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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Properties of random-walk in infinite and finite two-dimensional space: probability of two particles being in the same location at time t

I have been told that one of the property of the continuous-time random walk in two dimensions is that: $$\int_{Z} \, G(z, t | p_1) \, G(z, t | p_2) \,dz = \,G(p_1,p_2,2t)$$ where ...
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Do Stochastic Differential Equation models conserve energy?

I have recently started looking into stochastic models of chemical reaction systems, particularly the Chemical Langevin Equation (CLE) SDE model (e.g. here). One thing I'm trying to understand is ...
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Where can I get a good quality video (preferably slow motion / high frame rate) of Brownian motion particles for tracking their positions?

I am trying to analyse how good the Langevin equation fits actual experimental data by tracking the position of Brownian motion particles from video footage. However, I was unable to get my hands on a ...
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2D-Random Walk or just Simple Math?

I'm looking into a problem that involves the following sum (which is commonly reffered to as the decoherence-factor): $$r(t)=\sum_{\substack{z_l=0,1\\1\leq l \leq N}}|b_{z_1z_2\dots z_N}|^2e^{-2i\...
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Probability distribution of particle diffusion system with a source and absorbing boundaries

Consider a simple 1D particle diffusion process described by the SDE $dx=\sigma dW$, where $dW$ is a Wiener process. The forward Fokker-Planck equation can then be written as $$ \frac{\partial P(x,t)}{...
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Fokker-Planck: uniqueness and convergence to stationary distribution

Consider the Langevin equation ($N$-dimensional) with nonlinear drift term, but expressible as a gradient of a function $U(\vec{x})$. Namely, consider the stochastic process described by the set of ...
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Meaning of $\langle X(t')X(t'') \rangle$?

Context My background is not in physics so I am not very familiar with the $\langle \rangle$ notation. I am trying to understand the following in a paper that I am reading (Berglund AJ., PhysRevE., ...
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Physical interpretation of a multi-time (more than 2) autocorrelation function: non-Gaussian diffusion

In non-equilibrium statistical mechanics, the time-autocorrelation functions become the cornerstone of various theories and models. One such important autocorrelation is the velocity autocorrelation ...
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Regarding calculation the moments of a random variable whose probability distribution obeys the Fokker Planck equation

I was going through Van Kampen's Stochastic Processes in Physics and Chemistry, and I was trying to solve the exercises from Chapter 8 about the Fokker Planck equation (just in case context could help ...
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Cosmology context - MCMC code : Recomputation of covariance matrix after each point accepted

I am working on a MCMC code (basically with Metropolis-Hastings) and I would like to understand different important points. We always mention the covariance matrix which is used in the computation of ...
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Second-order Dirac equation

I'm wondering if one of you could tell me about the following equation: $$\partial_t \Psi = i \sigma_z m - \sigma_y k \partial_x \Psi + i \sigma_y k' \partial_{xx}\Psi$$ where $m, k,k'$ are real ...
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Is there consensus among physicists that reality is fundamentally deterministic? [duplicate]

Does Heisenberg’s Uncertainty Principle mean that the universe cannot deterministically be predicted by observers, or does it mean that the universe is inherently indeterministic, meaning that the ...
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Bells jump process on the lattice, simple example

At the moment i am reading the paper of Vink, "Quantum mechanics in terms of discrete beables". (http://www.psiquadrat.de/downloads/vink93.pdf) Here, in section III, Vink uses Bells beable ...
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Statistically stationary, periodic random process

As I understand in a statistically stationary process, the statistics are invariant under a shift in time. It is natural to assume that the statistics are periodic in a periodic random process. If ...
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Why there isn't a factor of $1/n!$ in the transition probabilities of this reaction?

