Partition Definition and 307 Threads

Homework Statement Please see P2 in http://panda.unm.edu/pandaweb/graduate/prelims/SM_S09.pdf "Starting with \mathbb{Z} (z_1,z_2) above, derive expressions for the gas pressure..." Homework Equations The Attempt at a Solution To find the pressure at the top and the bottom of...

Homework Statement Compute the partition function Z = Tr(Exp(-βH)) and then the average number of particles in a quantum state <nα > for an assembly of identical simple harmonic oscillators. The Hamiltonian is: H = \sum _{k}[(nk+1/2)\hbar - \mu nk] with nk=ak+ak. Do the calculations once...

Homework Statement Let f: R -> R, x -> x^2 What does the partition for the equivalence relation of this function look like? Homework Equations The Attempt at a Solution Uh...I have no idea. Sorry, the book only has examples of like integers from modulo n, if anybody could...

And why are the partitions not equal to one value? Why x1, x2, ... , xk, ... , xn-1, xn ? And why |the norm| -> 0 ? I was just curious if there is some specific logic behind it or if it is just there to discuss things in general. Thanks a lot. P.S.: Norm is the partition having the...

Hello all, I have some trouble understanding the partition function. In wikipedia it is written that the partition function needs to be calculated with the multiplicity of the states: z=SUM[g(E)exp(-BE)] where g(E) is the multiplicity of the states corresponding to energy E. It is...

The example which I'll use to illustrate my problem is not a homework question but something I've found in a book and already know the answer to. The grand partition function, G, is defined as SUM(over i)[exp(-B(Ei-yNi))] where B=1/kT, y is the chemical potential and Ei is the energy of the...

i need to show that the average value of the energy is -(1/Z)(dZ/dBeta)= -(d/dBeta)Ln(Z) where Z is the partition function i know how to do the first part, i don't know how to show this is equal to the derivative w/ respect to beta of lnZ. i think my math is wrong when taking Ln(Z) Beta =...

Homework Statement N sites 3 possible situations: empty with energy = 0, occupied by A with energy = E1, occupied by B with energy = E2. fugacities: for A = 10^-5, for B = 10^-7 T = 37 C 1) if no B find E1 such that 90% of sites occupied by A 2) with B find E2 such that 10% of sites...

Homework Statement The problem is simple: two compartments, allowed to exchange heat with environment (canonical ensemble) are allowed to mix. Show change in U,P and S.Homework Equations Z_{total} = \frac{1}{N!} Z_{1}^{N} Z_{1} = e^{-\beta E_{j}} The Attempt at a Solution I know how to...

Homework Statement QN1. A system consists of N weakly interacting subsystems. Each subsystem possesses only two energy levels E1and E2, each of them non-degenerate. Obtain an exact expression for the heat capacity of the system. QN2. A system possesses three energy levels E1 =E , E2 =...

The guiding premise of this thread is the following proposition: If fractals play a role in the behavior of partitions, then maybe, just maybe, they play a role also in the positioning of the primes; and if they do, then who is to say that the two, prime numbers and partition numbers, cannot at...

Hi, has anyone else seen this news item http://www.physorg.com/news/2011-01-math-theories-reveal-nature.html on how to crack partition numbers using fractals? It came out on Thursday the 20th Jan 2011. They gave a tantalising glimpse and said the full new theory will be revealed on Fri 21 st...

Homework Statement I'm looking for a closed form expression for the partition function Z using the Canonical Ensemble Homework Equations epsilon_j - epsilon_j-1 = delta e Z = Sum notation(j=0...N) e^(-beta*j*delta e) beta = 1/(k_B*T) t = (k_B*T)/delta e N is the number of excited...

Does anyone recognize this expression for the pressure: p(T,\mu) = T s^*(T,\mu) where s^* is the extreme right singularity in the Laplace transform of the grand canonical partion function. If someone knows this, I am curious in the derivation, and in what cases it is applicable. (In the...

Hi all, Firstly, I'm not sure where to post this thread, but I'm hoping here is the right place. My questions developed through reading Verlinde's paper on entropic gravity: http://arxiv.org/abs/1001.0785" However my questions are with the introductory thermodynamic ideas he presents...

