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.
520 questions
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Time-domain algorithm to generate noise with Power Spectral Density (PSD) $S(\omega,T) \propto \coth(\omega/T)$ sequentially
I need to numerically generate colored noise with a Power Spectral Density (PSD) $S(\omega,T) \propto \coth(\omega/T)$ (quantum thermal noise). I cannot use standard Spectral Synthesis (Inverse FFT) ...
- 29
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4
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Mathematical rigor behind renormalization [closed]
I struggle to understand how theories that are based on renormalization can be considered mathematically rigorous. I understand how renormalization works for non-abelian theories, through loop ...
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0
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25
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What is the physical role of the control operator C in quantum filtering equations?
I am studying Vassili N. Kolokoltsov's paper "On the Mathematical Theory of Quantum Stochastic Filtering Equations for Mixed States" and need to understand the role of the control operator $...
2
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The physical meaning of the "coupling operator"
I am reading Vassili N. Kolokoltsov's paper arXiv:2505.14605, "On the Mathematical Theory of Quantum Stochastic Filtering Equations for Mixed States", and having trouble understanding the ...
2
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3
answers
70
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Complex systems with noise
A complex system is typically defined as a system composed of many interacting components whose collective behavior cannot be easily inferred from the behavior of the individual parts. The whole ...
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1
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Stochastic calculus clarification
In an introductory statistical physics class, the overdamped Langevin equation was introduced as: $\frac{dx}{dt} = \frac{1}{\gamma}\xi$, where $\xi$ is the white noise representing the fluctuations. ...
- 310
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2
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469
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Compute probability current from stochastic path integral
I would like to compute the probability current associated with a stochastic differential equation, say
$$
\frac{\mathrm{d} X}{\mathrm{d} t}
=
v
+
\sigma \xi(t)
$$
where $v$ is a drift velocity, $\xi$ ...
- 335
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65
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Sharp error bounds for approximating a state-dependent SDE by a constant-noise surrogate in a bounded basin with double limits
Consider the state-dependent SDE on the basin $ B_\delta = [m^* - \delta, m^* + \delta] $:
$$dM_t = b(M_t) \, dt + \sigma(M_t) \, dW_t, \qquad b(m) = -(m - \tanh(Am)), \quad \sigma(m) = \sqrt{\frac{1 -...
3
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1
answer
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How to derive Fourier-Laplace transform of random walk master equation on 2D lattice (Part 2)?
This question is linked with this question and is related to this paper.
The Fourier-Laplace transform is given by:
$$P(q,r,s)=\sum_{t=0}^{\infty}\sum_{m,n=-\infty}^{\infty}\frac{e^{iqm+irn}}{(1 + s)^{...
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2
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How to derive Fourier-Laplace transform of random walk master equation on 2D lattice (Part 1)?
I have certain gaps in clearly understanding the derivation given in this paper. Suppose a particle moves on a 2D lattice randomly. The probability of going in any one direction outb of four available ...
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1
answer
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Why does a higher concentration gradient lead to faster rates of diffusion?
I was reading Fick's law. I was wondering why does a higher concentration gradient lead to faster rates of diffusion? How does having 20 particles on one side of a permeable membrane differ to another ...
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How to incorporate boundary conditions in mean field descriptions while deriving macroscopic equations from microscopic stochastic processes?
The question is linked to this question.
The microscopic stochastic processes are defined using homogeneous jump probabilities between sites. The assumption will be broken when we have physical ...
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1
answer
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Are mean field descriptions, where macroscopic equations are approximations of microscopic stochastic processes, not valid with boundary conditions?
I am not a physicist. However, I am looking into some diffusion dynamics for my research. I am interested in diffusion in crowded environments and for that I reading a this paper. Here to find the ...
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2
answers
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Why is no flickering observed in polarized filters?
Consider a computer screen which emits horizontally polarized light and a polarized filter. If you rotate the filter 45° relative to the screen, then it will let through roughly 50% of the light from ...
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1
answer
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Is there a physics equation for this problem?
I have a cylindrical cup with volume 250 ml;
The cup is full of hot water at $99°$ Celsius;
At $t =0$, I place a tea bag with 100 mg of tea in the center of the cup.
Question: Is there some equation ...
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1
answer
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Ensuring the stability and validity of a numerical solution for the Fokker-Planck equation
I am developing a finite-difference numerical scheme for solving the Fokker–Planck equation. The scheme is validated by comparing its solutions with histograms constructed from trajectories of the ...
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1
answer
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Likelihood of the data in a Ornstein-Uhlenbeck-like process
I am interested in finding the likelihood for the location of a given number of particles at time = 1 in a process that resemble (or is) a Ornstein-Uhlenbeck (OU) process.
