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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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What is the mean of the stochastic differential equation $dX=K dt + sigma X dW$ and how to find it? [migrated]

I have the stochastic differential equation, $$dX=k dt + \sigma XdW,$$ which I expect to have just the mean $kt$, since taking the expectation of the SDE gives E[dX]=E[k dt] due to the brownian term ...
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What is the correlation between Brownian noise's low frequency components and the actual movement of particles?

I do have some crude training in mathematics, but I'm not a physicist or engineer. So I'd appreciate a simple not too technical explanation. I conceptually understand how hitting a piece of wood will ...
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Helmholtz decomposition of flow at non-equilibrium steady state

I'm trying to work through Karl Friston's mathematical derivation of the Free Energy Principle from Langevin Dynamics (see this paper). I'm confused about the part at the end of page 8 where he uses ...
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Using the functional derivative and delta function for proving The Fokker-Planck Equation

I am reading "Lectures on Phase Transitions" by Nigel Goldenfeld, specifically Chapter 8, where the Fokker-Planck equation is derived. I found the following part of the proof, but there are ...
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Factorization on increments in Markov chain

I'm trying to show the following property for a Markov chain: $$\left<[x(t+\tau)-x(t)][x(t'+\tau)-x(t')]\right> =\left<x(t+\tau)-x(t)\right>\left<x(t'+\tau)-x(t')\right> $$ Where $t\...
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Method of inserting random numbers in the numerical calculation of mean-squared displacement for brownian particle

I am trying to plot Mean-Squared-Displacement for a passive Brownian particle. For that I'm using the discretized over-damped version of the Langevin equation as: $$x(i+1)=x(i)+\sqrt{\frac{2.k_BT.\...
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Boundary-condition-changing Operators for Free Boson BCFT with Dirichlet Boundary Conditions (or more general BCFTs)?

Is there any literature about boundary-condition-changing (b.c.c.) operators for the Free Boson with Dirichlet Boundary Conditions? The b.c.c. operators I'm interested in would replace boundary ...
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Viscous stress, and closed dynamic equations for density and velocity fields of an Ornstein-Uhlenbeck process?

Context I am trying to write simple derivation of hydrodynamic definition of viscous stress $\partial^{2}_{x}v$ based on OU-process which is \begin{align} \dot{x} &= v \\ \dot{v} &= -\gamma v +...
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Self-similarity of the diffusion equation

I am going through this book Simulation of Complex Systems. In the chapter on Brownian Dynamics, we considered a "free diffusion" given by the Stochastic differential equation: $$\dot{x}(t)=...
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Does Equipartition hold in overdamped dynamics?

We start with the Langevin equation $$m\frac{\mathrm{d}^{2}x}{\mathrm{d}t^{2}} = -\Gamma \frac{\mathrm{d}x}{\mathrm{d}t} +\sqrt{2\Gamma k_{B}T} \eta(t). $$ Now, we know that at $t \gg m/\Gamma$, the ...
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Why the transition probability in the master equation approach just the rate$*dt$ for a simple birth process?

I am modeling a process of an exponentially growing population of cells as $\frac{dn}{dt}=\lambda n$. To account for the intrinsic noise in the birth process of these cells, I write down the ...
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Evaluating the PDF of an active Ornstein-Uhlenbeck process

I am going through this paper: Inertial Self-Propelled Particles. For a free-particle, when there is no external potential, the authors have written the stochastic differential equations as - $$\dot{\...
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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 ...
seeker_after_truth's user avatar
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When is a Schrodinger equation equivalent to a Fokker-Planck equation?

