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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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Different definitions of resolvent in matrix model

When I study the matrix models, I get confused of different definations of resolvent. After we define the partition function as $$Z=\int[dM]e^{-NTrV(M)},$$ where $V(M)$ is a matrix valued function of $...
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Degeneracy of states in an ensemble of $N$ harmonic oscillators in $d$ dimensions

In the canonical ensemble consisting of $N$ independent harmonic oscillators in $d$ spatial dimensions one has to evaluate sums like $$ \sum_n \ldots e^{-\beta E_n} = \sum_E g_E \ldots e^{-\beta E} $$ ...
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Deriving partition function for various ensembles

I'm wondering the derivation of following partition functions corresponding to various ensembles: $$ \begin{aligned} \Xi(V, T, \mu) & =\sum_N Q(N, V, T) e^{\beta \mu N} \\ \Delta(p, T, N) & =\...
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Partition function of $N$ distinguishable particles

Suppose there are $N$ distuinguishable particles. Each particle can have energy either $\epsilon_1$ or $\epsilon_2 $. $ n_1$ particles have energy $\epsilon_1 $and $n_2$ particles have $\epsilon_2$ i....
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Distinguishability in Maxwell-Boltzmann statistics

Per the Wikipedia page on Maxwell-Boltzmann statistics, the mean occupation number describes the average number of particles in the i-th single-particle state for distinguishable particles. To be ...
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The definition on vacuum-vacuum amplitude with current in chapter of External Field Method of Weinberg's QFT

I'm reading Vol. 2 of Weinberg's QFT. As what I learnt from both P&S and Weinberg, the generating function is defined as $$ Z[J] = \int \mathcal{D}\phi \exp(iS_{\text{F}}[\phi] + i\int d^4x\phi(x) ...
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Calculation of canonical partition function for fermion system with degenerate energy levels

I'm having trouble in visualising the generalized version of the question asked here. We have a system with levels whose energies are $0, \epsilon, 2 \epsilon, ..., n\epsilon$, and the number of ...
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The definition of the path integral

I still have big conceptual questions about the path integral. According to (24.6) of the book "QFT for the gifted amateur" from Lancaster & Blundell the path integral is equal to $$Z =\...
Frederic Thomas's user avatar
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What are exactly the loop correction to the potential? [duplicate]

I am struggling with the concept of quantum effective action, but first recall the definition : given a Wilsonian effective action $W[J]$ of our theory, the quantum effective action is just $$\Gamma[\...
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The partition function of a particle in a magnetic field diverges. Why?

Using the symetric gauge $\mathbf{A} = \tfrac{B}{2} (-y, x, 0)$, the stationary states wave functions of a quantum particle in a constant and homogeneous magnetic field are $$\tag{1} \psi_{n m}(r, \...
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Is there any meaning or statistical distribution associated with the Jacobi's $\theta$ functions?

The Jacobi's theta functions $$\theta_1(0,\tau )=0$$ $$\theta_2(\tau) =\sum_{n\in \mathbb{Z}} q^{(n+\frac{1}{2})^2 /2 }$$ $$\theta_3(\tau) =\sum_{n\in\mathbb{Z}} q^{n^2/2}$$ $$\theta_4(\tau) =\sum_{n\...
ShoutOutAndCalculate's user avatar
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Yang-Lee circle theorem and the distribution of zeros

I am reading a note on Yang-Lee's zeros. The note first focuses on the grand canonical partition function $\Xi(T, z)$, where $z$ is the complex fugacity conjugate to the number of particles. If the ...
user31415926's user avatar
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Making sense of stationary phase method for the path integral

I am trying to understand this paper/set of notes. I have already seen the following related question: Does the stationary phase approximation equal the tree-level term? but had some trouble following ...
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Partition function of Hydrogen atoms problem

I know there are several questions asking this problem, but I found this problem has not been solved yet to me. I will repeat the problem and state my view. Consider the statistical mechanics of a ...
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Interpreting generating functional as sum of all diagrams

The generating functional is defined as: $$Z[J] = \int \mathcal{D}[\phi] \exp\Big[\frac{i}{\hbar}\int d^4x [\mathcal{L} + J(x)\phi(x)]\Big].$$ I know this object is used as a tool to generate ...
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Is average total energy of two objects is the sum of their individual average energies?

In the context of Boltzmann's distribution, Schroeder states that an average is defined as $$\bar{x}=\frac1Z\sum_sx(s)e^{-\beta E(s)}$$ Where $\beta=1/kT$ and E(s) is the energy corresponding to the ...
GedankenExperimentalist's user avatar
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Homogeneity Restrictions on the Distribution of states in Thermodynamic Systems

The expected energy in the canonical ensemble is given by \begin{equation} \begin{split} \langle E \rangle &= \frac{\displaystyle\sum_{i=1} E_i e^{-\beta E_i}}{\displaystyle\sum_{i=1} e^{-\beta ...
Lodin Ellingsen's user avatar
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Examples of Path integral $\neq$ Partition function?

