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. E

    Details regarding the high temperature limit of the partition function

    My main question here is about how we actually justify, hopefully fairly rigorously, the steps leading towards converting the sum to an integral. My work is below: If we consider the canonical ensemble then, after tracing over the corresponding exponential we get: $$Z = \sum_{n=0}^\infty...
  2. ergospherical

    I Average magnetic moment of atom in magnetic field ##B##

    from the partition function - am trying to show that ##\langle \mu \rangle = \beta^{-1} (\partial \log Z / \partial B)## where ##Z## is the canonical partition function for one atom, i.e. ##Z = \sum_{m=-j}^{j} \mathrm{exp}(\mu_0 \beta B m)##, and ##\mu = \mu_0 m##. The average...
  3. H

    Partition function for a spin i

    ##Z = \sum_{-i}^{i} = e^{-E_n \beta}## ##Z = \sum_{0}^j e^{nh\beta} + \sum_{0}^j e^{-nh\beta}## Those sums are 2 finites geometric series ##Z = \frac{1- e^{h\beta(i+1)}}{1-e^{h\beta}} + \frac{1-e^{-h\beta(i+1)}}{1-e^{-h\beta}}## I don't think this is ring since from that I can't get 2 sinh...
  4. G

    I How Is the Partition Function of BaTiO3 Calculated in a Cubic Lattice?

    I have a cubic lattice, and I am trying to find the partition function and the expected value of the dipole moment. I represent the dipole moment as a unit vector pointing to one the 8 corners of the system. I know nothing about the average dipole moment , but I do know that the mean-field...
  5. LCSphysicist

    Partition function of modified Ising model

    $$H = - J ( \sum_{i = odd}) \sigma_i \sigma_{i+1} - \mu H ( \sum_{i} \sigma_i ) $$ So basically, my idea was to separate the particles in this way:: ##N_{\uparrow}## is the number of up spin particles ##N_{\downarrow}## "" down spin particles ##N_1## is the number of pairs of particles close...
  6. H

    I Partition function for continuous spectrum

    Let's say that we have a one-particle Hamiltonian that admits only a continuous spectrum of eigenvalues ##E(k)=\alpha k^2## parameterized by asymptotic momentum ##\mathbf{k}## (assuming the eigenfunctions become planewaves far from the origin), would the partition function then be $$Z=\int...
  7. Dario56

    I Derivation of the Canonical Ensemble

    One of the common derivations of the canonical ensemble goes as follows: Assume there is a system of interest in the contact with heat reservoir which together form an isolated system. Heat can be exchanged between the system and reservoir until thermal equilibrium is established and both are...
  8. Simobartz

    I Hamiltonian formalism and partition function

    In hamiltonian formalism we have the generalized coordinates ##q_i## and the conjugates moments ##p_i##. For a dipole in a give magnetic field ##B## the Hamiltonian is ##H=-\mu B cos \theta## where ##\theta## is the angle between ##\vec \mu## and ##\vec B##. Can i consider ##\theta## or ##cos...
  9. S

    I How Does Electron Spin Affect the Partition Function in Saha's Equation?

    Hey, I have a question about proving Saha's equation for ionizing hydrogen atoms. The formula is \frac{P_{p}}{P_{H}} = \frac{k_{B} T}{P_{e}} \left(\frac{2\pi m_{e} k_{B}T}{h^2} \right)^{\frac{3}{2}}e^{\frac{-I}{k_{B} T}} with P_{p} pressure proton's, P_{H} pressure hydrogen atoms, m_{e}...
  10. M

    Partition Function for system with 3 energy levels

    I determined the partition function of the particle A, B and C. C should be the same as B. I then considered the situation, where all particles are in the system at the same time, and drew a diagram of all possible arrangements: The grey boxes are the different partitions, given that we...
  11. M

    I Partition function of mixture of two gases

    I have a question about statistical physics. Suppose we have a closed container with two compartments, each with volume V , in thermal contact with a heat bath at temperature T, and we discuss the problem from the perspective of a canonic ensemble. At a certain moment the separating wall is...
  12. Dom Tesilbirth

    How to find the partition function of the 1D Ising model?

