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# 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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### Transition rate of two level system subjected to noise

(this question is simpler than its length implies. I did this on purpose to provide a nice complete development for future readers) The setup Suppose we have a two-level quantum system with ...
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Using conservation of particles in a control volume in phase space (in one dimension with no sources of particles or external forces), one can derive the formal transport equation \partial_t P(x,v,... 2 votes 0 answers 56 views ### Brownian Motion (Geometric, Fractional, Drift) I have been researching Brownian motion for a while and have come across terms/types of Brownian motion such as fractional, geometric, and Brownian motion with drift. I understand the physical meaning ... 2 votes 0 answers 87 views ### Does the Kalman filter incorporate a Heisenberg-like uncertainty principle? In the case of mechanical systems, applying the Kalman filter involves combining model based prediction (using an apriori known dynamical model) with real-world noisy observations of the positions and ... 2 votes 0 answers 46 views ### What are the possible and meaningful measurements on a quantum walk on a graph? In the contest of quantum walks, the graph is defined as G=\{V,E\} with V the set of vertices and E the set of edges. Thus, the Hilbert space is defined as the \mathcal{H}=\operatorname{span}\{\... 2 votes 0 answers 65 views ### Treat stochastically non-Hamiltonian perturbations Let us consider a classical dynamical system whose obserbvables A evolve according to the following equation of motion \begin{align} \dot A &= -\{H,A\}+f(q) \end{align} f(q) is a non-... 2 votes 0 answers 32 views ### 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 ... 2 votes 0 answers 69 views ### 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 ... 2 votes 0 answers 319 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 ... 2 votes 0 answers 151 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 ... 2 votes 0 answers 313 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 ... 2 votes 0 answers 42 views ### A micro-reversible stochastic process that models transitions between states with variable energies Suppose we have a system with 3 possible states A, B and C (there could be n states as well) with energies E_a(t), E_b(t) and E_c(t) that vary with time. If our system has a constant finite ... 2 votes 0 answers 123 views ### Overdamped Fokker-Planck equation with general force field (gradient + curl) I am looking for a general formulation of the Fokker-Planck equation for diffusing particles in a general force field F = -\nabla U + \nabla\times A in the overdamped regime (Smoluchowski equation). ... 2 votes 0 answers 193 views ### Expression of heat by the Brownian motion folks. I was reading a paper from PRE and I'm not sure what's going on about the following equation. So for the system composed of two heat baths governed by Brownian motion, the entropy change of the ... 2 votes 2 answers 661 views ### Random walk through Ising model Ising model consists of up spin and down spin or empty/filled space. Can we model random walk for different densities of packing through the Ising model? 1 vote 0 answers 29 views ### Adiabatic theorem for stochastic time-dependence I am trying to derive the adiabatic theorem when my time-dependent Hamiltonian is stochastic and I have a few questions. Usually, one starts with the Schrödinger equation \begin{equation} i\frac{d |\... 1 vote 0 answers 34 views ### 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 ... 1 vote 0 answers 59 views ### 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 ... 1 vote 0 answers 28 views ### 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, ... 1 vote 0 answers 26 views ### 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}\... 1 vote 0 answers 51 views ### Probability of branching times under a Ornstein–Uhlenbeck-Yule process According to Edwards, 1970 the probability density of the branching times in a Brownian-Yule branching process can be expressed as: \begin{equation} P(\mathbf{u'},n|\lambda,n_0,T)=\lambda^{n-n_0}\frac{... 1 vote 0 answers 66 views ### Why are realizations of the Stochastic Schrödinger Equation (apparently) not normalized? In "A Straightforward Introduction to Continuous Quantum Measurement" the Stochastic Schrodinger Equation (SSE) for continuous measurement of the observable X is given as d |\psi\rangle ...
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We know Langevin equation in the presence of an external applied force is given by, $$m\frac{dv(t)}{dt} = -\gamma v(t) + \eta(t) + \vec{F}$$ where $\eta(t)$ is delta correlated stationary white ...
We know that Fokker-Planck equation can be written as(with proper boundary and initial condition) $$\frac{\partial p(v,t)}{\partial t} = Lp(v,t)$$ where $L$ is known as the Fokker-Planck operator ...