For questions about probability, probability theory, probability distributions, expected values and related matters. Purely mathematical questions should be asked on Math.SE.

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1answer
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Which position and momentum distributions arise from some wave function?

Consider a particle in one dimension with wave function $\psi(x)$. The probability density function describing how likely it is to find it in a given position is given by ...
2
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1answer
55 views

Any fractal physical model that generates time series which demonstrate heavy-tailed (non-Gaussian) behavior in some form?

I know that fractal structures have power-laws in various forms "hidden" in them. I am looking for the most simple fractal model that I can find that generates time series with, say, ...
2
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1answer
58 views

Cosmological fluctuations: what is gaussian?

When we are speaking about gaussianity and non-gaussianity in a cosmological context, what is gaussian or non-gaussian in the CMB? What would a non gaussian CMB look like compared to a gaussian one? ...
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95 views

Free probability in Physics

Recently I have started reading some materials on non-commutative probability. IN this area mathematicians sometimes consider quantum theory as a non-commutative version of classical probability, with ...
4
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0answers
77 views

Help with deriving an asymptotic expression

Note: I don't know if this is the best place for this question, because it is very specific. However, I'm not sure of a better place to go (apart from one of the other SE's). If you have a ...
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0answers
93 views

Reference for stochastic processes which helps moving from a basic level to a measure theory one

I'm looking for a reference (books, notes, lectures) which helps a physicist to understand the language of measure theory in the context of stochastic processes (in particular markov chains). I've ...
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261 views

Electron hopping among molecules - Marcus equation

I'm running out of professors to talk to, and I need to clarify a couple of things for the sake of making a realistic model of electron travel through a mesh. This is about calculations of electron ...
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0answers
33 views

Is the conservation of probability in the Schroedinger's equation unique?

The Schroedinger's equation can be viewed as a diffusion equation with imaginary constants $a$ and $b$ satisfying, $$(1) \quad \Psi_t=a \cdot \Delta \Psi-b \cdot V(x,t) \cdot \Psi$$ However if $a$ ...
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97 views

What can I expect to see in a oscillator exhibiting bifurcation?

I have a program which aims to simulate a Josephson Bifurcation Amplifier. I am currently trying to obtain a plot of the probability of bifurcation as a function of the ratio between the driving and ...
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407 views

Probability density of Klein-Gordon equation

This may, perhaps, stir some healthy debate; at least I am having some "fun" thinking about it, hopefully I can solicit some outside views too. It is often regarded that the Klein-Gordon equation ...
3
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0answers
93 views

Are frequency and likelihood the same across the multiverse?

My probability text distinguishes between two interpretations of probability values: the frequency of occurrence "as percentage of success in a moderately large number of similar situations" (coin ...
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45 views

Measure-theoretic maths behind Born's probabilistic interpretation of Schrodinger's equation

I was reading a bit about Quantum Mechanics, Schrodinger's equation and its probabilistic interpretation (found this very insightful intro here https://plus.maths.org/content/schrodinger-1), my ...
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75 views

Statistical Mechanics: Most probable orientation of grain particle in gas chamber?

I'm in an introductory statistical mechanics course, and we've been posed the following situation: Long-shaped dust particle (so imagine something like a grain of rice) is placed in a gas chamber (so ...
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117 views

How to explain Tsirelson's inequality using extended probabilities?

How to explain Tsirelson's inequality using extended probabilities? Some people have tried explaining the Bell inequalities using extended probabilities. For instance, a pair of entangled photons ...
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0answers
41 views

Why does the preservation of transition probabilities imply the preservation of all quantum probabilities?

I have a question about symmetries in quantum mechanics. Let $H$ be a Hilbert space, and $\mathbb{P}H$ the corresponding projective Hilbert (ray) space. In quantum mechanics, a symmetry is usually ...
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0answers
29 views

Boltzman equation for a collisionless medium?

In the derivation of the Boltzmann equation (link to Wikipedia) for a collisionless gas it is assumed that: $$f(\vec r+\frac{\vec p}{m} \Delta t, \vec P + \vec F \Delta t, t + \delta t)=f(\vec r, ...
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27 views

Confusion about radioactivity

The following question is from General Problems on Physics by I.E Irodov 6.220. Find the decay constant and the mean lifetime of $^{55}\operatorname{Co}$ radionuclide if its activity is known to ...
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0answers
27 views

Probability current: what's a good definition?

