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 - naming a vector field

A Fokker Planck equation for the prob. density $\rho$ may be written in the form of a continuity equation $$\frac{\partial \rho(x,t)}{\partial t} = - \nabla \cdot \left[ g(x,t) \rho(x,t) \right].$$ ...
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Relation between *sigma algebra* and *sigma models*

Is there connection between sigma models in physics and sigma algebra in the probability theory? Background: I have never had to study the former, but I am somewhat familiar with the sigma-algebra in ...
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Evolving policy of Ising models?

Setup Let the Hamiltonian of the Ising model be $$ H_{J,h}(\sigma) = \Sigma_{i, j} J\sigma_i \sigma_j + \Sigma_{i} h \sigma_i.$$ Then the Gibbs partition function for the pair $(J,h)$ is given by $$ ...
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3D Coalescence of Particles under Brownian Motion [closed]

Consider a system of $N$ point particles undergoing Brownian motion on a $3D$ space with equal diffusion coefficients $D$ and an average spacing $L$. If we have a probability $p$ of these particles ...
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Probabilty Density for $x$

Suppose that I have the function $$x(t)=e^{-b(W(t))^2} \ \ \ \ \ \ \ \ (1)$$ where $W(t)$ is just a Wiener process (i.e. a Gaussian in general). I want to know what the probability density for $x$, $P(...
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Non-markovian random walk. Elephant random walk

When one does the numerics for the usual random walk, one might use the transition matrix in order to get the probability ar time $t$ of the process. As expected, the asymptotic behavior yields a ...
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Jacobian in dynamic path integral

I'm confused whether the Jacobian is needed in a path integral representation of a dynamical system, as I've seen multiple conventions in the existing literature. For simplicity, let's just consider ...
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Why do some aerosol particles never settle down?

Particles and liquid droplets below the size of 1 micrometer usually never settle down easily, and as their size decreases, it takes longer for them to settle down. Why is it like that? Does Stoke's ...
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Meaning of Fokker-Planck with non-differentiable and/or infinite potential

The Fokker-Planck equation for a diffusing particle in the potential $V$ is $$\partial_t p = -\nabla\cdot (p \nabla V) + D \Delta p.\tag{1}$$ In the literature, one often sees this formulation used ...
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Non-polynomial correlators for stochastic path integrals

What is the right way to evaluate a stochastic path-integral of the form: $$\int \mathcal{D}x \mathcal{D}\tilde{x} \left( \int_0^T \sin(x(t_2)) x(t_2) dt_2 \int_0^T \tilde{x}(t_1) dt_1 \right) e^{-\...
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SDE with drift multiplied by telegraph like random process

Let the stochastic process $\{X_t\}$ be defined by the following SDE (Ito's convention for discretization) $dX_t=\frac{1}{p}S_tg(X_t)dt+\sqrt{2}dW_t$ where $W_t$ is a standard Wiener process, $g: \...
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Probabilities for quantum random number generators

Consider a quantum random number generator (QRNG) X, which generates integers at random. (Apparently, due to quantum statistical properties, this type of generation is truly at random, see e.g. "...
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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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Fokker-Planck equation with time-dependent potential

Consider a Fokker-Planck (FP) equation where the advection term is a function of time, i.e. \begin{align} \frac{\partial P ( x , t )}{\partial t} = -\nabla \cdot \left[ -\mu \, P \, \nabla U (x,t) ...
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Why is Wiener process a homogenous but not a stationary stochastic process?

I found the claim that wiener process is an example of a stochastic process which is homogeneous but not stationary in the book 'The Theory of Open Quantum Systems' by Breuer and Petruccione. (Section ...
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What is the characteristics time scale of a quantum system, in context of the Markovian approximation?

In the theory of open quantum system, we make the markovian approximation when the timescale of the memory of the reservoir is small. But this timescale is measured with respect to the characteristic ...
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What does Schrödinger equation reduce to in the limit of a continuous position measurement?

If we measure position of a quantum particle, we force its wavefunction to collapse into a wavefunction whose probability density is given by a Dirac delta function (all the probability density of ...
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Brownian motion and the heat equation

Einstein showed that the Brownian motion provides a solution to the heat equation. As written here, the relation between the brownian motion and the heat equation can be shown by the taylor series. ...
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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 ...
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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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Why does a collection of radioactive atoms show predictable behaviour while a single one is highly random?

Well, we know that it is impossible to say exactly when a radioactive atom will go on decay. It is a random process. My question is why then a collection of them decays in a predictable nature (...
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On the definition of a stationary process

I have come across various ways one refers to a process as stationary and cannot seem to get the equivalency and the level of rigor in each of them: According to Stochastic processes in physics and ...
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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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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(...
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The Fokker-Planck equation and potential

Anybody know the Fokker-Planck equation with a potential term in the front? $$\frac{\partial P}{\partial t}=\left[V(x)-\partial A+\frac{1}{2}\partial B\partial B\right]P.$$ The above is the form of ...
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Derivation of Evolution Operator on finding Hu-Paz-Zhang (HPZ) Quantum Brownian Motion (QBM) Master Equation

I am trying to understand this paper by Hu, Paz, and Zhang about exact master equation of QBM in general environment. In the paper they used influence functional method introduce by Feynman and Vernon ...
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Treat stochastically non-Hamiltonian perturbations

Let us consider a classical dynamical system whose obserbvables $A$ evolve according to the following equation of motion \begin{align} \dot A &= -\{H,A\}+f(q) \end{align} $f(q)$ is a non-...
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Measure of Feynman path integral

