50

Physics models rarely hint at ontological level. Throwing dice can be modelled as deterministic process, using initial conditions and equations of motion. Or it can be modelled as stochastic process, using assumptions about probability. Both are appropriate in different contexts. There is no proof of "the real" model.


34

Throwing dice is just throwing dice. That's all. It's not stochastic, nor deterministic. It's just throwing dice. Now we model throwing dice as a process, and that's where the stochastic or deterministic side starts to play in. It is the process that is stochastic or deterministic, not the throwing of the dice. It's how we think about the throwing of ...


14

Look up Diaconis's work on flipping coins. While it is technically deterministic, what happens is that extremely small changes in the initial conditions flip the outcome. The same would be true of dice. When you shake them in your hand and throw, small changes would give different outcomes. What makes it seem random is that we can't control our hands well ...


12

Probability distribution of time until next event First we calculate the probability density that a time $t$ passes without any event happening. Divide $t$ into $N$ small intervals each of length $dt = t/N$. Defining $\lambda$ as the probability per time that the event occurs, the probability that no event occurs within any one short time interval is ...


11

There are several ways I can interpret the question, so my main focus is going to be on the autocorrelation of an Ornstein-Uhlenbeck (O-U) process. So what is an O-U process and how is it different from regular Brownian diffusion? Brownian diffusion The stochastic differential equation (SDE) for Brownian diffusion of a particle can be written as $$\mathrm{...


9

Thanks for the nice problem. First, let us scale the variables $f \rightarrow x$, $\zeta \rightarrow \sqrt{\Gamma} \, \xi$, and consider the Langevin equation $$ \dot x = -k \, x + \sqrt{\Gamma} \, \xi, \qquad (1) $$ where $\xi$ is a colored noise satisfying $\langle \xi(t) \, \xi(t') \rangle = e^{-\gamma \, |t-t'|}$. We wish to show that the Fokker-Planck ...


8

PREFACE After several edits, this answer provides a naive explanation of why your approach failed, how to fix it (naive-ish) and a completely different (but right) approach to solve the problem. Intro You are right: the diffusion coefficient should be $D=4pqD_0$, $D_0$ being the "normal one" (see below for derivations). I do not know precisely why your ...


8

Both are relevant, and "the misconception that Langevin equation is the universal stochastic differential equation for all kinds of noisy systems is responsible for the difficulties mentioned"* in your post. Take the SDE from Thomas' answer, $$\frac{dy}{dt} = A(y) + C(y)L(t)$$ where $L(t)$ is the noise term. Suppose we can turn the noise off, so we'd only ...


7

White noise is characterised by the autocorrelation function $$\langle \eta(t) \eta(t') \rangle = D \delta(t-t'),$$ (here I'm assuming that $\eta$ is dimensionless so $[D] = [t]$). This means that (1) the noise should be uncorrelated with itself at different times, but also that (2) the variance of $\eta(t)$ must be infinite. Condition (2) is the essential ...


7

It is a sticky question, and as van Kampen puts it, " no universal form of the diffusion equation exists, but each system has to be studied individually." https://link.springer.com/article/10.1007/BF01304217 (Unfortunately, I don't have full access to his paper, but you might be able to get it through your library.) Now, the main reason the question is ...


6

There is a connection between QFT and random walks. It turns out the generating function $P_{ji}(z)$ is equivalent to the correlation function for the free scalar field on a Euclidean lattice. The parameter $z$ in the generating function effectively ends up being related to the mass of the scalar field (actually to a combination of mass and temperature). ...


6

This question is a little too big, entire text books have been written to answer it. A standard reference is van Kampen, Stochastic processes in physics and chemistry. Roughly speaking, for a Markov process Master equation -> Kramers-Moyal expansion -> Fokker-Planck equation where the master equation gives the microscopic probabilistic rule for ...


6

The Poisson distribution describes the probability of a certain number ($n$) of unlikely events ($p\ll 1$) happening given $N$ opportunities. This is like doing a very unfair coin toss $N$ times, with the probability $p$ of the coin turning up heads. The number of heads would follow the binomial distribution: $$P(n|p,N) = ~^{N}C_n~p^n (1-p)^{N-n} =\frac{N!}...


5

Alpha particles are only significantly scattered by nuclei. Electrons are so much lighter than an alpha particle that it is hard for the alpha particle to transfer much momentum to them. But nuclei are small. The radius of a nucleus is of the order of $10^{-5}$ times the radius of an atom, so the cross-sectional area of the nucleus is of order $10^{-10}$ ...


5

Both Ito and Stratonovich stochastic PDEs can be used to derive a Fokker-Planck equation. Indeed, for simple one-dimensional processes the Ito process $$ dx = a\, dt + b\, dW(t) $$ is equivalent to the Stratonovich process $$ dx =\left( a\,-\frac{1}{2}b\partial_x b\right) dt + b\, dW(t) $$ The answer is then that both are physically reasonable, for a given ...


