Questions tagged [partition-function]

The partition function describes the statistical properties of a system in thermodynamic equilibrium, and is constructed to represent a particular statistical ensemble (microcanonical, macrocanonical, grand canonical).

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Quantum versus Classical Partition Function

I am confused as to which partition functions are classical and which are quantum. I am interested in the canonical ensemble. Several places I have seen the "classical" partition function as:...
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Ensemble associated with a system of Photons

In the book "Fundamentals of Statistical and Thermal Physics" by F. Reif it is said that : Photon Statistics is a special case of Bose-Einstein Statistics with no restriction on the total ...
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Should vibrational internal energy of a diatomic gas be considered when calculating below the characteristic temperature $\theta_{vib}$?

(this is a confusion raised from an exercise problem in Carter's Book:) For a diatomic gas, (take $N_2$ for example, which has $\theta_{vib}=3352k$), at room temperature: $\theta_{rot} \ll T=298k \ll ...
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Is it OK to do this manipulation at the partition function level? (auxiliary fields in quadratic gravity)

Background I am working with the following action in the Euclidean signature ($C^2$ is the Weyl quadratic term): \begin{equation} S_B = -\frac{1}{2\kappa^2}\int d^4x\sqrt{g}\left(2\Lambda_C+\zeta R-\...
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String partition function

Hey I am a bit confused because when reading physics papers I encounter the free energy as a generating function of topological invariants as integrals over certain compactified moduli spaces. For ...
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Ideal Fermi gas close to $T=0 \rm K$

I am reading about the behaviour of ideal Fermi gas close to $T=0K$ from Kardar's Statistical Mechanics. In the paragraph which I have highlighted, we have the inyegral representation of $f_m^{-}(z)$....
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High temperature limit of ideal quantum gas

I am reading ideal quantum gas from Kardar's Statistical Mechanics. $VII.35$ is the representation of pressure, number density and energy density in the form of $f_m^{\eta}$. $z=e^{\beta\mu}$ where $\...
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Quantum partition of two integer spins

I am trying to write the partition function for a simple system made of two interacting bosons of spin $S=1$ having the interaction Hamiltonian $$ \hat{H} = \vec{S}_1 \cdot \vec{S}_2 $$ I think I ...
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Partition function for the indistinguishable particles using symmetrization of states

In the derivation of the partition function for the the N particle ideal gas, the factor of $\frac{1}{N!}$ does not come naturally. We have to go for symmetrized and asymmetrized state. So, to derive ...
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How the Zero-point Energy of the System containing 2 Fermions in 3 Micro-Energy States is 1?

If we distribute 2 Fermions $\mathrm{(A,A)}$ in 3 Micro-Energy States (0,$\epsilon$,$2\epsilon$), the confirmation is given by : $$ \begin{array}{|c|c|c|c|c|} \hline 0 & \varepsilon & 2 \...
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Clarification regarding the terminology of Microstates

I would like to understand how microstates are defined or used in physics. Are microstates suppose to only mean eigenvalues of a given observable (or a generator of symmetry)? The reason for my ...
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Free particle in momentum representation

In Greiner, density operator for a free particle has been calculated in momentum basis. They consider a large box of vilume $V=L^3$ and periodic boundary condition. $$\phi_\vec{k}(\vec r)=\frac{1}{\...
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Derivation of partition function for $N$ identical quantum harmonic oscillators

What is the partition function $$\mathcal Z^{(N)}_\beta(H) : =\mathrm{Tr}\exp(-\beta H) \tag{Z} $$ $\left(\beta >0\right)$ for a system of $N$ indistinguishable and non-interacting bosons (e.g. ...
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Computing the partition funciton of 2 identical particles in a harmonic oscillator

Say I have two identical (fermionic) non-interacting particles in a 1D harmonic oscillator. I would like to compute the entropy of the system as the temperature $T$ varies, for which I need the ...
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Partition Function quantum identical particles

In Pathria and Beale Statistical Mechanics section 5.5, the book tries to compute the Partition function of a system of noninteracting, indistinguishable particles confined to a cubical box of volume $...
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Evaluation of thermal average with path integral

I want to evaluate the thermal average $$<\hat{\phi}(0)\hat{\phi}(0)\hat{\phi}(x)\hat{\phi}(x)>$$ with the path integral. $\phi$ is a real scalar field. In general: $$<\hat{\phi}(0)\hat{\phi}(...
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Sums in statistical mechanics

