positions?
I am trying to analyse how good the Langevin equation fits actual experimental data by tracking the position of Brownian motion particles from video footage. However, I was unable to get my hands on a ...
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Probability distribution of particle diffusion system with a source and absorbing boundaries
Consider a simple 1D particle diffusion process described by the SDE $dx=\sigma dW$, where $dW$ is a Wiener process. The forward Fokker-Planck equation can then be written as
$$
\frac{\partial P(x,t)}{...
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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-...
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Can Brownian particle itself theoretically exhibit thermal motion? [closed]
The reason that we observe a Brownian Motion of a particle is being commonly explained by collisions of molecules of the surrounding fluid (liquid, gas) that the Brownian particle is suspended in. ...
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Meaning of $\langle X(t')X(t'') \rangle$?
Context
My background is not in physics so I am not very familiar with the $\langle \rangle$ notation. I am trying to understand the following in a paper that I am reading (Berglund AJ., PhysRevE., ...
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Physical interpretation of a multi-time (more than 2) autocorrelation function: non-Gaussian diffusion
In non-equilibrium statistical mechanics, the time-autocorrelation functions become the cornerstone of various theories and models. One such important autocorrelation is the velocity autocorrelation ...
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The dimension analysis for the langevin force [duplicate]
In wikipedia(https://en.wikipedia.org/wiki/Langevin_equation), the langevin force formula is given as
$$\langle\eta_i(t)\,\eta_j(t')\rangle = {2\gamma\,k_B\,T\,\delta_{i,j}\,\delta(t-t')}.$$
However, $...
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Brownian motion and multi-scale stochastic processes
The Stokes-Einstein equation for the diffusion coefficient of small colloidal particles in suspension is canonically derived under the assumption that the primary motion of the particle is ...
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Qubit system coupled to a bath of quantum harmonic oscillators
It is well known that when we consider a probe harmonic oscillators (called system) that is coupled to a reservoir of N harmonic oscillators, i.e. the Hamiltonian is written as the following, the ...
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Are there any different ways to theorise that atoms exist?
I have read that Albert Einstein and some notable others theoretically proved atoms through Brownian motion. Are there any other perspectives or methods to theoretically or experimentally prove that ...
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Wiener process as the integral of a stochastic force
I have seen (in my lecture notes) the following definition for a Wiener process:
$$W(t)=\int _0 ^t dt'\eta(t') \tag{1}$$
where $\eta(t)$ is the stochastic force appearing in the Langevin equation for ...
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Coupled Brownian Fans
Two fans are immersed in different mediums in equilibrium at different temperatures. Given that $\gamma$ and $\gamma^{\prime}$ are the friction coefficients for each fan and $\theta$ and $\theta^{\...
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Does indoor odor smell travel up or down?
Does odor smell, let's say it's from caulk off-gasing, travel upward or downward in the air?
Are all odor smell lighter than air?
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Probability density function for a Brownian motion in confined geometries
In the classical Brownian motion, the probability density function of finding a given particle in $x$ at time $t$, given that it is at $y$ can be expressed as:
$$p(t,x,y)=\frac{1}{2\pi \sigma^{2} t} e^...
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How does a virus fall down in static air?
If we drop a virus from a height, in static air, will it fall to the ground like a lead ball, a balloon, or like a virus? How will it fall to the bottom? Like a Brownian particle? It will not float ...
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Do air particles "fly"? If not, how do they stay afloat? [duplicate]
I was reading my old physics textbook (from middle school), and it mentioned something about the idea of having non-existing attractive forces between particles like air. "We would live in a very ...
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Can Brownian motion explain why white smoke moves around in the air?
Can Brownian motion explain why white smoke (fume) generated by chemical reactions moves around in the air?
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Random Walk of Thermal Electrons
The drift velocity of electrons in a typical electronic circuit might be measured in mm per second. In contrast, the thermal velocity of electrons is in the vicinity of km per second.
Because of the ...
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Is the Navier-Stokes equation valid in $d=2$ spatial dimensions?
In this article, the authors study the time behaviour of the velocity-velocity correlation function of a particle in a gas. If the gas is immersed in $d$ spatial dimensions, they find that
$$
C(t)=\...
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Where is the flaw in this brownian ratchet design?
As I understand, Brownian ratchets do not work because the paddle, which is supposed to keep the gear spinning in a single direction, also oscillates with thermal noise. A the paddle can dissipate the ...
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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 ...
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Asymmetric Random walk with a pause [closed]
In the non-equilibrium statistical mechanics framework, there are two basic paradigms for defining the dynamics of the system: the Langevin and Fokker-Planck equations for diffusion processes and the ...
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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 ...
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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 ...
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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 ...
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How to get marginalized Fokker-Planck equation for the time-dependent Gaussian velocity distribution?
I have come across the term "Marginalized Fokker-Planck equation! ", which I have never heard of and could not find any resource online. The equation reads as following
$$
\frac{\partial}{\...
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Probablity of Cauchy jump between two position
I have some doubts about how to calculate the probability $P(x,t)$ of finding a particle with a certain initial uniform distribution $ P(x,0)=\rho (x) $ and typical displacement $ x^*=Dt $.
My idea ...
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Second order brownian motion $\ddot{x}(t) = \xi(t)$
I'd like to solve for the pdf of position $$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 $$
for the second order Brownian motion given by a Langevin-type ...
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Is there any shortcoming of the Langevin equation which is solved by its generalization?
The ordinary Langevin equation describing the velocity $v(t)$ of a Brownian particle of mass $M$ in a fluid bath in equilibrium at a fixed temperature reads $$M\frac{dv}{dt}=-M\gamma v(t)+\zeta(t)+F_{\...
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Boltzmann Law in moving fluids?
In my research, I am concerned with the analysis of systems which operate essential like this:
There is a tube, say of radius $r$. In this simplification in can be infinitely long. Air moves along it ...
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