Questions tagged [probability]

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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Renormalisation and the Fisher-Rao metric

The renormalisation group (I'm talking about classical, statistical physics here, I'm not familiar with field theory too much) can be thought of as a flux in a space of possible Hamiltonians for a ...
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Simulating the spatial distribution of water droplets from a dripping tap

I saw this pattern under a leaky tap. (Recreated images) The pattern was interesting because it looked like a probability distribution. Bigger droplets lie in the centre, and smaller ones scattered ...
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9 votes
1 answer
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Which position and momentum distributions arise from some wave function?

Consider a particle in one dimension with wave function $\psi$. The probability density function describing how likely it is to find it in a given position is given by $f(x)=\left|\psi(x)\right|^2$. ...
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6 votes
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What properties would a physical biased die have?

Imagine a physical biased six-side die. How much and what kind of bias we could possibly introduce by moving it's center of mass? What would be the exact mechanism describing the relation of center of ...
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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 ...
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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 ...
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5 votes
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"Synchronization" Probability of Multiple Waves with Varying Frequencies

Update 1: I've done some digging and I think this is related to signal coherence, namely, that I'm seeing a coherence time of ~3 σ, which is consistent with the definition where Ct=1/Δv where Δv is ...
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Ehrenfest theorem and correlation among observables at the quantum scale

I am studying quantum mechanics and I encountered the famous Ehrenfest Theorem, which states that given an observable $A$, its expectation value time evolution is governed by $\partial_t\langle A\...
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4 votes
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Fokker-Planck equation for 2D SDE

Consider the following two-dimensional SDE \begin{align*} \mathrm{d}\mathbf{X}(t) &= {\mathbf{f}(\mathbf{X}(t))}\mathrm{d}t+\mathrm{d}\mathbf{W}(t)\\ \end{align*} where $\mathbf{X}(t)=\begin{...
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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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4 votes
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172 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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4 votes
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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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Why do some factorized states have different probability than others in terms of Clebsch-Gordan coefficients?

When adding a spin-1 to a spin-1/2, we have a six-dimensional Hilbert space spanned by the factorized states $$ \left \{ \left | j_1=1, m_1 \right \rangle \otimes \left |j_2= \frac{1}{2}, m_2 \right \...
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3 votes
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Indistinguishability and different pure state decompositions of mixed states in non-simplex convex set of states in Quantum Statistics

In statistical physics (mechanics), the transition from Maxwell-Boltzmann statistics to Bose-Einstein and Fermi-Dirac statistics was motivated by classically inexplicable phenomena such as Bose-...
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Finding total flux of probability current through a sphere

For a wavefunction: $$\Psi(\textbf{x}) = e^{ikz} + \dfrac{f(\theta)}{r}e^{ikr}$$ Where $z = r\cos(\theta)$. The probability current $J$ is then given by: $$J(\textbf{x}) = J_1(\textbf{x}) + J_2(\...
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Is there an open quantum system analogue of the equilibration time bounds for classical ergodic Markov chains?

Background For classical ergodic discrete Markov chains, we can bound the time taken to reach the stationary distribution to the spectral properties of the transition matrix. I will outline this ...
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Reconstruct quantum state from probabilities

Given a quantum state, the Born rule lets us compute probabilities. Conversely, given probabilities, can we reconstruct the quantum state? I think the answer is almost trivially positive but how ...
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A proof for this equivalent version of the Infrared Bound/Gaussian Domination

Consider the Ising Model in the $d$-dimensional discrete torus with side lengh $L$, denoted by $\mathbb{T}_L $, with nearest neighbors interaction (with interaction parameter $J$, no magnetic field, ...
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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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Finding the probability that a $1s$ electron in ${}^3$H remains in the $1s$ state after beta decay to ${}^3$He with the sudden approximation

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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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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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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2 votes
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73 views

What is the probability that a random walk forms (almost) a circle?

Given is a random walk of a particle in 3d (such as an atom in a liquid). The particle proceeds randomly (in 3d), with an average straight displacement length a. Is there a way to get a probability ...
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2 votes
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Interpretation of probability in Statistical Mechanics

In statistical mechanics, in particular the canonical ensemble, the probability of the system to have a particular state is given by : $$P_i=\frac{e^{-\beta E}}{Z}$$ Here $Z$ is the partition function ...
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3 answers
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Probability Waves vs. Amplitude Waves

It is often asserted, and it is common knowledge, that the waves associated with a particle are probability waves. This seems reasonable. But what about $E=hf$? This does not seem to be about ...
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How to compute the standard deviation of a free fermionic random correlation matrix

I am reading this paper on free random fermions. That is, fermions governed by $$ H = \sum_{ij} t_{ij}c^{\dagger}_ic_j\quad \longrightarrow \quad H = \sum_k E_k d^{\dagger}_kd_k, $$ with $t_{ij}$ ...
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2 votes
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71 views

What is the field value of a quantized fermionic field?

I'm trying to make an analogy with the phonon field. While preparing this answer I've learned that for a chain of atom-like entities, we have a probability density of the phonon field configuration: ...
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1 answer
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Why do we ignore higher-order derivatives of entropy with energy in deriving the Boltzmann distribution?

