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34

This ratchet-like Maxwell's demon has the same problem as all of the other ones: the door/coil mechanism itself will heat up, and become useless. Before thinking about this one, think about the simpler scenario where there's just a door, that opens if a fast particle hits it hard enough. Since particles have energy on the order of $kT$, the door must ...


12

This problem can be solved with noise-shaping. Since the shape of the spectrum is known, it can be used as a base for the power spectral density: $$ P(f,T)=\frac{ 2 h f^3}{c^2} \frac{1}{e^\frac{h f}{k_\mathrm{B}T} - 1} $$ where $k_\mathrm{B}$ is the Boltzmann constant, $h$ is the Planck constant, and $c$ is the speed of light. This outputs the relative ...


10

The key to this is the physical principle that the quantity you're asking about (delay between noise and noise cancelling) carries dimensional information (i.e. it's a time) and therefore it has to depend on the specific situation. The simplest case is trying to cancel out a pure note, with a sinusoidal waveform, then the delay can be as long as you want: ...


8

This is indeed a problem, which is dealt with using seismic isolation via Internal Seismic Isolation and Hydraulic External Pre-Isolators. The original LIGO isolation systems were passive - described here as shock absorbers - and included a single pendulum system, but the Advanced LIGO (aLIGO) upgrades added actuators that counteract vibrations in various ...


7

The spectral density, or spectral function, describes the coupling between a small quantum system that is coupled to a larger environment. In many cases, this environment can be modelled effectively as a system of free bosonic or fermionic modes, with Hamiltonian (working in units with $\hbar = 1$) $$ H_B = \sum_k \omega_k b_k^{\dagger}b_k. $$ The mode ...


7

The position of the mass, as a function of time, will simply be a filtered version of the random noise 'input' signal. To see this in the frequency domain, take the (magnitude of the) Fourier transform of both sides and rearrange: $$|X(\omega)| = \frac{1}{\sqrt{\left(1 - \omega^2\right)^2 + \frac{1}{Q^2}\omega^2}}|N(\omega)|$$ For $\omega = 1$, we have $$...


6

There are a couple of main sources of intrinsic error (that is, not associated with counting photons from your source) which CCD's have. The first is as you have already mentioned called read noise. Here is a reasonable definition of read noise (taken from Romanishin's free pdf on Photometry): After an integration (exposure), the CCD must be read out to ...


6

The situation you are describing is an example of Fresnel diffraction (or near-field diffraction). In general, when a wave propagates every point of the wave front can be thought of as its own source of waves traveling in all directions (called Huygens construction). It turns out that neighboring point sources along an infinite straight wave front reinforce ...


6

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{...


5

I think you've just derived the Stefan-Boltzman law for a one-dimensional system. The T^4 comes from three dimensions. The more dimensions the quanta can populate the higher power of T you get.


5

The threshold theorem says that if the error rate is below the threshold, a quantum algorithm with T locations (breadth times depth) can be made fault-tolerant with a blow-up (in both number of qubits and circuit size) by a factor which is a polynomial in the log of T. This is not enough to change BQP.


5

Treating the signals as time series: If the first signal $S_1$ has a noise component $N_1$ added to it, then the noisy signal is $S_1+N_1$, similarly the second signal is $S_2+N_2$, so the difference signal would be $(S_1+N_1)-(S_2+N_2)$ and its signal to noise ratio would be $\langle(S_1-S_2)^2\rangle\over\langle(N_1-N_2)^2\rangle$ If the signals are ...


5

The idea that frequency modulated signals are more resilient to noise than amplitude-modulated ones is somewhat of a myth. Both are susceptible to noise: the demodulation sequence (including the human hearing and sight senses) reacts slightly differently to the effects of noise so that. It can be shown that if there is additive Gaussian noise with ...


4

Apart from motor and bearing noise, most of the acoustic power comes from the eddy swirls following the trailing edge of the blade after it passes by. There is also an outward pulse of air as the leading edge of each blade pushes forward cutting the air. The trailing eddies produce a broad spectrum of random noise, modulated by the fan blade frequency. ...


4

I didn't see the episode, but it may be referring to "Phreaking", by which the signals from a CRT monitor can be listened-in on (it uses high frequency changing currents to display the information, so these will inevitably result in some RF radiation from which this information can in principle be extracted). Wikipedia article has a bit more info.