Homework Statement Let S(n,r) denote the number of elements of A_n of rank r. Then S(n,r) satisfies the recursion S(n,r)=(n-r)S(n-1,r-1) + S(n-1,r) Verify this formula for n=4 and r=0,1,2,3,4, using the values S(3,0) = 1 S(3,1)=3 S(3,2)=1 Homework Equations The Attempt at a...

Homework Statement Calculate the grand partition function for a system of N noninteracting quantum mechanical oscillators, all of which have the same natural frequency \omega_0. Do this for the following cases: (i) Boltzmann statistics; (ii) Bose statistics. Homework Equations The...

Homework Statement Consider a system of N noninteracting particles in a container of cross-sectional area A. Bottom of the container is rigid. The top consists of an airtight, frictionless piston of mass M. Neglect the potential energy of the molecules of gas. Construct the partition...

Homework Statement The partition function of a given gas can be written z=(\frac{V-Nb}{N})^{N}(\frac{mk_{B}T}{2\pi\hbar^2})^{\frac{3N}{2}}e^{{\frac{N^2a^2}{Vk_{B}T}} Homework Equations lnz= Nln(\frac{V-Nb}{N})+\frac{3N}{2}ln(\frac{mk_{B}T}{2\pi\hbar^2})+\frac{N^2a^2}{Vk_{B}T}}...

Hi! I am for the moment reading a course in statistical physics where the author has definied not less then three diffrent partitionfunctions. W, Z an Z which are called the microcanonical partitionfunction, canonical partitionfunction (?) and the grand canonical partitionfunction. I...

Homework Statement We had a lecture about partition function, canonical ensemble etc. Can someone explain to me how this work out this formula Homework Equations we are supposed to find the mean energy and preasure of a gas with given partition function The Attempt at a Solution...

Homework Statement Having a unidemsional array of N oscillators with same frequency w and with an anharmonic factor ax^4 where 0 < a << 1 Calculate, up to the first order of a, the partition function. Homework Equations For one oscillator...

If we have a partition \mathcal{P}=\{A_1,A_2\} of some set A, then we can talk about the sigma-algebra generated by this partition as \Sigma=\{\emptyset, A_1,A_2,A\}. How can I define this concept more generally? Here is what I have: A partition \mathcal{P} of some set A generates the...

Homework Statement In part a) to this question I calculated the partition function which is Z = 1 + 3/e + 5/e^2 Homework Equations I can't find an equation relating U to Z. The Attempt at a Solution If someone has an explanation or a link to an equation that would be great...

1. I can't seem to get the same answer my textbook does, basically I need to calculate E (average energy) from the Partition function (Z) which is defined as: E=(-1/Z)*(dZ/dBeta) Where Z=(1/1-exp(-Beta*h*f)) (where h and f are constants and beta=1/kT for simplicity) So for my...

Hi, I've got this homework problem on my statistical physics module and I'm really unsure about it as this stuff is all new to me. I have an "atomically flat" solid substrate in contact with a gas of molecular mass m, and the two are in thermal equilibrium. The substrate has a total of M sites...

I have a really conceptual question on vibrational partition function for a diatomic molecule.If we consider a diatomic molecule, we write : Energy of simple harmonic oscillator=E_{i}=(n + 1/2) h\nu.We plug this eqn. into Z_{vib}=\sum e^{-\beta\epsilon_{i}}. Now , my question, is that the...

So I've been wrestling with something I was reading in a stat mech text. It's the derivation of the partition function for an ideal gas but I imagine the technique is used again. The author starts with the partition function for a single particle but then approximates the sum as an integral...

Hello. I come across a problem: how to calculate the partition function of a classical spring whose energy is 1/2kx^2, and use thermodynamics to show that the force on the spring is linearly proportional to its elongation x? I got stuck at the first step. What is the energy of the spring, is...

Basic question: meaning of "partition of R into maximal connected intervals" What does the phrase "partition of R into maximal connected intervals" mean? The full sentence: "Let I_1, I_2, ... ,I_m be the partition of R into maximal connected intervals with disjoint interiors."

Homework Statement I. Finding the partition function Z. II. If the middle level (only) is degenerate, i.e. there are two states with the same energy, show that the partition function is: Z = (1+exp(\frac{-\epsilon}{k_{B}T}))^{2} III. State the Helmholtz free energy F of the...

I am wondering how one would construct the grand partition function of a composite system of solid and gas with the same chemical potential energy. I would think to begin with the partition function for a single particle and sum over it's energy states (available in the solid and the gas)...