In Grainder handbook of stochastic methods page 241, a chemical bistable system is given by We want to solve this as a birth-death master equation. My question is about the transition probabilities $...
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Brownian motion and multi-scale stochastic processes

The Stokes-Einstein equation for the diffusion coefficient of small colloidal particles in suspension is canonically derived under the assumption that the primary motion of the particle is ...
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Integral expression for covariance matrix in diffusion process

Consider the Fokker-Planck equation $$\frac{\partial \rho}{\partial t} = \sum_{i,j=1}^2 \mathbf{\Gamma}_{ij}\frac{\partial}{\partial x_i}(x_j \rho) + \mathbf{D}_{ij}\frac{\partial^2 \rho}{\partial x_i ...
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Calculating the 2D power spectrum of an axis-averaged Gaussian Random Field along a given axis given the 3D power spectrum

Suppose we are given a $3D$ Gaussian Random Field inside a unit cube $M \subset \mathbb{R}^3$, such that $\phi(\mathbf{x}) : M \rightarrow \mathbb{R}$, with a given $3D$ power spectrum $P_{3D}(k)$. We ...
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What is the probability that a random walk forms (almost) a circle?

Given is a random walk of a particle in 3d (such as an atom in a liquid). The particle proceeds randomly (in 3d), with an average straight displacement length a. Is there a way to get a probability ...
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If you have a gas with $n$ particles, can you model this as a random walk of a single particle in $3n$-space?

If you have $n$ particles in a box that undergo diffusion, this is basically a random walk of $n$ particles. Can this exactly be modeled by a single random walk in $3n$ space? Does the variance of ...
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(Detailed) Fluctuation Theorems/Relations and their implied symmetry

I'm currently reading up on non-equilibrium statistical mechanics, in particular so-called fluctuation theorems or fluctuation relations. In Section 3.1.2 of arXiv:1205.4176, the author introduces the ...
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In the Langevin dynamics: neglecting inertia. A mathematician trying to understand physics terminology

If we write the Langevin equation: for a particle with mass $m$, position $x$ and velocity $v$, with some damping coefficient $\gamma$, $$ m dV(t)=-\gamma V(t)dt+dW(t) ,~~~~~~~dX(t)=V(t)dt.$$ Then as $...
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How does dye move in water?

My understanding is that dye moves through water primarily through diffusion. The introduction to these lecture notes seems to confirm: If you we put a drop of red dye in water, it will slowly ...
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What is the Fokker-Planck collision operator and how is it derived?

On page 7 of Goldston and Towner (1981) they state that "The Fokker-Planck collision operator for pitch-angle scattering is given by $$\left.\frac{\partial f}{\partial t}\right|_c=\frac{\nu_{ii}}{...
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Power-Spectrum for Self-Organised Criticality

In 1987 Bak, Tang and Weisenfeld authored a paper (link) on Self-Organised Criticality, on how minimally stable self-organised systems propagate the perturbations it is subjected to, scale-freely - ...
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Conceptualizing time series data of fluctuating sizes of particle aggregates

I am working with simulation data (a time series of positions) of aggregating particles. I want to look at the overall distribution of aggregate size. A colleague calculated the number of aggregates ...
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Fokker-Planck for the Wiener process [duplicate]

Suppose I have the Fokker-Planck equation for a system (Wiener process) as follows $$\frac{\partial}{\partial t}p(w, t|w_0, t_0)=\frac{1}{2} \frac{\partial^2}{\partial w^2} p(w, t |w_0, t_0)$$ And I ...
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Onrstein-Uhlenbeck Process with power-law-correlated noise

It's a copy I posted as: https://math.stackexchange.com/questions/4356212/orstein-uhlenbeck-process-with-power-law-correlated-noise Consider a noise-driven drifting system given by the Langevin Eq: $$\...
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Wiener process as the integral of a stochastic force

I have seen (in my lecture notes) the following definition for a Wiener process: $$W(t)=\int _0 ^t dt'\eta(t') \tag{1}$$ where $\eta(t)$ is the stochastic force appearing in the Langevin equation for ...
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How many flips does a tossed macroscopic coin need to go through until the coinflip's result becomes indeterministic?

A coinflip is a macroscopic event and is deterministic in nature. A coin-flipping machine that operates at the greatest physical precision possible would be able to predict the coinflip's result (...
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Does indoor odor smell travel up or down?

Does odor smell, let's say it's from caulk off-gasing, travel upward or downward in the air? Are all odor smell lighter than air?
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How does a virus fall down in static air?

If we drop a virus from a height, in static air, will it fall to the ground like a lead ball, a balloon, or like a virus? How will it fall to the bottom? Like a Brownian particle? It will not float ...
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How are single photons correlated inside the "coherence time" of the laser? (And how would this affect a random number generator?)

