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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Correlation function from Laplace transform of distribution function?

I know the Laplace transform of a (time-dependent) probability distribution for a random variable $x(t)$: $$ \hat{P}(s,t) = \int_0^\infty dx P(x,t) e^{-sx},$$ but in my particular application this ...
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Randomly stopped dynamics of $x(t)$: How can I find $\text{var}\{ x(t) \}$?

Consider the simple dynamical equation $$ \dot{x}(t) = u H(t-\tau),$$ where the timescale $\tau$ is an exponentially distributed random variable $\tau \sim \omega \exp\{\omega \tau\}$ and $H(t) = 1-\...
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Is there a way to argue that the appearance of the dependence of past states is stochastic behavior?

Suppose in a particular set of observations that you observe what appears to be a correlation of future results based on past results. Is there a way to be certain that such a phenomena isn't just the ...
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35 views

Second quantisation for dynamical systems

The paper "Perturbative approach to an $A + B \rightarrow C$ reaction-diffusion system", (Z. Phys. B 96, 137-144 (1994)), by Conrad and Trimper, applies the Fock Space formalism for the master ...
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Why do we interpret the first term of the Fokker-Planck equation as drift?

With the derivation of the Fokker-Planck equation we get: $$\frac{\partial}{\partial t}P(x,t)=-\frac{\partial}{\partial x}(A(x,t)P(x,t))+\frac{1}{2}\frac{\partial^2}{\partial x^2}(B(x,t)P(x,t))$$ We ...
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55 views

Where does the Master equation for the derivation of the Fokker-Planck equation come from?

I'm participating in an introductory course for biophysics. We briefly discussed the derivation of the Fokker-Planck equation and used the so-called Master equation as a starting point. $$ \frac{\...
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Stochastic version of the Kirchoff circuit law

I assume this question could be written in a non-technical jargon, but I will try to be as simple as possible. The Kirchoff circuits law assert that the sum of inward and outward currents at a node ...
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1answer
61 views

Solution to diffusion equation of a random walk

In my class of statistical physics, we studied the classic problem of random walk for the discrete case. In the end, we made the changes necessary for the master equation to be in the continuous ...
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Is throwing dice a stochastic or a deterministic process?

As far as I understand it a stochastic process is a mathematically defined concept as a collection of random variables which describe outcomes of repeated events while a deterministic process is ...
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22 views

First-passage time of a 1d marked Poisson (shot noise) process

Given a marked Poisson process in one dimension $Y(t)=\sum_{\{t_i,a_i\}}g(t−t_i,a_i)$ so that $𝑌(𝑡)$ is a sum of impulses arriving as a Poisson process and the impulses $𝑔$ belong to a continuous ...
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Question about the Markovian property of the velocity of a Brownian particle following Langevin equation

I'am now studying Langevin model and Fokker-Planck equation with the lecture notes by Borghini Topics in Nonequilibrium Physics (NB: PDF). On page 92, he talks about the Markovian property of the ...
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35 views

How is heat dissipation rate the product of force and velocity?

Let $q$ be heat dissipation to midium, $F$ be the force to a particle, and $\dot{x}$ is the velocity of it. According to the equation (8) in Seifert 2005, $\dot{q} = F \dot{x}$ holds. How does this ...
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Are particles in a perfect fluid in random motion?

A perfect fluid has no heat conduction, but it exerts pressure in all directions (according to stress-energy-momentum tensor). If it does not conduct heat, then it means it does not have random ...
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What does stochastic nature of work (quantum scale) really mean?

Fluctuation theorems are (also) concerned with defining work in the non-equilibrium regime. Now I've read that in regimes where Fluctuations become very strong (which I assume are the non-...
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41 views

Order of phase transition in random walks

If we consider a random walk with step size distribution $P(s)\sim s^{-\gamma}$, we know the order of $\langle s^2\rangle$ changes at $\gamma=3$, while the order of $\langle s\rangle$ changes at $\...
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What happens if you modulate a Hamiltonian with white noise?