In particular, I am ...
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0
answers
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What are realistic parameters values for a molecule in a viscous medium?
I have been scouring through the internet to find any source that explicitly states the constant parameters used for simulating a simple molecular system like the one I'm interested in currently, ...
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1
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Does the stochastic model of radioactive decay imply decay will go on even beyond the heat death of the universe?
I am not a physicist, but a mathematician with a pop-science level of understanding of physics, so forgive me my naivete.
Recently realized I hold the following assumptions:
The universe is infinite, ...
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votes
1
answer
180
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Choosing the timestep when applying the Euler-Maruyama scheme
I want to find the numerical solution to a stochastic differential equation of the following form:
\begin{align}
\dot{q} &= p, & \dot{p} &= -\gamma p - \frac{dV}{dq} + \xi(t),
\end{align}
...
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3
votes
0
answers
122
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Path integral for a run-and-tumble particle
I am familiar with the path integral formalism for stochastic differential equations of the form (in 1d for simplicity)
\begin{equation}
\dot{x}(t) = f(x(t)) + \sqrt{2 D} \ \xi(t).
\end{equation}
It ...
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1
vote
1
answer
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Spin evolution under random-Haar unitary
Consider a spin state $|\psi\rangle$ under the random-Haar unitary evolution $u \in U(2)$. Note that we draw a random-Haar unitary once, and repeatedly act on $|\psi\rangle$ so that at any time step $...
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Where can I find information about Random Processes's Characteristic Functionals and Volterra series?
At first, hi everyone.
I'm having difficulties looking for information about the so-called 'characteristic functional' of a random process.
What I know about this concept is what follows:
when you ...
1
vote
1
answer
139
views
Average momentum of a heavy particle subjected to Brownian motion?
As far as I understand it, in the Maxwell-Boltzmann equation, the diffusion equations, Langevin eq., etc. one calculates the average energies, momenta, diffusions, etc. for particles in a gas all ...
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2
votes
2
answers
218
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Noise term in ODE
Imagine that I am in a lab measuring a certain force that is time dependent, e.g. there is a spring subjected to changes in temperature, which results in a time-dependent stiffness,
$$F(t)=k(t)\delta,$...
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3
votes
1
answer
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What does it mean to be in thermal equilibrium in context of Brownian Particle?
Suppose we start with the standard Langevin equation for a free Brownian particle in an infinite medium:
\begin{equation}
m \frac{dv}{dt} = -\gamma v + \sqrt{2 \gamma k_B T} \, \eta(t),
\end{equation}
...
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1
vote
1
answer
172
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Effective temperature for a stochastic particle
Suppose $x(t)$ exhibit a random motion. Often, we may want to understand the ratio
$$
\frac{D}{\mu}
$$ where
$$
D \approx \frac{{\rm var}(x(t))}{t} \: \mbox{ when } \: t \: \mbox{ is large}
$$
and
$$...
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4
votes
1
answer
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Pair Correlation Function for ideal polymer chain
I am reading Rubinstein's polymer physics book and on p. 78 it says that if $m$ is the number of monomers within range $r$ of an arbitrary monomer, then
$$
m \approx (r/b)^2
$$
where $b$ is the bond ...
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votes
2
answers
528
views
How the Kramers escape rate relates to the diffusion?
Consider a Brownian motion in a one-dimensional potential $U(x)=U_0\cos\left(\frac{2\pi}{a}x\right)$.
What is the diffusion constant of this process?
My first thought is to find the Kramers escape ...
3
votes
3
answers
179
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Density loss at boundary in Fokker-Planck equation
I'm trying to study a particle undergoing Brownian motion, with initial position $x_0$ and a reflection boundary at $x = L$:
$$dX = \nu dt + \sigma dW$$
As far as I know, the probability density ...
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1
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0
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Finding Degeneracies in Quasienergy of 4D-Quantum Walk
Preface: I am an undergrad Physics student who may have bitten off a touch more than I can chew but im not ready to give this up just yet. I have omited much of the gory details for the sake of ...
0
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1
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181
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Solutions to Schrodinger Equation with Feynman-Kac Formula
I am currently taking a course on stochastic differential equations, and part of the course is doing a project on applications of stochastic differential equations and I am curious if there is a ...
0
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0
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Fokker-Planck equation in the generalized $P$-representation of a two-level system
The Fokker-Planck equation (FPE) for a single transmon is well-known, and one can even obtain analytical expressions for the steady-state distribution (https://arxiv.org/abs/1606.08508) using the ...
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Supersymmetric interpretation of some non-equilibrium systems
There is a known connection/isomorphism between supersymmetric quantum mechanics and stochastic dynamics of systems that satisfy detailed balance at equilibrium, i.e. with an evolution equation of the ...