I am evaluating the partition function of a system of particles and incurred in sums like $$ S(a)=\sum_{k=0}^\infty (2k+1)^\frac{\kappa}{2}e^{-(2k+1)a} $$ being $\kappa\in\mathbb{Z}$ and $a=m_0\beta$ ...
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Eigenvalue of transfer matrix using Monte Carlo in 2D Ising model

Montecarlo is an algorithm capable of numerical estimation of any quantity which can be written as the average of a state function like, for example, the magnetization or the internal energy in the 2D ...
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Proving Relationship between Statistical Weight $Ω$ and Partition Function $Z$

How can I prove that $$\frac{\partial(T \ln Z)}{\partial T}=\ln \Omega,$$ without using the relation $S=k\ln Ω$? where $Z$ is the Partition Function, $T$ is the Absolute Temperature, $Ω$ is the ...
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How to Derive the 0 Temperature Action of Bose-Hubbard System?

I am currently reading chapter 8 of the textbook A Modern Approach to Critical Phenomena. This chapter deals with the Bose Hubbard Model and is filled with many equations that aid the analysis of this ...
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Partition function for bosons with path integral

In this book the partition function for bosons is defined in eq. 2.17 as: $$Z=\mathrm{Tr}[e^{-\beta (H-\mu_i N_i)}]=\sum_a\int d\phi_a\langle\phi_a|e^{-\beta(H-\mu_i N_i)}|\phi_a\rangle$$ The ...
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Chemical Potential of an electron or positron in a pair production reaction

In the Thorne and Blandford Modern Classical Physics text there is an exercise that walks you through deriving the density-temperature limit for pair production in plasmas. To do this they consider a ...
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Ratio of particles in energy states

For a system of N weakly interacting distinct particles distributed in 2 energy states and degeneracies given by $\epsilon_{1} = O(g_{1} = I)$ $\epsilon_{2} = 2 \epsilon (g_{2} = 4)$ where $epsilon = ...
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Partition function for weakly interacting gases

I'm studying the Susskind lecture on statistical mechanics. A potential energy of pairwise interactions has been defined: $$\sum_{n>m}U(|x_n -x_m|)$$ We want to calculate the partition function, ...
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Partition Function of a Hydrogen Atom - How it is derived?

It is well-known that the Partition Function of Hydrogen Atom diverges if we calculate in naive manner. And I could find the partition function named Brillouin-Planck-Larkin partition function which ...
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Correlation Function and Generating Functional in QED

Peskin and Schroeder (1995, p.82 and p.292) define the two-point correlation function of a $\phi^4$ theory as $$\langle \Omega|T\{\phi(x)\phi(y)\}|\Omega\rangle\tag{4.10}$$ and the generating ...
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Geometric Entropy & the Partition Function for Conical Deficits

In their paper, Callan and Wilczek claim to derive from the thermal entropy $$S_\text{thermal} = -\left(\beta\frac{\partial}{\partial\beta}-1\right)\ln(\mathcal{Z})$$ a geometric entropy which is ...
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Partition function of a double-stranded dna molecule [closed]

I have a doubled-stranded dna molecule. The molecule has N links, each of which can be one of two states: a closed state and an open state with energy $\epsilon$. A link can be open only if the link ...
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Question on "proof" of the holographic feature of euclidean quantum gravity

In a preceding question, where more details about what I'm doing here are given, I pointed out the possibility for euclidean quantum gravity to exhibit a holographic property. Technically there is no ...
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Canonical and Grand Canonical Ensemble with degeneracy considered

I know that for the canonical ensemble: $Z=\Sigma_n e^{-\beta E_n}$ By using the Lagrange Lagrange multiplier method, one can find for the probability of the system being in an microstate: $Pr_n=\frac ...
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Thermodynamic potential and partition function

I am a bit confused by the relation between thermodynamic potential and partition functions. From my understanding, we can generate all thermodynamical quantities by taking partial derivatives to the ...
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Feynman Rules from Generating Functional

For the following Lagrangian: $$\mathcal{L}= \overline \psi \left(i \gamma^{\mu}D_{\mu} - m \right)\psi -\frac{1}{2}\left(F_{\mu\nu}\right)^2,$$ I'm trying to find the Feynman rules. I know that the ...
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Quenched and annealed disorder in a combinatorial problem

For a research project I'm dealing with a combinatorial problem which I am modeling as a disordered system. For some context, the problem is the TSP, and the disorder enters through the weights on its ...
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Derivation of the Grand Canonical Partition Function for Fermions

Regarding the derivation on this page: http://lampx.tugraz.at/~hadley/ss2/fermigas/thermo/thermo.php I'm stuck with the summation over macrostates {$q$} being the same as the sum over microstates {$...
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Cluster Expansion and Partition Function for a Collection of Interacting Particles in Presence of an External Potential

I was watching a lecture on statistical mechanics (link here) in which the instructor explains the statistical mechanics of weakly interacting gas molecules (no external potential) using the standard ...
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In Srednicki's quantum field theory, page 71. Why is $Z_1 = W_1$?

