All Questions
Tagged with correlation-functionsstochastic-processes
24 questions
0votes
0answers
26views
Characterization of Markovianity, Gaussianity, and color for noise processes
Consider a noise process $\xi(t)$ that has some statistics in time. There are various ways to characterize such a process, 3 being Markovianity (independence from history), Gaussianity (Gaussian ...
2votes
1answer
125views
Find correlation function $\langle R^2(t_1) R^2(t_2)\rangle$ for 2d stochastic dynamics of polymer
The problem: Consider 2d stochastic dynamics \begin{equation} \partial_t R_\alpha = \sigma_{\alpha \beta} R_\beta \end{equation} \begin{equation} \langle \sigma_{\alpha \beta}(t_1) \sigma_{\mu \nu}(...
- 103
0votes
0answers
37views
References on getting the correlation function in a 3D Markov Random Field?
Does anyone know where to look to find analytical formulae for the correlation function of the Ising model on a 2D or 3D lattice (assuming toroidal or infinite is easier?), or, even better, a ...
1vote
1answer
92views
Phase-amplitude stochastic differential equations
In the book of $\textit{The Quantum World of Ultra-Cold Atoms and Light: Book 1 Foundations of Quantum Optics}$ by Peter Zoller and Crispin Gardiner on page 75, they derive the phase-amplitude ...
- 105
0votes
0answers
88views
Meaning of $\langle X(t')X(t'') \rangle$?
Context My background is not in physics so I am not very familiar with the $\langle \rangle$ notation. I am trying to understand the following in a paper that I am reading (Berglund AJ., PhysRevE., ...
- 55
1vote
0answers
54views
Physical interpretation of a multi-time (more than 2) autocorrelation function: non-Gaussian diffusion
In non-equilibrium statistical mechanics, the time-autocorrelation functions become the cornerstone of various theories and models. One such important autocorrelation is the velocity autocorrelation ...
- 3,134
0votes
0answers
61views
Power-Spectrum for Self-Organised Criticality
In 1987 Bak, Tang and Weisenfeld authored a paper (link) on Self-Organised Criticality, on how minimally stable self-organised systems propagate the perturbations it is subjected to, scale-freely - ...
- 138
3votes
1answer
470views
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 ...
- 4,683
5votes
1answer
282views
Ornstein–Uhlenbeck process: joint probability as a Gaussian
The problem Consider a stochastic process with the following three properties: The process is Markov, meaning that $p(x_n,t_n|x_{n-1},t_{n-1},\ldots x_1, t_1) = p(x_n,t_n|x_{n-1},t_{n-1}).$ The ...
- 25.6k
1vote
1answer
907views
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 ...
1vote
2answers
1kviews
What does it mean to have delta-correlated process physically?
I am reading about Langevin dynamics, and I see the following equation: $$\frac{dx}{dt} = -\frac{1}{\xi} \frac{\partial U}{\partial x} + g(t)$$ Then, they claim that the average $$\langle g(t) \rangle ...
- 719
4votes
3answers
302views
Are stationarity, Markovianity and Gaussianity sufficient conditions to ensure that the random force on a Brownian particle is delta correlated?
In the Langevin model, if we make the assumption that the random force $\eta(t)$ acting on the Brownian particle is a stationary, Markovian, and gaussian process, does it automatically ensure that the ...
- 27.6k
5votes
4answers
436views
Is there really an inconsistency with the original Langevin equation (as claimed in the book Nonequilibrium Statistical Mechanics - V. Balakrishnan)?
I am reading the book Nonequilibrium Statistical Mechanics by V. Balakrishnan. In chapter $17$ (page $244$) he argues that the original Langevin equation has inconsistencies and should, therefore, be ...
- 27.6k
0votes
1answer
252views
Dirac delta function and stochastic processes
It is given to us some white noise as $A z(t)$ and the autocorrelation of $A z(t)$ is given as $\phi(t)= A^2 \delta(t)$ where $\delta(t)$ is the Dirac delta function Now one signal with $y(t)= B \cos(...
0votes
1answer
488views
Expression of Dirac Delta Correlation
spatio-temporal white noise $\xi(x,t)$ is often expressed as $$\langle\xi(x,t)\rangle=0,$$ $$\langle\xi(x_1,t_1)\xi(x_2,t_2)\rangle=\delta(t_2-t_1)\delta(x_2-x_1).$$ Now I understand that the first ...
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