# Tag Info

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For Brownian motion, Langevin equation, Fokker-Planck equations, Stochastic process.. from the viewpoint of physicists, the following are standard references: Brownian Motion: Fluctuations, Dynamics, and Applications The Fokker-Planck Equation: Methods of Solutions and Applications Handbook of Stochastic Methods: for Physics, Chemistry and the Natural ...

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The article makes no sense. Einstein realized that matter was composed out of atoms, so the number of collisions of a Brownian particle with the surrounding molecule is finite in a finite period of time. However, for times $t$ much longer than the typical scale between the collisions, the particle moves by a distance scaling like $\sqrt{t}$. It follows that ...

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The differential is used to specify that the number is for a "differential range", which is a way to remind you that the notions involved are somewhat fuzzy. Let me give a purely mathematical example. Suppose I tell you that I am going to pick an arbitrary real number between 0 and 10, with the likely hood of a number being picked being proportional to the ...

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There are tons of papers on the connection between quantum processes and probability theory (though I don't understand why you single out coherent states - they don't play a special role in this connection). The theory of stochastic processes and the theory of quantum processes are the commutative and noncommutative side of the same coin, with many ...

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Many different types of connection can be made between stochastic states over commutative algebras of observables and quantum states over noncommutative algebras of observables. As Arnold says, there is a substantial literature. One approach is to construct both classical and quantum models in a formalism that accommodates both; within the structured ...

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Since the comment answered your question I'll just go ahead and set out a more generalised version. It's straightforward to simplify things back down to your case. Consider the following continuity equation: $$\dot{N}(x,t) = -\nabla\cdot\vec{\Gamma}(x,t) + S(x,t),$$ where $\vec{\Gamma}(x,t)$ is the flux (in your case $\vec{\Gamma}(x,t)= -D\nabla N(x,t)$, ...

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The autocorrelationfunction is actually some kind of measure for memory. The $C_X(t')= \langle X_s(t)X_s(t+t')\rangle$ should be compared with the statistical correlation (Wikipedia: http://en.wikipedia.org/wiki/Correlation_and_dependence). With the statistical correlation one measures the dependency between two veriables. If two variables are independent ...

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The basic idea is illustrated on page 64 of the paper you cited. In this case, "directed transport" refers to a bias in the direction the wheel can turn, which could then be used to lift the weight and do work. A full explanation of this example can be found in section 2.1, beginning on page 64. This can be generalised to other situations. Let's ...

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If you describe the combined system of the molecules of the liquid and the Brownian particle and you know the mechanism of the collisions and all initial conditions, then it is deterministic. If you want to describe only the Brownian particle, then you would do so by a stochastic processes (called Brownian motion or the Wiener process) and it would be ...

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This is a bit strange. The Langevin equation $$\frac{dv}{dt}~+~\beta v~=~\frac{F}{m}$$ for the motion of a free particle under a stochastic force $F$ evaluates the velocity as an average or in an interval. The stochastic force has a Gaussian probability distribution $\langle F(t)F(t’)\rangle~=$ $2\beta kT\delta(t~-~t’)$, which is also a Markov ...

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