    Attempt at a solution: \begin{aligned}Z=\sum ^{N}_{r=0}C\left( N,r\right) e^{-\beta \left[ -NJ+2rJ\right] }\\ \Rightarrow Z=e^{\beta NJ}\sum ^{N}_{r=0}C\left( N,r\right) e^{-2\beta rJ}\end{aligned} Let ##e^{-2\beta J}=x##. Then ##e^{-2\beta rJ}=x^{r}##. \begin{aligned}\therefore Z=e^{\beta...
  13. raisins

    I Phase space integral in noninteracting dipole system

    Hi all, Consider a system of ##N## noninteracting, identical electric point dipoles (dipole moment ##\vec{\mu}##) subjected to an external field ##\vec{E}=E\hat{z}##. The Lagrangian for this system is...
  14. anaisabel

    Grand partition function (Volume divided into N spaces)

    equation i need to proof. the N in here, is the avarege number of particles, N0 is the total number of particles,V is total volume, v0 I am not quite sure what it is because it isn't mentioned in the homework, but I am assuming it is the volume of which space.
  15. mjmnr3

    Partition function of a particle with two harmonic oscillators

    Here is the solution I have been given: But I really don't understand this solution. Why can I just add these two exponential factors (adding two individual partition...
  16. D

    Rotational partition function for CO2 molecule

    Hello fellow physicists, I need to calculate the rotational partition function for a CO2 molecule. I'm running into problems because I've found examples were they say this rotational partition function is: ##\zeta^r= \frac T {\sigma \theta_r} = \frac {2IkT} {\sigma \hbar^3}## Where...
  17. S

    Probability of a state given the partition function

    If my partition function is for a continuous distribution of energy, can I simply say that the probability of my ensemble being in a state with energy ##cU## is ##e^{-\beta cU} /Z##? I believe that isn't right as my energy distribution is continuous, and I need to be integrating over small...
  18. Diracobama2181

    Classical Canonical Partition Function in Two Dimensions

    For a single particle, $$Z=\frac{1}{h^2}\int_{-\infty}^{\infty} e^{-\beta \frac{P^2}{2m}}d^2p \int e^{-U(r)}drd\theta= \frac{1}{h^2}(\frac{2\pi m}{\beta}) 2\pi [\int_{0}^{r_0}e^{U_0}dr+\int_{r_0}^{R}dr]$$ $$ =\frac{1}{h^2}(\frac{2\pi m}{\beta}) 2\pi [e^{U_0}(r_0)+(R-r_0)]=\frac{\pi...
  19. LCSphysicist

    Estimate the partition function by analyzing a graphic

    I am not sure, but since the partition function Z is just the sum of all Boltzmann Factor We can just add: (some terms don't appear in the image, by the way, the estimative is nice, the result is above ANS) But i didn't understand what the author did: While i didn't even care about the...
  20. T

    Exploring the Grand Partition Function for an Einstein Solid

    $$Q_{(\alpha, \beta)} = \sum_{N=0}^{\infty} e^{\alpha N} Z_{N}(\alpha, \beta) \hspace{1cm} (3.127)$$ Where ##Q## is the grand partition function, ##Z_N## is the canonical partition function and: $$\beta = \frac{1}{kT} \hspace{1cm} \alpha = \frac{\mu}{kT} \hspace{1cm} (3.128)$$ In the case of an...
  21. SchroedingersLion

    A Lennard Jones, 3 particles, partition function

    Greetings, similar to my previous thread (https://www.physicsforums.com/threads/lennard-jones-potential-and-the-average-distance-between-two-particles.990055/#post-6355442), I am trying to calculate the average inter-particle distance of particles that interact via Lennard Jones potentials...
  22. PGaccount

