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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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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Unruh radiation "observed" by a particle undergoing Brownian motion

The Unruh effect is a basic result from the theory of quantum fields in curved space-time (a sort of precursor to quantum gravity) which can be thought of as describing how the ground state of ...
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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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Thermal noise in suspensions of mesoscopic particles

By considering the steady state mean-squared-displacement of a particle undergoing Langevin dynamics in a harmonic potential well, Einstein was able to relate the diffusion constant of a colloidal ...
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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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Definition of heat when the temperature is changing in stochastic thermodynamics

I am currently studying stochastic thermodynamics, where the heat for a Brownian particle is defined by $$ dQ = -\gamma \dot{x} dx +\eta(t)dx, $$ where $\eta(t)$ is a white noise, with correlation $\...
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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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Multidimensional discretized Wiener measure for Langevin eq

How do I generalize the discretized Wiener measure in the case of the multidimensional Langevin equation: $$ dx^\omega=f^\omega(x,t) + \sum ^d_{\alpha=1} g^\omega_\alpha(x,t) dB^\alpha(t) \quad \omega=...
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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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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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What is the best software to simulate a Brownian Ratchet?

I'm trying to simulate a Brownian Ratchet, but don't know the best approach to tackle the problem. These are my inputs for the simulation that will each be varied one by one: Number of particles in ...
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Why this perpetuum mobile won't work?

Design of this perpetuum mobile is based on brownian motion. When you place a small particle ($3\ \mathrm{\mu m}$) in liquid, you can see it moving randomly, because it gets hit by moving molecules. ...
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Movement of a biological cell in water-Probability of collision?

Firstly, I apologize, as I am a systems biology scientist, so quite naïve when it comes to physics and mathematics. There is a chance that my question is deemed as very simple, but help would be ...
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What is the velocity in the Langevin equation?

The Langevin equation is a stochastic differential equation for the velocity of one degree of freedom performing Brownian motion. It is supposed to describe the motion of a big particle at a much ...
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Pressure tensor in brownian dynamics

I need to calculate the pressure autocorrelation function in a Brownian dynamics particle simulation to get the viscosity (which is proportional to the time integral of this function): $$ \left \...
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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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Langevin equation autocorrelation function

Langevin equation of a free Brownian particle has the solution of the form: $$v(t)=v(0)e^{-t\gamma}+\dfrac{1}{m}\int_0^t e^{-\gamma(t-\tau)}\eta(\tau)d\tau$$ where $\langle \eta_i(t) \eta_j(t')\...
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Can density be spatially dependent? [closed]

I was reading Gardiner's Handbook of stochastic Methods of Physics, Chemistry and Natural sciences. In page 4, he was discussing Brownian motion and referencing Einstein's own work on the subject. It ...
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Brownian Motion (Geometric, Fractional, Drift)

I have been researching Brownian motion for a while and have come across terms/types of Brownian motion such as fractional, geometric, and Brownian motion with drift. I understand the physical meaning ...
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Forces acting on particle in brownian motion

I need to simulate the motion of a small particle (100nm rigid sphere) in water. For the purposes of this I'm only interested in the forces acting on the particle, not its position. I need to ...
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Does the Kalman filter incorporate a Heisenberg-like uncertainty principle?

In the case of mechanical systems, applying the Kalman filter involves combining model based prediction (using an apriori known dynamical model) with real-world noisy observations of the positions and ...
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What is the frequency of Gaussian white noise in a Bode plot?

The design of control systems, particularly for SISO systems, is made convenient by tools such as Bode plots and the corresponding Nyquist stability criterion. These concepts allow engineers to design ...
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4 votes
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Why do some aerosol particles never settle down?

Particles and liquid droplets below the size of 1 micrometer usually never settle down easily, and as their size decreases, it takes longer for them to settle down. Why is it like that? Does Stoke's ...
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1 vote
1 answer
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Meaning of Fokker-Planck with non-differentiable and/or infinite potential

The Fokker-Planck equation for a diffusing particle in the potential $V$ is $$\partial_t p = -\nabla\cdot (p \nabla V) + D \Delta p.\tag{1}$$ In the literature, one often sees this formulation used ...
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Is Thibado’s Graphene Brownian Capacitor Charger Perpetual Motion of the Second Kind?

In Fluctuation-induced current from freestanding graphene (peer-reviewed version on Phys. Rev. E, note: behind a paywall) Thiabado, et al, report the extraction of work from brownian motion. The ...
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Can an atom first obey Schrodinger equation and after obey the heat equation?

This question is related to this other question. I first consider an atom in a cat's superposition. It obeys the Schrödinger equation. The idea would be to make it weakly interfere with a thermal bath ...
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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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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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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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