# statisical physics Definition and Topics - 33 Discussions

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1. ### A Cl's : sum into a chi^2 when we have a sum of chi^2

1) If I take as definition of ##a_{lm}## following a normal distribution with mean equal to zero and ##C_\ell=\langle a_{lm}^2 \rangle=\text{Var}(a_{lm})##, and if I have a sum of ##\chi^2##, can I write the 2 lines below (We use ##\stackrel{d}{=}## to denote equality in distribution)...
2. ### I Exceptions to the postulate of equal a priori probabilities?

Not sure if this is the appropriate forum for this, hopefully if it isn't someone can move it to a more appropriate place. The fundamental postulate of equal a priori probabilities in statistical physics asserts that all accessible microstates states in an ensemble happen with equal...
3. ### A Computing a variance in astrophysics context

Below the error on photometric galaxy clustering under the form of covariance : $$\Delta C_{i j}^{A B}(\ell)=\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}}\left[C_{i j}^{A B}(\ell)+N_{i j}^{A B}(\ell)\right]$$ where ##_{\text {sky }}## is the fraction of surveyed sky and ##A, B##...
4. ### I Understanding 2 equivalent formulations of both data set measures

I have two independant experiments have measured ##\tau_{1},\sigma_{1}## and ##\tau_{2},\sigma_{2}## with ##\sigma_{i}## representing errors on measures. From these two measures, assuming errors are gaussian, we want to get the estimation of Ï and its error (i.e with a combination of two...
5. ### Question in Bose-Einstein statistics

iam not getting why in bose statistics the number of ways to arrange ni particles in gi degenerate states is = (gi+ni-1) ? and why do we divide by ni factorial , and gi factorial .
6. ### A Quantum statistical canonical formalism to find ground state at T

For my own understanding, I am trying to computationally solve a simple spinless fermionic Hamiltonian in Quantum Canonical Ensemble formalism . The Hamiltonian is written in the second quantization as $$H = \sum_{i=1}^L c_{i+1}^\dagger c_i + h.c.$$ In the canonical formalism, the density...
7. ### Statistics physics problem -- atom is in the ground state or excited state?

I am learning for my exam in particle physics. One topic is statistical physics. There I ran into this question: Consider an atom at the surface of the Sun, where the temperature is 6000 K. The atom can exist in only 2 states. The ground state is an s state and the excited state at 1.25 eV is a...
8. ### I What is the relation between chemical potential and the number of particles?

Chemical potential is defined as the change in energy due to change in the number of particles in a system. Let we have a system which is defined by the following Hamiltonian: $$H = -t \sum_i^L c_i^\dagger c_{i+1} + V\sum_i^L n_i n_{i+1} -\mu \sum_i^L n_i$$ where ##c^\dagger (c)## are creation...
9. ### A A paradox for bosons with non-degenerate states?

The BE-distribution for the case of only one state per energy level (gi = 1) is ni = 1 / (exp(ui - μ)β - 1) This is a reasonable and well defined distribution as far as I can see. On the other hand the number of possibilities to realize a given distribution of bosons among k energy levels with...
10. ### Pressure in spin-1/2 fermions

Homework Statement I have to find the mean Energy $<E>$ and pressure of a system of N fermions with spin 1/2. The energy per particle is \varepsilon = \frac{p^2}{2m}. \endu{equation} Homework Equations The relevant equations are the degeneracy of the system: ...
11. ### I Deriving the Boltzmann distribution

I was reading the derivation of boltzmann distribution using the reservoir model. lets call the reservoir by index R and the tiny system by index A. In the derivation they proposed that the probability for being at energy e (for A) is proportional to the number of states in reservoir. I didn't...
12. ### Partition Function of N particles in an assymetrical box

Homework Statement Consider a gas sufficiently diluted containing N identical molecules of mass m in a box of dimensions Lx, Ly, Lz. Calculate the probability of finding the molecules in any of their quantum states. Calculate the energy of each quantum state εr, as a function of the quantum...
13. ### Classical Swarm Dynamics Introductory Book

I am searching for a good book that could be used as an introduction to the dynamics of swarms and maybe their self organization. I am a physics junior and would like something that is decent for that level.
14. ### Effusing gas onto the interior of an evacuated sphere

Homework Statement A gas effuses into a vacuum though a small hole of area A. Show that if the particles effused into an evacuated sphere and the particles condensed where they collided that there would be a uniform coating. (7.6 of Blundell and Blundell) Homework Equations Angular...
15. T

### Pressure caused by beam of molecular oxygen

Homework Statement A beam of molecular oxygen containing 1010 molecules/cm3 and average speed of 500 m/s strikes (elastic collision) a plate at an angle of 30º with the normal direction. Calculate the exerted pressure on the plate. Homework Equations P = Impluse x Flux The Attempt at a...
16. ### Number of individual states with the same occupation numbers

Homework Statement A state of a system of many noninteracting particles can be specified by listing which particle is in which of the accessible single particle states. In each microscopic state we can identify the number of particles in a given single particle state ##k##. This number is...
17. ### I [Stat Phy] What does exhausting the states of a system mean?

