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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Is this a valid alternative definition of the delta function?
The delta function can be defined as:
$$
\delta(x) = \int_{-\infty}^{\infty} e^{-2\pi i k x} \, dk
$$
Loosely speaking, I can understand this because unless $x=0$, the complex exponential oscillates ...
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Random walk average meeting time [closed]
Imagine having a random walk of $N$ steps ($N$ is large, may be approximated to be infinity), but all the points are 1 unit length apart and they have to stay this unit length apart. Every timestep $\...
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Stochastic thermodynamical entropy production on growing state space
Suppose we have a stochastic process for a biased random walker starting at a position $x_0 = 0$. At a discrete time $n$, it can either move a distance $s^n$ to the right with probability $p$ or a ...
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Is anything truly stochastic? [duplicate]
If everything in the universe happens according to rules, thermodynamic or otherwise, then how would anything (or any choice) ever be stochastic?
Multiple choices might be probable, but in any instant ...
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Novikov's theorem for time-independent functional
$\textbf{Introduction}$
Given a Gaussian-colored noise
\begin{equation}
\begin{split}
&\langle z(t)\rangle=0,\\
&\langle z(t)z(t')\rangle=C(t-t').\\
\end{split}
\end{equation}
A given ...
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Fokker-Planck: Uncertainty Propagation
I am interested in propagating the probability density function along the sampled trajectories having parametric uncertainty $a$. However I discovered that the fokker-planck equation for that system ...
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Is the Boltzmann distribution 'memoryless'? What is the physical interpretation?
The chance of occupying state with energy $E_i$ is given by the Boltzmann distribution:
$$ P(E_i)= \frac{1}{Z} \exp \left( \frac{-E_i}{kT} \right) $$
The exponential distribution is a common ...
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Itô correction on the Schrödinger equation
$\textbf{Introduction:} $
Given a time-dependent Hamiltonian $\mathcal{H}(t)$, and a state $|\psi(t)\rangle$, one would like to solve the Schrödinger equation
\begin{equation}
i\frac{d}{dt}|\psi(t)\...
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Is the solution of a stationary Fokker-Planck equation the Boltzmann's distribution?
I've seen that if we consider a stationary closed system such that $\partial_x J=0$, so that $J=const.$ and, since it must vanish at infinity, $J=0$, the solution of the Fokker-Planck equation $\...
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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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Is the invariant distribution of the underdamped Langevin equation the Bolztmann distribution?
Essentially what the question asks. I've seen proofs that in the overdamped limit, Langevin dynamics relaxes distributions to the Boltzmann distribution in the $t \to \infty$ limit; but what happens ...
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Derivation of Fokker-Planck equation from Langevin equation
I have been trying for a long while to wrap my head around this step in the derivation of the Fokker-Planck equation in Appendix 8 of Nigel Goldenfeld's "Lectures On Phase Transitions And The ...
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A simple picture for work fluctuation relations?
One possible formulation of the second law of thermodynamics is that the work extracted during the change of a thermodynamic system between two thermodynamic states is at most equal to the free energy ...
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A simple picture for the fluctuation-dissipation theorem?
In condensed matter and materials physics it is often assumed that the response of a condensed phase to some perturbation is determined by the fluctuations of the system at equilibrium (without ...
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Average of a time derivative
Given a probability distribution P(x,t), we can take the average of a time-dependent quantity x(t) as
\begin{equation}
\overline{x(t)}=\int dx x(t) P(x,t)
\end{equation}
My question is: what if I have ...
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What does it mean for a classic field to be defined in terms of stochastic parameters?
I'm writing a bachelor's thesis related to inflationary cosmology and I don't quite understand some things about a paper I've been reading called Signals of a Quantum Universe. Specifically, the paper ...
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Finding back a simple SDE from its solution
I'm trying to self-learn Kurt Jacob's Stochastic Processes for Physicists: Understanding Noisy Systems. I've followed Chapter 3, where I saw how to derive that the solution to the SDE
$$
dx=\left(c+\...
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Stochastic forces, their relation to Probability and active matter
Recently, I've been trying to read up a lot on active matter, and I realize most of what I read are papers that are relatively new. But to dive into the topic as well, I am unable to find much backing ...
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Issue deriving Fokker-Planck equation starting from Boltzmann's equation
I was trying to derive the Fokker-Planck equation starting from the Boltzmann's equation and I run into some issue while trying to do so.
Starting from Boltzmann and using the notation $f \equiv f(x, ...
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What is relation connecting $H$ (generating matrix of a continuous time random walk) and the laplacian matrix of the graph?
In the paper Unstructured Search by Random and Quantum Walk by TG Wong, the time evolution of the states of continuous time random walk (CTRW) is given as:
$$\vec{p}(t)=\mathrm{e}^{Lt/\vert\vert L\...
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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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Kinetic Monte-Carlo Simulation to solve master equation
Kinetic monte Carlo can be used to simulate Master equation for Markov processes.For simple 1D random walk the master equation is -
$$\frac{\partial P(n,t)}{\partial t}=pP(n-1,t)+qP(n+1,t)-(p+q)P(n,t)$...
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The distribution function of active Brownian process
I have a problem with the derivation of the distribution function of the stationary state using the system`s propagator as it has been mentioned in equation number 14.
