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12
votes
2answers
264 views

Random Walk Randomly Reflected

Hi I am not specialist in probability so I will not be surprised if the answer for this question is just a simple consequence of well known results from the random walk theory. In this case, I will ...
5
votes
2answers
883 views

What is the probability density function over time for a 1-D random walk on a line with boundaries?

If a single particle sits on an infinite line and undergoes a 1-D random walk, the probability density of its spatio-temporal evolution is captured by a 1-D gaussian distribution. \begin{align} ...
5
votes
2answers
151 views

Counting of brownian particles: Point Process

Imagine a point process defined by the passage time of purely brownian particles through a given point (in 1D), line (2D) or plane (3D). I'm interested in the variance of the counts (number of ...
3
votes
3answers
135 views

What does “an average over noise” mean in Zwanzig's book

This is a very specific question about Robert Zwanzig's book Nonequilibrium Statistical Mechanics. Specifically, what is he talking about in equation 1.25 on page 10 that he calls "an average over ...
3
votes
2answers
167 views

Detailed balance condition for coupled Langevin equation

Suppose $a$ and $m$ are real variables and they satisfy the following two coupled Langevin equations: $$ \dot{a}=F_a(a,m)+\eta_a(t);\quad\dot{m}=F_m(a,m)+\eta_m(t); $$ where $\eta_a$ and $\eta_m$ are ...
3
votes
1answer
60 views

Introducing Randomness into Lagrangian Mechanics

Let's say at $t_o$ we have a ball rolling along a (rigid) tight rope. Is there anyway that we can solve for the trajectory of the ball knowing that at some $ t' $ there will be a random constraint ...
3
votes
0answers
40 views

Stochastic process generating fractional diffusion

One way to generate Brownian motion is as follows: Define a waiting time probability distribution $\psi(t)$ and step length probability distribution $\lambda(x)$. Require also that $\langle \psi ...
2
votes
1answer
88 views

Rate of probability loss from absorbing boundary

The following is the solution to the 1D diffusion equation with diffusion coefficient D, initial particle position $x_0$, and a perfectly absorbing boundary at $x=0$ (s.t. $P(x=0)=0$). $$ ...
2
votes
0answers
86 views

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 ...
1
vote
1answer
205 views

Describe Ising model dynamics in stochastic differential equation or stochastic process

The Ising model is described by the Hamiltonian $$ H(\sigma) = - \sum_{<i~j>} J_{ij} \sigma_i \sigma_j -\mu \sum_{j} h_j\sigma_j, $$ and is treated extensively by equilibrium statistical ...
1
vote
1answer
539 views

Autocorrelation and Power density spectrum : Continuous Markov Process

I've been reading through the paper from Gillespie on Brownian motion and Johnson Noise (DOI, PDF). He considers $X_s(t)$, a zero-mean stochastic variable, that is stationary in the sense that all of ...
1
vote
2answers
93 views

Resources on Master Equations

Presently I am reading about "Introduction to dynamical process theory and simulation" which uses the notion of Master Equations to solve Markov process. I am very new to this. Can someone provide me ...
1
vote
1answer
24 views

Examining the presence of persistent domain from time series data

There are three variables, $X_t$, $Y_t$, and $Z_t$ that are dependent of each other, and I have the time series data of those variables from replicated experiments. The stochastic dynamics look quite ...
0
votes
1answer
27 views

From exponential distribution to Poisson distribution

We know that the exponential distribution characterises the probability distribution for the waiting time between two consecutive Poisson events. Then I think if we fix a time interval $T$ then we ...
0
votes
1answer
43 views

bimolecular reaction master equation

I was wondering how to write down the deterministic rate equation for a bimolecular reaction with similar particles. e.g. $A \rightarrow^{k_+} B +B$ and $B +B \rightarrow^{k_-} A$ now the rate ...
0
votes
1answer
121 views

Is this hypo-theoretical model of future prediction feasible? [closed]

First let me state that I am not, nor ever have I been, a physics student. I am working on an idea for a book I'm writing. This is a thought experiment that posits the existence of a computer system ...
0
votes
0answers
49 views

Importance sampling for Coulomb potential

The integral I have to solve is: $$I=\int\int d \mathbf{r}d \mathbf{r}' \frac{\Phi(\mathbf{r})\Phi(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|}$$ It is a six-dimensional integral which I am going to ...
0
votes
1answer
32 views

Reference for passive scalar

I would like to learn about turbulence of a scalar advected by a random velocity field. I know that this problem can be solved analytically and that the statistics of the scalar field are ...