Questions tagged [partition-function]

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Indistinguishability in statistical mechanics

I have two questions about using the concept of indistinguishability to determine the partition function in statistical mechanics, like for instance when determining the partition function of an ideal ...
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Computing braket with exponentials when derivating the classical partition function from the quantum function

I am currently reading David Tong's notes on statistical physics (Page 33) where, just before finishing the derivation of the classical partition function, he obtains the following equation $$ Z=\int{...
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Canonical partition function

I have a question regarding the addition of a constant energy in the Hamilton when we compute the Canonical partition function. In my script it is said that even if we add a constant value of energy, ...
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Ideal gas partition function

I am studying how to calculate the density of states and the partition function of $N$ non-interacting particles. My question is why the integral of the momentum, in the density of states calculation, ...
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How to construct a diagrammatic interpretation out of $Z[J]$?

I am aimed at understanding how to derive the Feynman rules out of a generating functional $Z[J]$, which depends on the set of coordinates $x=(x_1,...,x_n)^T \in \Bbb R^n$ and Grassmann variables $\...
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What is the single particle translational partition function for a single gas atom in unknown volume?

In the standard expression for the ideal gas translational partition function, we have a V for the volume. How do we compute this partition function for a gas when the volume is unknown, like the ...
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Confusion in Partition function

I encounter a problem in understanding the bosonic grand partition function. Suppose I have a harmonic oscillator, where the energy levels are $E_{s} = \hbar \omega \Big( s + \frac{1}{2} \Big)$. Then, ...
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1answer
126 views

$N$ bosons in two energy levels

What is the partition function for $N$ bosons in a two state system with $E_1=0$ and $E_2= E$? I know that bosons don't obey the Pauli exclusion principle and they are indistinguishable but I am ...
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21 views

Proof of the continuous limit for a rotor

I have a rotor with hamiltonian $$ \hat{H}= \frac{ \hat{L}^2 }{2I} $$ then the spectrum can be write as $$ \epsilon_j = \frac{j(j+1) \hbar^2 }{2I} \quad \quad j=0,1,2,3,4,... $$ with degeneration ...
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Average radius of gyration of an ideal chain anchored to a wall at two ends

Suppose you have an n-monomer ideal chain with both ends constrained to be at the same point on an infinte wall (as shown below). Using the method of images, the partition function can be shown to be ...
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Partition function

I am trying to calculate the partition function of a system that has 2 bosons (spin 0) in thermal equilibrium with a reservoir. The particles are independent and each one admits as possible orbitals ...
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Why should $\lim_{V\to\infty} \frac{1}{V} \ln Q(z, V, T)$ have a finite limit?

In the book Intro. Statistical Physics by K.Huang, on page 174, it is given that In the thermodynamic limit $V \rightarrow \infty,$ we expect that: $$ \frac{1}{V} \ln Q(z, V, T) \underset{V \...
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Question about the Partition Function For 2 Fermions In a 2-Level System

So say we have $2$ indistinguishable electrons that can occupy $2$ energy states $E=0$ and $E=\Delta$. If we only had one fermion then are both states doubly degenerate since an electron can be spin ...
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Calculating generating functional with stationary phase approximation

Let's say that I have a generating functional $Z[J]$ defined as: \begin{equation*} Z[J]=\int \mathcal{D}\phi\,e^{iS[\phi]+i\int d^4x\,J\phi}.\tag{1} \end{equation*} I want to use the stationary phase ...
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Entropy of a gas of $N$ distinguishable Particles

Suppose I have a container contain $N$ particles, all of which are distinguishable. How would I calculate the entropy of this gaseous system. The Hamiltonian of the system is simply the sum $\sum_{i=1}...
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1answer
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Wick rotation for calculation of the heat kernel for massive scalar field in curved spacetime

Let $(\mathcal{M},g)$ be a pseudo-Riemannian manifold. I am interested in the calculation of $\det(\square_g+m^2)$, more precisely in the evaluation of the partition function: \begin{equation*} Z[g]=\...
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When does QFT perturbation theory stop being valid?

When introduced to the concept of perturbation theory in Quantum Mechanics we split the hamiltonian $H= H_0 + \delta H$ where $\delta H$ is small in some manner, ie if say $\epsilon$ is the relevant ...
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Density function in microcanonical ensemble

From the Gibbs canonical ensemble, we have the density of states: $$ \rho(\mathbf{q}, \mathbf{p})=Z^{-1} \mathrm{e}^{-\beta H(\mathbf{q}, \mathbf{p})} $$ With $Z(T,V,N)$ the partition function: $$ Z(...
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Derivation of fermionic partition function, how does commutation work?