I was reading the *Statistical Physics An Introductory Course* by Daniel J.Amit and need some help to understand a certain passage: In an isolated composite system of two paramagnetic system: System a with ##N_a## spins and a magnetic field ##H_a ## System b with ##N_b## spins and a...
18. ### Average magnetic moment of the system

I was reading the statistical physics textbook and was really confused with the notation: I don't understand the last part of the section. Why is that \sum_{\sigma = \pm1} \sigma P(\sigma) equals to \left< \sigma \right>? And what does \left< \sigma \right> actually mean? Is it the average...
19. A

### Statistical Physics: Quantum ideal gas

Homework Statement I'm reading the book about Statistical Physics from W. Nolting, specifically the chapter about quantum gas. In the case of a classical ideal gas, we can get the state functions with the partition functions of the three ensembles (microcanonical, canonical and grand...
20. ### Classical Best Statistical Mechanics books for studying for qualifier?

Does any one have any good books, or other references, that they would recommend for studying for the thermodynamics & statistical mechanics portion of graduate qualifying exams? I didn't have any undergrad Stat Mech and my grad prof/class was really not good, to the point that I didn't really...
21. ### Question about studying statistical mechanics before or after MQ?

Hi i would like to understand if it is advisable to study statistical mechanics before of the MQ (with the classical stat. mec.), or after the MQ all together ?? Thank you
22. ### Entropy of a generalised two-state quantum system

Hi. This is the problem I'm trying to solve: A system may be in two quantum states with energies '0' and 'e'. The states' degenerescences are g1 and g2, respectively. Find the entropy S as a function of the Energy E in the limit where the number of particles N is very large. Analyse this...
23. ### A Classical gas with general dispersion relation

i'm trying to understand the solution to this problem: http://physweb.bgu.ac.il/COURSES/StatMechCohen/ExercisesPool/EXERCISES/ex_2065_sol_Y13.pdf (link to the problem and the solution of it) All my questions come from the partition function: 1) From where the term (2*pi)^d comes from?, I...
24. ### Expectation values as a phase space average of Wigner functions

Hi. I'm trying to prove that [\Omega] = \int dq \int dp \, \rho_{w}(q,p)\,\Omega_{w}(q,p) where \rho_{w}(q,p) = \frac{1}{2\pi\hbar} \int dy \, \langle q-\frac{y}{2}|\rho|q+\frac{y}{2}\rangle\,\exp(i\frac{py}{\hbar}) is the Wigner function, being \rho a density matrix. On the other hand...
25. ### Relation between the matrix elements of the density matrix

Hi. I must prove that, in general, the following relation is valid for the elements of a density matrix \rho_{ii}\rho_{jj} \geq |\rho_{ij}|^{2}. I did it for a 2x2 matrix. The density matrix is given by \rho = \left[ \begin{array}{cc} \rho_{11} & \rho_{12} \\ \rho^{\ast}_{12} & \rho_{22}...
26. ### I Decomposing a density matrix of a mixed ensemble

I'm trying to solve a problem where I am given a few matrices and asked to determine if they could be density matrices or not and if they are if they represent pure or mixed ensembles. In the case of mixed ensembles, I should find a decomposition in terms of a sum of pure ensembles. The matrix...
27. ### Range in solid surface for liquid-liquid phase separation

I am investigating the nucleation on solid surface during liquid-liquid phase separation, I know the free energy change is and I don't know if it is correct to get Can someone calculate it to have a check?
28. ### Temperature and volume fraction of a liquid mixture

I have known the interaction parameter of a certain liquid mixture which has the phase behaviour can be described by the lattice model. , is it possible for us to know the temperature at the critical point? Besides, if we know the temperature at the critical point, can the volume fraction of...
29. ### Isotropic material fitted by Ornstein-Zernike form

I have known what Ornstein-Zernike equation is. I try to plug in the form as follow to the isotropic materials: Still, I cannot show the pair correlation function as follow. Can anyone know what I have missed?
30. ### Derive approximate expression by regular solution theory

The question is about to derive an approximate expression by regular solution theory, It is difficult for me to find relevant source on this question. However, the question to me is so vague that I do not know how to answer. What I have tried is to search what the interaction parameter is...