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.

Filter by
Sorted by
Tagged with
2
votes
1answer
53 views

Where can I find Einstein's proof of the existence of atoms?

as the question states, where i can i find einstein's proof of the existence of atoms, and also, what math pre-requisites do i need to understand it deeply enough to be able to replicate it.
0
votes
1answer
31 views

Resources on an intuitive understanding of the Girsanov Transformations

My current project involves the use of Girsanov transformation. Can anyone suggest me some resources for an intuitive understanding of the same. The pages I have been referring to (Wikipedia), deals ...
1
vote
0answers
25 views

What is the role of the density distribution function by Liouville equation in statistical physics?

Constant density is a solution of Liouville equation which says that total derivative of density distribution functions with respect to time is zero, and it is the distribution function in ...
0
votes
0answers
32 views

Work done by the drift term of Ornstein–Uhlenbeck process

Consider a particle obeying the Ornstein–Uhlenbeck process: $$ dx_{t}=\theta (\mu -x_{t})\,dt+\sigma \,dW_{t}, $$ where $x_t$ is the position of the particle at time $t$, $W_t$ denotes the Wiener ...
0
votes
0answers
11 views

Reference Request: An introductory book on Kinetics, similar to Schroeder's on thermodynamics [duplicate]

I come from a statistics background and don't have much knowledge of physics. I need to gain bit of knowledge in statistical physics for one of the projects that I work on. I started looking at Reichl'...
0
votes
0answers
35 views

Text Recommendation: Random Walks (for physicists)

I am an incoming graduate student in Theoretical Physics in the Netherlands, and I would like to know if any of you could recommend texts on random walks with applications to physics. My university ...
0
votes
0answers
48 views

Reference request: Langevin dynamics of particle in high dimensional quenched random potential

I would like to have some reference (either papers or books) for the study of the Langevin dynamics (out of equilibrium, non asymptotically) of a particle $X(t)\in \mathbb{R}^N$ in a quenched random ...
1
vote
0answers
41 views

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 ...
0
votes
0answers
19 views

Using the Martin-Siggia-Rose (MSR) formalism for oscillator with general non-harmonicity

I am wondering if using the Martin-Siggia-Rose (MSR) formalism can be convenient/treatable for calculating correlation functions [or their spectral densities] of a linear [underdamped] oscillator with ...
0
votes
0answers
16 views

Laplace transform of triple convolution involving Heaviside function — stems from multi-state random walk

I'm a bit stuck on this problem stemming from a multi-state random walk. I have a function of the form $$C(t) = \int_0^t dt' \theta(t'-T)A(t')B(t-t')$$ and I'd like to calculate its Laplace transform....
0
votes
0answers
49 views

Symplectic time-integrator for Fokker-Planck equation

Is there a way to use a symplectic time-integrator for the numerical solution Fokker-Planck equation of the form: $$\partial_{t}p = -\nabla \cdot (\underline{v}p - D\nabla p)~?$$ where $p = p(\...
0
votes
0answers
32 views

Non-Markovian noise coupled to atomic system

I want to calculate the density matrix element's average over all the realization of Gaussian colored noise when the atomic system is coupled to the said noise. I know how to do it for atomic energy ...
0
votes
0answers
24 views

Can I form a generating function for the Laplace transformed cumulants?

Setup: I have the double Laplace transform $\hat{\tilde{p}}(\eta,s)$ of a probability distribution $p(x,t)$ of finding a particle at position $x$ at time $t$, defined by $$ \hat{\tilde{p}}(\eta,s) =...
0
votes
1answer
36 views

Random quantum walk related to quantum computing

I am an undergraduate doing research on QC/QI. My current topic to learn is continuous-time quantum walks, but first I must learn the random quantum walk. That being said, I was wondering if someone ...
0
votes
0answers
12 views

Question on the correlation function of dichotomous Markov noise

Setup: A two-state switching process $I(t)$ between two values $\Delta_1$ and $\Delta_2$ with rates $\alpha$ and $\beta$ can be represented by the transition probabilities $$ P_{ij}(t) = \frac{1}{\...
0
votes
0answers
13 views

Dynamics subject to an alternating step/rest sequence with two different resting intervals

Consider that a particle is either in motion (with velocity $u$) or at rest: $$ \dot{x}(t) = u S(t) $$ where $S(t)$ is a switching process between $0$ and $1$. This switching is defined as follows. ...
1
vote
0answers
54 views

Correlation function from Laplace transform of distribution function?

I have a time-dependent random process $x(t)$ which takes on two values $\Delta_1$ and $\Delta_2$. I know the Laplace transforms of the (time-dependent) probabilities $\hat{p}_{ij}(s)$ of these values ...
2
votes
1answer
57 views

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-\...
1
vote
0answers
36 views

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 ...
0
votes
0answers
36 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 ...
1
vote
2answers
91 views

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 ...
1
vote
1answer
63 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{\...
0
votes
0answers
26 views

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 ...
0
votes
1answer
74 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 ...
25
votes
5answers
5k views

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 ...
0
votes
0answers
26 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 ...
1
vote
0answers
51 views

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 ...
0
votes
1answer
36 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 ...
1
vote
0answers
53 views

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 ...
1
vote
0answers
27 views

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-...
0
votes
0answers
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 $\...
2
votes
0answers
72 views

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 $W_t$ ...
2
votes
0answers
70 views

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 ...
0
votes
1answer
56 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 ...
0
votes
1answer
76 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{...
2
votes
2answers
172 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 ...
2
votes
1answer
39 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 ...
2
votes
0answers
259 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 ...
0
votes
1answer
67 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 ...
1
vote
2answers
221 views

Can I apply the standard Runge Kutta 4th order method to the Langevin Equation?

If I have a Langevin Equation with an external force term (which may be time dependent), is it possible for me to apply the standard 4th order Runge Kutta algortihm to solve it numerically? Edit: I ...
1
vote
0answers
33 views

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 ...
1
vote
0answers
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 ...
1
vote
0answers
57 views

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 ...
0
votes
0answers
26 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 ...
2
votes
1answer
246 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 ...
1
vote
0answers
139 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 ...
2
votes
0answers
72 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 ...