Partition function Definition and 182 Threads

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...

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 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...

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...

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 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

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...

Hello everybody, I registered here hoping to finally find a fundated answer about what I by myself seem not be able to figure out. Question in short: We have calculated a list of wavenumbers for some molecular systems. How do you get the vibrational partition function from that? My...

Hey, If I have a canonical partition function with: Z=\frac{1}{h}Z_{Pot}\cdot Z_{Kin}. Can i callculate immediately the average potential Energy, by: \bar{U}=-\frac{\partial}{\partial \beta}\ln(Z_{pot}) ?

Why in case of Ising model ##H=-J\sum S_iS_{i+1}## we calculate canonical partition function?

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...

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...

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?

This is from self-study coursework rather than homework. I hope it's ok in this forum. I'm following a statistical mechanics lecture on youtube, and the professor is deriving the average energy as a function of the partition function. He goes: -1/Z dZ(beta)/d beta = -dlnZ(beta) / d beta where...

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...

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...

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...

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)...

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...

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.

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...

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 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 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...

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...

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...

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...