Basically, we know that the ...
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Autocorrelation of Brownian motion - question about derivation
The calculation of the autocorrelation function for particles undergoing Brownian motion is described in chapter 7.6 of Chaikin and Lubensky (See picture below). There are couple of things which I ...
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Anomalous diffusion in a system of confined interacting particles
I have a system of charged particles confined parabolically. I want to analytically study anomalous diffusion in such a system using Langevin Dynamics. In a system of non-interacting particles there ...
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Power spectrum normalization
First off, I post this question in Physics since it stems from a physics problem, but it may be more pertinent to signal processing; sorry if it's the wrong place.
TLDR: what are the correct ...
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Any restrictions on drift vector in Fokker-Planck equation?
The most general Fokker-Planck equation for a probability density $f$ over phase space is
$$\partial_t f = -\partial_i(u^i f) + \frac{1}{2} \partial_i \partial_j (D^{ij} f)$$
where $u^i$ is the drift ...
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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 ...
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Adiabatic theorem for stochastic time-dependence
I am trying to derive the adiabatic theorem when my time-dependent Hamiltonian is stochastic and I have a few questions. Usually, one starts with the Schrödinger equation
\begin{equation}
i\frac{d |\...
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I need help with the solution of the general Fokker-Planck equation derived here
$p(\mathbf{x},t\vert\mathbf{y},t)=\delta\left(\mathbf{z}-\mathbf{y}\right)\tag{3.5.7}$
For a small $\delta t$, the solution of the Fokker-Planck equation will still be on the whole sharply peaked and ...
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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 ...
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How to take the average of a stochastic differential equation?
I am solving a set of stochastic differential equations and I need some feedback about if what I am doing is correct.
Given a vector $\boldsymbol{C}(t)=(C_+(t),C_-(t))^T$, we can writte a set of ...
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Adiabatic theorem with stochastic variables
Suppose a system which is driven by a stochastic complex variable $\alpha$(t). Looking at the eigensystem, both eigenvectors and eigenvalues are then stochastic variables. In my case, after building a ...
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Searching for steady state of a 2d non-linear Fokker-Planck equation without detailed balance
I'm studying a system given by two stochastic processes, $x(t)$ and $y(t)$ where $x(t)$ is an Ornstein-Uhlenbeck process with spring constant $\alpha$ and $y(t)$ is ruled by
$$\frac{d y}{dt} = -\alpha ...
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Is there sufficient "content" in the field of econophysics to write a substantial undergraduate thesis/project on? [closed]
Okay, maybe the title is somewhat misleading. My university calls this a BSc Project, but it is limited to between 4000 and 6000 words, so it isn't particularly long. Anyhow, one of the projects ...
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Spectral representation of a white stationary process
I am trying to better understand the spectral representation of stochastic processes. From the book "Spectral Analysis for physical applications" by Walden and Persival:
The spectral ...
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How can I form an advection-diffusion SDE to obtain the desired discretization?
Suppose that $\mathbf{s}(t)\in S$ denotes the spatial location of a process at time $t$. Further, let $\mathbf{x}(\mathbf{s}(t))$ denote covariates at the location $\mathbf{s}(t)$. My goal is to write ...
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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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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 ...
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Under what conditions is mean square displacement $\text{MSD}(t)=4Dt+v^2t^2$ a valid asymmetric random walk model?
I am reading the paper Actin dynamics drive microvillar motility and clustering during brush border assembly by Meenderink et al. (2019). In this paper, the authors fit the mean square displacement (...
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Controllability of stochasic bilinear system in 2-dimension
I have a question about controllability and I am thankful for any reply.
A linear system $\dot{x} = Ax + Bu$ is controllable iff it satisfies the well-known Kalman rank condition. Here, ...
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How can imparted energy be a stochastic quantity?
It may be a silly question, but I have a dosimetry course and it started by defining deposed energy and imparted energy and for both it says that they're stochastic quantities. The mathematical ...
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Distribution of density operators under Stochastic Master Equation
Stochastic master equations (SME) are used in studies of open quantum systems. The general form of an SME is:
\begin{align}
\tag{1} d\tilde{\sigma}(t) = - i [H, \tilde{\sigma}(t) ]dt + \frac{1}{2}\...
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Cormorant takeoff patterns
I was watching Our Planet recently and one scene in particular stood out to me: https://youtube.com/clip/UgkxyggGjQg-0_uV2N-n_m0t_1NK2hNyEqyW.
There are plenty of references that explain, to varying ...
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How often does a molecular machine run in reverse?
I was reading the Wikipedia article on Stochastic thermodynamics, and came across this statement in the overview:
When a microscopic machine (e.g. a MEM) performs useful work it generates heat and ...
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Euclidean Time Feynman Path Integral as Stochastic Differential Equation
For a quantum system with Lagrangian $L(x, \frac{dx}{dt})$ we can represent the action of a path $\mathbf{x}$ as $$S(\mathbf{x}) = \int_0^{t} L(\mathbf{x}(s), \mathbf{\frac{dx}{dt}}(s)) ds.$$ Then, ...
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How to understand the Fourier transform of a stationary random function?
A stationary random function like $v(t;w)$ is not periodic and not squared integrable, why we can do a Fourier transform to it, which is a very common process to analyze turbulence. How will a ...
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