Questions tagged [stochastic-processes]
A stochastic process is a random process evolving with time , i.e., a time sequence representing the evolution of some system represented by a variable whose change is subject to a random variation.
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Fokker-Planck equation for overdamped motion: how to define the average velocity
Consider the Langevin equation in the overdamped regime,
$$
0 = -\gamma \dot{\mathbf{x}} -\nabla U(\mathbf{x}) +\boldsymbol{\eta}(t) \,
$$
where $\boldsymbol{\eta}$ is the usual white-noise term, $U$...
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Is throwing dice a stochastic or a deterministic process?
As far as I understand it a stochastic process is a mathematically defined concept as a collection of random variables which describe outcomes of repeated events while a deterministic process is ...
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Derive Poisson distribution from probability per time of event
Suppose we have a probability per time $\lambda$ that something (e.g. nuclear decay, random walk takes a step, etc.) happens.
It is a known result that the probability that $n$ events happen in a time ...
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What is the relation between Langevin and Fokker planck equation?
What is corresponding fokker planck equation for,
$\frac{df(t)}{dt}=-kf(t)+\zeta(t)$
where, $\zeta(t)$ is random noise.
In particular, how will the fokker planck equation will look like if $\zeta(t)...
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Suggestion on good stochastic processes book for self-teaching
I am a first year physics grad student and I am looking for a book on stochastic processes. I have learned basic statistics and probability in my undergraduate. Recently I read by N.G. VAN KAMPEN and ...
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Describe Ising model dynamics in stochastic differential equation or stochastic process
The Ising model is described by the Hamiltonian
$$
H(\sigma) = - \sum_{<i~j>} J_{ij} \sigma_i \sigma_j -\mu \sum_{j} h_j\sigma_j,
$$
and is treated extensively by equilibrium statistical ...
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Physical meaning of the power spectrum: information it gives about the frequency content of a noise
Consider a stationary random variable $F(t)$ representing the random force on a Brownian particle in a fluid. Suppose the autocorrelation function is given by $$\langle F(0)F(t)\rangle=Ce^{-\gamma|t|}$...
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In the Langevin dynamics: neglecting inertia. A mathematician trying to understand physics terminology
If we write the Langevin equation: for a particle with mass $m$, position $x$ and velocity $v$, with some damping coefficient $\gamma$,
$$ m dV(t)=-\gamma V(t)dt+dW(t) ,~~~~~~~dX(t)=V(t)dt.$$
Then as $...
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Topological entropy in Markov chains
Given a finite Markov chain, how do I find the topological entropy $h_T$?
Furthermore, I should compare it with the Shannon entropy $h_S$ and show that $h_T\leq h_S$. Is this a general fact?
This ...
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Stochastic process vs high dimensional chaos in models
I'm trying to figure out what are the theoretical and practical, implications and limitations, when a high-dimensional chaotic process is modeled as a random process. I understand how low-dimensional ...
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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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Are there any experimental or statistical tests to check deterministicity or stochasticity of a dynamical system?
Are there any simple experimental or statistical tests to check whether a dynamical system is deterministic?
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Why is Johnson noise a Gaussian process?
Noise processes in engineering and physics are frequently assumed to be Gaussian processes.
This allows use of convenient analytical techniques.
The question then arises as to why natural processes ...
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Diffusion coefficient for asymmetric (biased) random walk
I want to obtain a Fokker-Planck like equation by taking the continuous limit of a discrete asymmetric random walk. Let the probability of taking a step to the right be $p$, and the probability of ...
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Partition function for Gaussian white noise
Problem
I'm trying to understand, motivate, or derive from first principles, the partition function for gaussian white noise, namely
$$
Z = \int \mathcal{D}\eta(t)\exp\left[-\frac{1}{2D}\int dt \eta(...
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What is the proper way to model diffusion in inhomogeneous media (Fokker-Planck or Fick's law) and why?
I'm quite confused with the following problem. Normally a one-dimensional Fokker-Planck equation is written in the following form:
$$\frac{\partial \psi}{\partial t}=-\frac{\partial}{\partial x}(F\...
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Schrodinger equation in term of Fokker-Planck equation
From Wikipedia on the Fokker-Planck equation:
$$\tag{1}\frac{\partial }{\partial t}f\left( x^{\prime },t\right) ~=~\int_{-\infty}^\infty dx\left( \left[ D_{1}\left( x,t\right) \frac{\partial }{\...
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Why is the Fokker-Planck equation only valid for the forward and backward velocities but not for the flux velocity?
I noticed that the Fokker-Planck equation is often only written for the forward velocity $\vec b$ and the backward velocity $\vec b^*$:
\begin{align}
\partial_t \rho + \nabla (\vec b \rho) &= D \...
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Are stationarity, Markovianity and Gaussianity sufficient conditions to ensure that the random force on a Brownian particle is delta correlated?