Feynman path integral for non-relativistic case is defined as: $$\int\mathcal{D}[x(t)]e^{iS/\hbar}$$ where $$\int \mathcal{D[x(t)]}=\lim_{N\rightarrow\infty}\Pi_{i=0}^{i=N}\bigg(\int_{-\infty}^{\infty}...
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Small time solution to Fokker-Planck equation

In reference to this note, a specific Focker-Planck equation with initial condition $W(\rho, t=0)=\delta(\rho-1)$ have the solution $$W\left(\rho,t\right)=\dfrac{e^{-\frac{t}{4}}}{\sqrt{\pi}t^{\frac{3}...
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Meaning of $\frac{1}{\sqrt{dt}}$ in stochastic forcing

I am running a 2D fluids simulation with a stochastic forcing $f$ in a doubly-periodic box, i.e. solving $$ \frac{\partial \nabla^2 \psi}{\partial t} = J(\psi,\nabla^2 \psi) +f,$$ where $J$ is a ...
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Fokker-Planck equation in $N$-dimensions: a doubt regarding 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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Estimate time for a simple quantum evolution process

Consider the Hamiltonian given by the sum of two projectors $$H=-\gamma N P_s-P_w,$$ where $|s\rangle=\sum_{j=1}^N |j\rangle/\sqrt{N}$ is the uniform state on the $N$ orthonormal nodes $|j\rangle$, ...
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How to understand scale expansion in Fokker-Planck equation Part 2

The background of the question is given in How to understand scale expansion in Fokker-Planck equation. Using Taylor expansion $\bar{a}_{k}(x)$ as a funciton of $\epsilon$ around $\bar{x}_{0}(t)$, we'...
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How to understand scale expansion in Fokker-Planck equation

I read the book Statistical Methods in Quantum Optics 1 (Master equations and Fokker-Planck equations), published by springer. In Chapter $5$, to do scale expansion, the writer introduces a system-...
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Euler-Maruyama scheme

Can the Euler-Maruyama method be used to simulate Langevin equations for non-Gaussian white noise? I need to evaluate a Langevin equation of the form $$ dx= a(x)dt+D \eta dt$$ where $\eta$ is a non-...
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Application for computing an empirical mean from Markovian observations

Not being a physicist, I am trying to understand the applications of Markov chains in physics. I am looking for a few canonical examples in physics (possibly with references) where the scientist ...
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Why do we have different probability densities in the forward and backward Fokker-Planck equations?

For a system involving randomness, we can find a probability distribution $\rho$ that obeys the forward Fokker-Planck equation: \begin{align} \partial_t \rho + \nabla (\vec b \rho) &= D \nabla^2 \...
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Relationship between Markovianity and linear response function

I consider a two level system interacting with a continuum of field. The Hamiltonian in Schrodinger picture, under the rotating wave approximation is: $$H=-\hbar \omega_0 \sigma_z + \sum_k \hbar \...
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How can we interpret a system in which the probability distribution obeys the forward and the backward Fokker-Planck equation simultaneously?

For a system involving randomness, there is no longer a unique derivative and hence no longer a unique definition of velocity. But for the forward (Ito) derivative, we can find a probability ...
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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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Is the continuity equation valid for a diffusion current?

On the one hand, we have the diffusion equation: \begin{align} \frac{\partial\rho}{\partial t}&=D \nabla^2 \rho \end{align} and on the other hand, we have Fick's first law: \begin{align} \vec J = ...
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Why is the continuity equation only valid for the flux velocity but not for the osmotic velocity?

The continuity equation $$ \partial_t \rho + \nabla (\vec v \rho) = 0 , $$ can be derived from the Fokker-Planck equations for the forward- and backward velocity ($b,b^\star)$: $$ \partial_t \...
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What's the meaning of a continuity equation with $\nabla^2 \rho$ on the right-hand side?

I stumbled upon a continuity equation with a $\nabla^2$ term on the right-hand side: $$ \partial_t \rho + \nabla (\vec b \rho) = D \nabla^2 \rho , $$ where $b$ denotes the forward velocity and $D$...
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Why is $\left(P_{j+1}-P_{j}\right)$ proportional to the transmission rate from $(j+1)$th state to $j$th state

In the book of Zwanzig, Nonequilibrium statistical physics, at page 63, while giving an example of how to use Master equation, he states that A common application of master equations is in the ...
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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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How is the Mean Squared Displacement (MSD) affected by the drift?

Given a stochastic process $X(t,\omega):\mathbb{R}^+ \times\Omega \rightarrow \mathbb{R}^n$ that satisfies the following classical stochastic differential equation (SDE) in the It$\hat{\text{o}}$ ...
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Calculating entropy in truncated Wigner

I'm trying to get some reasonable measure of the entropy of a system modelled by the truncated Wigner method. The Wigner function contains all the information about a density matrix. So, I figure it ...
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Deriving the Bloch Equation of qubit purification for any measurement angle

I am trying to derive the Bloch vector $dr$ for a measurement of a observable in any arbitrary direction $\theta$. For context this is the setup and derivation I have for continuous measurement along ...
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Can a gas molecule theoretically have zero velocity?

According to Maxwell's speed distribution law gas molecule can have speed which lies between zero to infinity. But in the graph of the distribution curve it seems to touch zero velocity. So can a gas ...
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The justification for stochastic time evolution equation (in stochastic thermodynamics)

I came across an equation in the context of stochastic thermodynamics, specifically in the paper "ensemble and trajectory thermodynamics - a brief introduction": The time evolution of the state is ...

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