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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Discrete simulation of a Levy flight

I am trying to construct a discrete simulation of Levy flight in 1D and am wondering what is the best way to do so. For example, for pure diffusive random walk, one may assign probability of $1/2$ to ...
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Brownian noise variance

I have a question on a Brownian noise mean square which I get from the exercise (10-4) reference [p493, Athanasios Papoulis and Unni Krishna Pillai, “Probability, Random Variables and Stochastic ...
Pierre Polovodov's user avatar
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How to go from probability distribution to transitions probability distribution

For the past few days I have been studying Advanced statistical mechanics. I am studying a Wiener process in general. Such a process is a non-stationaty time-independent Gaussian process. The ...
luki luk's user avatar
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1 answer
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Is it possible to have an anisotropic temperature to a Brownian motion?

Resolving the Langevin equation. Tenperature is a scalar, is there a way to make it into vector?
John Paul Maquiling's user avatar
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Onsager relation in the Casimir Paper

My question is about the paper On Onsager's Principle of Microscopic Reversibility by Casimir (see page 346, second column). The relations between forces and currents have the form $$\dot x_1 = l_{11}...
Testina's user avatar
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Asymptotic form of solution to biased random walk

(Cross post from math.stackexchange) Consider a continuous time biased random walk on a 1D lattice. The random walker walks with rate $k_\mathrm{R}$ to the right and with rate $k_\mathrm{L}$ to the ...
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Energy exchanges between a Brownian fluid and particles

In the context of the dynamics of polymeric models, and specifically the dumbbell model, one of the forces acting on a dumbbell spring is said to result from "a time smoothed Brownian force" ...
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How to calculate mean squared displacement (MSD) value as function of Tau (lag-time)

I'm doing a research on Brownian motion (in 2D) and I want to calculate the MSD values in order to find the diffusion coefficient $D$. However, online I find different approaches on how to calculate ...
Naitzirch's user avatar
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Langevin dynamics with position dependent random force

Suppose I want to solve for some Langevin dynamics. Let us consider a single particle: $$m\ddot {\mathbf r}=-\eta\dot{\mathbf r}+\mathbf F$$ where $\mathbf r$ is position, $\eta$ is viscosity force ...
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Path Integrals and Brownian Motion with non-standard kinetic term

Recently I've been thinking about $1d$ Path Integrals of some theories with non-standard Lagrangians. The adjective non-standard meaning that the Lagrangian $\mathcal{L} \neq \frac{1}{2} m \dot{x}^2 - ...
Physicist in disguise's user avatar
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Partition function for fractional Brownian motion with $H < 1/2$

Recently I was interested in computing the logarithmic derivarivative $Z'(H)/Z(H)$ of the following partition function: $$ Z(H) = \int e^{-S_H(x)} \mathcal{D} x, \quad \text{where} \quad S_H(x) = A(H) ...
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Path integral derivation of exact identity for bosonic field

Let $\eta(t)$ be a non-dynamical Euclidean Gaussian bosonic field with partition function $$ Z=\int D[\eta]\exp\left(-\frac{1}{2\sigma^2}\int_0^t \mathrm{d}\tau\,\eta(\tau)^2\right) $$ so that $\left\...
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How does Einstein's paper on Brownian motion actually prove atoms exist?

Reading through Einstein's Brownian motion paper "On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat", it seems the final ...
Pecan Lim's user avatar
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How is this paper published in Physical Review E not claiming to break the second law of thermodynamics?

