Partition Definition and 307 Threads

Homework Statement Can you partition the positive integers in such a way that if x, y are member of A, then x+y is not a member of A. x and y have to be distinct. That is, {1, 2, 3} cannot be in the same set, since 1+2 = 3, but 1 and 2 can be, since 1+1=2, but 1 and 1 are not distinct...

I am practicing for my math exam next week and I came across this problem: A set has 200 elements in it. It is partitioned into three subsets so that the second and third subsets have the same number of elements. If four times the number of elements in the second subset is three times as many...

Hi, it is well established that octanol is the preferred solvent for partition coefficient studies in environmental chemistry, however, some studies ui.e Noble A. Parition coefficients (n-octanol-water) for pesticides) J Chromat. 642 3-14, mention that estimating toxicity based on solubility in...

Regarding the recent discovery by Ken Ono and colleagues of the fractal structure of partition numbers for primes: a great lever of intuition would be to see a diagram, or any presentation of the numbers that reveals this fractal structure. Perhaps the fractal structure is somehow hidden in a...

Hey there, Just wondering where I can get a nice treatment of this with derivations. I could swear I read about this in Jon Cardy's Scaling and renormalization in statistical physics but I can't find it again so maybe I was wrong.

Hi, I was studying for my final exam on statistical physics and a doubt raised on my head that was truly strong and disturbing (at least, for me), and that I couldn't answer to myself by now. The doubt is: Given that we have in d dimensions a fermion non interacting gas, the statistical...

Homework Statement A certain magnetic system contains n independent molecules per unit volume, each of which has four energy levels given by 0, ##Δ-gμ_B B##, ##\Delta##, ##\Delta +gμ_B B##. Write down the partition function, compute Helmholtz function and hence compute the magnetization ##M##...

I don't understand how could be a particular configuration described by eqn. (29.4). Why is it said a particular configuration? I know that this is the grand Boltzmann factor for one particle in energy state ##E_1##. But how does it describe configuration of the system?

Homework Statement I don't quite follow the solution to this problem (problem 2.11 in Bergersen's and Plischke's textbook), here are the quoted problem and its solution: problem: solution: Homework EquationsThe Attempt at a Solution My problem is with the solution to (a), it seems they...

Homework Statement System of two energy levels, E_0 and E_1 is populated by N particles, at temperature T. The particles populate the levels according to the classical (Maxwell-Boltzmann) distribution law. (i) Write an expression for the average energy per particle. Homework EquationsThe...

Hey all, ran into a game theory problem I can't solve. A and B have a set of 10 random numbers from 1-10, players can make so called "piles", a pile has a goal number from 1 to 10, if 6,3 are on the table, a pile of nine may be started, the pile is added to by adding sets of numbers that sum to...

Why do they introduce the partition function. I have seen it in the derivation of the Boltzmann distribution. But I don't know the physical significance of it here? And how do they get to (L.11) after that? I get everything until L.7. Including L.7. The rest of the proof is here just in case...

Homework Statement Homework Equations I have question for (a) section. The Attempt at a Solution I have two answer for the question but I can't figure out which one is right. (1)Since the partition function is to sum up all the state in the system, I write down the answer (2)In other...

We have a partition function ## \displaystyle Z=\frac{1}{N! \: h^{f}} \int dq\: dp \:e^{-\beta H(q,p)} ## And we obtain, for example, the pressure by ##\displaystyle p = \frac{1}{\beta} \frac{\partial\: \ln Z}{\partial V}##. So if we do the transformation ##Z \rightarrow a Z## where ##a >0##...

Hi, maybe someone can help me with this problem? Homework Statement A system consist of N Atoms that have a magnetic moment m. The Hamiltonian in the presence of a magnetic field H is $$\mathcal{H}(p,q) - mH \sum_{i=1}^N cos(\alpha_{i})$$ where ##\alpha_i## is the angle between the magnetic...

Homework Statement With the Hamiltonian here: Compute the cananonical ensemble partition function given by ##\frac{1}{h} \int dq dp \exp^{-\beta(H(p,q)}## for 1-d , where ##h## is planks constant Homework EquationsThe Attempt at a Solution I am okay for the ##p^2/2m## term and the...

Looking for the heat capacity of ideal gas due to rotational degrees of freedom. If the temperature of the gas is much higher than the temperature corresponding to the energy differential between states,the partition function can be written as the integral over the density of states. If the...

