Partition Definition and 307 Threads

I am going through Spivak's Calculus on manifolds. I am on the chapter now regarding partitions of unity. I understand the construction of it, but why exactly is a partition of unity useful? Why do we care about it?

Hello! (Wave) The following pseudocode is given: partition(A,p,r){ x<-A[r] i<-p-1 for j<-p to r-1 if A[j]<=x then i<-i+1 swap(A[i],A[j]) swap(A[i+1],A[r]) return i+1 I have to describe the function of partition at the array: $A= \langle 13, 19, 9, 5, 12...

Hi all This is probably a naïve question to ask, but I am puzzled by it and need an answer. The first time I encountered the term 'partition function' that was in context of Boltzmann distribution. But the same formulas of manipulating a partition function ( to obtain free energy, temperature...

Given an element a in a group G, class(a) = {all x in G such that there exists a g in G such that gxg^(-1) = a} class(b) = {all x in G such that there exists a g in G such that gxg^(-1) = b} so let's say y is a conjugate of both a and b, so it is in both class(a) and class(b), does that mean...

Hi all, I am struggling to grasp the sense of the partition function. First of all, I had a look at a couple of derivations (which the relevant Wikipedia page follows) in which the concept of heat"energy of a thermal bath" is invoked. Well this is already confusing me: if the thermal bath has an...

Homework Statement An ideal monatomic gas at the temperature T is confined in a spherical container of radius R. There are N molecules of mass m in the gas. Molecules move in a spherically symmetric potential V(r) where r is the distance from the center of the container. The potential V(r) is...

Hi. 1) Does the Grand canonical partition function ##\Xi## behave in a similar fashion to the good old canonical partition function ##Z##? Do you calculate thermodynamical quantities (entropy, hemholtz free energy etc.) in similar fashions? 2) Is it possible for some kind soul here to...

Hi I am struggling immensely with understand some aspects of chemical thermodynamics: 1) Let's say I have a solid with N atoms and am examining the ionization of individual atoms, and I am supposed to think of the electrons as ideal gasses. Or, 2) a solid or liquid is in thermal equilibrium...

First off, I'm glad I'm finally a member on this board. It has helped me TREMENDOUSLY over the past few years with various problems I've had. You guys/gals are awesome and hopefully I can make some contributions to this site. 1. Homework Statement A. Write down the partition function for...

Homework Statement Consider two particles, each can be in one of three quantum states, 0, e and 3e, and are at temperature T. Find the partition function if they obey Fermi Dirac and Bose Einstein statistics. Homework EquationsThe Attempt at a Solution I have obtained solutions to both : FD...

So I'm trying to use the excitation law of Boltzmann and the ionisation law of Saha to calculate stuff about what percentage of a quantity of hydrogen in ionised and in what energy state. I have the temperature and energy levels values, so i still need the: -statistical weight (degeneracy)...

What is the energy partition in the laser-induced plasma? Based on the atomic emission spectra from the plasma, we can infer the electron temperature and electron density from Boltzmann-plot and Stark broadening, I wonder how can I calculate the total energy in this plasma?

Hi guys, I'm studying a classical ideal gas trapped in a one-dimensional harmonic potential and I first want to write out the partition function for a single particle. This, I believe, requires two Gaussian integrations, like so: Z=\int_{-\infty}^{\infty} d\dot{x}...

Definition/Summary Let A be a non-empty set. A collection P of non-empty subsets of A is called a partition of A if 1) For every S,T\in P we have S\cap T=\emptyset. 2) The union of all elements of P is A We say a partition P is a collection because it is a "set of sets". The elements...

My Calculus tool is coming along. The only thing left is to write some it's helper functions, such as the one described below: void CalculusWizard::partitionEquation(const std::string & eq, std::string & eq1, std::string & eq2, CalcWizConsts::eqOps & oper) { /* Given an equation eq, partion...

In the semi-classical treatment of the ideal gas, we write the partition function for the system as $$Z = \frac{Z(1)^N}{N!}$$ where ##Z(1)## is the single particle partition function and ##N## is the number of particles. It is semi-classical in the sense that we consider the...

I have a partition function in euclidean quantum field theory. I have a parameter, let's say a charge, that I can change in the action that define the partition function. I found that for small charge the partition function is positive, but there is a critical charge, above the one the...