Srednicki defines here: https://arxiv.org/abs/hep-th/0409035 on p.71 a "Z1", which is $exp(iW_1)$ where $iW_1$ is the sum of all connected diagrams with sourceless ...
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Partition function for specific potential

Considering a gas of N classical particles, in the canonical essemble, with the following potential: $$ V(|\overrightarrow{r_i} - \overrightarrow{r_j}|) = \alpha |\overrightarrow{r_i} - \...
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Higher derivatives of the log-partition function?

I need higher derivatives of the log-partition function $Z(z)=\log \sum_i \exp(z_i)$, has anyone derived the formula? Looking at concrete values of derivatives up to order 8, evaluated at $z=(1,1,1)$ ...
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Partition function of given system

I am given rotational free energy of a diatomic molecule as $$F_{rot}=-NkT ln[\frac{1}{π}(\frac{2πIkT}{\hbar^2})^{\frac{3}{2}}]$$ The question is if I take the partition function from it would it give ...
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Extra factor in the canonical ensemble expression?

I'm trying to understand the derivation of the canonical ensemble, I will now present the derivation I'm following and then I will explain what the problem is. We have a system $\Sigma=\Sigma_s+\...
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Canonical partition function of a system composed of 1 particle in a box. Transition from quantum mechanical expression to classical

While the expression for the classical canonical partition function is derived in my notes, there is a small detail that goes unexplained: Assuming the particle in the box represents an isolated ...
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Partition function in terms of particle states VS microstates

I'm a computational chemist trying to understand more deeply the concepts of statistical mechanics. I followed Susskind's lectures. He starts by dividing a system in N subsystems; for my brain to like ...
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Propagator in Path Integrals

I am reading Section I.2 in Zee's QFT in a Nutshell. The amplitude for a particle to start at position $I$ and end at $F$ is (eq. (I.2.6)): $$ \langle q_f|e^{-iHT}|q_I\rangle=\int Dq(t)\ e^{i\int_0^T ...
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Wouldn't a simple scalar field fix the non-renormalizability of gravity?

It is well known that quadratic gravity is renormalizable. On the other hand it is possible to transform the partition function of Einstein-Hilbert + free minimally coupled complex scalar field into a ...
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Canonical Partition function of a Boltzmann gas, for non-interacting particles

In our lecture, for the canonical partition function of a Boltzmann gas made out of $N$ particles which do not interact it's given: $$Z=\frac 1 {N!}\Sigma_{\{\vec P_i\}}e^{-\beta\Sigma_{i=1}^N \frac{\...
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Why the modularity of 2d CFT is SL$(2,\mathbb{Z})$?

I believe there is a short and quick answer about this. But I don't have clear intuition. The partition function of 2d CFT obey the SL$(2,\mathbb{Z})$ invariance, which is a consequence of conformal ...
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QFT generating functional and Green function and propagator

I am confused about why does the generating functional gives the propagator by differentiation, and why that propagator is the Green function. I understand how to take the functional derivative like ...
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Free Energy vs. Partition Function in QFT

The partition function of QFT is defined as $$Z=\int\mathcal{D}\varphi e^{iS[\varphi]}.$$ Now, it is a general fact that this formal path integral can be computed perturbatively as (sketchy) $$Z=\sum_{...
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In 2d CFT, if the partition function is $Z=\text{Tr}q^{L_0-c/24}$, where is the analogue of temperature?

In 2d CFT, restricting to the holomorphic part of the theory, the torus partition function is given by $$Z(q)=\text{Tr}q^{L_0-c/24}$$ where $L_0$ is the element of the Virasoro algebra, i.e. mode of ...
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What is the relation between the partition function from Stat. Mech. And the Path Integral? [duplicate]

Beside the fact that they look identical when you take imaginary time in the path integral formulation. I understand we doing statistics and we are just integrating over all states with a relative ...
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