I am taking my first course in statistical mechanics, one point that I don't really get is the justification for ignoring higher-order derivatives of entropy w.r.t energy. We began the course by ...
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2 votes
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81 views

Overlap of state subspaces in QM

In many-body QM we are rarely able to solve exactly for (some or all of) the eigenstates of the Hamiltonian. In some fields of condensed matter physics there has been a successful "business" ...
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2 votes
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Probability of two electrons of different energy levels contained in a single infinite potential well being found in same region

What is the probability of two electrons in a single infinite potential well centered at 0, one in the ground state, the other in the first excited state, being in the same region? I know by the Pauli ...
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2 votes
1 answer
69 views

Link between rotating wave approximation and stimulated emission and absorption?

In my lecture notes deriving the probability density of a two energy level atom we arive at the following equation: $$c_f(t) = \frac{1}{2} \Omega \left[\frac{1-e^{i(\omega + \omega_0)t}}{(\omega + \...
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Why does the correlation depend linearly on the angle, in the classical case of Bell experiments?

I am struggling to understand why we would predict the decay in correlation between the two measurements in a Bell Inequality experiment to be linear (see red line in this image from Wikipedia). I ...
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What does it mean to integrate with respect to matrices?

In Random matrix theory, the following definition of a partition function for an ensemble is common. $$Z=\int dM \exp [-N Tr(M^2)]$$ where $M$ is a Random matrix of dimension $N \times N$. In general,...
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2 votes
1 answer
67 views

Inequality for quantum probability

Let $H$ be a separable Hilbert space for a quantum mechanical system then $$w (x, y) = {{\langle y \mid x\rangle\langle x \mid y \rangle} \over \langle x \mid x \rangle\langle y \mid y \rangle}$$ is ...
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2 votes
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52 views

List of Replica Symmetry results for different models?

Does anyone know of a good source that might have a list of problems or models along with what kind of replica symmetry they are conjectured to have? I am aware of some of the more famous results, e....
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Calculating the probability of one particle being in a certain state in two-particle system

Let's say I have the two-particle state $$|\psi\rangle=\frac{|H\rangle_a|H\rangle_b+|V\rangle_a|V\rangle_b}{\sqrt{2}}$$ where $H$ is horizontally polarized and $V$ is vertically polarized. And I ...
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Can we write the Quantum Fidelity between two density operators in terms of Quasi-Probability Distributions: $P$, $Q$ and $W$?

Quantum Fidelity between two density operators, $\hat{\rho}$ and $\hat{\sigma}$, is given by $F(\hat{\rho},\hat{\sigma})=\left(Tr\sqrt{\sqrt{\hat{\rho}}\hat{\sigma}\sqrt{\hat{\rho}}}\right)^2$, where $...
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2 votes
1 answer
65 views

Probability of a system in the canonical ensemble

In the canonical ensemble, we have the state of system $x_s$ and the state of the environment $x_e$. The probability of the total system is $$P(x_s,x_e)= const.$$ and that is independent of the states ...
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Probability at temperature in system has energy

Salutations, I'm starting in statistical mechanics and reviewing some related studying cases I would like to understand what occurs in small systems with normal modes of vibration, for example, a ...
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Time and Quantum Mechanics

my question is about the nature of time across classical/macro and quantum scales. I understand that the 2nd law of thermodynamics and entropy increase has a lot to do with our understanding of time ...
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given an AC field, how many photons are there?

Say you are applying a time varying potential across double quantum dots in the form of $V_{ac}$cos($2\pi f t$). We know that each photon has an energy $E_{photon}= hf$. Is it correct to say that ...
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2 votes
1 answer
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Can local hidden variables models be ruled out without invoking expectation values?

Bell's inequalities, in their standard form, are a statement about the limitations faced by a probability distribution that can be written as $$p(a,b|x,y)=\sum_\lambda p(\lambda) p(a,b|x,y,\lambda)=\...
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2 votes
1 answer
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Particle ensemble performing shm, calculate amplitude pdf

Consider the shm for a single particle. Then the particle's position is given by (assume zero initial phase): $$x = a \times \sin(\omega t)$$ The infinitesimal probability of finding a particle ...
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Shift a Galton board's distribution by manipulating pegs

My question is related to this one but is simpler and more specific. It's also related to this this question which doesn't answer my question. In a standard Galton board, balls are dropped onto a ...
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Does Bell's theorem have anything to say about the locality or realism of Quantum Mechanics?

In the original paper written by Bell, it's clear to me that what he's really trying to answer is what class of Classical Theories (that obey the laws of classical probability theory) can replicate ...
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2 votes
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605 views

Probability density to find the end of a double pendulum

Given infinite time, is there a way to find the probability to find the end of a double pendulum at a given position? I am looking at a system with no damping (constant energy). The bobs are point ...
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2 votes
0 answers
541 views

Distribution of Photon Inter-arrival Times

In case of a coherent light source (eg. laser beam) the number of photons counted in a given time interval can be modeled as a Poisson distribution. From the theory of Poisson processes, it is well ...
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2 votes
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360 views

Examples of "Ordered state from disordered state"

A thought experiment - Imagine two containers joined to each other with a stopcock between them. One container initially has only two molecules of a gas and other is empty. Now, if we open the ...
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2 votes
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2k views

Physical interpretation of a complex potential for a particle in quantum mechanics

In Griffiths' Quantum Mechanics, it is mentioned in a problem that For an unstable particle that spontaneously disintegrates with a lifetime $\tau$, the total probability of finding the particle ...
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2 votes
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133 views

Interpretation of Characteristic function of probability density function for a classical system

Can the characteristic function ( fourier transform ) of a probability density function in position space for a classical system be related to its momentum space probability density function in any ...
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