Homework Statement Consider a system at 290 K for which the energy of the nth state is En=n(.01 eV). Using 1+a+a2+a3...=1/(1-a) for a <1 where necessary, Find the value of the partition function. Homework Equations The Attempt at a Solution Z=∑e-βE_{}n β=1/κT I don't know...

Greetings, I have been studying stat mech lately, and while I have gotten good at using partition functions to solve problems, I wanted to check my interpretation of what a partition function is, and especially to contrast it with the number of states. So, I'm just looking for a yes or no to...

Homework Statement A molecule has 4 states of energy -1, 0,0 and 1. Find its partition function and limit of energy as T → ∞. Homework Equations The Attempt at a Solution Z = \sum_r e^{-\beta E} = e^{-\beta} + 2 + e^{\beta} U = -\frac{\partial ln(Z)}{\partial \beta} = \frac{e^{-\beta} -...

Can someone take a look at picture and show me how to derive the pressure from the grand partition function ?

Homework Statement If ##Z## is homogeneous function with property ##Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)## and you calculate Z(T,V,N). Could you calculate directly ##Z(\alpha T,\alpha^{-\frac{3}{\nu}}V,N)##. Homework Equations ##Z(T,V,N)=\frac{1}{h^{3N}N!}(2\pi m k...

It seems like they have missed out a factor of ##(2S+1)## in the final expression for grand potential? I'm thinking it should be ##(2S+1)^2## instead.

Hey! :) Let $f:[a,b] \to \mathbb{R}$ bounded.We suppose that f is continuous at each point of $[a,b]$,except from $c$.Prove that $f$ is integrable. We suppose that $c=a$. $f$ is bounded,so $\exists M>0$ such that $|f(x)|\leq M \forall x$ Let $\epsilon'>0$.We pick now a $x_0 \in (a,b)$ such...

Is there a way to derive entropy or free energy without using partition function?

Homework Statement Hey guys, Here's the question. For a distinguishable set of particles, given that the single particle partition function is Z_{1}=f(T) and the N-particle partition function is related to the single particle partition function by Z_{N}=(Z_{1})^{N} find the following...

Hello! I'm just beginning some volunteer research work on a project and I've been tasked with finding the partition function of a brain tumor that has magnetic nanoparticles injected into it. CT scans provide concentrations and distributions of the magnetic nanoparticles, but I'm quite lost...

Homework Statement The pressure of a non-interacting, indistinguishable system of N particles can be derived from the canonical partition function P = k_BT\frac{∂lnQ}{∂V} Verify that this equation reduces to the ideal gas law. The Attempt at a Solution I have a very poor...

Homework Statement For three systems A, B, and C it is approximately true that Z_{ABC}=Z_{A}Z_{B}Z_{C}. Prove this and specify under what conditions this is expected to hold. Homework Equations Z is the partition function given by Z=∑e^{-ε/KT} ε is energy, T is temperature and K is...

"A partition of a set X is a set of nonempty subsets of X such that every element x in X is in exactly one of these subsets." I was just wondering why the subsets must be nonempty. Is it just convention/convenient or is it because it would violate something else? Thanks!

in the paper written by Jose Torres-Hernandez in 1984 titled as : "Photon mass and blackbody radiation" in the first page he writes for the partition function: lnZ=λ \sum_{normal modes} e^{-βε_l} = \frac{-λπ}{2} \int_{ε_0}^∞ n^2 ln(1-e^{-βε}) \frac{dn}{dε}dε i really don't understand...

How to calculate <n_i ^2> for an ideal gas by the grand partition function (<n_i> is the occupation number)? In other words, I like to know how do we get to the formula <n_i>=-1/\beta (\frac{\partial q}{\partial\epsilon}) and <n_i ^2>=1/Z_G [-(1/\beta \frac{\partial }{\partial\epsilon})^2 Z_G]...

Why when the particles are nonlocalized, the single particle partition function is directly proportional to V, namely the volume of the system, and when the particles are localized, the single particle partition function is independent of V? (Pathria, Statistical Mechanics, chapter 4, section...

