# Tag Info

### Why is thermal noise Gaussian distributed in voltage, but Rayleigh distributed in amplitude?

Maybe this will help a bit. Consider two normally distributed variables $X_1,X_2\sim N(\mu=0,\sigma=1)$. Then $X_1,X_1^2,X_1^2+X_2^2$ and $\sqrt{X_1^2+X_2^2}$ will produce the following plots. Notice ...

### Why is thermal noise Gaussian distributed in voltage, but Rayleigh distributed in amplitude?

The amplitude is $\rho^2=x^2+y^2$, which is by definition positive, and which will obey Reyleigh distribution, if $x,y$ are a bivariate normal. Note also that detector is always a quadratic detector - ...
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1 vote

### Calculating average of a function of molecule's orientation (Euler angles)

Maybe this is just a silly comment, but it is too long to be a comment itself. @GiorgioP 's answer explains why the "$\sin$" has to be in your expression, but is still interesting to notice ...
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### Calculating average of a function of molecule's orientation (Euler angles)

The reason is the same why, in the case of the spherical coordinates, the surface area element contains the $\sin(\theta)$ factor: we would like to have a uniform measure on the unit sphere surface ...
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### Properties of random-walk in infinite and finite two-dimensional space: probability of two particles being in the same location at time t

In general, for this kind of stochastic processes, and $t_f > t > t_i$, we have : $$G(x_f,t_f|x_i,t_i) = \int G(x_f,t_f|x,t)G(x,t|x_i,t_i)\text dx$$ Representing the fact that the particle must ...
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### How does an exothermic reaction release energy?

The energy release din chemical reaction comes from the difference in binding energies of the reactants and the reaction products. Thus, if we consider two atoms $A$ and $B$ that can join in a ...
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### On-shell Poisson brackets and time derivative

According to Liouville's theorem $$\frac{d\rho}{dt}(S^t(p,q),t)=0\iff\frac{\partial\rho}{\partial t}(S^t(p,q),t)+\{H,\rho\}=0\tag{A}$$ where the Poisson Brackets are evaluated on the Hamiltonian flow ...
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### How does an exothermic reaction release energy?

Here is more information related to the basic question "How does an exothermic reaction release energy?" Classical thermodynamics accounts for the release of energy using such concepts as &...
• 6,446
Accepted

### Thermal noise phasor amplitudes are Rayleigh distributed. How are voltages at the antenna distributed?

Let $\mathbf z = \mathbf x + \mathfrak j \mathbf y$ where the random variates $\mathbf {x,y}$ are independent and normal distributed with zero mean, that is: $\mathbf {x,y} := \mathcal N (0, \sigma)$. ...
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### Do Stochastic Differential Equation models conserve energy?

My question is this: Does this model conserve energy at all points in time? Alternatively, how do stochastic models like this deal with energy conservation? You actually answer this question in the ...
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### How does an exothermic reaction release energy?

For example, one (or more) of the reaction partners is after the chemical reaction not in the ground state, but in an excited state. The electron will sooner or later fall back into the ground state ...
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Accepted

### Derive Canonical Ensemble from Maximum Entropy Principle

A good way to start the proof is to first select some basis such that the density matrix $$\rho=\sum_{j}{p_j|\psi_j\rangle \langle\psi_j|}$$ where $p_j>0$ and $\{|\psi_j\rangle\}$ is orthonormal. ...
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### The probability of a circular region being "invaded" by moving spheres as a function of time

I assume classical mechanics applies to this problem. If by the spheres "do not interact with each other" you mean no collisions, then only those spheres initially headed in a direction that ...
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### Why does adiabatic expansion occur in the carnot process?

In the second step, the system is separated from the reservoir and is now thermally insulated, resulting in another small adiabatic expansion. I ask myself, why is a little internal energy converted ...
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### Why does adiabatic expansion occur in the carnot process?

You are in control of the piston, the piston does not move spontaneously, the external pressure is set up (controlled) by you at every time so that the process is quasistatic. So the system is at ...
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### Why does adiabatic expansion occur in the carnot process?

Because both heat transfer steps must be isothermal to be reversible and generate no entropy. The Carnot cycle works by exploiting the temperature difference between two reservoirs. It uses reversible ...
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### Why does adiabatic expansion occur in the carnot process?

At the end of the reversible isothermal expansion the external pressure is deliberately further slowly reduced, but now according to $Pv^{\gamma}$ = constant, instead of $Pv$=constant, to allow the ...
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### How do solids transfer heat?

As the highly influential 20th century physicist, Rudolph Peirls said in his article "Quantum Theory of Solids" in 1951: "It seems there is no problem in modern physics for which there ...
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### Do Stochastic Differential Equation models conserve energy?

Physically motivated Langevin equations usually contain dissipative terms alongside the noise terms, whose parameters are related via the fluctuation-dissipation theorem (of which the Einstein ...
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### Do we need atomic theory to do thermodynamics?

Thermodynamics and statistical physics are two different ways of looking at the same thing. The former indeed does not require atomic theory, see, e.g., here and here. Also relevant: Timeline of ...
• 39.2k
1 vote
Accepted

### Explaining internal energy from a macroscopic perspective

What you need really is a proper starter-course in thermodynamics, with a good textbook (and reader can guess which one I recommend!) The logic goes as follows. Various experiments (especially those ...
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### Explaining internal energy from a macroscopic perspective

The concept of internal energy is introduced in the first law of thermodynamics, based on Joule's work. Based on his experiments with heating and mixing water with a paddle wheel, he realized the ...
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• 28.4k
1 vote
Accepted

### Probability of collision between two particles (Statistical Mechanics)

$\iiint P(\vec{r}) dV = {1 \over N} \iiint n(\vec{r}) dV = {1 \over N} \iiint dN(\vec{r}) = 1$ Where $N$ is the total number of particle in $dV$. Now these functions can be generalized to the ...
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### Energy dissipated by friction and entropy

If the initial temperature and the final temperature of the block are both T, then applying the complete version of the first law of thermodynamics to the block gives $Q=-K$, where Q is the amount of ...
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