In the Langevin model, if we make the assumption that the random force $\eta(t)$ acting on the Brownian particle is a stationary, Markovian, and gaussian process, does it automatically ensure that the ...
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How to derive the backward Fokker-Planck equation from a forward Fokker-Planck equation (with state-dependent diffusion coefficient)?
I am interested in a system with state-dependent diffusion coefficients. This paper by Lau and Lubensky derives the correct Forward FPE in this case:
$$\partial_tP(x,t) = \frac{\partial}{\partial x} ...
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What is the velocity in the Langevin equation?
The Langevin equation is a stochastic differential equation for the velocity of one degree of freedom performing Brownian motion. It is supposed to describe the motion of a big particle at a much ...
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Density density correlations of a simple Brownian particle [closed]
Suppose, I have a particle satisfying the equation
\begin{equation}
\frac{dX}{dt}=\eta(t)
\end{equation}
Where $\langle \eta(t)\eta(t')\rangle=\delta(t-t')$. I can now define a density like $\rho(x,...
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Lagrangian description of Brownian motion?
I'm interested in the existence of a Lagrangian field theory description of Bronwnian motion, does such a thing exist? Given a particle of some spin $\sigma$, which has a Lagrangian associated with ...
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Langevin Equation - Stochastic Differential Equation. What are the subtleties?
I am trying to find out the motion of a particle in 3D governed by the Langevin equation, numerically.
Anyway, the Langevin equation is given by
$$m \ddot{x} = -(6\pi a\nu) \dot{x} + F_b $$
where $...
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Why does propagator for SDE involve response variable?
Being trained as a mathematician I am trying to understand stochastic field theory (i.e. field theory applied for dynamics of stochastic processes, e.g. SDE) and I have a difficulty at one point.
...
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Integrating Langevin Dynamics equations in computer simulation
In a Langevin Dynamics simulation the following equation is solved numerically :
$$
m_i\frac{d^2r_i}{dt^2}=F_{int}-\gamma\frac{dr_i}{dt}+R(t)$$
$$\langle R(t)\rangle=0 \quad \quad \langle R(t)R(t')\...
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Stochastic resonance
I am trying to look for a stochastic resonance in a system described by Langevin equation and a periodic forcing. While I can simulate an SDE numerically I have no idea how to calculate the 'signal to ...
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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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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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Driven harmonic oscillator with thermal Langevin force. How to extract temperature from $x(t)$?
Suppose you have driven harmonic oscillator (parameters: mass,gamma,omega0) by a deterministic force Fdrive (a sine wave say). Now suppose that you add stochastic Langevin force FL which is related to ...
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Overdamped Fokker-Planck equation with general force field (gradient + curl)
I am looking for a general formulation of the Fokker-Planck equation for diffusing particles in a general force field $F = -\nabla U + \nabla\times A$ in the overdamped regime (Smoluchowski equation).
...
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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:
$$
\lim_{\...
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Is there any shortcoming of the Langevin equation which is solved by its generalization?
The ordinary Langevin equation describing the velocity $v(t)$ of a Brownian particle of mass $M$ in a fluid bath in equilibrium at a fixed temperature reads $$M\frac{dv}{dt}=-M\gamma v(t)+\zeta(t)+F_{\...
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Is interchanging the orders of averaging operation with integral operation allowed?
In the book of Zwanzig, Nonequilibrium statistical physics, at page 6, after explaining Langevin equation Brownian motion, to show that $<v^2> = 3/2 k_B T/m$ consistent with the Langevin ...
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Numerical integration of overdamped Langevin equation: explicit methods?
Assume we have a generic overdamped Langevin equation
$$
\frac{d {\bf{x}} }{dt} = {\bf{f}}({\bf{x}}) + B {\bf{w}}(t)
$$
where $\bf{f}$ is a deterministic external (and fixed) force field (non ...
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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,
$$
\alpha (t) = \frac{\...
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Expectation Value of Wiener Process
I want to perform this integral,
$$\int^t_0\int^{t'}_0 \delta(s-s')dsds'$$
I know that the result should be min(t,t'), as it is the expectation value of the wiener process. I just want to know how to ...
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Probability of branching times under a Ornstein–Uhlenbeck-Yule process
According to Edwards, 1970 the probability density of the branching times in a Brownian-Yule branching process can be expressed as:
\begin{equation}
P(\mathbf{u'},n|\lambda,n_0,T)=\lambda^{n-n_0}\frac{...
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From exponential distribution to Poisson distribution
We know that the exponential distribution characterises the probability distribution for the waiting time between two consecutive Poisson events. Then I think if we fix a time interval $T$ then we ...
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Dynamical interpretation of reflecting boundary conditions in the Fokker-Planck equation
Background:
For a particle driven by the dynamical equation
$$ \dot{x}(t) = a(x,t) + b(x)\xi(t),$$
where $\xi(t)$ is a Gaussian white noise, the probability distribution of position $x$ is governed by ...
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