# 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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### 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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### Mechanical pressure under periodic confinement

In this paper the authors define a mechanical pressure for self-propelled particles that are confined by a potential $V(x)$ as follows: $$P = \int\limits_0^\infty \rho(x)V'(x)\mathrm{d}x$$ I ...
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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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### 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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### 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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### Why is the equilibrium measure of a conductor the same as the hitting distribution of Brownian motion from infinity?

It is a theorem that for a conductor, the equilibrium distribution of charge on its boundary is the same as its harmonic measure: the location where a Brownian motion started from far away first meets ...
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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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### Existence of atoms - Einstein or Smoluchowski

When Einstein's seminal work on Brownian motion is discussed, Smoluchowski's name often comes up as having derived more or less the same results as Einstein, but from the perspective of kinetic theory....
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### Diffusion in porous media - interface flow

i have been working on a problem lately where i think i m missing some basic understanding: I consider the diffusion of a macromolecule in porous media, which i see as Brownian motion through the ...
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### How to apply Novikov Theorem in Deriving Fokker-Planck Equation?

I am looking at this paper Smoluchowski diffusion equation for active Brownian swimmers, in which they describe using something called Novikov's theorem to take correlations between a random force and ...
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