Homework Statement In the real world, most oscillators are not perfectly harmonic. For a quantum oscillator, this means that the spacing between energy levels is not exactly uniform. The vibration levels of an ##H_2## molecule, for example, are more accurately described by the approximate...

Homework Statement Hi I have the following definition for the partition function of ##N## particles in ##s## dimensions: I am looking at computing the partition function for this Hamiltonian: The solution is here: Homework Equations above The Attempt at a Solution I don't...

Homework Statement Calculate the partition function, the entropy and the heat capacity of a system of N independent harmonic oscillators, with hamiltonian ##H = \sum_1^n(p_i^2+\omega^2q_i^2)## Homework Equations ##Z = \sum_E e^{-E/kT}## The Attempt at a Solution I am not really sure what to...

What is the difference between micro canonical Partition function and canonical Partition function? Is the mathematical expression of the above two Partition function are same? If it is then why?? [emoji29]

Homework Statement Consider the case when the three Ising spins are replaced by quantum spins 1/2's with a Hamiltonian H=-J(s1.s2+s2.s3+s3.s1) calcualte the quantum partition function Homework Equations Partition function is the sum of E^(-H*B) where B is 1/kt The Attempt at a Solution...

The canonical partition function in classical statistical mechanics is calculated by ## Q_N(V,T)=\frac 1 {N! h^{3N}}\int e^{-\beta H(\mathbf q,\mathbf p)}d^{3N}q \ d^{3N}p ##. The ## \frac 1 {N!} ## is there to prevent the Gibbs paradox. But now consider a system of N particles that have no...

There doesn't seem to be a forum that is specifically about statistical mechanics, so I'm posting this question here. I apologize for the long-winded introduction, but I think it's needed to provide context for my question: If you have a discrete collection of single-particle energy levels...

Hi, I'm trying to calculate the partition function for a certain system and I arrived at an expression for the partition function $Z$, and have been stuck here for two weeks at the least. This is not a homework problem. If this is the wrong place to post a question like this, could you please...

Homework Statement For a diatomic gas near room temperature, the internal partion function is simply the rotational partition function multiplied by degeneracy ##Z_e## of the electronic ground state. Show that the entropy in this case is ## S = Nk\left[ \ln \left(...

Why is the partition function ##Z[J]=\int\ \mathcal{D}\phi\ e^{iS[\phi]+i\int\ d^{4}x\ \phi(x)J(x)}## also called the generating function? Is the partition function a q-number or a c-number? Does it make sense to talk of a partition function in classical field theory, or can we define...

of size N into an K subgroups. I've been trying for hours to do this and still haven't found a solution. Example: The array {A,B,C} of size N=3 and I want all the move combinations that make it into K=1 subgroups. The only such subgroup is the one with all the elements, and I can get that with...

Hey! :o A trucking company has two same trucks. The luggage space is a cuboid with floor area $2m\times 5m$ and height $3m$. The packages that the comany has to deliver are also cuboid. The packages must be distributed into the two trucks. The problem is defined as followed: $n$ packages...

Hello everyone, How can I calculate the partition function of N classical electrons (forgetting about the spin) in a box of volume V with Hamiltonian (The Hamiltonian is missing a factor of 1/(2m)) ? I tried calculating the partition function of one electron first in the canonical ensemble but...

Homework Statement Exercise 4 in the upload titled Dok1.pdf. Write down an expression for the canonical partition function for N ideal Na2 gas molecules, when the rotational contribution is treated classically, and all inner degrees of freedom are treated quantum mechanically. Use this and...

I was reading about partition function. I noticed that there are two approaches toward partition function. The first approach: Suppose we are dealing with a closed system where the system is composed of heat bath (R) and inside it there is a very small system (E), the two systems are in thermal...

Homework Statement Find the horizontal force across the partition in a water tank with two compartments, one filled with water to x height and the other with water to y height. I've worked out the thrust acting on the partition as a result of density, height, gravity and average depth, so I...

Homework Statement Homework Equations Rd = retardation factor = (1 + (ƥbKd / n)) log(Koc) = -0.55logS + 3.64 Kd = Koc*foc The Attempt at a Solution Part C is the only part I feel sure on. I simply plugged 1480 mg/L into the log(Koc) equation above to solve for Koc. Then I multiplied Koc...

