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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Calculating entropy in truncated Wigner

I'm trying to get some reasonable measure of the entropy of a system modelled by the truncated Wigner method. The Wigner function contains all the information about a density matrix. So, I figure it ...
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Deriving the Bloch Equation of qubit purification for any measurement angle

I am trying to derive the Bloch vector $dr$ for a measurement of a observable in any arbitrary direction $\theta$. For context this is the setup and derivation I have for continuous measurement along ...
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73 views

Can a gas molecule theoretically have zero velocity?

According to Maxwell's speed distribution law gas molecule can have speed which lies between zero to infinity. But in the graph of the distribution curve it seems to touch zero velocity. So can a gas ...
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The justification for stochastic time evolution equation (in stochastic thermodynamics)

I came across an equation in the context of stochastic thermodynamics, specifically in the paper "ensemble and trajectory thermodynamics - a brief introduction": The time evolution of the state is ...
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Are there any experimental or statistical tests to check deterministicity or stochasticity of a dynamical system?

Are there any simple experimental or statistical tests to check whether a dynamical system is deterministic?
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185 views

Is the Feynman's path integral a density?

The Feynman-Kac path integral formula is used to solve parabolic equations related to stochastic processes. Considering the probabilistic expression, the solution is indeed not a density. However, ...
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In mean-field theory, why are the collisions of particles in the mean-field neglected?

Mean field theory is a tractable framework for analyzing parameters of a continuum or an infinite number of identical micro-particles or agents. The former has been treated extensively in statistical ...
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Unbounded time derivative of the brownian motion and Newtonian dynamical models

The second order dynamics model $dx_s = v_s ds$, $d v_s = dw_s$ where $w_s$ is s standard Brownian motion is an example of Langevin dynamics. Note that the Brownian motion in this case models a white ...
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Can we deduce that particles behave as Brownian motions if the collection obeys the Einstein model?

The density dynamics of a continuum of particles with the dynamics $$dx^i_s = d w^i_s,$$ where $dw^i_s$, $0 \leq s$, $i \in \mathcal{N}$ is a standard Brownian motion, are given by the diffusion PDE $$...
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49 views

Variance of Simulated Langevin Equation

I simulated (by Matlab) the Langevin equation for optical-trapped particle in very short time "steps" And I got this white noise figure.. The question is how I can calculate the variance (or in ...
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93 views

What is a real world example of noise excitation in the dynamics of macro objects (other than to model sensor noise)?

The literature on stochastic processes (Ornstein–Uhlenbeck, Langevin) is not very clear as to the motivation behind using the Brownian motion or other types of noise in the dynamics. Are there any ...
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Path integrals 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\...
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Fokker-Planck linear potential

I am struggling with finding a solution to the following Fokker Planck equation with linear potential: $$\partial_{t}P(x,t)=k\partial_{x}P(x,t)+D\partial_{x}^{2}P(x,t)$$ Can anyone help me please? ...
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What's the meaning of the osmotic velocity?

To describe a random-walking particle, we can use the stochastic differential equation \begin{equation} dx(t)= b\Big(x(t),t \Big) dt + dw(t) , \end{equation} which is also known as the Langevin ...
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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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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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When we should use Langevin equation and when we should use Fokker-Planck equation?

As everyone knows that we can go from Langevin equation to Fokker-Planck equation which gives the evolution of probability density function. But what I don't understand is what is exactly the main ...
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an example of stationary markov chain which doesn't satisfy the detailed balance

I understand that if a markovian process satisfies the detailed balance relation:$\pi_{i}p_{ij}=\pi_{j}p_{ji}$, then it is stationary. So detailed balance relation is a sufficient condition. But is ...
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Random fields in physics - how do they work?

I'm trying to get an intuitive understanding of what random fields are. Wikipedia's article gives a formal definition (which is in very mathematical language), but also says various much more ...
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64 views

Expression of Dirac Delta Correlation

spatio-temporal white noise $\xi(x,t)$ is often expressed as $$\langle\xi(x,t)\rangle=0,$$ $$\langle\xi(x_1,t_1)\xi(x_2,t_2)\rangle=\delta(t_2-t_1)\delta(x_2-x_1).$$ Now I understand that the first ...
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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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Martin-Siggia-Rose action corresponding to Langevin equation

What is the Martin-Siggia-Rose-DeDominicis-Janssen action corresponding to the overdamped Langevin stochastic equation $$\frac{d\mathbf{x}}{dt} = -\mathbf{\nabla}V + \mathbf{\eta}$$ where V is the ...
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State equation to state-space

I have a one question about convert state equation to state-space model. I can not create a state-space model because of $g$ term from given this state equations... $\dot{x_{1}} = \dot{z} = x_{2};$ ...
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93 views

A discrete time Ornstein-Uhlenbeck type process

Let's consider a stochastic process $$X_{t+1}=X_{t}+\Delta X_t$$ where $\Delta X_t$ is a Gaussian with mean $-\lambda X_t \Delta t$ and variance $2\Delta t$ where $\lambda$ and $\Delta t$ are ...
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How to derive the backward Fokker-Planck equation from a forward Fokker-Planck equation (with state-dependent diffusion coefficient)?

I am interested in a system with state-dependent diffusion coefficients. This paper by Lau and Lubensky derives the correct Forward FPE in this case: $$\partial_tP(x,t) = \frac{\partial}{\partial x} ...
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Solving a continuous space, discrete time walk

I posted this in the math forum but thought that putting it here might be helpful too. So here goes: Consider a random walk starting from position $x_0$ on the $1D$ line that takes steps (of step ...
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What does convergence to equilibrium for the Fokker-Planck equation mean?

I am a math major who recently started to study thermodynamics seriously. I have some confusing points while studying it, so I'd appreciate it if you'd correct me and give me some answers. (1) As ...
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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.
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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 ...
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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 ...
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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 ...
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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'...
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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 ...
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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 ...
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62 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 ...
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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 ...
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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....
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57 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(\...
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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 ...
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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) =...
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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 ...
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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}{\...
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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. ...
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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 ...
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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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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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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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