    I Partition function of quantum mechanics

    In quantum mechanics, we have the partition function Z[j] = e-W[j] = ∫ eiS+ jiOi. The propagator between two points 1 and 2 can be calculated as ## \frac{\delta}{\delta j_1}\frac{\delta}{\delta j_2} Z = \langle O_1 O_2 \rangle## The S in the path integral has been replaced by S → S + jiOi...
  23. snatchingthepi

    Partition function from the density of states

    I'm given the following density of states $$ \Omega(E) = \delta(E) + N\delta(E-\Delta) + \theta(E-\Delta)\left(\frac{1}{\Delta}\right)\left(\frac{E}{N\Delta}\right)^N $$ where $ \Delta $ is a positive constant. From here I have to "calculate the canonical partition function as a function of $$...
  24. P

    Partition Function for Spin-1 One Dimensional Ising Model

    $$H=-J\sum_{i=1}^{N-1}\sigma_i\sigma_{i+1}$$ There is no external magnetic field, so the Hamiltonian is different than normal, and the spins $\sigma_i$ can be -1, 0, or 1. The boundary conditions are non-periodic (the chain just ends with the Nth spin) $$Z=e^{-\beta H}$$...
  25. A

    Partition function of 2 bosons in two energy level 0 and E

    For 2 bosons each of which can occupy any of the energy levels 0 and E the microstates will be 3 0 E a a aa - - aa the partition function is therefore $$z=1+e^{-\beta E}+e^{-2\beta E}...(1)$$ Another approach to do.. The single particle partition function is $$z=1+e^{-\beta E} $$...
  26. maajdl

    A Getting structure data from a partition function?

    Hello, From wikipedia, this is the partition function for a "classical continuous system": This is the pillar of classical statistical physics, but it can be seen as a mere kind of "mathematical transform" . It can be used even without thinking to statistics or temperature. If we focus only...
  27. T

    Solving Ising Spin 1 Model w/ Transfer Matrix Method

    I did the first part using the transfer matrix method: $$ Z = Tr(T^{N}) $$ In this case, the transfer matrix will be $$ T(i,i') = \begin{pmatrix} e^{\beta J} & 1 & e^{-\beta J}\\ 1 &1 &1 \\ e^{-\beta J} & 1 & e^{\beta J} \end{pmatrix} $$ To get the trace of $T^N$, you find the...
  28. W

    I Grand Canonical Partition function

    Hi everyone, I understand that the grand-canonical partition function is given by $$Z = \sum_i e^{-\beta(E_i - \mu N_i)}$$ Is there any interpretation to the quantity ##E_i - \mu N_i## here? In the canonical ensemble this would simply be energy of the ##i##th state, so I suppose this would be...
  29. T

    Stat-Mech problem: pressure from a partition function

    Homework Statement A vessel having a volume ##V## initially contains ##N## atoms of dilute (ideal) helium gas in thermal equilibrium with the surroundings at a temperature ##T##, with initial pressure ##P_{i} (T ,V ) = \frac{NRT}{V}## . After some time, a number of helium atoms adhere to the...
  30. L

    A Partition function for a driven oscillator?

    I've seen the partition function calculated for the SHO before in a thermodynamics course in order to calculate entropy. Is it possible to calculate it for a driven harmonic oscillator?
  31. S

    Canonical partition function of an ideal gas (unit analysis)

    Homework Statement Basically the units of the Canonical Partition Function within the logarithms should be zero Homework Equations The Attempt at a Solution N here is a number so we ignore the left logarithms, applying a "Unit function " for the terms within the logarithm...
  32. J

    Where can I find rotational/vibrational temperature data for ethane?

