What is Partition function: Definition and 213 Discussions

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. J

    How Does the Grand Partition Function Apply to Electron Occupancy in Defects?

    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...
  2. V

    Derivative of the partition function Help

    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 =...
  3. S

    Grand partition function - find occupation energies

    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...
  4. P

    Partition function, particular energy for microstate

    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...
  5. Q

    Partition function and heat capacity

    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 =...
  6. H

    Closed form expression for the partition function Z using the Canonical Ensemble

    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...
  7. E

    Laplace transform of the grand canonical partition function

    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...
  8. I

    Partition function in polymers and entropic forces

    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...
  9. H

    Quantum harmonic oscillators - grand partition function

    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...
  10. E

    Construct a partition function for the system

    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...
  11. R

    How Is the Partition Function Used to Calculate Energy and Pressure in Gases?

    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}}...
  12. H

    Partition function in Statistical Physics

    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...
  13. R

    Mean energy and preassure inolving partition function

    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...
  14. C

    Partition function of classical oscillator with small anharmonic factor

    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...
  15. D

    Finding the energy of a system using the partition function

    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...
  16. P

    Calculating average every from partition function

    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...
  17. T

    The grand canonical partition function and a gas/2D solid?

    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...
  18. M

    Vibrational Partition Function

    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...
  19. homology

    Partition function approximation

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  20. L

    Partition function of a classical spring

    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...
  21. H

    Partition Function for Thermodynamic System

    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...
  22. U

    Grand Partition Function of Solid-Gas System

    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)...
  23. M

    Partition Function for Phonons

    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...
  24. B

    How do I deal with huge exponents in the partition function?

    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...
  25. M

    Partition function of rotating molecule

    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...
  26. B

    Using the partition function find the entropy

    Question is on attachment file.
  27. C

    Generating functional (or partition function)

    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?
  28. W

    Fermi Energy, Temp & Wave Vector Calc for Protons & ^3He

    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...
  29. W

    Partition function for electrons/holes

    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...
  30. C

    Quick partition function question. (Stat. Mech.)

    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...
  31. N

    Partition Function of CO2 at 1000K - Database?

    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.
  32. T

    N particles, Partition Function and finding U and Cv

    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)...
  33. P

    How Does the Grand Partition Function Compare in Fermi and Bose Systems?

    Homework Statement a) Suppose particles can be absorbed onto a surface such that each absorption site can be occupied by up to 6 atoms each in single-particle quantum state \psi_{\it i} with an absorption energy \varepsilon_{i}. Write down the grand partition function for one site. b) If...
  34. P

    Integration problem to calculate partition function of a gase in a blackbody

    Homework Statement This is the integration i have to solve I=\int x^{2}In(1-exp(-ax))dx integration is from zero to infinity The Attempt at a Solution I know that it should be solved with integration by parts so u=In(1-exp(-ax)) du=[a exp(-ax)] / [1-exp(-ax)] dv=x^{2}dx...
  35. P

    Grand Partition Function Question

    Homework Statement a) Suppose particles can be adsorbed onto a surface such that each adsorption site can be occupied by up to 6 atoms each in single-particle quantum state \psi_i with an adsorption energy \epsilon_i. Write down the grand partition function for one site. b) If...
  36. N

    Statistical Physics: Partition function and fermions

    Homework Statement Hi all. The partition function for fermions is (according to Wikipedia: http://en.wikipedia.org/wiki/Partition_function_(statistical_mechanics)#Relation_to_thermodynamic_variables_2) given by: Z = \prod\limits_i {\left( {1 + \exp \left[ { - \beta \left( {\varepsilon _i -...
  37. P

    What is the method for expanding the partition function Z_{\pi}?

    Hello! I've got a problem. If you scroll down to page 118 on the following link http://books.google.com/books?id=ntrPDA6zE1wC&dq=Quarks+bound+by+chiral+fields&pg=PP1&ots=_29vGOurGs&source=bn&sig=h4vGPNBbKX14DpoOyt4tuRZHkRc&hl=de&sa=X&oi=book_result&resnum=4&ct=result#PPA118,M1 there you...
  38. I

    Is there a closed-form formula for the Partition Function p(n)?