Are there any systems we know of whose partition function is not simply Wick rotation of the path integral? Does anyone know of any examples?
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Partition function with Hamiltonian depending on parameter

Given a Hamiltonain $H(p,q)$, I know that the classical partition function for a single particle is given by an integral over the phase space $$ Z_1 = \frac{1}{h^3} \int e^{-\beta H(p,q)} d^3pd^3q $$ ...
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Partition function for a $SO(3, 1)$ invariant "Hamiltonian"

Suppose, I look at the $SO(3, 1)$ generalization of $H = \frac{p^2}{2m}$, i.e. $$H = \lambda P^{\mu}P_{\mu}$$ where $P^{\mu}P_{\mu}$ is a $SO(3, 1)$ invariant object and $\lambda$ is some dimensionful ...
Dr. user44690's user avatar
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Probability and Single Particle Partition function

I have a fundamental question in statistical mechanics that I can't wrap my head around. When we speak about the probability of a SYSTEM being in macrostate with energy E we calculate this by: $$ P(E) ...
Lugerfan's user avatar
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Why non-extensive terms cancel out in the low temperature expansion of 2d Ising model?

I'm currently reading David Tong's lecture on Statistical Physics, and I cannot understand the logic in the following paragraph about the low temperature expansion of 2d Ising model. It can be ...
Jason Chen's user avatar
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2 answers
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Partition function for independent particles

I am trying to understand Section 3.8.3, "Independent particles", of Piers Coleman's Introduction to Many-Body Physics (self-study, mathematics background). He considers "a system of ...
gilgamesh's user avatar
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Partition function for a classical two-particle oscillator: Infinite limits?

Previous page: What is the partition function of a classical harmonic oscillator? Whenever I see the partition function of a classical two-particle oscillator, $$Z(\beta) \, = \, \int dx \int dp ~ e^{-...
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Grand canonical partition function for Bose-Einstein statistic and Maxwell-Boltzmann statistics for indistinguishable particles

If we have N non-interacting particles and N assumes the values 1 and 2.They can be found in three energy levels $\epsilon_{l}=\epsilon*l$ with l = 0, 1, 2. I tried to write the function as Z(µ, T)=$\...
Mario's user avatar
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$(\mathcal{S}\mathcal{T})^3=\mathcal{S}^2=+1$ mistake in CFT big yellow book?

In Conformal Field Theory Philippe by Di Francesco, Pierre Mathieu David Sénéchal Sec 10.l. Conformal Field Theory on the Torus eq.10.9 says the modular transformation $\mathcal{T}$ and $\mathcal{S}$ ...
zeta's user avatar
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Should I partial trace the hamiltonian or partition function for a reduced system?

Suppose I have a quantum spin model, let's say e.g. the quantum transverse field model with hamiltonian $H$, on some lattice of particles, with partition function $\text{Tr}(e^{- \beta H})$ and I do ...
user3397129's user avatar
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Are partition functions invariant under Bogoliubov transformations?

Consider a Hamiltonian $H(a_i, a^{\dagger}_i)$ as a function of some ladder operators $a_i, a^{\dagger}_i$. Now, consider a partition function $H(a'_i, a'^{\dagger}_i)$ where $a', a'^{\dagger}$ are ...
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Constrained partition function(s) of a multi cluster system

Can a system have multiple partition functions? I am trying to find the partition function of two cluster system with specific constraints. For the sake of brevity, I am consider a simple case. The ...
RAJENDRA PRASAD G's user avatar
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2 answers
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Fixing total particle number for Canonical Ensemble vs Grand canonical ensemble

For a (non-interacting) gas of bosons (any gas for that matter), the total particle number has to be a finite value and in the canonical description this is ensured by writing a constrained sum for ...
Lost's user avatar
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Partition function for fractional Brownian motion with $H < 1/2$

Recently I was interested in computing the logarithmic derivarivative $Z'(H)/Z(H)$ of the following partition function: $$ Z(H) = \int e^{-S_H(x)} \mathcal{D} x, \quad \text{where} \quad S_H(x) = A(H) ...
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What is the gravitational path integral computing?

What is the gravitational path integral (which roughly goes like $\int [dg]e^{iS_{\text{EH}}[g]}$) computing? Usually, path integrals arise from transition amplitudes such as these: $\lim_{T\to\infty}\...
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What does the matrix mean in matrix models?