Homework Statement Show that the partition function for the harmonic oscillator with an additional force H = \hbar \omega a^{\dagger} a - F x_0 (a + a^{\dagger}) is given by \frac{e^{\beta \frac{F^2 x_{0}^2}{\hbar \omega}}}{1-e^{\beta \hbar \omega}} and calculate \left<x\right> = x_0...

Hi, How did they break down the following summation? When finding the vibrational partition function ofa diatomic molecule it was approximated that the energy levels of the vibrational part of the diatomic molecule were harmonic and therefore the energy equation for a harmonic oscillator was...

Hi! The following image is taken from my note in Stat Mech. Please excuse my ugly handwriting... I copied this from my professor's note on a whiteboard, and I'm not so sure if it is correct. The equations for Z1 (partition function before mixing) and Z2 (partition function after mixing) seems...

Homework Statement The first excited state of the helium atom lies at an energy 19.82 eV above the ground state. If this excited state is three-fold degenerate while the ground state is non-degenerate, find the relative populations of the first excited and the ground states for helium gas in...

Homework Statement Consider a gas in equilibrium with a surface. The surface can adsorb the gas molecules onto any of M independent, distinguishable sites. The molecular partition function for an adsorbed molecule is q(T) ≡ exp[−β A_surface]. a) Assume that the adsorbed molecules are...

Determine the partition coefficient of chlorpromazine in DMSO and n-pentane I'm a physics student and doing a course in biophysics. I would really appreciate it, if you would take some time and provide some hints as to how to design a more concrete plan. Especialy, how do I find out the...

I am reading an article by Tachikawa on the Nekrasov partition function ("A review on instanton counting and W-algebras"). The article is meant to be pedagogical but I have some trouble with what is supposed to be "baby" examples. The first one involves susy QM on \mathbb{C}^2 . He says the...

Homework Statement Ground state energy is set at 0. E_n=\left(1-\frac{1}{n+1}\right)\in with no degeneracy (\Omega(n)=1); (n=0,1,2...) Write down the partition function and look for its limit when kt \gg \in\\ kt \ll \in Homework EquationsThe Attempt at a Solution Partition function for this...

Hi all This is a fairly basic QFT question but it's bothering me. And Peskin/Schroeder fails at explaining basic stuff, so here I am. After calculating Z for a particular theory I know this can be used to calculate all kinds of correlation functions. Itself, however, is the probability...

Homework Statement Homework Equations Partition function = ##\frac{z_{i+1}}{z_i} ## ##z = \Sigma_{j=1}^\infty g_j e^{\frac{-(E_j - E_i)}{KT}}## ##g_j = 2(j^2)## The Attempt at a Solution I should get 2, but I keep getting ##2 + 8e^7.8 + ... ## I used ##K = 1.38 \times 10^-23## and I converted...

Homework Statement Consider a zipper of N links, each of which can either be open or closed with associated energy 0 if closed and ##\epsilon## if open. a) Suppose the N links are independent, compute the partition function of the system and the average number of open links b)Now assume that...

Homework Statement Why is it that the microcanonical partition function is ##W = Tr\{\delta(E - \hat{H})\}##? As in, for example, Mattis page 62? Moreover, what's the meaning of taking the Dirac delta of an operator like ##\hat{H}##? Homework Equations The density of states at fixed energy is...

Hello! (Wave) Consider a nonuniform partition $a=t_0< t_1< \dots < t_{\nu}=b$ and assume that if $h_n=t^{n+1}-t^n, 0 \leq n \leq N-1 $ is the changeable step, then $\min_{n} h_n > \lambda \max_{n} h_n, \lambda>0$ independent of $n$. Show a bound of the error of Euler method analogous to...

Hi everibody, the other day in a stadistical physics lesson we were studyng Fermi Dirac and Bose Einstein stadistics and comparing it to the classical Maxwell Boltzmann's. We learned that in the quantum stadistics for indistinguishable particles the partition function of the whole system...

Homework Statement A container has movable(without friction) piston on top. The container and the piston are all made of perfectly insulating material allowing no heat transfer between outside and inside the container. The container is divided into two compartments by a rigid partition made of...

This question is in regards to the degeneracy of states for an Argon atom with just one missing electron. For hydrogen the problem of finding the partition function depends on finding the the ionized state of hydrogen divided by the non-ionized state... (please see Saha equation ->...