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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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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Why the relaxation time of a globular cluster is referred to the center when solving the Fokker-Planck equation? [closed]
I'm currently studying Stellar Dynamics and I have a problem with the concept of relaxation time which appears in the Fokker-Planck equation.
Basically, solving the Fokker Planck equation, we obtain a ...
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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 -
$$\frac{\partial P(n,t)}{\partial t}=pP(n-1,t)+qP(n+1,t)-(p+q)P(n,t)$...
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Going from stochastic Schrödinger equation to master equation
I am currently reading the book "Quantum measurement and control" by Wiseman and Milburn (https://doi.org/10.1017/CBO9780511813948) and something is really bugging me in the chapter on ...
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The distribution function of active Brownian process
I have a problem with the derivation of the distribution function of the stationary state using the system`s propagator as it has been mentioned in equation number 14.
Basically, we know that the ...
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Autocorrelation of Brownian motion - question about derivation
The calculation of the autocorrelation function for particles undergoing Brownian motion is described in chapter 7.6 of Chaikin and Lubensky (See picture below). There are couple of things which I ...
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Anomalous diffusion in a system of confined interacting particles
I have a system of charged particles confined parabolically. I want to analytically study anomalous diffusion in such a system using Langevin Dynamics. In a system of non-interacting particles there ...
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Power spectrum normalization
First off, I post this question in Physics since it stems from a physics problem, but it may be more pertinent to signal processing; sorry if it's the wrong place.
TLDR: what are the correct ...
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Any restrictions on drift vector in Fokker-Planck equation?
The most general Fokker-Planck equation for a probability density $f$ over phase space is
$$\partial_t f = -\partial_i(u^i f) + \frac{1}{2} \partial_i \partial_j (D^{ij} f)$$
where $u^i$ is the drift ...
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Frictionless Brownian Motion
The Langevin equation for a Brownian particle without the friction term is:
\begin{equation}
m\dot{v}=F(t)
\end{equation}
Where $F(t)$ is the random force acting on the Brownian particle due to ...
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Adiabatic theorem for stochastic time-dependence
I am trying to derive the adiabatic theorem when my time-dependent Hamiltonian is stochastic and I have a few questions. Usually, one starts with the Schrödinger equation
\begin{equation}
i\frac{d |\...
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I need help with the solution of the general Fokker-Planck equation derived here
$p(\mathbf{x},t\vert\mathbf{y},t)=\delta\left(\mathbf{z}-\mathbf{y}\right)\tag{3.5.7}$
For a small $\delta t$, the solution of the Fokker-Planck equation will still be on the whole sharply peaked and ...
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Phase-amplitude stochastic differential equations
In the book of $\textit{The Quantum World of Ultra-Cold Atoms and Light: Book 1 Foundations of Quantum Optics}$ by Peter Zoller and Crispin Gardiner on page 75, they derive the phase-amplitude ...
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How to take the average of a stochastic differential equation?
I am solving a set of stochastic differential equations and I need some feedback about if what I am doing is correct.
Given a vector $\boldsymbol{C}(t)=(C_+(t),C_-(t))^T$, we can writte a set of ...
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What does it mean to have delta-correlated process physically?
I am reading about Langevin dynamics, and I see the following equation:
$$\frac{dx}{dt} = -\frac{1}{\xi} \frac{\partial U}{\partial x} + g(t)$$
Then, they claim that the average $$\langle g(t) \rangle ...
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Fokker-Planck equation from Langevin equation in stochastic inflation
I'm reading this paper by Starobinsky and Yokoyama where they give the coarse-grained equation of motion,
$$
\dot{\bar{\phi}}({\bf x},t ) = -\frac{1}{3H}V'(\bar{\phi}) + f({\bf x},t)
$$
where $f({\bf ...
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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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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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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
$$\frac{d y}{dt} = -\alpha ...