In chap. 3 in these notes on kinetic theory, Tong shows that the Fokker-Planck operator for a particle undergoing overdamped Langevin dynamics in a potential $V$ is equivalent to a Schrodinger ...
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Interpretation of a stochastic Schrodinger equation

Suppose we have a linear stochastic Schrodinger equation (SDE) describing the evolution of a system in a finite-dimensional Hilbert space: $$ d\psi(t) = \left(-iH(t) - \frac{1}{2}\sum_{j=1}^dR_j^{\...
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Fokker-Planck equation for colored noise Langevin equation

I was wondering if there is a way to derive the Fokker-Planck equation for Langevin equation with a general colored noise: $$m \ddot{x} = -\frac{\partial V(x)}{\partial x} - \gamma \dot{x} + F(t)$$ ...
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How to calculate the thermal radiations of a seawater ball in the microwave range

I'm seeking for guidance on how to calculate the farfield spectral irradiance of the thermal radiations of an object made of a material with known complex permittivity ($\epsilon_r=\epsilon^{'}_r+j\...
Lionel Chemin's user avatar
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Discrete simulation of a Levy flight

I am trying to construct a discrete simulation of Levy flight in 1D and am wondering what is the best way to do so. For example, for pure diffusive random walk, one may assign probability of $1/2$ to ...
Brownian_Motion's user avatar
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White noise fluctuation amplitude

I'm trying to understand better noise processes, and have a very basic question. Suppose I have a stochastic process characterised by white noise, namely $$ \langle X(t) \rangle = \overline{X} \,;\...
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Brownian noise variance

I have a question on a Brownian noise mean square which I get from the exercise (10-4) reference [p493, Athanasios Papoulis and Unni Krishna Pillai, “Probability, Random Variables and Stochastic ...
Pierre Polovodov's user avatar
5 votes
1 answer
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Onsager-Machlup functional and the Boltzmann distribution

I've been looking into path integral representations of stochastic processes lately and came across the Onsager-Machlup functional description of the Langevin equation. In the overdamped case, where ...
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How to go from probability distribution to transitions probability distribution?

For the past few days I have been studying Advanced statistical mechanics. I am studying a Wiener process in general. Such a process is a non-stationaty time-independent Gaussian process. The ...
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Misunderstanding of deriving gillespie's algorithm

I recently studied various stochastic methods. In this lecture note (Stochastic and Diffusive Processes lecture note by Dr. Luca Donati) and this book(Markov Processes: An Introduction for Physical ...
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Asymptotic form of solution to biased random walk

(Cross post from math.stackexchange) Consider a continuous time biased random walk on a 1D lattice. The random walker walks with rate $k_\mathrm{R}$ to the right and with rate $k_\mathrm{L}$ to the ...
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Ornstein-Uhlenbeck process autocorrelation

In a Kubo's classic 1966 paper about the Fluctuation-Dissipation theorem (https://iopscience.iop.org/article/10.1088/0034-4885/29/1/306), I found the following issue which confused me: The author ...
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Limit for big system size of Fokker-Planck eigenfunctions

I am learning how to use diagonalization methods applied to Fokker-Planck equations with Gardiner's book and these notes. The idea is to find the probability density, $ P[X_t\in[x,x+dx]]=\rho_t \, dx$,...
Javi's user avatar
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In Langevin equation of stochastic oscillator, can the system has only fluctuation force with no fraction?

I am new to fluctuation statistical mechanics, In all examples that I encountered, the Langevin equation always has the fluctuation force accompanied by a friction force. $$\frac{d^2x}{dt^2}+\gamma \...
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Energy exchanges between a Brownian fluid and particles

In the context of the dynamics of polymeric models, and specifically the dumbbell model, one of the forces acting on a dumbbell spring is said to result from "a time smoothed Brownian force" ...
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Alternatives to the Fokker-Planck equation for deriving the probability distribution associated with Langevin dynamics

I was wondering if there are any other means of obtaining exact (or analytical approximations) of the phase space probability density for a system evolving according to Langevin dynamics. The typical ...
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Relation between hamiltonian perturbation theory (classical) and the Fokker Planck drift and diffusion coefficients?

Suppose I have a hamiltonian of the form $$ H(q,p) = H_0 + \epsilon H_1(q,p) $$ In perturbation theory we approximate the solution to the equations of motion as a power series in $\epsilon$: $$ q(t) = ...
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344 views

How to calculate mean squared displacement (MSD) value as function of Tau (lag-time)

I'm doing a research on Brownian motion (in 2D) and I want to calculate the MSD values in order to find the diffusion coefficient $D$. However, online I find different approaches on how to calculate ...
Naitzirch's user avatar
5 votes
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107 views

Entanglement island distance = quantum width of BH horizon?