Consider the Hamiltonian $f(t)H,$ where $H$ is time-independent and $f(t)$ is classical white noise. Then I would write a Schrodinger equation $$\mathrm{d}\psi=-iH\psi\ \mathrm{d}W_t,$$ where $W_t$ ...
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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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1answer
45 views

Diffusion 2D on a surface : diffusion coefficient and surface friction

We have a particle that is diffusing actively (meaning that the source of energy is a motor; the diffusion is like a Brownian motion, the only difference is that the diffusion coefficient is much ...
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1answer
69 views

References in diffusion of quantum state

I would like to know if there are books, articles or any other type of references where a (heuristic) derivation of the equation: \begin{eqnarray} \textrm{d}|\psi(t)\rangle=-\frac{i}{\hbar}H_{\textrm{...
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2answers
167 views

The mean of Langevin equation

I have a very basic question regarding the mean of the Langevin equation. So we have an equation of the form: $$\dot{v}(t)=-\beta v(t)+ \xi (t)$$ Where $\xi (t)$ is a Gaussian white noise with an ...
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1answer
38 views

Are there any continuous-time stochastic processes in which transition probabilities are discontinuous functions over time?

In stochastic processes, like homogeneous Markov processes, Poisson processes, Queueing systems etc., the functions that represent (transition) probabilities are continuous over time. This is also ...
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257 views

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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1answer
64 views

Statistics of 1D discrete random walks

I have already asked this question in Math.SE. Let $P(n)$ be a probability distribution on the integers. Suppose a random walker starts off at the origin and, at every positive integer time, takes a ...
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What is the decoherence rate and the thermal de Broglie wavelength in quantum Brownian motion?

I know that when the thermal de Broglie wavelength is on the order of the interparticle distance, the gas must be treated as a Fermi gas or a Bose gas, depending on the nature of the gas particles. I ...
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60 views

Phase fluctuations electromagnetic field

The electric field strength is given by: $$E(t)=E_0 \exp(i(\omega t + \phi(t))),$$ where $\phi(t)=\sqrt{2D} \ \Gamma(t)$. $D$ is the diffusion constant and $\Gamma(t)$ the line width. We have to ...
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Advected Dirac comb with random number of teeth which are born and die

I'm looking for a topic which I struggle to put into words. It's a reasonable consideration which I expect has been carefully studied. I hope someone can tell me the name of it and offer some guidance ...
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25 views

Correlation function and power spectrum of discrete time Gaussian noise summed with a time delayed version of itself

Suppose we have a process $\zeta(n) = \xi(n) + \xi(n + 1)$ Where $\xi(n)$ is discrete time white noise process, where the values taken at different times are from identically distributed Gaussian ...
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1answer
168 views

Derivation of diffusion equation from Fokker-Planck equation

I need your help, could you please explain me the sentence "The diffusion equation is the Fokker-Planck equation for the Brownian motion". I have tried to use some assumption and transform a ...
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130 views

Density Fluctuation in N-Particle Brownian Motions

I am studying spatial population movement and would like to model the density fluctuation by assuming a Brownian movement for each individual. Because the total number of individual ($N$) is large but ...
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71 views

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 ...
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A hydrodynamic theory for systems with rich microscopic detail?

I've been looking at various models of stochastically interacting particle systems. Let's take for example the totally asymmetric simple exclusion process on a 1D lattice with some initial conditions. ...
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1answer
82 views

Brownian motion from two gaussian noise processes

Consider some brownian motion for which we obtained the following solution for the langevin equations $$ u\left(t\right)=e^{-\alpha t}\int_{0}^{t}e^{\alpha s}\left(\xi\left(s\right)-\xi'\left(s\right)...
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1answer
81 views

Density density correlations of a simple Brownian particle [closed]

Suppose, I have a particle satisfying the equation \begin{equation} \frac{dX}{dt}=\eta(t) \end{equation} Where $\langle \eta(t)\eta(t')\rangle=\delta(t-t')$. I can now define a density like $\rho(x,...
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1answer
153 views

Langevin equation. What is the meaning of temperature?

