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.
265
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For diffusion with a drift, what if the drift velocity is not a constant but varies with time?
Recently, I read some materials about diffusion with drift, and found that the mean squared displacement (MSD) of this process will be:
$$MSD(t)=4Dt+v^2 t^2$$
Question:
What if the drift velocity is ...
- 1
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vote
1
answer
50
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Finding equilibrium solution from Fokker-Planck equation for planer Brownian rotator
In Andrew Zwanzig's Nonequilibrium Statistical Mechanics, the author has given the expression of the equilibrium density distribution function for a planer Brownian rotator.
Given the equations of ...
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0
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46
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Pressure at a point around the corner in a conical fask [duplicate]
I have gone through this two very informative links in understanding pressure.
Weight of fluid in a conical container act entirely on the base?
Pressure is isotropic
But in a long conical flask which ...
1
vote
1
answer
44
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What is the correlation between Brownian noise's low frequency components and the actual movement of particles?
I do have some crude training in mathematics, but I'm not a physicist or engineer. So I'd appreciate a simple not too technical explanation.
I conceptually understand how hitting a piece of wood will ...
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1
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42
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Method of inserting random numbers in the numerical calculation of mean-squared displacement for brownian particle
I am trying to plot Mean-Squared-Displacement for a passive Brownian particle. For that I'm using the discretized over-damped version of the Langevin equation as: $$x(i+1)=x(i)+\sqrt{\frac{2.k_BT.\...
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1
answer
99
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Viscous stress, and closed dynamic equations for density and velocity fields of an Ornstein-Uhlenbeck process?
Context
I am trying to write simple derivation of hydrodynamic definition of viscous stress $\partial^{2}_{x}v$ based on OU-process which is
\begin{align}
\dot{x} &= v \\
\dot{v} &= -\gamma v +...
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1
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70
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Self-similarity of the diffusion equation
I am going through this book Simulation of Complex Systems. In the chapter on Brownian Dynamics, we considered a "free diffusion" given by the Stochastic differential equation: $$\dot{x}(t)=...
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Does Equipartition hold in overdamped dynamics?
We start with the Langevin equation
$$m\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}} = -\Gamma \frac{\mathrm{d}x}{\mathrm{d}t} +\sqrt{2\Gamma k_{B}T} \eta(t). $$
Now, we know that at $t \gg m/\Gamma$, the ...
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35
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Diffusion from a rod with constant concentration
Suppose I have an infinite rod that is suddenly brought into a medium where some substance starts to diffuse radially outwards from the rod. During this, the concentration in the rod is kept constant....
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1
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1
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141
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Mean squared displacement of a free Brownian particle moving in harmonic potential
For a free Brownian particle moving under harmonic potential ($\frac{1}{2}m\omega^2x^2$), the equation of motion can be written as,
$$m\ddot{x}=-m\omega^2 x-m\gamma\dot{x}+R(t)\;,$$
where, $\gamma$ is ...
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0
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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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1
answer
55
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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 ...
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1
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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 ...
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1
answer
40
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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?
2
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52
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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
$$\dot x_1 = l_{11}...
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1
answer
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Asymptotic form of solution to biased random walk
(Cross post from math.stackexchange)
Consider a continuous time biased random walk on a 1D lattice. The random walker walks with rate $k_\mathrm{R}$ to the right and with rate $k_\mathrm{L}$ to the ...
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Energy exchanges between a Brownian fluid and particles
In the context of the dynamics of polymeric models, and specifically the dumbbell model, one of the forces acting on a dumbbell spring is said to result from "a time smoothed Brownian force" ...
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1
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499
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How to calculate mean squared displacement (MSD) value as function of Tau (lag-time)
I'm doing a research on Brownian motion (in 2D) and I want to calculate the MSD values in order to find the diffusion coefficient $D$.
However, online I find different approaches on how to calculate ...
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0
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30
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Langevin dynamics with position dependent random force
Suppose I want to solve for some Langevin dynamics. Let us consider a single particle:
$$m\ddot {\mathbf r}=-\eta\dot{\mathbf r}+\mathbf F$$
where $\mathbf r$ is position, $\eta$ is viscosity force ...
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1
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Path Integrals and Brownian Motion with non-standard kinetic term
Recently I've been thinking about $1d$ Path Integrals of some theories with non-standard Lagrangians. The adjective non-standard meaning that the Lagrangian $\mathcal{L} \neq \frac{1}{2} m \dot{x}^2 - ...
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1
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130
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Partition function for fractional Brownian motion with $H < 1/2$
Recently I was interested in computing the logarithmic derivarivative $Z'(H)/Z(H)$ of the following partition function:
$$
Z(H) = \int e^{-S_H(x)} \mathcal{D} x,
\quad \text{where} \quad
S_H(x) = A(H) ...
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votes
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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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3
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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 ...
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147
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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, ...
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1
answer
102
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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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1
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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{...
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1
vote
1
answer
88
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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 ...
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1
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63
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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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3
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617
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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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3
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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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51
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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'...
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2
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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 ...
3
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0
answers
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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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285
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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&...
4
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1
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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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0
answers
212
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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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0
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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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2
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0
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363
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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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5
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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 $t_1 < t_2$:
$$W_{t_2} =...
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5
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1
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523
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Mean square displacement of brownian particle at small times
Starting from the Langevin equation for a 1D Brownian particle, the mean square displacement (MSD) is calculated:
$$\langle r^2 \rangle = \frac{2k_BT}{ζ}[t+τ_B(e^{-t/τ_B}-1)]$$
When $t\rightarrow \...
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0
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Use equipatition theorem when studying Brownian motion [duplicate]
Not a physicist so excuse my ignorance.
I am currently studying introductory topics on Brownian mechanics. Utilizing the Langevin equation for a Brownian particle submerged in a fluid with no external ...
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2
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174
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If charged particles have Brownian motion, would this motion be associated with (or produce) heat or electricity?
If we have charged particles having Brownian motion, would this motion be associated with (or produce) heat or electricity? Would it produce electromagnetic radiation (and if it would produce it, what ...
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0
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An invitation to brownian traps to capture or destroy viruses and bacteria
I'm interested for general feedback about modern brownian traps for viruses and bacteria. In my rememberings a related reference is a column added to the article in Spanish [1] (I don't remember well ...
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0
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An elementary random walk model to incorporate non-Gaussianity
I am preparing a talk for young students to introduce heterogenous dynamics in complex fluids and give them a flavour of non-Gaussianity in displacements which are defined by,
$$
\alpha (t) = \frac{\...
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3
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2
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782
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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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0
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1
answer
502
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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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4
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3
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148
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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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1
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186
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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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1
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1
answer
243
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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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