    I am aware that there are several generator functions for the Partition Function p(n), but does anyone know if a closed form formula exists for this function?
  39. M

    Partition Function of a Single Magnetic Particle

    Homework Statement For a magnetic particle with an angular momentum "quantum number", j, the allowed values of the z component of a particles magnetic moment are: µ = -jδ, (-j + 1)δ, ..., (j-1)δ, jδ δ is a constant, and j is a multiple of 1/2 Show that the partition function of a...
  40. MathematicalPhysicist

    Grand Canonical Partition function question.

    The question: A system consists of N sites and N particles with magnetic moment m. each site can be in one of the three situations: 1. empty with energy zero. 2. occupied with one particle and zero energy (when there isn't magnetic field around). 3. occupied with two particles with anti...
  41. T

    Partition function problem (Reif)

    Hi, I am trying to work through some of the problems in Reif, not for homework, but to prepare for a final. I was hoping someone could show me how to work through the first problem of chapter seven in Reif. 7.1 - Consider a homogeneous mixture of intert monatomic ideal gases at absolute...
  42. M

    Meaning of Zeros of Partition function

    given a partition function of the form Z[u]= \prod Z_{i} [u] Z_{i} [u] = \sum_{n=-\infty}^{\infty}e^{iuE_{n}^{i}} what is the meaning of zeros ? i mean the values that make Z[u]=0 and how could we calculate these zeros ??
  43. P

    Microcanoncal partition function

    Homework Statement Does anyone know the mathematical definition of the microcanonical partition function? I've seen \Omega = {E_0\over{N!h^{3n}}}\int d^{3N}q d^{3N}p \delta(H - E) where H=H(p,q) is the Hamiltonian. This looks like a useful definition. Only thing is I don't know what E_0...
  44. P

    Partition Function of N Particles: Is Z=(Z_1)^N?

    Homework Statement If we have a system of N independent particles and the partition function for one particle is Z_1, then is the partition function for the N particle system Z=(Z_1)^N? Homework Equations The Attempt at a Solution I'm pretty sure that this is true for a classical...
  45. H

    Entropy and Energy Density from a Partition Function

    Homework Statement Partition Function: Z = 1/N! [8 pi V (kT/hc)^3]^N There are several parts to this but here are the parts I'm struggling with: a) Compute the entropy of the system S. b) Computer the energy density u and compare the result with the corresponding results for a...
  46. H

    Equation of State from Partition Function

    Homework Statement Find the partition function for a two-dimensional nonrelativistic classical gas. Find the equation of state. Calculate the specific heat at constant volume cv and the entropy S. Homework Equations The partition function is Z = (A^N / N!) [(2 pi m k T / h^2)^N]...
  47. Q

    Why Is the Partition Function Crucial in Thermodynamics?

    I have been messing up the partition function for thermo all semester, and now it's really starting to bite me with the carry-down error (i.e. mess up Z for (a) which in turn will mess up S in (b) and so on). I was looking at Reif Problem 9.1 and was wondering if someone could please explain...
  48. nicksauce

    Magnetic system, partition function

    Homework Statement A certain magnetic system has N independent molecules per unit volume, each of which as 4 distinct energy levels: 0, \Delta - \mu_BB, \Delta, \Delta + \mu_BB. i) Write down the partition function, and hence find an expression for the Hemholtz function ii) Use this...
  49. M

    Zeros of the partition function

    given a partition function Tr[e^{-BH}] or Z(B)= \int_{P}dx dp e^{-BH(p,q)} is there any meaning for its zeros ? , i mean what happens in case the partition function Z(B)=0 for some 'B' or temperature B=1/kT do these zeros have a meaning ?? thanks
  50. E

    Entropy and partition function

    Is it possible to obtain the relation S = \log Z + \langle U \rangle /T directly from the Boltzmann distribution? Edit: It seems that we can if we use the VN entropy: S = -\Sigma p_i \log p_i This suggests that the entropy of a single microstate should be s = -\log( \frac{e^{-\epsilon...
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