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Is there sufficient "content" in the field of econophysics to write a substantial undergraduate thesis/project on? [closed]
Okay, maybe the title is somewhat misleading. My university calls this a BSc Project, but it is limited to between 4000 and 6000 words, so it isn't particularly long. Anyhow, one of the projects ...
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Spectral representation of a white stationary process
I am trying to better understand the spectral representation of stochastic processes. From the book "Spectral Analysis for physical applications" by Walden and Persival:
The spectral ...
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How can I form an advection-diffusion SDE to obtain the desired discretization?
Suppose that $\mathbf{s}(t)\in S$ denotes the spatial location of a process at time $t$. Further, let $\mathbf{x}(\mathbf{s}(t))$ denote covariates at the location $\mathbf{s}(t)$. My goal is to write ...
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Full translation of the paper written by M Smoluchowski in 1906 "Zur kinetischen Theorie der Brownschen Molekular Bewegung und der Suspensionen"
May I ask if the full translation of the classic paper on Brownian motion and SDE written by M Smoluchowski in 1906 "Zur kinetischen Theorie der Brownschen Molekular Bewegung und der Suspensionen&...
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Fokker-Planck equation for overdamped motion: how to define the average velocity
Consider the Langevin equation in the overdamped regime,
$$
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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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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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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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.$$ Then, ...
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Sufficient and necessary conditions on random walk to obtain standard diffusion equation
In the simplest random walk model that is generally considered, the probability of the finding the particle at time $t$ in $x$, $P(x,t)$ is given by,
$$
P(x,t) = \frac{1}{2}\big[ P(x-a, t-\tau) + P(x+...
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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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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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The "Algorithm" for Derivation of the Master Equation
I'm trying to come up with a master equation for a spin system. So, I'm trying to understand in general how can we derive the master equation and the Lindblad operators if we know the system, bath and ...
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How to evaluate the integral of velocity autocorrelation function for calculating the diffusion coefficient
I am reading Kubo's review article "The fluctuation-dissipation theorem" (http://www-f1.ijs.si/~ramsak/km1/kubo.pdf)
Could someone help me with how Eq. 2.5 is derived? I am confused with how ...
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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 $t_1 < t_2$:
$$W_{t_2} =...
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Magnetic field modeling with noises
I am trying to make a 3d grid of a magnetic field with some noises (which will be added to the ordinary field) for a computer simulation. I have the formula for the ordinary field, also I am using a ...
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Why is it legitimate to use the Poisson law for likelihood computation in particle physics? (background events are not rare)
In experimental particle physics at colliders, there are a high number of collisions of incoming particles, for example protons at LHC. Once protons collide in a given collision event, those protons ...
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Dirac delta correlated white noise 3 time points [closed]
If I know that there is a noise which is delta correlated that is $\langle f(t)f(t') \rangle =\delta(t-t')$, can I say something about $\langle f(t)f(t')f(t'') \rangle $?
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Why are realizations of the Stochastic Schrödinger Equation (apparently) not normalized?
In "A Straightforward Introduction to Continuous Quantum Measurement" the Stochastic Schrodinger Equation (SSE) for continuous measurement of the observable $X$ is given as $$ d |\psi\rangle ...
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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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References for understanding Martin-Siggia-Rose (MSR) formalism towards path integral
Can anyone help me understand Matin-Siggia-Rose-deJensen formalism? To be more specific, I'm studying this paper, and in section 5, the authors go all about classical limit and the usage of MSRJD ...
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Probability density of delta-correlated Gaussian white noise
The delta correlated Gaussian white noise $\eta(t)$ used in the study of stochastic processes is defined as the "derivative" of the Weiner process whose conditional probability density is ...
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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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Master equation for system of N discrete states
I am given a system with $N$ discrete states $1,2,...,N$. Between this states transitions with rates $\omega_{n\rightarrow m}$ take place, which are non-zero for $m = n \pm1$.
From state $1$ only the ...
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