I'm learning what a matrix model means, for example, in Yang–Mills matrix models, IKKT matrix model and BFSS matrix model. I have consulted a large amount of information but still not sure what the ...
Errorbar's user avatar
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Computation of Density for an Ideal Gas

I'd like to get some help deriving the following result: $$ \langle \rho(\mathbf{q}) \rangle = \frac{N}{V}$$ where $$\rho(\mathbf{q}) = \sum_{i}^{N}\delta(\mathbf{q}_i-\mathbf{q}) $$ and $\mathbf{v} = ...
Claudio Menchinelli's user avatar
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Statistical mechanics of a gas in a rotating cylinder

The theoretical premises that allow us to study the statistical mechanics of a substance in a rotating equilibrium are not completely clear to me. For example, consider a gas ($N$ non-interacting ...
Alessandro Tassoni's user avatar
2 votes
1 answer
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Strange Wick rotation in the computation of string partition function

In order to compute the one-loop vacuum-to-vacuum amplitude for the bosonic string, one runs into \begin{equation} Z(\tau) = V_D (q \bar{q})^{-D/24} \int \frac{d^Dk}{(2 \pi)^D} \exp({- \pi \alpha^\...
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Thermodynamic free energy of interacting system

This question concerns an interacting system's thermodynamic free energy $\Omega$. Generally speaking, The action $S$ for an interacting system has the following form: \begin{equation} S(\phi,\psi) = ...
Mass's user avatar
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Why does Gibbs' factor appear in the partition function of Maxwell-Boltzmann statistic?

In the assumption of Maxwell-Boltzmann statistics, particles are treated as distinguishable. So for identical particles with discrete energy levels, there are degeneracies for any specific occupation ...
Hsu Bill's user avatar
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Entropy and partition function for system with normally-distributed microstates

I'm looking at an example system which has a very large number of microstates (effectively infinite - not possible to enumerate exhaustively, but possible to sample from). The energies of the ...
Alex I's user avatar
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3 answers
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Truly classical limit of classical ideal gas

It's well-known that a classical statistical mechanical treatment of an ideal gas leads to a fundamental equation which is not extensive. For $N$ particles, one arrives at a per-particle partition ...
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Physical interpretations of compressibility definitions

I can compute the thermodynamic compressibility in terms of a partition function as $$ \kappa = \frac{1}{\langle n \rangle^2} \frac{\partial n}{\partial \mu} = - \frac{1}{\langle n \rangle^2} \frac{\...
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Computing partition function of Potts model on a "thin" 2D grid

I am looking for an exact way of evaluating the partition function of the $q$-state Potts model on a $2D$ grid of size $N \times H$: $$ \mathcal{Z}(\beta) = \sum_{\left(\sigma_{i, j}\right)_{\substack{...
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Partition function of atoms adsorbed on a surface

I have the following problem in basic statistical physics, and the following doubts on my interpretation. An ideal gas ($N$ atoms, chemical potential $\mu$) is in contact with an adsorbent lattice ($...
emonGuinness's user avatar
1 vote
1 answer
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Dynamics of a discrete stochastic classical model

I was going through Dijkgraaf (https://arxiv.org/abs/0803.1927) and I came across the following two equations $$ \frac{\mathrm{d}}{\mathrm{d} \tau} P_{\alpha}(\tau)=\sum_{\substack{\beta \\ \beta \...
Thomas Shelby's user avatar
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1 answer
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Local Density Profile From Grand Canonical Partition Function

I have grand canonical partition function $Z_{\lambda}$. From the statistical mechanics, we can calculate average number of particles $\langle N \rangle$ as $$\langle N \rangle = \lambda \frac{\...
이영규's user avatar
2 votes
1 answer
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External/Background Fields Meaning

(I'll work in the Euclidean for convenience) In the path integral formulation of QFT given a field $\phi$, or a set of them if you want to, we have that the partition function is given by: $$Z[J] = \...
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Canonical Partition Function - Why does the probability for a system to be in a state not take into consideration the Helmholtz exponential component?

It is known that the equation to calculate the probability for a system to be in a specific energy level is $\sum_s{P_s} = 1 = e^{\beta F} \sum_s e^{-\beta E_s}$, where the sum component is now ...
Entangled Being's user avatar
1 vote
1 answer
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Semiclassic limit of a QFT in Zinn-Justin

I am reading the Zinn-Justin book "Quantum Field Theory and Critical Phenomena" and i have come across a perplexing point. Given the partition functional, in Euclidean QFT: $$Z[J, \hbar] = \...
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Chemical potential of ideal solution in grand canonical ensemble

I am attempting to derive the chemical potential of an ideal solution within the grand canonical ensemble. I know that the final answer should be (https://en.wikipedia.org/wiki/Ideal_solution) $$\mu_j=...
Greenhorn3.14's user avatar
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Is the Euclidean generating functional $Z_{E}[J]$ identified with original Minkowskian generating functional $Z[J]$?

In quantum field theory, it is common to perform wick rotation $t\rightarrow -i\tau$ and get Euclidean generating functional $Z_{E}[J]$. When I first studied QFT, I just saw this a magic trick to ...
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