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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Probability for a brownian motion to exceed a value in finite time [closed]

What the probability for for a brownian motion $X(s) $ to exceed a distance 'a' in any moment in an interval of time $ [0, t] $? $Prob(sup\; X(s) \geq a) $ I need to compute this using weiner path ...
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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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35 views

Langevin equation for anharmonic oscillator

I am trying to get my head around how to solve the following physical problem as an analogy for another field. Reading around the problem suggests a Langevin formulation. Consider two equal masses, $A$...
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2answers
215 views

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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91 views

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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13 views

Markov chains and Brownian Motion [migrated]

I am trying to understand the concept of Markov chains and its classes. If I am considering the model of Brownian motion on $\mathbb{Z}$, then the transition matrix of the Markov chain would be $$ \...
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28 views

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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2answers
149 views

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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What is the actual meaning of fact used in langevin's theory that averaging and differentiation are commutative?

I am studying Langevin's theory of Brownian motion and in the derivation I came across a statement(1) that differentiation and averaging are commutative i.e $\overline{\frac{d}{dt}x^{2}}=\frac{d{\bar{...
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38 views

Root mean square of $N$ random Walkers

Lets say we have $N$ (independent) random walkers where root mean square of one of the walkers is $x_{rms}$ . What is the relation between $xN_{rms}$ and $x_{rms}$ where $xN$ is the average of all the ...
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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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48 views

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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2answers
78 views

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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53 views

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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2answers
71 views

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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1answer
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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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40 views

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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1answer
90 views

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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52 views

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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45 views

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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38 views

Fractional order infinitesimals and Brownian motion

In this highly interesting answer by Ron Maimon here, under the section of fractional order infinitesimals, he explains fractional order infinitesimals by usage of an example from brownian motion. ...
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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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1answer
32 views

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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66 views

Canonical Partition function of constant gravitational potential: what is the normalization?

I'm imagining a system of particles subject to a constant gravitational field, say $V(x)=g x$. The Hamiltonian for one particle will be $$H=\frac{1}{2}m v^2 + gx ,$$ where the first term is the ...
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327 views

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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85 views

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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59 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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38 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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29 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
41 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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1answer
93 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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54 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
64 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
298 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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1answer
57 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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32 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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1answer
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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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1answer
222 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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94 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
113 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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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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38 views

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
90 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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1answer
80 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
59 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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1answer
183 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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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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1answer
29 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 ...