Homework Statement what is the weight percent concentration of SO2 in the exsolved aqueous fluid phase that existed with solution if sulphur has ppm by weight of 90. partition coefficient= 47 Homework Equations partition coefficient- 47= concentration of aqueous fluid/ concentration of...

In looking at phonons, and their energy, I came across the Partition function. THis was needed to calculate the internal energy of the solid. But howcome the Partition function is used, and not the GRAND Partition function? The number of phonons is not conserved, I know that, but isn't N, the...

I recently installed ubuntu netbook remix on my newly bought netbook to try it out. I created a new partition on my HDD from the free disk space to install ubuntu on. So right now I have dual boot ubuntu and windows xp. The ubuntu netbook remix turned out to be more buggy than I thought and I'm...

Hello everyone, I have passed my integral calculus class and it's been a little while so I don't really remember everything. Can anyone help me out with this? Homework Statement Let f(x) = sqrt(x), x E [0,1] and P=\left \{ 0,\left ( \frac{1}{n} \right )^{2}, \left ( \frac{2}{n} \right...

Homework Statement This is just a general question, not a "problem" Homework Equations Z = sum(e^Ej/kT) The Attempt at a Solution I'm working on a problem in which I'm asked to find the probabilities of an electron in a hydrogen atom being at one of three energies. The...

Hello I am working through a textbook here, struggling to follow a mathematical step. We are deriving the partition function Q due to pure rotation of a system containing molecules with quantum rotation energy levels: E = h2J(J+1) / 8pi2I Where J is the rotational quantum number, J = 0,1,2...

Hi, I've been trying to find out the ways in which an odd integral number can be represented by smaller integers such that the even integers occur an even number of times. So for example, the number 15 has the following representations: 15: 2+2+2+2+2+2+3 (even integer 2 occurs even number...

Homework Statement For n>14 such that n is an integer, prove that a square can be partitioned into n smaller squares... Homework Equations None... The Attempt at a Solution I was thinking this would be somewhat of an induction proof because we are working our way up to n. So far...

Question is on attachment file.

I am reading a book (Di Francesco's "CFT", pg 337) in which it is given that if we take the operator that translates the system along some direction (which is a combination of time and space) as 'A', then the partition function is just trace(A). How do we get this?

Homework Statement Calculate fermi energy, fermi temp and fermi wave vector. a)Protons with n= 1.0E43 m^{-3} b) ^{3}He in liquid He (atomic volume= 46E^-3 m^3 Homework Equations E_f=\frac{h^2}{8 m} (\frac{3 n}{\pi V})^\frac{2}{3} T_f= \frac{E_f}{k_B} The Attempt at a Solution I get the...

Homework Statement By shining and intense laser beam on to a semiconductor, one can create a collection of electrons (charge -e, and effective mass me) and holes (charge +e, and effective mass mh) in the bulk. The oppositely charged particle may pair up (as in a hydrogen atom) to form a gas of...

Physical interpretation of the partition function. Consider a single-particle quantum system whose states are labeled with an index i = 1, 2, 3, ... with corresponding energies E1, E2, E3, ... . Set the zero of energy at the ground state energy so that E1 = 0. Argue that, if the absolute...

Does there exist a database for the Partition function at different temperatures? As I understand it it only varies with temperature and is otherwise the same for the species. I am looking for the partition function for C02 at T=1000K.

Hi All - If I have a potential energy surface with two energy states, one higher than the other, where I can make the assumption that both potential wells can be approximated via harmonic potentials with force constants kA and kB then would the partition function for the system be...

Homework Statement Consider a system of N identical particles. Each particle has two energy levels: a ground state with energy 0, and an upper level with energy epsilon . The upper level is four-fold degenerate (i.e., there are four excited states with the same energy epsilon ). (a)...

Homework Statement For a mole of nitrogen (N2) gas at room temperature and atmospheric pressure, compute the following: U, H, F, G, S and μ. The internal partition function is purely rotational, and the rotational constant ε for N2 is 0.00025 eV. The electronic ground state is not...

Homework Statement Partition the following collection of groups into subcollections of isomorphic groups. a * superscript means all nonzero elements of the set. integers under addition S_{2} S_{6} integer_{6} integer_{2} real^{*} under multiplication real^{+} under multiplication...