When deriving the fermionic partition function with coherent states $|\psi\rangle$ we make the following step $$ \mathcal Z=\int d(\bar\psi,\psi)\ e^{-\sum_i\bar\psi_i \psi_i}\sum_n\langle n|\psi\...
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Grand Canonical Ensemble for finite particle system

Suppose I have a system of a finite number of lattice sites, Say M, and binding energy of the particle to be $-\epsilon$. The system is in equilibrium and the chemical potential is $\mu$. Multiple ...
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Specific volume of the coexisting phase at this first-order phase transition

Given the grand partition function $$\mathcal{Z}=(1+z)^V(1+z^{\alpha V})$$ where $z=e^{\mu\beta}$. I'm asked to find the specific volume of the coexisting phase at the first-order phase transition. I ...
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Partition Function of a Hydrogen Atom

I'm trying to figure out how to write the partition function as a function of temperature for just a single hydrogen atom with a bound electron. I know the Energies will be $$E_n=\frac{-13.6eV}{n^2}$$ ...
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Diagrammatic representations of generating functionals $Z[J]$, $W[J]$, and $\Gamma[\varphi]$

The book Boulevard of Broken Symmetries by Adriaan Schakel gives an excellent, if not exceedingly brief, overview of the path integral approach to perturbation theory. In particular, pages 47-58 give ...
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2 dimensional Ising model: How do we visualize the Hamiltonian for interacting spins, but with no external magnetic field?

Suppose we don't have any external magnetic field, so that the Hamiltonian is given by $H=-J\sum_{i,j}s_is_j$. If we have an $n\times n$ 2D lattice of spins. Then does the $H$ correspond to one whole ...
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Gibbs information and information theory

In the study of statistics, a given family of probability densities depending smoothly upon a parameter $\theta$ can be expressed in the form $p(x,\theta)=exp\left[c(x)+\sum_{r}\theta^{r}S_{r}(x)-\psi(...
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Reading off energy levels from series expansions of partition functions

I'm having trouble understanding exactly how to approach this practice problem. It seems pretty straight forward but I just don't see the solution. The canonical partition function $Z(\beta)=\sum_k e^{...
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1answer
87 views

Closed form of partition function in $0+0$-dimenional $\phi^4$ theory

The problem: In one of McGreevy's excellent exercises in QFT, we are given the $0+0$ dimensional partition function $$Z=\int_{-\infty}^{+\infty}dq\ e^{-S(q)}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$, ...
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1answer
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Average number of particles in a certain energy level in the Canonical Ensemble

A quantum system has $r$ discrete energy levels $\varepsilon_1,\varepsilon_2,\varepsilon_3,...,\varepsilon_r$ and $N$ particles distributed in these levels, with the number of particles at each level ...
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2answers
65 views

Why do we need effective action $\Gamma$ given the connected generating functional $W$?

I have just learnt the path integral formalism in QFT, up to the point where we computed the generating functionals $\mathcal{Z}[J] := Z[J]/Z[0]$, $W[J]$, and $\Gamma[\varphi]$. Here $J(x)$ is the ...
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1answer
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Properly accounting for indistinguishability of (homonuclear) diatomic molecules in “internal” partition function

I have always liked Schroeder's take on the partition function being a product of translational and internal degrees of freedom: $$ Z_1 = Z_{\text {trans}} Z_{\text{int}} $$ where $Z_{\text{int}}$ can ...
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Does Bohr van Leeuwen theorem not exclude the possibility of explaining paramagnetism classically?

We have the Bohr-van Leeuwen theorem which tells that magnetism cannot be explained classically. The proof is simple; it turns out that classically the partition function is independent of the ...
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1answer
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Perturbative expansion and path integrals

We’ve started studying path integrals and perturbative expansion. We wrote the action as $S[x]= S_0[x] +S_{int}[x]$ where the first term is the action for the model which we can solve exactly, while ...
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Calculating canonical partition function and pressure of ideal gas under uniform gravitational field

The problem as given to us is this: I am mostly interested in the a and b parts. For a I have the following working out: The Hamiltonian is given by: $$ H = \sum_{i=1}^N \frac{\vec{p_i}^2}{2m} + \...
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First order correction due to dipolar interaction

Considering N atoms with localized moment $\vec\mu$ interacting with the magnetic field $B$ is expressed by $$ \mathscr{H} = - \sum^{N}_{i = 1} \mu m_i B$$ where $m_{i} \in {-l, -l + 1 ,..., l -1 , l}...
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Question about the measure in the partition function of a lattice Yang-Mills theory

This can seem like a dumb question but the partition function of a lattice pure gauge field theory in euclidean space is: \begin{equation} Z=\int \prod_{x,\mu} dU_\mu(x)\,e^{-S_W[U_\mu(x)]}\,\,\,,\,\,\...
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Does fixed $N$ constraint reduce grand canonical ensemble to canonical ensemble?