    Hi, Where would I find data for rotational/vibrational temperatures for a particular molecule (ethane)? I tried googling but had no luck. Also can you compute the moments of inerta for a particular (simple) molecule?
  33. Mentz114

    I Derivation of the partition function

    Starting from the definition of energy levels ##e_n## and occupations ##a_n## and the conditions ##\sum_n a_n = N## (2.2) and ##\sum_n a_n e_n = E## (2.3) where ##N## and ##E## are fixed I'm trying to find the distribution which extremizes the Shannon entropy. Using the frequency ##f_n=a_n/N##...
  34. J

    Entropy and the partition function for nitrogen

    Homework Statement I'm attempting to calculate the translational entropy for N2 and I get a value of 207.8 J/Kmol. The tabulated value is given as 150.4 and I am stumped as to why the decrepancy. T = 298.15 K and P = 0.99 atm and V = 24.8 L R = 8.314 J/Kmol[/B] Homework Equations Strans =...
  35. J

    Vibrational entropy and the partition function

    Homework Statement I'm asked to compute the molar entropy of oxygen gas @ 298.15 K & 1 bar given: molecular mass of 5.312×10−26 kg, Θvib = 2256 K, Θrot = 2.07 K, σ = 2, and ge1 = 3. I'm currently stuck on the vibrational entropy calculation. Homework Equations [/B]S = NkT ∂/∂T {ln q} + Nk...
  36. H

    I Help with a partition function calculation

    <Re-opening approved by mentor.> Hi, I've always wondered why when calculating the partition function for a quantum system, we only sum over the eigenstates and not superimposed states. Thus I decided to actually try summing over all normalized states and see what would happen. Feedback is...
  37. H

    I Help with partition function calculation

    Hi, would very much appreciate it if I could get some help for something I was trying to calculate. I'm not very good at latex so I've just attached images of my attempt. I would very much appreciate it if you could look over the images I've taken and provide some feedback, thank you. I've been...
  38. D

    Partition Function for N Quantum Oscillators

    Homework Statement For 300 level Statistical Mechanics, we are asked to find the partition function for a Quantum Harmonic Oscillator with energy levels E(n) = hw(n+1/2). No big deal. We are then asked to find the partition function N such oscillators. Here I am confused. Homework Equations...
  39. R

    Harmonic Oscillator and Volume of Unit Cell in Phase Space

    Long time no see, PhysicsForums. Nevertheless, I have gotten myself into a statistical mechanics class where the prof is pretty brutal and while I can usually manage, this problem finally has me stumped. I'd like to be nudged in the right direction, not outright given the answer if possible. I...
  40. T

    I Partition Function Derivation: Where Did I Go Wrong?

    Self-repost from physics.SE; I underestimated how dead it was. So this follows Schroeder's Intro to Thermal Physics equations 6.1-6.7, but the question isn't book specific. Please let me be clear: I know for a fact I'm wrong. However, it feels like performing seemingly allowed manipulations, I...
  41. FranciscoSili

    Partition Function of N particles in an assymetrical box

    Homework Statement Consider a gas sufficiently diluted containing N identical molecules of mass m in a box of dimensions Lx, Ly, Lz. Calculate the probability of finding the molecules in any of their quantum states. Calculate the energy of each quantum state εr, as a function of the quantum...
  42. D

    Deriving Fermi-Dirac Distribution misunderstanding

    Homework Statement The actual question was deriving Bose-Einstein, but I got confused on the F-D example. I'm basically following the method given here. Homework Equations [All taken directly from the above link] Taylor series: The Attempt at a Solution So after that third equation...
  43. Ventrella

    A Examples of fractal structure in prime partition numbers?

    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...
  44. diegzumillo

    A Zeros of the partition function (Yang-Lee and Fisher zeros)

    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.
  45. Iliody

    I Doubt about partition functions in QFT and in stat Mechanics

    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...
  46. C

    Find magnetic susceptibility using partition function

    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##...
  47. MathematicalPhysicist

    Partition Function at a Fixed Pressure

    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...
  48. patrickmoloney

    Mean energy using the partition function

    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...