Partition function Definition and 214 Threads

In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless, it is a pure number.
Each partition function is constructed to represent a particular statistical ensemble (which, in turn, corresponds to a particular free energy). The most common statistical ensembles have named partition functions. The canonical partition function applies to a canonical ensemble, in which the system is allowed to exchange heat with the environment at fixed temperature, volume, and number of particles. The grand canonical partition function applies to a grand canonical ensemble, in which the system can exchange both heat and particles with the environment, at fixed temperature, volume, and chemical potential. Other types of partition functions can be defined for different circumstances; see partition function (mathematics) for generalizations. The partition function has many physical meanings, as discussed in Meaning and significance.

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  1. U

    Partition function of simple system

    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} -...
  2. L

    Statistical mechanics. 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...
  3. U

    Partition Function, Grand Potential

    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.
  4. C

    Thermodynamics without partition function

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

    Weird form of entropy using grand partition function for a system

    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...
  6. E

    Partition Function of a brain tumor

    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...
  7. M

    Deriving Ideal Gas Law through partition function

    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...
  8. N

    Proof: Partition Function of 3 Systems A, B, & C

    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...
  9. S

    Partition Function in Torres-Hernandez "Photon Mass" 1984

    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...
  10. H

    How to get mean occupation numbers by Grand partition function?

    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]...
  11. H

    Relationship between single particle partition function and V

    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...
  12. P

    Partition function as a description of the system

    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}...
  13. I

    I don't understand partition function

    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...
  14. A

    Finding the Partition Function in Paramagnetism

    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} =...
  15. S

    Finding the partition function

    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 =...
  16. Einj

    Partition function for position-independent hamiltonian

    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...
  17. C

    Partition Function For a Single Molecule

    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...
  18. R

    Hamiltonian as applied to the grand canonical partition function

    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...
  19. marcus

    Holonomy spinfoams: convergence of 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...
  20. A

    Partition function for one particle

    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
  21. U

    Field Theory Partition Function

    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...
  22. A

    Partition function vs config integral

    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...
  23. G

    Vibrational partition function - Calculate from several frequencies

    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...
  24. A

    Partition function: [itex] Z=Z_{kin}*Z_{pot} [/itex]

    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}) ?
  25. Y

    Partition function for hard spheres on a lattice

    Hi everyone, I'm reading some lecture notes on statistical physics and thermodynamics and I'm stuck at an expression for a partition function which I really don't understand. The chapter is on mean field theory and the discussion is about hard spheres on a lattice. The interaction of the hard...
  26. L

    Ising model canonical partition function

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

    Partition function related to number of microstates

    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...
  28. E

    Partition Function of a Composite System (Product Rule and Temperature)

    Homework Statement Show that the partition function for a composite system, let's call it '3', composed of systems '1' and '2' is the product of the partition functions of '1' and '2' independently.Homework Equations Kittel defines partition functions using the fundamental temperature τ (which...
  29. A

    Grand Canonical Partition Function and Adsorption Statistics

    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...
  30. Jalo

    Statistical mechanics - Partition function of a system of N particles

    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...
  31. C

    What does zero partition function physically mean?

    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?
  32. T

    Can't understand one step in derivation (partition function)

    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...
  33. U

    Verifying the Partition Function of the Quantum Harmonic Oscillator

    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...
  34. S

    Grand partition function Z of a system

    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...
  35. B

    Find the Factor Increase of the Total Partition Function

    Homework Statement By what factor does the total partition function (excluding electronic) increase when 20 m^3 of Neon at 1.00 atm and 300 K is allowed to expand by 0.0010%? Homework Equations translational partition function qt= (V×[(2∏mkT)]^3/2])/ (h^3), vibrational partitition...
  36. M

    Reif Ch7, Decomposition of partition function

    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...
  37. J

    Partition function to find expected occupancy of a lattice defect

    Homework Statement An impurity can be occupied by 0, 1 or 2 electrons. The impurity orbital in non-degenerate, except for the choice of electron spin. The energy of the impurity level is \epsilon, but to place the second electron on the site requires an additional energy \delta \epsilon...
  38. L

    Why is Q=q^{N} only valid for distinguishable particles?

    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...
  39. P

    Grand Canonical Partition Function for Simple System

    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...
  40. W

    Electronic partition function for molecule with degeneracies

    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...
  41. K

    Path integral and partition function

    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)...
  42. Z

    How Is the Strange Partition Function Derived in Superconductivity Theories?

    While reading an article about superconductivity I found out a strange partition function which I don't know how to re-obtain. The partition function is given by: Z=-\prod_{\omega,\mathbf{k}} (\omega^2 + E(\mathbf{k})^2) where the sum over \omega runs over Matsubara frequencies and...
  43. D

    Partition function lennard jones potential

    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...
  44. E

    Pendulum & Partition Function Problem

    [b]1. A pendulum of mass m hangs from a weightless string of length l The string makes an angle θ with the vertical Find (i) <θ> (ii) <θ^2> (iii) <v> (iiii) <v^2> Homework Equations The Hamiltonian in terms of θ and the angular momentum L= H= L^2/2ml^2 + mgl(1-cosθ) The...
  45. M

    How Does Volume Affect the Partition Function of a Diatomic Molecule?

    I need some help with this problem: Consider a diatomic molecule closed in a cubic container of volume V which hamiltonian is: H=\frac{p_1^2}{2m}+\frac{p_2^2}{2m}+\frac{K}{2}| \vec r_2 - \vec r_1|^2 where \vec r_1, \vec r_2 are the positions of the two atoms. a) Find the partition function...
  46. T

    Harmonic oscillator partition function

    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.
  47. L

    How Does Particle Type Affect Partition Functions and Energy States?

    Homework Statement 1. If the system, which has N identical particles, only has two possible energy states E=0,e(e is an energy) ,what's the ensemble average of E? 2. Find the partition function which has two identical Fermion system if the energy states only have E=0,e.Homework Equations I...
  48. G

    Partition Function: Understanding Z in Statistical Physics

    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...
  49. Truecrimson

    Gas pressure in gravitational field from the partition function

    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...
  50. V

    Quantum Mechanics, Simple harmonic oscillator, partition function

    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...
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