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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Equivalence of quantum state diffusion and heterodyne trajectory
According to Breuer-Petruccione, the SDE quantum trajectory evolution for heterodyne detection
$$d\psi=-iH\psi dt-\frac{\gamma}{2}\left(C^\dagger C-\langle C^\dagger \rangle_{\psi} C+\frac{1}{2}\...
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Deriving the Path Integral Representation of the Fokker-Planck Equation
Suppose $$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t$$ is a 1D nonlinear stochastic differential equation ($dW_t$ is typically assumed to be Brownian). According to wikipedia the distribution of $X_t$ at ...
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Rationale for definition of input/output operators in quantum Langevin equations
I am going through Gardiner & Zoller's 'Quantum Noise', and following the derivation of the Langevin equations in terms of input/output operators. Let the bath Hamiltonian be:
$$H_B=\hbar\int_{-\...
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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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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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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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Thermodynamic equilibrium or thermal equilibrium and equipartition theorem
In all derivations of the equipartition theorem I can find a thermodynamic equilibrium distribution is used to show it's validity.
But more vague sources (physics.stackexchange answer by Luboš Motl, ...
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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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Why is it a disadvantage for Langevin dynamics to be non-reversible?
In Molecular Dynamics, it is commun to use a Langevin thermostat to maintain a constant temperature. However, it is often written that the Langevin Thermostat not being time reversible is a ...
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Has the theory of fractional quantum mechanics been experimentally validated in any way?
I'm referring to the generalisation of quantum mechanics developed by Nick Laskin: http://arxiv.org/abs/quant-ph/0206098
He suggests that if a particle's trajectory is integrated over Lévy paths ...
user29305
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Reference for stochastic processes which helps moving from a basic level to a measure theory one
I'm looking for a reference (books, notes, lectures) which helps a physicist to understand the language of measure theory in the context of stochastic processes (in particular markov chains).
I've ...
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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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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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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 $T$ ...
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A simple picture for the fluctuation-dissipation theorem?
In condensed matter and materials physics it is often assumed that the response of a condensed phase to some perturbation is determined by the fluctuations of the system at equilibrium (without ...
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Ito-Stratonovich drift term for spatial white noise
Suppose I have a Langevin equation with multiplicative noise of the form
$$ \dot{x} = f(x) + g(x)\eta(t) $$
where $ \eta(t) $ is a Gaussian white noise with zero average, unit strength, and delta ...
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How often does a molecular machine run in reverse?
I was reading the Wikipedia article on Stochastic thermodynamics, and came across this statement in the overview:
When a microscopic machine (e.g. a MEM) performs useful work it generates heat and ...
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References for understanding Martin-Siggia-Rose (MSR) formalism towards path integral
Can anyone help me understand Matin-Siggia-Rose-deJensen formalism? To be more specific, I'm studying this paper, and in section 5, the authors go all about classical limit and the usage of MSRJD ...
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What is a stochastic electromagnetic wave?
In statistical optics we always talking about stochastic electromagnetic wave but I am not able to understand how this wave is different from electromagnetic wave
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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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Does the fluctuation-dissipation (FD) relation $\sigma_E^2=k_BT^2C_V$ arise as a special case of the FD theorem?
The classical fluctuation-dissipation theorem states that the power spectral density $S_{\eta\eta}(\omega)$ of a classical random variable $\eta(t)$ is given by $$S_{\eta\eta}(\omega)=\frac{2k_BT}{\...
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Understanding $D \partial^2 P(x,v,t)/\partial x^2$ as a type of collision term
Using conservation of particles in a control volume in phase space (in one dimension with no sources of particles or external forces), one can derive the formal transport equation
$$ \partial_t P(x,v,...
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Stochastic resonance
I am trying to look for a stochastic resonance in a system described by Langevin equation and a periodic forcing. While I can simulate an SDE numerically I have no idea how to calculate the 'signal to ...
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Transition rate in systems without thermal noise
I've been lately reading about Transition State Theory (TST) and different methods to estimate the transition rates between metastable states in the context of chemical reactions using the review ...
- 1,533
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Connections between Lindblad equation and black holes
There have been some investigations about wave function collapse caused by gravitation. And as is known, Lindblad equation can be regarded as a formal description of stochastic collapse of wave ...
