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12

So, whenever I want to find a nice introduction to a concept in physics, I check the American Journal of Physics, as it is full of articles with clever descriptions of phenomenon appropriate for presentation in university courses. In this case, this yields many results. In particular, I found the following three articles very helpful: The mathematics of ...


10

I would say that one experiment that demonstrates the atomic nature of things is the observation of Brownian motion. But it is not the experiment itself that convinces that things are made of atoms, rather its theoretical explanation given by Einstein in one of his 1905 papers (actually Einsteins work for his PhD was on the subject of atomic theory and there ...


6

I once heard Uhlenbeck give a lecture on this to high school students over the Christmas break at the Rockefeller Univ. years ago. He recounted a published argument he attributed to Einstein around 1905 (I think), which was that atoms were real if you could count the number of them/mole (Avogadro's number) many different independent ways, and you always ...


5

A thought experiment: after N steps, each of which create a change in angle $\Delta \theta$, we should end up with a normal distribution of angles with a standard deviation of $\frac{\sigma}{\sqrt{N}}$. When you change the step length, you therefore need to scale the standard deviation by the square root of that change, so that after moving the same distance ...


5

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 ...


5

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 ...


4

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 $\vec{\Gamma}(x,t)= -D\nabla N(x,t)$, ...


4

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 ...


4

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 ...


4

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 ...


4

No experiments prove any theory. Experiments can only refute theories.


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 ...


3

Apparently the search term I was missing was "Brownian motion". With that, I found several leads. They contradict each other somewhat, but I can at least post a partial answer: Geisler - Sound to Synapse: Physiology of the Mammalian Ear: Estimates for the first of these sources, the pressure fluctuations due to the Brownian motion of air molecules ...


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

Here's how I would come to some intuition for it. I would think about the rate of "probability flow" into a region by integrating the equation over a region in space. For now, let's suppose that no diffusion occurs at all, since that is more complicated (although directly doable and understandable). Then $$\int_a^b dx\frac{\partial p(x,t|x_0)}{\partial t}=\...


3

When the coils are moved around randomly through brownian motion inside the liquid, wouldn't this induce some current? Yes. Wouldn't this basically resemble a magnetic brake Accounting for the fact that copper has a nonzero resistance, then the thermal energy of the water causes the coils to move, this causes induced current, and this current is ...


2

The history of atoms is definitely intertwined with quantum mechanics. There are many features of the quantum theory that make atomic nature of our world apparent. But here I'd like to state an earlier result. Thomson's 1897 discovery of the electron not only showed that atoms exist but also that they have substructure.


2

This is not a answer to your question but a close cousin to it, perhaps you will find it of interest. The Schrodinger equation can be analytically continued to give the Heat Diffusion equation. t->-i*t Google can point you further elaborations and references.


2

Some kinds of mutation provide an example of this kind of indeterminacy. UV light can be bad for our health. One of the reasons is that, when we are exposed to sunlight, UVB photons are absorbed by double bonds in pyrimidines, which break open, become reactive, and dimerize (photo-dimerization). This damages the DNA in the same way that it would damage a ...


2

It sounds like you're trying to solve the Langevin Equation. This is a model of Brownian motion where the particle experiences stochastic kicks at discrete time intervals. Your force, in this case, is a random variable you draw from a distribution each time step (instead of being given by an explicit formula). For the simplest case, the "kicks" are ...


2

Here is the intuitive explanation: When a particle is moving, it will "run into" things. Thus, the "random force" from impacting another particle is not completely random: it is in part correlated to the motion of the particle before the collision - the force of the impact is more likely to be in a direction opposite to the current motion than any other ...


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 ...


2

Here is another reason why a ratchet should not work: it would define a directional arrow of time even in thermal equilibrium. To see this look at the modified ratchet in the graph (the triangular piece is attached to a spring). the brownian particle would move easier to the right as it can push the triangular piece down, but if it is at the right and ...


2

Let us use the definition $\langle\eta(s)\eta(t)\rangle=\Gamma\gamma^2\delta(t-s)$. First of all, $C(s,t)$ depends on $t$ because $$C(s,t)=\Gamma\min(s,t).$$ It is clear, from causality, that $\frac{\delta x(t)}{\delta \eta(s)}=0$ if $t<s$. If $t>s$, compute the difference $\delta x(t)$ caused by two realisations of noise that differ only at time $s$ ...


1

The diffusion is caused by collisions between the molecules due to the random thermal motion of each molecule. So you have a blob of oxygen in a box of nitrogen then they are all moving with a random velocity from the Maxwell velocity distribution. When two molecules collide, they change velocity and so they change both speed/direction. They then collide ...


1

You need to scale the standard deviation so that the rate of curvature will remain the same. You are summing a bunch of normally distributed random variables, $X_i\sim N(0,\sigma^2)$, to obtain the angle $X$ after $n$ steps: $$ X=\sum_{i=1}^n X_i $$ This will also be normally distributed with variance $n\sigma^2$. Now consider a larger sum: $$ X=\sum_{i=1}^{...


1

First remark, the average hitting time is finite because the volume is finite. None of what I write would make sense in an infinite system. Let us consider that the target is a ball of radius $a$ at the center of the sphere and let us call $T(\vec r)$ the average hitting time for a Brownian particle starting at position $\vec r$ from the origin. $T(\vec r)$...


1

In a volume $V$, a large number number of particles starting to move according to Brownian motion at time $t=0$ from the position $x_0\in V$ will exit from $V$ at a rate that depends on time, on the geometry of the volume $V$, and on the position $x_0$. For instance in one dimension, particles diffusing from $x_0>0$ will enter the region $x<0$ at time $...


1

No - thermal equilibrium means no heat transfer (essentially when the temperature of the gas inside the box remains constant but not necessarily zero). The atoms will continue to move, vibrate, rotate, etc... but do so now at a constant temperature. Also, just to be clear, an individual atom does not have a temperature - you need a large collection of ...


1

The quantity you are dealing with is the local time of the Brownian motion at one point. It is rather difficult to deal with local times usely. What you want to compute is the distribution of the quantity $$\lambda(\vec r_0)=\int_0^t\delta^{(d)}\left(\vec r_0-\vec B_u\right)\mathrm du.\tag{1}$$ Note that $\lambda$ is a spatial density of time, it has ...



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