4

It seems that the confusion is due to some unfortunate notation. As the OP states, Fano noise is due to the variance in photoelectron production per incident photon, and this should indeed be signal-dependent. However, the author also states that the total noise is given by: $$\tag{1} \sigma^2_\mathrm{TOTAL} = \sigma^2_\mathrm{READ} + \eta_i F_F + \eta_i S ...


3

First, dB means nothing by itself. You need to give a reference level, like dBW or dB SPL. We'll assume dB SPL. Second, noise measurements from a point source like this require a distance measurement to be meaningful, since the level drops off with distance. We'll assume you're measuring at the same distance in both cases, and the fans are equidistant ...


3

$\newcommand{\bra}[1]{\langle #1 |}$ $\newcommand{\ket}[1]{| #1 \rangle}$ $\newcommand{\braket}[2]{\langle #1 | #2 \rangle}$ $\newcommand{\bbraket}[3]{\langle #1 | #2 | #3 \rangle}$ Although the question asks specifically about a harmonic oscillator, we can understand the meaning of the spectral density by considering a somewhat more general problem. ...


3

I don't think you really need an answer. The answer is yes and moreover what you have done is a pretty sound model of the effect of noise on the damped oscillator. I'm assuming that you have normalised frequencies so that the oscillator's natural frequency $\omega_n$ is one unit. The only factor you haven't mentioned and which you seem to have overlooked is ...


3

A single measurement like this has a lot of noise on it - and random signal is always going to have some random correlation. You should definitely not pay too much attention to the stuff that is in the tail of the correlation distribution - it's all noise. The fact that the built in function does not produce negative values is related to you only looking at ...


3

"What is noise and how can it be used in physics simulations" could fill a large volume (or more). But let me give you a simple example. In nature there are many stochastic processes - where events occur "with a certain probability". An example is radioactive decay. If you want to model the properties of radiation emitted by a radioactive source surrounded ...


2

OK, a little bit like zephyr's answer, I started with white noise, took the FFT, and then scaled each frequency component up or down according to the square-root of the power at that frequency, then did inverse-FFT to get the time-domain signal. An equivalent approach would have been to generate the FFT directly by giving each component the appropriate ...


2

In statistical mechanics and thermodynamics you are describing systems with an extremely large number of possible variables or degrees of freedom, so describing EXACTLY what happens becomes impossible. Instead, you describe the average. To do this, you consider all physically possible configurations of your system, and say they are all equally probable. ...


2

Mechanical noise is a form of energy loss, which ultimately also will end as heat: the acoustic waves will be absorbed by different kinds of substances which will vibrate more causing friction which will ultimately cause a temperature rise. Note that the acoustic power is often extremely low, often no more than a few mW, and when those get absorbed by a ...


2

I'm no ANC expert but I'm pretty sure the limit you're talking about would have something to do with the Haas effect (also called precedence effect). from Everest's Master Handbook of Acoustics: "...Haas found that in the 5 to 35 msec delay range the sound from the delayed loudspeaker has to be increased more than 10dB over the direct before it ...


2

The heat equation comes from two very intuitive ideas: the rate of heat flow is proportional to the temperature difference, and the conservation of energy. First, from Newton's law of cooling or Fourier's law we get that the flow of heat is proportional to the gradient of the temperature: $$\mathbf{j}_{\text{heat}}=-k \nabla T$$ where $k$ is the thermal ...


2

Pink noise is not going to sound like voices; it sounds a lot more like water splashing in a fountain. The high frequencies are small, and it will not just sound like a garbled version of your voice. The reason is that the amplitude distribution you find in your voice cannot be maintained--voice doesn't have a 1/f distribution. You can maintain the phase ...


2

Letting $\mathbf{F}$ and $\mathbf{F}^{-1}$ be the forward and inverse discrete Fourier transform, the cyclic autocorrelation of a signal $A$ is given by $$S(A)=\mathbf{F}^{-1}\left[\mathbf{F}(A)\mathbf{F}(A)^*\right].$$ Let the low-passed signal $A_L$ be $$A_L=\mathbf{F}^{-1}\left[\mathbf{F}(A)\mathbf{F}(L)\right]$$ where $L$ is the low-pass filter in the ...


2

I do have the book, but not in front of me, so I am guessing from the form of equations. A Brownian particle can be represented by the stochastic differential equation $$m\dot{v} = -\xi v +\varepsilon$$ where the last term is the stochastic term, which is assumed to behave like $\langle\varepsilon\rangle = 0$, $\langle\varepsilon(t)\varepsilon(t')\rangle = \...



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