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0
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60
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Characterization of Markovianity, Gaussianity, and color for noise processes
Consider a noise process $\xi(t)$ that has some statistics in time. There are various ways to characterize such a process, 3 being Markovianity (independence from history), Gaussianity (Gaussian ...
- 1,002
2
votes
1
answer
152
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How do 1-dimensional random walks correspond to different interpretations of stochastic integration?
Very short summary of my problem: I want to intuitively understand why/how random walks with "random barriers"/"random traps" correspond to the Itô/Hänggi-Klimontovich ...
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0
votes
1
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179
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Discrete Quantum walk on a directed graph
A Coined discrete time quantum walk can only be implemented on a reversible graph (if $(i,j)$ is an edge there exist a path from $j$ to $i$). As far as I understand this is because any operator in ...
2
votes
0
answers
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views
Is the thermostat in langevin dynamics the random force?
Langevin MD appears to be described by
$$
M \frac{dV}{dt} = - \zeta MV + F(R) + \theta(t).
$$
Here (1) $V$ and $R$ are solute particle velocity and position, (2) $- \zeta MV$ is a systematic ...
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0
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Assumptions in the Langevin equation for optical cavities
I am struggling to interpret the Langevin equation for the outputs of an optical cavity. I am using Quantum Noise by Gardiner & Zoller, and Quantum Optics by Walls & Milburn as primary guides. ...
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Obtaining the local velocity profile of a Fokker-Planck Equation
Consider the following Fokker-Planck Equation:
$$\frac{\partial \rho (\textbf{r})}{\partial t} + \nabla \cdot \textbf{j} (\textbf{r}) = 0 \tag{1}$$
Where $\textbf{j}$ is the probability flux given by:
...
1
vote
1
answer
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Liouville Equation Derivation
On p. 20 in the book Theory of Open Quantum Systems by H.P. Breuer, the derivation of the infinitesimal generator for a deterministic Markov process involves taking the time derivative of a delta ...
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Worked solutions to the Kramers-Moyal Expansion
I am reading Risken and it's got to the Kramers-Moyal forward expansion and it derives a very formal general solution.
Are there any sources that explicitly solve this for concrete examples (...
4
votes
2
answers
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Managing the integration when calculating the variance of the position $\langle(x-\langle x \rangle)^2\rangle$ in the Langevin equation
Given the Langevin equation
$$\text{d}v=-\gamma v\text{d}t+\text{d}W,$$
where $v$ is the speed, $\text{d}W$ is a Wiener process with $\langle \text{d}W\rangle=0$ and $\langle (\text{d}W)^2\rangle=\...
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0
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Solution of quadratic stochastic differential equation
One can write down the solution of a linear stochastic differential equation in Ito convention of the form
$$
d\vec{x} = F\vec{x}dt+G\vec{x}dW
$$
where $G,F$ are constant matrices and $dW$ is a Wiener ...
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0
answers
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Modeling the motion of a particle on a surface with spatially random friction?
I had this thought in the shower one day. Assume we have a particle of mass $m$ at the top of an inclined plane of length $L$ and angle $\theta$. What if instead of some constant or deterministic ...
1
vote
1
answer
252
views
Can I take the functional derivative of a stochastic functional (Ito calculus)?
I am currently faced with an action (quantum field theory) containing a term of the form
$$
S \sim \int_0^T\text{dt} \,\xi(t)\cdot (a+ib+c-id)
$$
Here $\xi(t)$ is the "derivative" of a ...
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votes
2
answers
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views
Why does the root mean square represent the spread of a random walk?
I am trying to answer a question inspired by Sethna's Entropy, Order Parameters, and Complexity. The gist of the question is the following.
Suppose a photon is created at the center of the Sun, and ...
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1
vote
0
answers
58
views
Continuous Sample Paths for distributions determined by Schrodinger equation?
Let us first recall how Brownian motion is described in pure mathematics. For all $t\in [0,\infty)$, we have a probability distribution $\mathscr D_t:= \text{Normal}(0,t)$, and random variables $B_t$ ...
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vote
0
answers
134
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Homodyne feedback equation: Stratnovich form to Ito form
I am trying to understand the paper of equation 7 of Phys. Rev. Lett Vol 70, 548 (1993), Quantum theory of optical feedback via homodyne detection by H.M. Wiseman & G.J. Milburn.
(You can also ...
- 3,812
0
votes
0
answers
95
views
Derivation of the Fokker-Planck equation
A nonlinear dynamical system is considered
$$ dx/dt = f(x) + z(x)p(t), $$
where $p(t)$ Gaussian noise with zero mean and exponential correlation function.
How I can derivation of the Fokker-Plank ...