This paper has the following abstract: We theoretically consider a graphene ripple as a Brownian particle coupled to an energy storage circuit. When circuit and particle are at the same temperature, ...
Davis Yoshida's user avatar
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Which textbooks for random (Brownian) motion derivations? [duplicate]

I am a chemist currently trying to dive in to details of random motions. I have been studying Einstein-Smoluchowski treatment, and want to learn further (Fokker-Plank's equation and Langevin treatment ...
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Probability density function (pdf) of a Wiener process

I am working through a book right now in which there is a short introduction to Brownian motion and Wiener processes. I assume it is not treated nearly as rigorous as in mathematics but still more of ...
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Mean displacement of an active brownian particle [closed]

I am facing difficulty in a relatively simple integral. I have an active brownian particle with the equation of motions: $$ \frac{dx}{dt}= v \cos\phi+\sqrt{2D_T}\zeta_x $$ $$ \frac{d\phi}{dt}= \sqrt{...
Nine_Gardens's user avatar
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Computing friction tensor for a colloidal particle in (moving) harmonic potential

I am having difficulties reproducing the result for the friction tensor presented in equation (19) of this article. Here is my understanding: Consider the over-damped Langevin dynamics of a particle ...
cidrolin's user avatar
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Hamiltonian function of a system of particles governed by Langevin equation of motion

I have a system of particles which interact among themselves via some pairwise additive potential ( position dependent ) and I am also considering the collision of the particles with background ...
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The distribution function of active Brownian process

I have a problem with the derivation of the distribution function of the stationary state using the system`s propagator as it has been mentioned in equation number 14. Basically, we know that the ...
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Autocorrelation of Brownian motion - question about derivation

The calculation of the autocorrelation function for particles undergoing Brownian motion is described in chapter 7.6 of Chaikin and Lubensky (See picture below). There are couple of things which I ...
proton's user avatar
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Anomalous diffusion in a system of confined interacting particles

I have a system of charged particles confined parabolically. I want to analytically study anomalous diffusion in such a system using Langevin Dynamics. In a system of non-interacting particles there ...
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3 answers
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On the Brownian motion and Law of Large numbers

According to the Law of Large Numbers, if I throw a million coins, I expect to observe half of them face head, and the other half tail. Why doesn’t this apply to (1-dimensional) Brownian motion? A ...
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Is Brownian motion truly random?

We say that Brownian motion is caused by the random collisions of particles. But let's consider an ionized gas; in that case, there's a nonzero net charge on the atom. Doesn't this mean the ...
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About Particle Density and PDF in Einstein's Paper on Brownian Motion

In Einstein's paper, he deduced that the diffusion equation is satisfied for the number of particles per unit volume ($f(x, t)$), i.e $$\frac{\partial f}{\partial t}=D\cdot \frac{\partial ^2 f}{\...
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Conserved quantities in overdamped dynamics

I have an implementation of over-damped Brownian dynamics, with particles that follow the version of the Newtons law where the inertia is absent. This is a common thing to do at micrometer scale. $m x'...
user13831679's user avatar
1 vote
2 answers
132 views

Frictionless Brownian Motion

The Langevin equation for a Brownian particle without the friction term is: \begin{equation} m\dot{v}=F(t) \end{equation} Where $F(t)$ is the random force acting on the Brownian particle due to ...
Μπαμπης Ποζουκιδης's user avatar
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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 ...
llangford1's user avatar
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225 views

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 ...
CafféSospeso's user avatar
1 vote
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244 views

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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9 votes
5 answers
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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} =...
Sandip's user avatar
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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 \...
Christos's user avatar
2 votes
0 answers
56 views

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 ...
Christos's user avatar
1 vote
2 answers
141 views

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 ...
1 vote
0 answers
79 views

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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3 votes
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711 views

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 ...
Master Drifter's user avatar
4 votes
3 answers
142 views

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 ...
RKerr's user avatar
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4 votes
1 answer
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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 ...
U_flow's user avatar
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1 answer
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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 ...
CafféSospeso's user avatar
1 vote
1 answer
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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 ...
CafféSospeso's user avatar
21 votes
4 answers
4k views

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 ...
Pavel Borisov's user avatar
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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 ...
3 votes
1 answer
179 views

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-...
Andrew Steane's user avatar
-1 votes
1 answer
96 views

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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