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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Is there consensus among physicists that reality is fundamentally deterministic?

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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Reversible vs irreversible operators in stochastic transport theory

The particle in a potential driven by a fluctuating Gaussian white noise force $\Gamma(t)$ has equation of motion $$ \dot{v} = -\gamma v +f(x) +\Gamma(t) $$ and its probability $P(x,v,t)$ to have ...
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Dynamical interpretation of reflecting boundary conditions in the Fokker-Planck equation

Background: For a particle driven by the dynamical equation $$ \dot{x}(t) = a(x,t) + b(x)\xi(t),$$ where $\xi(t)$ is a Gaussian white noise, the probability distribution of position $x$ is governed by ...
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A random walk with different PDF for odd and even steps

Suppose I have a random walker that jumps $2N$ steps on $1D$ space, the position after jumping is $$X=\sum_{i=1}^{2N} \delta x$$ and for even $i$ we have PDF $f(\delta x)=\frac{e^{-(\delta x^2)/2}}{\...
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1 answer
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Expectation Value of Wiener Process

I want to perform this integral, $$\int^t_0\int^{t'}_0 \delta(s-s')dsds'$$ I know that the result should be min(t,t'), as it is the expectation value of the wiener process. I just want to know how to ...
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3 votes
1 answer
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The Random Walk in Feynman's Lectures of Physics [duplicate]

The distances in the random walk are unit positive or negative. In the calculation for the distance, $D_N$ traveled after $N$ steps, the author uses $D_N^2$ instead of $D_N$. From The Feynman Lectures ...
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How to get marginalized Fokker-Planck equation for the time-dependent Gaussian velocity distribution?

I have come across the term "Marginalized Fokker-Planck equation! ", which I have never heard of and could not find any resource online. The equation reads as following $$ \frac{\partial}{\...
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Multidimensional discretized Wiener measure for Langevin eq

How do I generalize the discretized Wiener measure in the case of the multidimensional Langevin equation: $$ dx^\omega=f^\omega(x,t) + \sum ^d_{\alpha=1} g^\omega_\alpha(x,t) dB^\alpha(t) \quad \omega=...
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1 answer
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Probablity of Cauchy jump between two position

I have some doubts about how to calculate the probability $P(x,t)$ of finding a particle with a certain initial uniform distribution $ P(x,0)=\rho (x) $ and typical displacement $ x^*=Dt $. My idea ...
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Difficulty bounds in a white noise model

Suppose we make a measurement in a real/virtual experiment for which the noise is given by a white noise (Gaussian) model. Suppose also that we have a very good filter or machine learning (ML) model ...
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How "Mean free passage time" depend on the temperature? [closed]

The mean free passage time in the Kramer's potential escape problem is given by $$ \tau_{MFPT} = \frac{2\pi \gamma} {\omega_{min} \omega_{max}} exp^{\beta({U_{max} - U_{min}})} = \frac{2\pi \gamma} {\...
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Derivation of probability density of isolated polymers

I am reading Introduction to Polymer Physics by Doi, and in his proof for the probability distribution for ideal polymers of length $N$ and end-to-end vector $\mathbf{R}$, he does the following: \...
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