This review on quantum random number generation describes how random numbers can be generated from a simple optical setup. In one example setup they give: weak light travels through a beam splitter ...
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Displacement root mean square for diffusion and random walks

For 1D random walks we have $$x_{rms}=\sqrt{\frac{l^{2}}{\tau } t}\tag{23}$$ (in this lecture) as well as for 2D case we have $$r_{rms}=\sqrt{\frac{l^{2}}{\tau } t}\tag{19}$$, where $l$ is length of ...
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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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Path Integral for Fokker-Planck equation

As per Wio, the special case of the Fokker-Planck equation (in SDE form) \begin{equation*} dX = f(x)dt + \sqrt{2D} dW_t \end{equation*} has the path integral representation in the Ito scheme as \...
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Higher-order Langevin noise correlation

Supposing Langevin noises are white noise, we know that the noises F are Gaussian and higher-order noise correlations, $\langle F_{t1}F_{t2}...F_{tn}\rangle$ can be decomposed by the second-order ...
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Interacting particle systems

I am novice to interacting particle systems. I was reading a book on the same. The book gives an example of a population system of $n(t)$ particles at time $t$. The births rates are $\lambda$ births ...
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Ornstein–Uhlenbeck process: joint probability as a Gaussian

The problem Consider a stochastic process with the following three properties: The process is Markov, meaning that $p(x_n,t_n|x_{n-1},t_{n-1},\ldots x_1, t_1) = p(x_n,t_n|x_{n-1},t_{n-1}).$ The ...
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Random Walk of Thermal Electrons

The drift velocity of electrons in a typical electronic circuit might be measured in mm per second. In contrast, the thermal velocity of electrons is in the vicinity of km per second. Because of the ...
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Can deterministic and stochastic models be used to models different parts of the same phenomenon?

A biological phenomenon including two different causally dependent phenomenon occurs in the cell. (Phenomenon2 initiates as a result of phenomenon1). Phenomenon1 and phenomenon2 are linked with ...
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Is quantum mechanics stochastic or probabilistic?

Is quantum mechanics stochastic or probabilistic? Is the universe fundamentally deterministic? Indeterminism in Quantum Mechanics is given by another "evolution" that the wavefunction may ...
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The results of different discretization rules in stochastic integral

I have learned how to calculate different discretization rules in differential forms, but some papers prefer not to express the random terms in differential forms like $$\frac{dx}{dt}=f(x,t)+\xi(t).$$ ...
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Furutsu-Novikov Formula Generalisation

The Furutsu-Novikov formula gives the expectation value of a zero mean Gaussian process $z(t)$ and a functional of that process $R[z]$: $$\langle{z(t') R[z]}\rangle = \int^{t}_0 \mathrm{d}s K_2(t',s) \...
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Is Nelson's stochastic mechanics wrong, and if so, why?

Is it [Nelson's stochastic interpretation of QM, and other similar theories] wrong? I honestly do not know but would be very happy to be educated and/or referred to a paper describing an experimental ...
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Correlation of position and velocity in Brownian motion

There are two definitions of the term "Brownian motion": a physical science definition based on how things such as Brownian particles move, and a mathematical definition as a certain ...
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Asymmetric Random walk with a pause [closed]

In the non-equilibrium statistical mechanics framework, there are two basic paradigms for defining the dynamics of the system: the Langevin and Fokker-Planck equations for diffusion processes and the ...
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Definition of heat when the temperature is changing in stochastic thermodynamics

I am currently studying stochastic thermodynamics, where the heat for a Brownian particle is defined by $$ dQ = -\gamma \dot{x} dx +\eta(t)dx, $$ where $\eta(t)$ is a white noise, with correlation $\...
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Interpreting distance in random walk

I've recently started reading about the random walk, from different sources across the internet, and there is this small detail that I'm not being able to wrap my head around. Suppose we have, a ...
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Chaotic and Ordered Random Boolean Newtorks with a fixed in-degree k and a probability p

I'm working with Random Boolean Networks, I made a python program to show the dynamics of the networks. Before coding the program I study the theory and it says that the in-degree k and the ...
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