I am doing scattering problems with a really weird hamiltonian (e.g., only first-order in derivatives). I don't know how to define a probability current: I've looked at answers on this site, and they ...
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0answers
74 views

Probability flux

I was reading a text on Quantum Mechanics in which it said that $$\int{d^3 x \, j(x,t)} = \frac{\langle p\rangle}{m},$$ where $\langle p\rangle$ is the expectation value of the momentum operator at ...
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64 views

Applicability of perturbation theory

Consider some system in some initial state $|k^{(0)}\rangle$. The probability that such a state makes a transition to some other state $|m^{(0)}\rangle$ can be computed to various orders in time ...
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152 views

Relaxation time approximation in Drude model apparant paradox

In the Drude model of the free electron gas to explain the conduction of a metal, the relaxation time approximation that the electron has a collision in an infinitesimal time interval $dt$ is ...
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64 views

Noise of a fiber optic gyro

I have not much experience with noise handling or calculations, furthermore in my researches couldn't find a similar problem, so here is my attempt: Having a fiber optic gyro that is rated with a ...
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0answers
41 views

Why can't we use the Neyman-Pearson likelihood ratio directly?

If you have a bunch of events and would like to choose a cut to distinguish background and signal, you can take the likelihood ratio $$ \lambda(\vec x) = \frac{f(\vec x| s)}{f(\vec x| b)} $$ and the ...
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0answers
97 views

Do doomsday arguments influence doomsday hypotheses?

The doomsday argument supposes that in the absence of any other knowledge, if we know the age of something now, we may assume that we are seeing it in the middle of its lifetime and then calculate our ...
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0answers
45 views

Books on collision probability and collision processes

Are there any books specifically on collision processes between atoms and molecules and collision probability? I would like to get an overview of the factors that determine collision probability ...
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67 views

probability of sequence for given rate constants

lets consider a copolymer, $C_{r,s}^A$ containing r number of A monomers and s number of B monomers with A at the reactive end of the polymer. The equilibrium constant for A-A, A-B, B-A, and B-B bonds ...
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113 views

Probability and the propagator

Due to the Wiki article, "...In quantum mechanics and quantum field theory, the propagator gives the probability amplitude for a particle to travel from one place to another in a given time, or to ...
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975 views

tritium beta decay - probability of being in 1s state

Hydrogen-like wavefunctions have the form: $$R_{10} = \left(\frac{Z}{a_0}\right)^{\frac{3}{2}} 2\space e^{-\frac{Zr}{a_0}}$$ $$Y_{00} = \frac{1}{\sqrt {4\pi}} $$ where $a_0 = \frac{4\pi \epsilon_0 ...
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141 views

Quantum Rigid Rotor Perturbation

As the title says, I have a rigid rotor with a perturbation given below $$H=\frac{L^2}{2I}-\alpha B L_z.$$ So I know that the eigenvalues of $H$ will be $\ell(\ell+1)/2I -\alpha B m$ where $m$ is our ...
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135 views

Ising model. What is large fluctuations of magnetization?

My background is in mathematics. I have studied the Ising model in $\mathbb{Z}^2$. The main model of statistical mechanics. Yesterday, I was reading the preliminary notes of the book Statistical ...
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87 views

Equivalence of simple formulations of qubit entanglement

I'm reading some very elementary treatments of quantum computation and am unsure about the correspondence among "definitions" of qubit entanglement. One definition states that (1) the bits of a ...
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0answers
58 views

Modeling the probability of a photodiode measuring photons targets at a neighbor

In current digital cameras, sensors are arrays of photodiodes which "transform" photons energy to electrons. I am aware that the probability of a photon to generate an electron is modeled by a Poisson ...
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0answers
20 views

Microstates, Distribution of Particles, and the Probability of an Empty Compartment

If I have a closed system composed of $N$ particles and $p$ compartments, the total number of microstates available to that system is $$ p^N $$ Now say I want to find the probability that any one of ...
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20 views

how to use generating function to solve coupled linear master equations?