Hi, Let, $$\hat{H} = a\hat{S_x} + b\hat{S_z}$$where Sx,Sz are the spin operators, a,b constants. Assume the system is coupled to a reservoir. For clarity, Let $$\hbar=\beta=1$$ The density matrix is $$ρ=\frac{e^{-β\hat{H}}}{Z}= \frac{1}{Z}...

is the Boltzmann factor the probability of a particular state of a system? can someone explain the partition function to me (qualitatively please!), we've been using it in class and i don't get it. we derived what the partition function was for a general system. usually when learning physics...

Homework Statement I was given a Hamiltonian H = -\muB\sumcos\alpha_{i} where the sum is over i from i = 1 to i = N I need the partition function given this Hamiltonian. Homework Equations The Attempt at a Solution I tried using the classical approach where Z_{N} =...

Given [a,b] a bounded interval, and f \in L^{p} ([a,b]) 1 < p < \infty, we define: F(x) = \displaystyle \int_{a}^{x} f(t) dt, x \in [a,b] Prove that exists K \in R such that for every partition: a_{0} = x_{0} < x_{1} < ... < x_{n} = b : \displaystyle \sum_{i=0}^{n-1} \frac{| F(x_{i+1}) -...

Homework Statement Consider a solid of N localized, non-interacting molecules, each of which has three quantum states with energies 0, ε, ε, where ε > 0 is a function of volume. Question: Find the internal energy, Helmholtz free energy, and entropy. Homework Equations Z =...

Hi everyone. Suppose I have an Hamiltonian which doesn't depend on the position (think for example to the free-particle one H=p^2/2m). I know that the classical partition function for the canonical ensemble is given by: $$ Z(\beta)=\int{dpdq e^{-\beta H(p,q)}}. $$ What does it happen to...

Hi, If I have a body which is freely vibrating with kinetic energy given by, say, Ekin=(1-1/10sin(ωt))cos2(ωt) what can be said about the potential energy? Of course the total energy should be constant but how big is it, in other words what is the potential energy? Thanks!

Homework Statement Polymers, such as rubberbands, are made of very long molecules, usually tangled up in a configuration that has lots of entropy. As a very crude example of a rubber band, consider a chain of N molecules which we call links, each of length l. Imagine that each link has only...

I am curious, are there any calculators that calculate integer partitions with the stipulation that the calculator only calculate unique partitions. For example, if I want to calculate the number of ways to sum to 5 using 3 integers I have the following unique sums: 1 1 3 1 2 2 I would...

This is spivak's calculus 2nd edition #12b. The question in part a defines Pn as a partition of [a,b] into n equal intervals. The question in part b states: Find an integrable function f on [0,1] and an ε > 0 such that U(f,Pn)-∫0 →1 (f) > ε for all n. I'm made no progress on this at all. Part...

Does anyone know of any REALLY good derivations of the grand canonical partition function(T,V,μ) from the hamiltonian. I am using the graduate level thermodynamics book by tester and there appears to be some algebric manipulation that occurs going from the ensemble to the partition function...

This looks like a significant step forward. The paper is clearly written and gives a brief historical account of progress in spin foams over the past half-dozen years or so: an understandable review that places its results in context. The authors, Hellmann and Kaminski, discover a problem with...

Does anyone know what a D-domain is and a D-domain partition?

Does the energy distribution of one particle also follow the Boltzmann distribution. I.e. can you get the energy distribution for a single particle by calculating its partition function and writing: P(E) = exp(-E/T)/Z

Remember that the number of partitions of n into parts not exceeding m is equal to the number of partitions of n into m or fewer parts? Does anyone know much about the number of partitions of n into m parts not exceeding m? Thanks! Edit: Or more generally, the number of partitions of n into j...

The 'partition function' in QFT is written as Z=\langle 0 | e^{-i\hat H T} |0\rangle, but I'm having a difficult time really understanding this. I'm assuming that |0\rangle represents the vacuum state with no particles present. If that's the case, and the Hamiltonian acting on such a state would...

In classical statistical physics we have the partition function: Z=Ʃexp(-βEi) But my book says you can approximate this with an integral over phase space: Z=1/(ΔxΔp)3 ∫d3rd3p exp(-βE(r,t)) I agree that x and p are continuous variables. But who says that we are allowed to make this...