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Continuous Measurement equations
In a physics text, "Quantum Measurement Theory and it's Applications" by Kurt Jacobs, it describes the idea of a "continuous measurement" (measurement taking place over time $T$): $$dy = x_{true}dt + \...
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Transition rate of two level system subjected to noise
(this question is simpler than its length implies.
I did this on purpose to provide a nice complete development for future readers)
The setup
Suppose we have a two-level quantum system with ...
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Derivation of Kramer's equation
For the derivation of Kramer's equation we use the multivariable Fokker-Planck equation:
$$\frac{\partial P}{\partial t} = \frac{\partial (P A_{1})}{\partial x} + \frac{\partial (P A_{2})}{\partial v}...
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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 ...
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answers
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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 ...
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Any restrictions on drift vector in Fokker-Planck equation?
The most general Fokker-Planck equation for a probability density $f$ over phase space is
$$\partial_t f = -\partial_i(u^i f) + \frac{1}{2} \partial_i \partial_j (D^{ij} f)$$
where $u^i$ is the drift ...
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Geometric Brownian Motion versus Ornstein-Uhlenbeck process
The Geometric Brownian Motion model is a continuous-time stochastic process in which a particle move according to a random fluctuations (Wiener process) and a drift term.
The corresponding stochastic ...
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Is there any paper/experiment on the deterministic behavior of coin-tossing?
I do not have background in Physics but Statistics. I am working on a small project on philosophy of probabilistic modeling.
Of course, we often model coin-tossing as if it comes from a Bernoulli ...
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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 ...
user85598
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votes
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answer
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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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votes
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answer
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The "Algorithm" for Derivation of the Master Equation
I'm trying to come up with a master equation for a spin system. So, I'm trying to understand in general how can we derive the master equation and the Lindblad operators if we know the system, bath and ...
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White noise approach to Feynman integrals: why do we use this renormalization?
I start by saying that I know very little about Feynman integrals so please bear with me.
In Kuo's book "White noise distribution analysis" or in Hida, et al. "White noise analysis"...
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Brownian Motion (Geometric, Fractional, Drift)
I have been researching Brownian motion for a while and have come across terms/types of Brownian motion such as fractional, geometric, and Brownian motion with drift. I understand the physical meaning ...
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Does the Kalman filter incorporate a Heisenberg-like uncertainty principle?
In the case of mechanical systems, applying the Kalman filter involves combining model based prediction (using an apriori known dynamical model) with real-world noisy observations of the positions and ...
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What are the possible and meaningful measurements on a quantum walk on a graph?
In the contest of quantum walks, the graph is defined as $G=\{V,E\}$ with $V$ the set of vertices and $E$ the set of edges. Thus, the Hilbert space is defined as the $\mathcal{H}=\operatorname{span}\{\...
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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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Path integral 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\pi (...
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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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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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Analytical solution to damped harmonic oscillator - Fokker-Planck equation
In the paper "Numerical solution of two dimensional Fokker-Planck equations" (available at: https://doi.org/10.1016/S0096-3003(97)10161-8), the authors quote an analytical solution to the damped ...
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Detailed Balance Violation and Fokker-Planck Equation
Suppose I have a system with N sites, and each site
can be modified (M) or anti-modified
(A). Transitions between these two states are in part random, and in part auto-regulated by recruitment of At ...
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2
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Exact solution for non-linear Fokker-Planck equation
I'm searching for exact (analytical) results for FP equation in 2 variables (such as $x$ and $p$ in 1D) with a steady state. Kramer's like (with force due to confining potential, such as harmonic ...
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votes
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answers
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A micro-reversible stochastic process that models transitions between states with variable energies
Suppose we have a system with 3 possible states A, B and C (there could be $n$ states as well) with energies $E_a(t)$, $E_b(t)$ and $E_c(t)$ that vary with time.
If our system has a constant finite ...
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votes
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answers
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Overdamped Fokker-Planck equation with general force field (gradient + curl)
I am looking for a general formulation of the Fokker-Planck equation for diffusing particles in a general force field $F = -\nabla U + \nabla\times A$ in the overdamped regime (Smoluchowski equation).
...
2
votes
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answers
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Expression of heat by the Brownian motion
folks. I was reading a paper from PRE and I'm not sure what's going on about the following equation. So for the system composed of two heat baths governed by Brownian motion, the entropy change of the ...
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