Partition Definition and 265 Threads

Hi, I have a question about the partition function. It is defined as ## Z = \sum_{i} e^{-\beta \epsilon_{i}} ## where ##\epsilon_i## denotes the amount of energy transferred from the large system to the small system. By using the formula for the Shannon-entropy ##S = - k \sum_i P_i \log...

Homework Statement Consider a two dimensional surface on a three dimensional crystal. This surface has M positions that can adsorb particles, each of which can bind one particle only and an adsorption does not affect the adsorption on nearby sites. An adsorbed particle has energy ε and an...

Hi there, I am reading an article, but I faced the following problem, and I am wondering if it is well known fact. If the integral of a function on some interval is infinity, can we partition this interval into countable disjoint (in their interiors) subintervals such that the integral...

I'm not sure if this is the right place to ask.. Anyway. Assume we have some integral I with 0 and 2 as limits. I = 3∫xexdx from 0 to 2. What exactly do we have to do to find the partition points (and what are they?) but using the composite trapezoidal rule? I = 25.1671683 upon computing...

Homework Statement Imagine a system with N distinguishable particles. Each particle may be in two states of energy: -ε and +ε. Find the the partition function of the system Homework Equations The Attempt at a Solution I know that I have to find the partition function for a...

Homework Statement Are the following subsets partitions of the set of integers? The set of integers divisible by 4, the set of integers equivalent to 1 mod 4, 2 mod 4, and 3 mod 4. Homework Equations The Attempt at a Solution Yes, it is a partition of the set of integers...

Is there a physical process in thermodynamics that results the value of the partition function as zero? When partition function is zero, then free energy becomes infinity, and it also yields negative entropy (at least within the system). Are there physical meanings of these?

I have a moderately sized music collection of 76GB which I would like to access from both ubuntu and windows. What is the best way to do this? My thoughts were... have ubuntu on one partition, windows on a second, and stick shared stuff in a third one. Is there software for both ubuntu and...

\int^{b}_{a}f(x)dx = \int^{c}_{a}f(x)dx + \int^{b}_{c}f(x)dx If one of the integrals on the right-hand-side is known to diverge, must the integral on the left also necessarily diverge? BiP

Hello guys, I work on a final project study discussed the use of optimization methods. The project in question consists in partitioning a grid dimensions NXM grids in dimensions 3 X 3 (which share no box) to the extent possible, otherwise find grids of dimensions 3 x 3 which are dependent...

sorry if i posted this topic here..let P1 and P2 be a path partition of a graph.is it possible that P1 and P2 to have the same end vertices?

I've derived Z for the quantum harmonic oscillator and was wondering if anyone could verify I did everything correctly. I don't have any experience working with exponential traces so I want to make sure I'm using them correctly. Z is defined as \mathcal{Z}= tr(e^{-\beta H}). So the natural...

The grand partition function Z of a system is given by formula: Z = Ʃ exp ((-Ei/KbT) + (μni/KbT)) where , 1, 2... i E i= are permitted energy levels, μ is the chemical potential, , 1,2... i n i= are number of particles of different types. Taking into account that averaged internal...

I've been working on this Linear Algebra problem for a while: Let F be a field, V a vector space over F with basis \mathcal{B}=\{b_i\mid i\in I\}. Let S be a subspace of V, and let \{B_1, \dotsc, B_k\} be a partition of \mathcal{B}. Suppose that S\cap \langle B_i\rangle\neq \{0\} for all i...

Homework Statement For a system A consists of two parts A' and A'' which interact only weakly with each other, if the states of A' and A'' are labeled respectively by r and s, then a state of A can be specified by the pair of numbers r,s and its corresponding energy E_{rs} is simply...

The probability of finding the system in microscopic state i is: p_{i}=\dfrac{1}{Q}e^{-\beta E_{i}} Where Q is the partition function. Assumption: molecule n occupies the i_{n}th molecular state (every molecule is a system). The total energy becomes...

Q: Suppose that the oscillation ω_f (x) of a function f is smaller than η at each point x of an interval [c,d]. Show that there must be a partition π of [c,d] s.t. the oscillation ωf([x_(k-1),x_k ])<η on each member of the partition. My solution (Rough sketch): This condition on x is...

If I have a totally ordered set and then create a noncrossing partition of that set it seems intuitively obvious that each block of the partition would be totally ordered as well. Can I assume this inheritance or do I need to prove each block is totally ordered? How would one go about proving...

Homework Statement I would like to calculate the grand canonical partition function (GCPF) for a system in which there are are m lattice sites. A configuration may be specified by the numbers (n_1, n_2, ... , n_m), where n_k = 1 if a particle occupies site k and n_k = 0 if no particle occupies...

For the given set A, determine whether P is a partition of A. A= ℝ, P=(-∞,-1)\cup[-1,1]\cup(1,∞) Is it correct to say that P is not partition? I don't understand why. Thank you

For the given set A, determine whether P is a partition of A. A= {1,2,3,4,5,6,7}, P={{1,3},{5,6},{2,4},{7}} Is it correct to say that P is a partition of A? Thank you

For the given set A, determine whether P is a partition of A. A= {1,2,3,4}, P={{1,2},{2,3},{3,4}} Is it correct to say that P is a partition of A? Thank you

Homework Statement A atom had a threefold degenerate ground level, a non degenerate electronically excited level at 3500 cm^-1(setting the energy orgin as the ground electronic state energy of the atom ) and a threefold degenerate level at 4700 cm^-1 . Calculate the electronic partition...

I have some confusions identifying the following objects: (1)Some transition amplitude involving time evolution(Peskin page 281, eqn 9.14): \langle\phi_b(\mathbf x)|e^{-iHT}|\phi_a(\mathbf x)\rangle=\int{\cal D\phi \;exp[i\int d^4x\cal L]} (2)Partition function(after wick rotation)...

Homework Statement Assume that it is appropriate to transfer the probabilities IP(F|L) and IP(F|T) from the police context to the insurance context. Define the following new events for the insurance context: L = “insurance claimant is lying”; T = “insurance claimant is truthful”; F =...

hi folks, I want to calculate the potential energy part of the partition function of 2 particles interacting via the Lennard-Jones potential. This partition function should be proportional to: \int_0^\infty exp(-\beta * 4((\frac{1}{r})^{12}-(\frac{1}{r})^6)) dr But this integral won't...

Homework Statement What exactly does partition coefficient tell us? if I have a low partition coefficient, does that mean its less soluble in that compound? Homework Equations The Attempt at a Solution I had .0555g Benzoic acid mixed with water and methylene chloride. It...

Well what is the partition function of harmonic oscillator with this energy E=hw(n+1/2) , n=1,3,5,... Z=e^(-BE) right? B=1/KT^2 How to expand this? Thank you.

Find the number of the partition of the alphabet {A,b.....Z} of the type (2,2,2,3,3,3,3,4,4) So I did 26!/(2!2!2!3!3!3!3!4!4!) = A REALLY BIG NUMBER then I took that number and dived it by (3!4!2!) and got 2.344 x10^17 which seems to big to be an answer. So I'm wondering if the number...

In my statistical physics class the partition function Z is used in the calculation of probabilities, and I even have a formula for it: Z=\sume-E/kT. While this is all very good I am having some trouble actually grasping what it is, qualitatively speaking. Would someone please be able to...

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 =...

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...

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...

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...