I am trying to solve a two dimensional continuous time and discrete state master equation. The master equation is linear and looks as follows, $\frac{\partial P_A(x,y,t)}{\partial t} = k_{11} ...
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0answers
17 views

Eigenvalues for correlation matrix which have the form of an harmonic function

I am trying to understand the written in the picture below. I took the matrix $C_{2 \times 2}$ which is: $$C=\left[ \begin{array}{} a& ace^{-\frac{|\phi_1-\phi_2|}{2}}\\ ...
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0answers
59 views

How does one intrepret probabilites in the many-worlds interpretation?

Let's say I flip a coin, and don't look. From the copenhagen interpretation, the state of the coin is: $\frac1{\sqrt2}(i|\text{heads} \rangle - |\text{tails}\rangle)$ If I observe the coin, there is ...
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0answers
35 views

What is the probabililty that a fair coin lands on its side?

This is a popular gag in movies, but I wonder how likely it really is. What is the probability that a uniform cylindrical coin (with radius $1$ and height $h$) lands on its side? If the ground were ...
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0answers
20 views

Entropic forces in Brownian motion

Reading Entropic forces in Brownian motion I'm having trouble to understand how the author makes a computation. He needs to calculate the number of ways a particle that is released from the origin can ...
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0answers
88 views

Problem in understanding Feynman's explanation of the Dirac-Delta function

This is quoted from Feynman's Lectures' Normalization of the states in $x$: We return now to the discussion of the modifications of our basic equations which are required when we are dealing with ...
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0answers
39 views

How is the probability density function in equilibrium state made equal to a dirac delta function?

While studying statistical mechanics, I stumbled upon the introduction of the dirac delta function in defining the probability density of the microstates and hence the conclusion of equal priori ...
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22 views

Construct bivariate symmetric (polynomial) Hilbert-Schmidt two-qubit volume functions over the unit square with certain properties

Construct bivariate symmetric polynomials (two-qubit volume functions) f(r,R) = f(R,r) >= 0 over [0,1]^2, with f(1,R) = f(r,1)=0, such that the univariate marginal (integrating over r or R) ...
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0answers
19 views

Can we say a certain yes to having probability independence for some events?

Mathematically it is just a question of assumption of proof to say that $P(A|B)=P(A)$ if A is independent from B. However, in real life is it possible to assume $P(A|B)=P(A)$ instead of ...
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0answers
43 views

On the probability of the existence of a similar observable universe

Let's assume a standard ($\Lambda$CDM + some simple inflation model) cosmology in an infinite universe. Really it doesn't matter much what cosmology we take, just that we're considering an infinite ...
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34 views

Are the random variables in a delay embedded phase space uncorrelated and independent?

Consider a smooth manifold $M=R^d$ embedded in a higher dimensional space $R^D$ using Takens Attractor reconstruction where $D > 2d+1$. Let, the Random Variable $X \in R^d$ have a Gaussian pdf and ...
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0answers
198 views

Average kinetic energy of molecules hitting a surface

I am trying to prove that the average kinetic energy of gas molecules hitting a containers surface is $2k_{B}T$ instead of the average for the entire gas, which is $\frac{3}{2}k_{B}T$, where $k_{B}$ ...
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0answers
74 views

What is the cause for the validity of Statistical Regularity?

My book writes: From experience it has been observed that the value of frequency ratio gradually approaches a definite constant number when the no. of trial becomes larger & larger. This ...
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0answers
97 views

Quantum mechanics and probability

I've done an intro course on QM and I'm now hoping to understand exactly how to use probability theory rigorously in solving problems. My question is: How do I do the same thing, or the closest thing ...
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0answers
65 views

Notation for Propability Amplitudes

I've recently stumbled upon a certain piece of notation that doesn't quite seem clear to me. When discussing the amplituhedron, my teacher mentioned the relation between the volume and the amplitude ...
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0answers
75 views

What is the probability that all the air ends up in the upper right corner of the room and we suffocate

Since someone commented this on this question(What is the probability of ice in boiling water?), I would like to ask what is the probability that all the air ends up in the upper right corner of the ...
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0answers
136 views

How to use The Schwarzchild Metric formula to get distribution representing “free-fall”

Given formula: How I can use to calculate distribution of points in space, so if i choose path which contains most of the points I get path that close to "free-fall path". As far as I know i should ...