In this recent paper, Bousso and Penington (B&P) finds that the protrusion distance outside the horizon of an entanglement island from a 4D Schwarzschild black hole is $\sqrt {l_P r_{hor}}$, where ...
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Path Integrals and Brownian Motion with non-standard kinetic term

Recently I've been thinking about $1d$ Path Integrals of some theories with non-standard Lagrangians. The adjective non-standard meaning that the Lagrangian $\mathcal{L} \neq \frac{1}{2} m \dot{x}^2 - ...
Physicist in disguise's user avatar
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Is there a way to simulate particle stopping stochastically, from a known $dE/dx$ curve?

I want to make a simulation of the stopping of Muons in liquid water that, instead of considering they lost exactly the amount of energy predicted by the $dE/dx$ curve at each iteration, sorts an ...
user2934303's user avatar
2 votes
0 answers
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Frequency Response of a Stochastic Oscillator Numerically

I am willing to obtain a frequency response plot for a stochastic oscillator governed by the following equation numerically. $$ \ddot{x}+2\Gamma \dot{x}+\omega_{0}^{2}x=f(t) $$ where $f(t)$ is a ...
Sourin Dey's user avatar
3 votes
2 answers
81 views

Can the solution of the Fokker-Planck equation exhibit negative values?

Is it possible for the solution of the Fokker-Planck equation to have negative values? I am referring to the mathematical aspect, irrespective of its physical interpretation. Additionally, considering ...
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Waiting/jumping times of non homogeneous process on a state chain (almost Markovian)

When considering a chain of states from 0 to N in continuous time with constant up and down going transition rates, the jumping or waiting times are exponentially distributed. Now consider the ...
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4 votes
1 answer
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Time-ordered matrix exponential quasi-static limit

Define a matrix differential equation $$\dot{X}=A(t)X(t),$$ where $X=[x1,x2,...]^T$ is a 1D vector and $A(t)$ is a complex-valued time-dependent matrix. This system can be solved by $$X(t)= \mathcal{T}...
J.Agusti's user avatar
3 votes
0 answers
112 views

Reduce multiplicative noise to additive noise with singular matrices

I have a stochastic differential equation as \begin{equation} \dot{\textbf{X}}=\textbf{A}\textbf{X}+\alpha(t)\textbf{B}\textbf{X}-\alpha^*(t)\textbf{B}^T\textbf{X} \end{equation} where $T$ ...
J.Agusti's user avatar
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Does the master equation break down for negative times?

I'm studying stochastic dynamics and have encountered the framework of the master equation for the study of continuous time Markov processes. First, I'll state some general definitions and then say ...
Felipe A. Barretto's user avatar
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2 answers
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What is a "necessary condition" to observe stationary probabilities? [closed]

In our Stat. Mech class the professor said that the detailed balance condition, \begin{align} K_{ij}P_{j}=K_{ji}P_{i} \end{align} is a sufficient but not necessary condition to observe stationary ...
Jaa na laurhy's user avatar
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Mathematical description of random noise

I am trying to mathematically model the noise affecting a certain physical quantity, say $X(t)$. Then, the noisy quantity would be $X'(t)$ which differs by some small value $\delta$ from the ideal ...
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Distribution sampling from a trajectory

Consider a sequence of random variables $\{X_1, X_2,.., X_n\}$ corresponding to regular measurements of a single observable over time obtained from a laboratory or computer experiment. Assume that the ...
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2 votes
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How to show random cluster models with non-integer $q$ have no local description?

It is known that the random cluster model with $q = 1$ corresponds to bond percolation, and $q = 2, 3, ... $ corresponds to the $q$-state Potts model. Both of these have a local description. What ...
tclin's user avatar
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1 vote
2 answers
155 views

Mean squared displacement of a particle on a biased random walk [closed]

Given a particle on a 1-D random walk with some drift velocity $\nu_d = \frac{\Delta x_d}{\Delta t}$, the position in at some time step j is given by $$x_j=x_{j-1}+k_j L + \Delta x_d$$ where $L$ is ...
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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 ...
JonasB's user avatar
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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 ...
Hitanshu Sachania's user avatar
1 vote
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55 views

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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