# 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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### Is this a valid alternative definition of the delta function?

The delta function can be defined as: $$\delta(x) = \int_{-\infty}^{\infty} e^{-2\pi i k x} \, dk$$ Loosely speaking, I can understand this because unless $x=0$, the complex exponential oscillates ...
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### Which textbooks for random (Brownian) motion derivations? [duplicate]

I am a chemist currently trying to dive in to details of random motions. I have been studying Einstein-Smoluchowski treatment, and want to learn further (Fokker-Plank's equation and Langevin treatment ...
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### Is the invariant distribution of the underdamped Langevin equation the Bolztmann distribution?

Essentially what the question asks. I've seen proofs that in the overdamped limit, Langevin dynamics relaxes distributions to the Boltzmann distribution in the $t \to \infty$ limit; but what happens ...
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### Derivation of Fokker-Planck equation from Langevin equation

I have been trying for a long while to wrap my head around this step in the derivation of the Fokker-Planck equation in Appendix 8 of Nigel Goldenfeld's "Lectures On Phase Transitions And The ...
1 vote
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### A simple picture for work fluctuation relations?

One possible formulation of the second law of thermodynamics is that the work extracted during the change of a thermodynamic system between two thermodynamic states is at most equal to the free energy ...
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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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### Average of a time derivative

Given a probability distribution P(x,t), we can take the average of a time-dependent quantity x(t) as \begin{equation} \overline{x(t)}=\int dx x(t) P(x,t) \end{equation} My question is: what if I have ...
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### What does it mean for a classic field to be defined in terms of stochastic parameters?

I'm writing a bachelor's thesis related to inflationary cosmology and I don't quite understand some things about a paper I've been reading called Signals of a Quantum Universe. Specifically, the paper ...
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### Hamiltonian function of a system of particles governed by Langevin equation of motion

I have a system of particles which interact among themselves via some pairwise additive potential ( position dependent ) and I am also considering the collision of the particles with background ...
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### Is there sufficient "content" in the field of econophysics to write a substantial undergraduate thesis/project on? [closed]

Okay, maybe the title is somewhat misleading. My university calls this a BSc Project, but it is limited to between 4000 and 6000 words, so it isn't particularly long. Anyhow, one of the projects ...
1 vote
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### Spectral representation of a white stationary process

I am trying to better understand the spectral representation of stochastic processes. From the book "Spectral Analysis for physical applications" by Walden and Persival: The spectral ...
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### How can I form an advection-diffusion SDE to obtain the desired discretization?

Suppose that $\mathbf{s}(t)\in S$ denotes the spatial location of a process at time $t$. Further, let $\mathbf{x}(\mathbf{s}(t))$ denote covariates at the location $\mathbf{s}(t)$. My goal is to write ...
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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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### Full translation of the paper written by M Smoluchowski in 1906 "Zur kinetischen Theorie der Brownschen Molekular Bewegung und der Suspensionen"

May I ask if the full translation of the classic paper on Brownian motion and SDE written by M Smoluchowski in 1906 "Zur kinetischen Theorie der Brownschen Molekular Bewegung und der Suspensionen&...
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### On the Fokker-Planck equation: deriving the transition PDF for small times

I report below (part of) page 73 of the book The Fokker-Planck Equation, by H. Risken We now derive an expression for the transition probability density for small $\tau$ \begin{equation}\tag{1} p(x,t+...
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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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### Changing sign at a Ornstein-Uhlenbeck process: mean, variance and likelihood

I am working with a multivariate Ornstein-Uhlenbeck process and its statistical properties (likelihood, expected values and variance). The Ornstein-Uhlenbeck process can be described as a random walk ...
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### Under what conditions is mean square displacement $\text{MSD}(t)=4Dt+v^2t^2$ a valid asymmetric random walk model?

I am reading the paper Actin dynamics drive microvillar motility and clustering during brush border assembly by Meenderink et al. (2019). In this paper, the authors fit the mean square displacement (...
1 vote
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### Controllability of stochasic bilinear system in 2-dimension

I have a question about controllability and I am thankful for any reply. A linear system $\dot{x} = Ax + Bu$ is controllable iff it satisfies the well-known Kalman rank condition. Here, ...
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### How can imparted energy be a stochastic quantity?

It may be a silly question, but I have a dosimetry course and it started by defining deposed energy and imparted energy and for both it says that they're stochastic quantities. The mathematical ...
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### Distribution of density operators under Stochastic Master Equation

Stochastic master equations (SME) are used in studies of open quantum systems. The general form of an SME is: \begin{align} \tag{1} d\tilde{\sigma}(t) = - i [H, \tilde{\sigma}(t) ]dt + \frac{1}{2}\...
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### Cormorant takeoff patterns

I was watching Our Planet recently and one scene in particular stood out to me: https://youtube.com/clip/UgkxyggGjQg-0_uV2N-n_m0t_1NK2hNyEqyW. There are plenty of references that explain, to varying ...
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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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For a quantum system with Lagrangian $L(x, \frac{dx}{dt})$ we can represent the action of a path $\mathbf{x}$ as $$S(\mathbf{x}) = \int_0^{t} L(\mathbf{x}(s), \mathbf{\frac{dx}{dt}}(s)) ds.$$ Then, ...
A stationary random function like $v(t;w)$ is not periodic and not squared integrable, why we can do a Fourier transform to it, which is a very common process to analyze turbulence. How will a ...