Consider a system of $N$ particles, subject to some interaction potential $U$ (e.g. Lennard-Jones) and to thermal noise. The equation of motion is given using the Langevin equation: $$m_i \ddot{\bf r}...
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1answer
73 views

Noise covariance matrix

I've been trying to understand this paper. The paper seems to be about analyzing noise properties of a superconducting coplanar waveguide microwave resonator. They use an IQ mixer to perform ...
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1answer
119 views

Does Irreversibly/no detailed balance implies there is no thermal equilibrium?

Consider the following transition matrix $$ T= \left[ {\begin{array}{cccc} \frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6}\\ \frac{1}{3} & \frac{1}{4} & \frac{1}{...
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1answer
174 views

Mean square displacement of a Langevin equation with inertia

Consider a 1D motion of a particle $$\ddot{x}(t)=-\gamma\dot{x}(t)+\eta(t)$$ where $\langle \eta \rangle=0$ and $\langle \eta(t)\eta(t') \rangle = \tilde{D} \delta(t-t')$. How can I obtain ...
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Stationary solutions: Fokker-Planck

I've a question about the stationary solutions of the FP equation. I know that for a differential stochastic equation like $$\frac{dx}{dt} = a(x) + \sqrt{2c}\eta $$ the FP equation is: $$\frac{\...
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1answer
132 views

Proof that the Wiener process is non-differentiable

I'm working through a proof showing that the Wiener process is non differentiable given as follows I am not quite sure where the 2 in front of the integral stems from though. Any help would be ...
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1answer
64 views

Deterministic vs stochastic approach

In a deterministic system, given by the system of differential equation $$\frac{dx_n}{dt}=F_n(x)$$ Which is ergodi, and mixing with respect to a $\rho^{inv}(x)$, in a limited subspace of $R^N$,show ...
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What is a zero dimensional stochastic system?

I happened to read a statement about Zero-dimensional stochastic systems in the context of Noise-induced phase Transitions (https://doi.org/10.1140/epjb/e2018-80624-9). I am not sure what does a zero-...
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1answer
75 views

Probability distribution of integrated position of a two state process with jump probability

Context This problem came up in the course of thinking about the statistics of the dispersive measurement signal coming from a superconducting qubit. Such qubits have finite excited state lifetimes, ...
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2answers
161 views

What causes viscosity of a fluid?

Consider a fluid like water. Intuitively I would say that its viscosity is caused by intermolecular interactions among its molecules. But the Einstein-Smoluchowski relation (and the Fluctuation-...
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Splitting the mean in the (Backward) Fokker-Planck

A method for deriving the Fokker-Planck equation is outlined below$^1$ (I believe it is a simplified version of the Kramers-Moyal expansion). Set: $$P(\vec x, t+\Delta t) =\left<\delta(\vec ...
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Balancing noise energy contributions with friction term

I'm looking for a result that would say something about the way in which to pick a proper constant $\sigma$ in generic models of the form: $$ \ddot{x} = - \gamma \dot{x} - \nabla V(x) + \sigma \xi(t) $...
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1answer
208 views

Topological entropy in Markov chains

Given a finite Markov chain, how do I find the topological entropy $h_T$? Furthermore, I should compare it with the Shannon entropy $h_S$ and show that $h_T\leq h_S$. Is this a general fact? This ...
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71 views

What are the assumptions underlying the master equation?

In Reichl, 2016; pg405 the author gives a derivation of the master equation - which I will outline below in my own words: We start with:$$P(n,t+\Delta t)=\sum_m^M P(m,t)P(n,t+\Delta t\mid m,t)\...
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42 views

How is BRST symmetry related to local integrals of motion?

I'm hoping someone can confirm or check my reasoning below: In this wiki, they describe caos in a classical system as the spontaneous symmetry breaking of a BRST. In this stackexchange, they clarify ...
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141 views

Brownian motion of a particle varying with time

I was thinking about a Brownian particle executing random motion in a liquid. Is there any time dependence if the particle travel some distance say $r$? well, obviously it is time dependence. i think ...
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What is a good reference for learning about the physicist's viewpoint of the dynamical $\Phi_3^4$ model?

I am a mathematician interested in stochastic PDEs. Recently, Martin Hairer introduced his theory of regularity structures to solve singular stochastic PDEs such as KPZ: $\partial_t u=\partial_{xx} u+(...