5

A gentle introduction to the basic ideas of stochastic processes - Stochastic Processes for Physicists: Understanding Noisy Systems by by Kurt Jacobs The following are excellent reference textbooks - The Fokker-Planck Equation by Hannes Risken Stochastic Methods by Crispin Gardiner For numerical solution of SDE the following are recommended - Numerical ...


5

Let $A_{n,t}$ be the event: exactly $n$ events happened over a time interval $t$. Then $$P(A_{n,t+\Delta t}) = P( A_{n,t} \cap A_{0,\Delta t }) + P(A_{n-1,t} \cap A_{1,\Delta t }) \, .$$ Using independence of occurrence and the definition $\lambda = \lim_{\Delta t\to 0} P(A_{1,\Delta t})/\Delta t$, taking the limit, and defining $P_n(t) \equiv P( A_{n,t})$,...


5

I think the answer is (surprisingly!) no for finite systems. Quantum mechanics evolves in a unitary way, which means that information is not lost. Evolution involves a selection step where candidate individuals are evaluated and removed; one can say the process of moving information from the environment into the genome occurs by making random variations ...


5

First, by the Chung-Fuchs theorem, any mean-zero one-dimensional random walk is recurrent. This tells you what the proper assumption on the step-distribution $P$ is. If, in addition, the step-distribution has finite variance $\sigma^2$, then the law of its excursions converges, after diffusive scaling, to the law of Brownian excursions (see, e.g., Annals of ...


4

You start by solving the differential equation. It is a first order, linear differential equation with constant coefficients. So the solution of the homogenous system is quite simple: $U(t) = c\cdot e^{-t/RC}$. Now we solve the particular system with variation of the constant $c$, which means we try the Ansatz $U(t) = c(t)\cdot e^{-t/RC}$. This give the ...


4

The Brownian motion $x(t)$ is non-differentiable, so a particular trajectory $x(t)$ can't extremize an action $S$ which would be a functional of $x(t)$ and its derivative, $\dot x(t)$, because the derivative isn't even well-defined and any expression of the type $\int [\dot x(t)]^2 dt$, the usual kinetic term in the action, diverges. (See e.g. middle of page ...


4

There are a couple of mistakes. Though $$ \delta r(\omega) = \frac{\eta(\omega)}{\gamma + i \omega} $$ is correct. However, if we assume the noise is $\delta$-correlated in the time domain, $$\langle \eta(t)\eta(t') \rangle = q\delta(t-t')$$ this leads to similarly $\delta$-correlated noise in the frequency domain $$\langle \eta(\omega)\eta(\omega')\...


4

The answer for this problem is given by Francesco Mezzadri for all classical compact groups. (I have mentioned that in my answer to a similar question on Mathoverflow) For $U(N)$ and $O(N)$, the answer is very simple based on the QR decomposition with a little extra care due to the non-uniqueness of the QR decomposition. The algorithm for $U(N)$ is given ...


4

Note: I'm using $$S_F(\omega) \equiv \int_{-\infty}^\infty dt e^{i \omega t} \left \langle F(0)F(t) \right \rangle$$ because I believe it is considerably more standard than the definition given in the question post. What physical information does the power spectrum $S_F(\omega)$ provide about the nature of the random variable $F(t)$ in this case? The ...


3

$R(t)$ is a function of time that represents complicated time-dependence of forces due to other molecules on the studied molecule. Since only correlation function is assumed, there is no single unique function $R(t)$ assumed; although not all, many functions would be appropriate. You can generate many of them in computer using Cholesky decomposition of ...


3

Hints: $\underline{(0) \Rightarrow (1)}$: Don't try to accomplish everything at once. Do it slowly in as many steps as you need to be sure that you are calculating correctly and understand everything. The trick is to integrate by part. Be very careful to keep track of what depends on $x$ and what depends on $x^{\prime}$. $\underline{(1) \Rightarrow (2)}$: ...


3

This is a standard result in statistics: the standard error of the mean produced by a sample of $N$ copies of a given system will scale with $1/\sqrt{N}$. The details of the proof depend on exactly what hypotheses are supplied, but every statistics textbook includes at least one such proof.


3

I strongly recommend two references: Young's paper (e-print), an illuminating conceptual review of entropy in dynamical systems; Pekoske's Master thesis, for its examples with step by step calculations. (Beware of the typo in page 36, though: $h_\mu(\sigma)\approx .075489$ should read $h_\mu(\sigma)\approx 0.32451$.) Topological Entropy $h_T$ how do I ...


2

The OP is correct in stating that the Fourier transform $$\xi(\omega) = \int\mathrm{d}t\, \mathrm{e}^{\mathrm{i}\omega t} \xi(t), $$ vanishes upon averaging over realisations, $\langle \xi(\omega)\rangle = 0$, so long as we assume that the noise is also zero on average in the time domain, $\langle \xi(t)\rangle = 0 $. However, the noise is not only ...


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