In condensed matter we were studying the Sommerfeld model where we had these expressions for a gas of fixed N electrons in a metal, where V is also constant but temperatre T is allowed to vary. We had:...
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Energy Mean Value of Quantum Gas [closed]

I was trying to solve a statistical mechanics exercise and got stuck. $\textbf{Attempt}$ The Hamiltonian matrix that describes the Quantum Gas is given by: $$H =\begin{bmatrix} -\epsilon & t &...
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Probability of a particle being at a given energy level in a discrete canonical set [closed]

According to Wikipedia, the canonical partition function for a system of discrete and non-degenerate energy levels at temperature $T$ is defined as $$Z=\sum_{i} \mathrm{e}^{-\beta E_{i}}$$ where $i$ ...
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How can one compute an effective probability at the critical point of a first order phase transition

In an excerpt from Finite-Size Scaling by John Cardy I found the following development: At a first-order transition, the correlation length ξ remains finite, and the finite-size scaling properties in ...
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1answer
54 views

Derivation operators as arguments or the Hamiltonian

In a book I am reading about QFT (Quantum field theory by Mark Srednicki ,page 48), I see the following equation: $$ \int \mathcal D p\mathcal D q \exp\left[i\int_{\mathbb R} dt (p\dot q - H_0(p,q)-...
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Partition function for 2D classical XY model

Studying the classical XY model (https://en.wikipedia.org/wiki/Classical_XY_model), I wish to compute the partition function: \begin{equation} Z=\int \mathrm{d}\mathbf{s}\; e^{-\beta H(\mathbf{s})} \...
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Partition function $\mathrm{tr} e^{-\beta H}$ for non-trace-class operator

For Hamiltonian $H$, the partition function is defined as $$Z=\mathrm{tr} e^{-\beta H}.$$ However, consider a free particle $H = -\Delta$ in $\mathbb R^1$. In this case, the operator $e^{-\beta H}$ is ...
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Periodic boundary conditions in Partition Function (Supersymmetry of stochastic dynamics)

In the context of supersymmetric theory of stochastic dynamics one introduces a partition function for a field satisfying a Langevin equation, starting from the partition function of the noise. As ...
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Statistical Mechanics Partition Function

We know from Boltzmann's Distribution for the canonical ensemble that $\overline{E}=\frac{-\partial{lnZ}}{\partial{\beta}}$, which for the translational degree of freedom comes out to be $\overline{E}=...
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1answer
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Integrating out bosons/fermions when the action is quadratic

Arovas and Auerbach, in their paper titled "Functional integral theories of low-dimensional quantum Heisenberg models" try to compute the free energy of $SU(N)$ models with the large $N$ ...
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Statistical Mechanics in rotating cylinder

I'm having trouble calculating the probability density of a particle being in a radius $r=r_0$ in a rotating cylinder. ($\omega, R, L$ are the frequency, radius, and height of the cylinder). First of ...
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1answer
38 views

Confusion about the canonical partition function and probabilities

I'm in a first course on statistical mechanics at the moment and I'm having trouble wrapping my head around an example problem involving the canonical partition function. The question setup has a ...
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1answer
38 views

Confused about partition function of identical particles

The total partition function of N identical, independent particles is $Z^N/N!$ where Z is the partition function of a single particle. To find out the correct magnetization due to N identical atoms of ...
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Canonical Partition function of constant gravitational potential: what is the normalization?

I'm imagining a system of particles subject to a constant gravitational field, say $V(x)=g x$. The Hamiltonian for one particle will be $$H=\frac{1}{2}m v^2 + gx ,$$ where the first term is the ...
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Assigning proper powers of $i$ to vertices of Feynman diagram

I'm reading the chapter 9 "the path integral for interacting field theory" of the Srednicki's QFT book. The lagrangian we are dealing with here is given by \begin{gather} \mathcal{L} = \...

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