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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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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38 views

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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70 views

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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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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Is the Feynman's path integral a density?

The Feynman-Kac path integral formula is used to solve parabolic equations related to stochastic processes. Considering the probabilistic expression, the solution is indeed not a density. However, ...
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In mean-field theory, why are the collisions of particles in the mean-field neglected?

Mean field theory is a tractable framework for analyzing parameters of a continuum or an infinite number of identical micro-particles or agents. The former has been treated extensively in statistical ...
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Unbounded time derivative of the brownian motion and Newtonian dynamical models

The second order dynamics model $dx_s = v_s ds$, $d v_s = dw_s$ where $w_s$ is s standard Brownian motion is an example of Langevin dynamics. Note that the Brownian motion in this case models a white ...
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Can we deduce that particles behave as Brownian motions if the collection obeys the Einstein model?

The density dynamics of a continuum of particles with the dynamics $$dx^i_s = d w^i_s,$$ where $dw^i_s$, $0 \leq s$, $i \in \mathcal{N}$ is a standard Brownian motion, are given by the diffusion PDE $$...
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Variance of Simulated Langevin Equation

I simulated (by Matlab) the Langevin equation for optical-trapped particle in very short time "steps" And I got this white noise figure.. The question is how I can calculate the variance (or in ...
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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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Path integrals for brownian motion in a harmonic potential

The problem is as follows: Use the path-integral formulation of stochastic dynamics for a particle in a harmonic potential $U(r)= \frac{1}{2}kr^2$ to show that $$P(x,t|x_0,t_0)=(\frac{\beta k}{2\...
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Fokker-Planck linear potential

I am struggling with finding a solution to the following Fokker Planck equation with linear potential: $$\partial_{t}P(x,t)=k\partial_{x}P(x,t)+D\partial_{x}^{2}P(x,t)$$ Can anyone help me please? ...
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What's the meaning of the osmotic velocity?

To describe a random-walking particle, we can use the stochastic differential equation \begin{equation} dx(t)= b\Big(x(t),t \Big) dt + dw(t) , \end{equation} which is also known as the Langevin ...
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Fokker-Planck equation for 2D SDE

Consider the following two-dimensional SDE \begin{align*} \mathrm{d}\mathbf{X}(t) &= {\mathbf{f}(\mathbf{X}(t))}\mathrm{d}t+\mathrm{d}\mathbf{W}(t)\\ \end{align*} where $\mathbf{X}(t)=\begin{...
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In what physical contexts do the Airy2 process or the Tracy-Widom distribution arise?

I'm currently writing a Master's thesis in mathematics on the Airy$_2$ process, and have read vague references that it comes up in statistical mechanics when dealing with large systems of an ideal ...
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When we should use Langevin equation and when we should use Fokker-Planck equation?

As everyone knows that we can go from Langevin equation to Fokker-Planck equation which gives the evolution of probability density function. But what I don't understand is what is exactly the main ...
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an example of stationary markov chain which doesn't satisfy the detailed balance

I understand that if a markovian process satisfies the detailed balance relation:$\pi_{i}p_{ij}=\pi_{j}p_{ji}$, then it is stationary. So detailed balance relation is a sufficient condition. But is ...
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Random fields in physics - how do they work?

I'm trying to get an intuitive understanding of what random fields are. Wikipedia's article gives a formal definition (which is in very mathematical language), but also says various much more ...
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74 views

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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Circuit quantization and energy dissipation

So when we do the procedure of circuit quantization we use the hamiltonian formalism which is only true when theres no dissipation. however we know real life circuits are dissipative , Im aware that ...
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88 views

Martin-Siggia-Rose action corresponding to Langevin equation

What is the Martin-Siggia-Rose-DeDominicis-Janssen action corresponding to the overdamped Langevin stochastic equation $$\frac{d\mathbf{x}}{dt} = -\mathbf{\nabla}V + \mathbf{\eta}$$ where V is the ...
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State equation to state-space

I have a one question about convert state equation to state-space model. I can not create a state-space model because of $g$ term from given this state equations... $\dot{x_{1}} = \dot{z} = x_{2};$ ...
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96 views

A discrete time Ornstein-Uhlenbeck type process

Let's consider a stochastic process $$X_{t+1}=X_{t}+\Delta X_t$$ where $\Delta X_t$ is a Gaussian with mean $-\lambda X_t \Delta t$ and variance $2\Delta t$ where $\lambda$ and $\Delta t$ are ...
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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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